Mercurial > dropbear
annotate libtommath/tommath.src @ 285:1b9e69c058d2
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| author | Matt Johnston <matt@ucc.asn.au> |
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| date | Wed, 08 Mar 2006 13:23:58 +0000 |
| parents | eed26cff980b |
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| rev | line source |
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1 \documentclass[b5paper]{book} |
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2 \usepackage{hyperref} |
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3 \usepackage{makeidx} |
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4 \usepackage{amssymb} |
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5 \usepackage{color} |
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6 \usepackage{alltt} |
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7 \usepackage{graphicx} |
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8 \usepackage{layout} |
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9 \def\union{\cup} |
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10 \def\intersect{\cap} |
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11 \def\getsrandom{\stackrel{\rm R}{\gets}} |
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12 \def\cross{\times} |
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13 \def\cat{\hspace{0.5em} \| \hspace{0.5em}} |
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14 \def\catn{$\|$} |
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15 \def\divides{\hspace{0.3em} | \hspace{0.3em}} |
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16 \def\nequiv{\not\equiv} |
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17 \def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} |
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18 \def\lcm{{\rm lcm}} |
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19 \def\gcd{{\rm gcd}} |
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20 \def\log{{\rm log}} |
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21 \def\ord{{\rm ord}} |
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22 \def\abs{{\mathit abs}} |
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23 \def\rep{{\mathit rep}} |
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24 \def\mod{{\mathit\ mod\ }} |
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25 \renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} |
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26 \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} |
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27 \newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} |
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28 \def\Or{{\rm\ or\ }} |
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29 \def\And{{\rm\ and\ }} |
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30 \def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} |
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31 \def\implies{\Rightarrow} |
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32 \def\undefined{{\rm ``undefined"}} |
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33 \def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} |
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34 \let\oldphi\phi |
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35 \def\phi{\varphi} |
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36 \def\Pr{{\rm Pr}} |
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37 \newcommand{\str}[1]{{\mathbf{#1}}} |
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38 \def\F{{\mathbb F}} |
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39 \def\N{{\mathbb N}} |
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40 \def\Z{{\mathbb Z}} |
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41 \def\R{{\mathbb R}} |
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42 \def\C{{\mathbb C}} |
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43 \def\Q{{\mathbb Q}} |
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44 \definecolor{DGray}{gray}{0.5} |
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45 \newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} |
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46 \def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} |
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47 \def\gap{\vspace{0.5ex}} |
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48 \makeindex |
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49 \begin{document} |
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50 \frontmatter |
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51 \pagestyle{empty} |
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52 \title{Multi--Precision Math} |
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53 \author{\mbox{ |
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54 %\begin{small} |
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55 \begin{tabular}{c} |
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56 Tom St Denis \\ |
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57 Algonquin College \\ |
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58 \\ |
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59 Mads Rasmussen \\ |
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60 Open Communications Security \\ |
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61 \\ |
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62 Greg Rose \\ |
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63 QUALCOMM Australia \\ |
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64 \end{tabular} |
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65 %\end{small} |
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66 } |
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67 } |
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68 \maketitle |
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69 This text has been placed in the public domain. This text corresponds to the v0.35 release of the |
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70 LibTomMath project. |
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71 |
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72 \begin{alltt} |
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73 Tom St Denis |
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74 111 Banning Rd |
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75 Ottawa, Ontario |
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76 K2L 1C3 |
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77 Canada |
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78 |
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79 Phone: 1-613-836-3160 |
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80 Email: [email protected] |
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81 \end{alltt} |
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82 |
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83 This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} |
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84 {\em book} macro package and the Perl {\em booker} package. |
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85 |
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86 \tableofcontents |
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87 \listoffigures |
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88 \chapter*{Prefaces} |
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89 When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. |
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90 They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' |
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91 Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which |
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92 perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps |
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93 others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give |
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94 back to society in the form of tools and knowledge that can help others in their endeavours. |
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95 |
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96 I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source |
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97 code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not |
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98 explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works |
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99 itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality |
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100 of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far |
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101 from relatively straightforward algebra and I hope that this book can be a valuable learning asset. |
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102 |
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103 This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora |
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104 of kind people donating their time, resources and kind words to help support my work. Writing a text of significant |
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105 length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, |
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106 comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg |
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107 were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to |
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108 continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. |
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109 |
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110 To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I |
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111 honour your kind gestures with this project. |
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112 |
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113 Open Source. Open Academia. Open Minds. |
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114 |
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115 \begin{flushright} Tom St Denis \end{flushright} |
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116 |
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117 \newpage |
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118 I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also |
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119 contribute to educate others facing the problem of having to handle big number mathematical calculations. |
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120 |
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121 This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of |
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122 how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about |
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123 the layout and language used. |
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124 |
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125 I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the |
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126 practical aspects of cryptography. |
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127 |
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128 Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a |
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129 great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up |
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130 multiple precision calculations is often very important since we deal with outdated machine architecture where modular |
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131 reductions, for example, become painfully slow. |
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132 |
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133 This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks |
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134 themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' |
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135 |
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136 \begin{flushright} |
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137 Mads Rasmussen |
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138 |
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139 S\~{a}o Paulo - SP |
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140 |
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141 Brazil |
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142 \end{flushright} |
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143 |
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144 \newpage |
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145 It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about |
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146 Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not |
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147 really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. |
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148 |
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149 At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the |
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150 sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real |
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151 contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. |
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152 Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. |
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153 |
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154 When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, |
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155 and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close |
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156 friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, |
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157 and I'm pleased to be involved with it. |
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158 |
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159 \begin{flushright} |
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160 Greg Rose, Sydney, Australia, June 2003. |
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161 \end{flushright} |
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162 |
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163 \mainmatter |
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164 \pagestyle{headings} |
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165 \chapter{Introduction} |
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166 \section{Multiple Precision Arithmetic} |
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167 |
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168 \subsection{What is Multiple Precision Arithmetic?} |
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169 When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively |
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170 raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can |
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171 reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. |
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172 Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple |
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173 precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} |
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174 of algorithms can be designed to accomodate them. |
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175 |
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176 By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in |
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177 the decimal system with fixed precision $6 \cdot 7 = 2$. |
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178 |
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179 Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in |
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180 schools to manually add, subtract, multiply and divide. |
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181 |
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182 \subsection{The Need for Multiple Precision Arithmetic} |
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183 The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation |
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184 of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require |
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185 integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a |
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186 typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and |
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187 Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. |
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188 |
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189 \begin{figure}[!here] |
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190 \begin{center} |
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191 \begin{tabular}{|r|c|} |
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192 \hline \textbf{Data Type} & \textbf{Range} \\ |
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193 \hline char & $-128 \ldots 127$ \\ |
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194 \hline short & $-32768 \ldots 32767$ \\ |
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195 \hline long & $-2147483648 \ldots 2147483647$ \\ |
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196 \hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ |
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197 \hline |
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198 \end{tabular} |
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199 \end{center} |
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200 \caption{Typical Data Types for the C Programming Language} |
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201 \label{fig:ISOC} |
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202 \end{figure} |
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203 |
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204 The largest data type guaranteed to be provided by the ISO C programming |
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205 language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they |
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206 see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is |
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207 insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be |
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208 trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, |
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209 rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by |
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210 extending the range of representable integers while using single precision data types. |
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211 |
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212 Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic |
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213 primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in |
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214 various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several |
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215 major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and |
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216 deployment of efficient algorithms. |
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217 |
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218 However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. |
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219 Another auxiliary use of multiple precision integers is high precision floating point data types. |
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220 The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. |
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221 Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE |
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222 floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small |
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223 (\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create |
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224 a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where |
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225 scientific applications must minimize the total output error over long calculations. |
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226 |
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227 Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). |
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228 In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. |
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229 |
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230 \subsection{Benefits of Multiple Precision Arithmetic} |
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231 \index{precision} |
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232 The benefit of multiple precision representations over single or fixed precision representations is that |
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233 no precision is lost while representing the result of an operation which requires excess precision. For example, |
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234 the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple |
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235 precision algorithm would augment the precision of the destination to accomodate the result while a single precision system |
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236 would truncate excess bits to maintain a fixed level of precision. |
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237 |
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238 It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic |
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239 curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum |
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240 size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the |
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241 integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard |
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242 processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not |
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243 normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. |
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244 |
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245 Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the |
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246 overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved |
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247 platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the |
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248 inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input |
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249 without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to |
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250 be written and tested once. |
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251 |
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252 \section{Purpose of This Text} |
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253 The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. |
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254 That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' |
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255 elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} |
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256 give considerably detailed explanations of the theoretical aspects of algorithms and often very little information |
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257 regarding the practical implementation aspects. |
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258 |
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259 In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For |
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260 example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple |
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261 algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning |
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262 the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple |
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263 as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not |
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264 discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). |
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265 |
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266 Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers |
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267 and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve |
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268 any form of useful performance in non-trivial applications. |
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269 |
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270 To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer |
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271 package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.org}} package is used |
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272 to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field |
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273 tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text |
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274 discusses a very large portion of the inner workings of the library. |
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275 |
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276 The algorithms that are presented will always include at least one ``pseudo-code'' description followed |
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277 by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same |
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278 algorithm in other programming languages as the reader sees fit. |
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279 |
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280 This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing |
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281 the reader how the algorithms fit together as well as where to start on various taskings. |
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282 |
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283 \section{Discussion and Notation} |
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284 \subsection{Notation} |
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285 A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent |
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286 the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits |
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287 of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer |
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288 $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. |
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289 |
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290 \index{mp\_int} |
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291 The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well |
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292 as auxilary data required to manipulate the data. These additional members are discussed further in section |
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293 \ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be |
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294 synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members |
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295 are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the |
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296 member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would |
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297 evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that |
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298 $a.length = 5$. |
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299 |
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300 For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used |
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301 to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is |
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302 a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to |
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303 mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These |
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304 algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple |
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305 precision algorithm to solve the same problem. |
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306 |
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307 \subsection{Precision Notation} |
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308 The variable $\beta$ represents the radix of a single digit of a multiple precision integer and |
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309 must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in |
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310 the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range |
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311 $0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the |
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312 carry. Since all modern computers are binary, it is assumed that $q$ is two. |
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313 |
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314 \index{mp\_digit} \index{mp\_word} |
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315 Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent |
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316 a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In |
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317 several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. |
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318 For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to |
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319 the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision |
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320 variable it is assumed that all single precision variables are promoted to double precision during the evaluation. |
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321 Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single |
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322 precision data type. |
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323 |
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324 For example, if $\beta = 10^2$ a single precision data type may represent a value in the |
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325 range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let |
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326 $a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written |
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327 as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. |
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328 In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit |
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329 in a single precision data type and as a result $c \ne \hat c$. |
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330 |
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331 \subsection{Algorithm Inputs and Outputs} |
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332 Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision |
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333 as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This |
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334 distinction is important as scalars are often used as array indicies and various other counters. |
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335 |
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336 \subsection{Mathematical Expressions} |
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337 The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression |
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338 itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression |
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339 rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when |
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340 the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, |
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341 $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a |
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342 fraction a real value division is implied, for example ${5 \over 2} = 2.5$. |
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343 |
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344 The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation |
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345 of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. |
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346 |
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347 \subsection{Work Effort} |
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348 \index{big-Oh} |
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349 To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all |
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350 single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. |
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351 That is a single precision addition, multiplication and division are assumed to take the same time to |
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352 complete. While this is generally not true in practice, it will simplify the discussions considerably. |
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353 |
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354 Some algorithms have slight advantages over others which is why some constants will not be removed in |
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355 the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a |
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356 baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these |
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357 would both be said to be equivalent to $O(n^2)$. However, |
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358 in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a |
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359 result small constant factors in the work effort will make an observable difference in algorithm efficiency. |
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360 |
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361 All of the algorithms presented in this text have a polynomial time work level. That is, of the form |
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362 $O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how |
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363 various optimizations will help pay off in the long run. |
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364 |
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365 \section{Exercises} |
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366 Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to |
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367 the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought |
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368 provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent |
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369 chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the |
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370 subject material. |
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371 |
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372 That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular |
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373 are encouraged to verify they can answer the problems correctly before moving on. |
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374 |
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375 Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of |
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376 the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these |
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377 exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the |
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378 scoring system used. |
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379 |
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380 \begin{figure}[here] |
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381 \begin{center} |
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382 \begin{small} |
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383 \begin{tabular}{|c|l|} |
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384 \hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ |
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385 & minutes to solve. Usually does not involve much computer time \\ |
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386 & to solve. \\ |
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387 \hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ |
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388 & time usage. Usually requires a program to be written to \\ |
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389 & solve the problem. \\ |
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390 \hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ |
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391 & of work. Usually involves trivial research and development of \\ |
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392 & new theory from the perspective of a student. \\ |
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393 \hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ |
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394 & of work and research, the solution to which will demonstrate \\ |
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395 & a higher mastery of the subject matter. \\ |
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396 \hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ |
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397 & novice to solve. Solutions to these problems will demonstrate a \\ |
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398 & complete mastery of the given subject. \\ |
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399 \hline |
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400 \end{tabular} |
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401 \end{small} |
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402 \end{center} |
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403 \caption{Exercise Scoring System} |
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404 \end{figure} |
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405 |
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406 Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or |
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407 devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level |
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408 are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These |
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409 two levels are essentially entry level questions. |
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410 |
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411 Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often |
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412 fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always |
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413 involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can |
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414 answer these questions will feel comfortable with the concepts behind the topic at hand. |
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415 |
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416 Problems at the fourth level are meant to be similar to those of the level three questions except they will require |
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417 additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide |
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418 the exact details of the answer until a subsequent chapter. |
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419 |
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420 Problems at the fifth level are meant to be the hardest |
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421 problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a |
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422 mastery of the subject matter at hand. |
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423 |
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424 Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader |
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425 is encouraged to answer the follow-up problems and try to draw the relevance of problems. |
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426 |
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427 \section{Introduction to LibTomMath} |
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428 |
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429 \subsection{What is LibTomMath?} |
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430 LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it |
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431 is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on |
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432 any given platform. |
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433 |
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434 The library has been successfully tested under numerous operating systems including Unix\footnote{All of these |
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435 trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such |
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436 as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such |
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437 as public key cryptosystems and still maintain a relatively small footprint. |
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438 |
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439 \subsection{Goals of LibTomMath} |
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440 |
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441 Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, |
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442 even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the |
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443 library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM |
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444 processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window |
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445 exponentiation and Montgomery reduction have been provided to make the library more efficient. |
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446 |
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447 Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface |
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448 (\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized |
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449 algorithms automatically without the developer's specific attention. One such example is the generic multiplication |
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450 algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication |
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451 based on the magnitude of the inputs and the configuration of the library. |
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452 |
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453 Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should |
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454 be source compatible with another popular library which makes it more attractive for developers to use. In this case the |
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455 MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits |
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456 in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument |
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457 passing conventions, it has been written from scratch by Tom St Denis. |
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458 |
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459 The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' |
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460 library exists which can be used to teach computer science students how to perform fast and reliable multiple precision |
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461 integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. |
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462 |
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463 \section{Choice of LibTomMath} |
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464 LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but |
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465 for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL |
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466 \cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for |
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467 reasons that will be explained in the following sub-sections. |
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468 |
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469 \subsection{Code Base} |
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470 The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional |
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471 segments of code littered throughout the source. This clean and uncluttered approach to the library means that a |
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472 developer can more readily discern the true intent of a given section of source code without trying to keep track of |
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473 what conditional code will be used. |
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474 |
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475 The code base of LibTomMath is well organized. Each function is in its own separate source code file |
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476 which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source |
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477 file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing |
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478 very hard. GMP has many conditional code segments which also hinder tracing. |
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479 |
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480 When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} |
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481 which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about |
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482 $50$KiB) but LibTomMath is also much faster and more complete than MPI. |
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483 |
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484 \subsection{API Simplicity} |
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485 LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build |
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486 with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the |
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487 functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided |
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488 which is an extremely valuable benefit for the student and developer alike. |
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489 |
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490 The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to |
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491 illegible short hand. LibTomMath does not share this characteristic. |
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492 |
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493 The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors |
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494 are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In |
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495 effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely |
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496 undersireable in many situations. |
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497 |
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498 \subsection{Optimizations} |
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499 While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does |
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500 feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP |
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501 and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few |
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502 of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP |
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503 only had Barrett and Montgomery modular reduction algorithms.}. |
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504 |
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505 LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular |
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506 exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually |
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507 slower than the best libraries such as GMP and OpenSSL by only a small factor. |
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508 |
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509 \subsection{Portability and Stability} |
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510 LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler |
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511 (\textit{GCC}). This means that without changes the library will build without configuration or setting up any |
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512 variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of |
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513 MPI has recently stopped working on his library and LIP has long since been discontinued. |
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514 |
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515 GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active |
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516 development and are very stable across a variety of platforms. |
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517 |
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518 \subsection{Choice} |
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519 LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for |
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520 the case study of this text. Various source files from the LibTomMath project will be included within the text. However, |
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521 the reader is encouraged to download their own copy of the library to actually be able to work with the library. |
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522 |
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523 \chapter{Getting Started} |
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524 \section{Library Basics} |
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525 The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, |
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526 a problem along with allowable solution parameters should be identified and analyzed. In this particular case the |
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527 inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written |
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528 as portable source code that is reasonably efficient across several different computer platforms. |
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529 |
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530 After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. |
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531 That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, |
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532 before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. |
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533 By building outwards from a base foundation instead of using a parallel design methodology the resulting project is |
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534 highly modular. Being highly modular is a desirable property of any project as it often means the resulting product |
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535 has a small footprint and updates are easy to perform. |
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536 |
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537 Usually when I start a project I will begin with the header files. I define the data types I think I will need and |
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538 prototype the initial functions that are not dependent on other functions (within the library). After I |
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539 implement these base functions I prototype more dependent functions and implement them. The process repeats until |
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540 I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as |
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541 mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to |
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542 why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the |
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543 dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the |
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544 mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development |
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545 for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. |
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546 |
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547 FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions. |
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548 |
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549 Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing |
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550 the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. |
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551 |
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552 It only makes sense to begin the text with the preliminary data types and support algorithms required as well. |
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553 This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. |
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554 |
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555 \section{What is a Multiple Precision Integer?} |
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556 Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot |
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557 be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is |
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558 to use fixed precision data types to create and manipulate multiple precision integers which may represent values |
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559 that are very large. |
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560 |
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561 As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system |
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562 the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits |
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563 (\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds |
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564 column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based |
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565 multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed |
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566 precision computer words with the exception that a different radix is used. |
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567 |
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568 What most people probably do not think about explicitly are the various other attributes that describe a multiple precision |
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569 integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, |
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570 that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in |
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571 its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper |
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572 arithmetic. The third property is how many digits placeholders are available to hold the integer. |
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573 |
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574 The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, |
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575 if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. |
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576 Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer |
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577 will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision |
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578 integer or mp\_int for short. |
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579 |
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580 \subsection{The mp\_int Structure} |
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581 \label{sec:MPINT} |
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582 The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for |
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583 any such data type but it does provide for making composite data types known as structures. The following is the structure definition |
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584 used within LibTomMath. |
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585 |
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586 \index{mp\_int} |
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587 \begin{figure}[here] |
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588 \begin{center} |
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589 \begin{small} |
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590 %\begin{verbatim} |
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591 \begin{tabular}{|l|} |
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592 \hline |
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593 typedef struct \{ \\ |
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594 \hspace{3mm}int used, alloc, sign;\\ |
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595 \hspace{3mm}mp\_digit *dp;\\ |
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596 \} \textbf{mp\_int}; \\ |
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597 \hline |
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598 \end{tabular} |
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599 %\end{verbatim} |
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600 \end{small} |
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601 \caption{The mp\_int Structure} |
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602 \label{fig:mpint} |
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603 \end{center} |
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604 \end{figure} |
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605 |
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606 The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. |
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607 |
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608 \begin{enumerate} |
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609 \item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent |
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610 a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. |
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611 |
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612 \item The \textbf{alloc} parameter denotes how |
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613 many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count |
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614 of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the |
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615 array to accommodate the precision of the result. |
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616 |
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617 \item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple |
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618 precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least |
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619 significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored |
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620 first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, |
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621 if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then |
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622 it would represent the integer $a + b\beta + c\beta^2 + \ldots$ |
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623 |
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624 \index{MP\_ZPOS} \index{MP\_NEG} |
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625 \item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). |
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626 \end{enumerate} |
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627 |
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628 \subsubsection{Valid mp\_int Structures} |
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629 Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. |
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630 The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). |
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631 |
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632 \begin{enumerate} |
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633 \item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated |
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634 array of digits. |
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635 \item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. |
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636 \item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, |
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637 leading zero digits in the most significant positions must be trimmed. |
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638 \begin{enumerate} |
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639 \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. |
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640 \end{enumerate} |
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641 \item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; |
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642 this represents the mp\_int value of zero. |
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643 \end{enumerate} |
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644 |
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645 \section{Argument Passing} |
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646 A convention of argument passing must be adopted early on in the development of any library. Making the function |
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647 prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. |
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648 In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int |
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649 structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. |
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650 Consider the following examples. |
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651 |
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652 \begin{verbatim} |
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653 mp_mul(&a, &b, &c); /* c = a * b */ |
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654 mp_add(&a, &b, &a); /* a = a + b */ |
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655 mp_sqr(&a, &b); /* b = a * a */ |
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656 \end{verbatim} |
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657 |
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658 The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the |
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659 functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. |
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660 |
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661 Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order |
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662 of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In |
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663 truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been |
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664 adopted. |
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665 |
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666 Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a |
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667 destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important |
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668 feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. |
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669 However, to implement this feature specific care has to be given to ensure the destination is not modified before the |
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670 source is fully read. |
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671 |
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672 \section{Return Values} |
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673 A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them |
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674 to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end |
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675 developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may |
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676 fault by dereferencing memory not owned by the application. |
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677 |
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678 In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for |
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679 instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor |
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680 will it check pointers for validity. Any function that can cause a runtime error will return an error code as an |
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681 \textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). |
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682 |
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683 \index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} |
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684 \begin{figure}[here] |
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685 \begin{center} |
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686 \begin{tabular}{|l|l|} |
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687 \hline \textbf{Value} & \textbf{Meaning} \\ |
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688 \hline \textbf{MP\_OKAY} & The function was successful \\ |
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689 \hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ |
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690 \hline \textbf{MP\_MEM} & The function ran out of heap memory \\ |
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691 \hline |
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692 \end{tabular} |
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693 \end{center} |
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694 \caption{LibTomMath Error Codes} |
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695 \label{fig:errcodes} |
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696 \end{figure} |
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697 |
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698 When an error is detected within a function it should free any memory it allocated, often during the initialization of |
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699 temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the |
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700 function was called. Error checking with this style of API is fairly simple. |
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701 |
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702 \begin{verbatim} |
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703 int err; |
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704 if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { |
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705 printf("Error: %s\n", mp_error_to_string(err)); |
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706 exit(EXIT_FAILURE); |
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707 } |
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708 \end{verbatim} |
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709 |
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710 The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal |
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711 and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. |
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712 |
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713 \section{Initialization and Clearing} |
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714 The logical starting point when actually writing multiple precision integer functions is the initialization and |
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715 clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. |
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716 |
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717 Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of |
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718 the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though |
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719 the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations |
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720 would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate |
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721 and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste |
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722 memory and become unmanageable. |
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723 |
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724 If the memory for the digits has been successfully allocated then the rest of the members of the structure must |
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725 be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set |
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726 to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. |
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727 |
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728 \subsection{Initializing an mp\_int} |
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729 An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the |
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730 structure are set to valid values. The mp\_init algorithm will perform such an action. |
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731 |
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732 \index{mp\_init} |
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733 \begin{figure}[here] |
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734 \begin{center} |
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735 \begin{tabular}{l} |
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736 \hline Algorithm \textbf{mp\_init}. \\ |
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737 \textbf{Input}. An mp\_int $a$ \\ |
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738 \textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ |
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739 \hline \\ |
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740 1. Allocate memory for \textbf{MP\_PREC} digits. \\ |
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741 2. If the allocation failed return(\textit{MP\_MEM}) \\ |
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742 3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ |
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743 \hspace{3mm}3.1 $a_n \leftarrow 0$\\ |
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744 4. $a.sign \leftarrow MP\_ZPOS$\\ |
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745 5. $a.used \leftarrow 0$\\ |
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746 6. $a.alloc \leftarrow MP\_PREC$\\ |
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747 7. Return(\textit{MP\_OKAY})\\ |
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748 \hline |
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749 \end{tabular} |
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750 \end{center} |
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751 \caption{Algorithm mp\_init} |
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752 \end{figure} |
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753 |
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754 \textbf{Algorithm mp\_init.} |
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755 The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly |
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756 manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly |
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757 a valid assumption if the input resides on the stack. |
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758 |
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759 Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for |
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760 the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} |
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761 name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} |
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762 used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest |
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763 precision number you'll be working with. |
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764 |
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765 Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow |
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766 heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack |
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767 memory and the number of heap operations will be trivial. |
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768 |
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769 Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and |
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770 \textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless |
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771 of the original condition of the input. |
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772 |
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773 \textbf{Remark.} |
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774 This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally |
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775 when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that |
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776 a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each |
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777 iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured |
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778 the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate |
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779 decrementally. |
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780 |
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781 EXAM,bn_mp_init.c |
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782 |
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783 One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It |
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784 is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The |
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785 call to mp\_init() is used only to initialize the members of the structure to a known default state. |
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786 |
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787 Here we see (line @23,XMALLOC@) the memory allocation is performed first. This allows us to exit cleanly and quickly |
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788 if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there |
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789 was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function |
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790 but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in |
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791 memory allocation routine. |
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792 |
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793 In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been |
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794 accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a |
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795 portable fashion you have to actually assign the value. The for loop (line @28,for@) performs this required |
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796 operation. |
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797 |
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798 After the memory has been successfully initialized the remainder of the members are initialized |
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799 (lines @29,used@ through @31,sign@) to their respective default states. At this point the algorithm has succeeded and |
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800 a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the |
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801 mp\_int structure has been properly initialized and is safe to use with other functions within the library. |
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802 |
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803 \subsection{Clearing an mp\_int} |
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804 When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be |
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805 returned to the application's memory pool with the mp\_clear algorithm. |
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806 |
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807 \begin{figure}[here] |
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808 \begin{center} |
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809 \begin{tabular}{l} |
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810 \hline Algorithm \textbf{mp\_clear}. \\ |
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811 \textbf{Input}. An mp\_int $a$ \\ |
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812 \textbf{Output}. The memory for $a$ shall be deallocated. \\ |
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813 \hline \\ |
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814 1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ |
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815 2. for $n$ from 0 to $a.used - 1$ do \\ |
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816 \hspace{3mm}2.1 $a_n \leftarrow 0$ \\ |
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817 3. Free the memory allocated for the digits of $a$. \\ |
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818 4. $a.used \leftarrow 0$ \\ |
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819 5. $a.alloc \leftarrow 0$ \\ |
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820 6. $a.sign \leftarrow MP\_ZPOS$ \\ |
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821 7. Return(\textit{MP\_OKAY}). \\ |
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822 \hline |
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823 \end{tabular} |
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824 \end{center} |
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825 \caption{Algorithm mp\_clear} |
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826 \end{figure} |
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827 |
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828 \textbf{Algorithm mp\_clear.} |
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829 This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that |
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830 if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal |
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831 is to free the allocated memory. |
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832 |
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833 The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this |
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834 algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid |
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835 digit pointer \textbf{dp} setting. |
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836 |
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837 Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm |
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838 with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. |
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839 |
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840 EXAM,bn_mp_clear.c |
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841 |
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842 The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line @23,a->dp != NULL@) |
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843 checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be |
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844 \textbf{NULL} in which case the if statement will evaluate to true. |
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845 |
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846 The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit. Similar to mp\_init() |
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847 the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. |
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848 |
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849 The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to |
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850 a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer |
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851 still has to be reset to \textbf{NULL} manually (line @33,NULL@). |
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852 |
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853 Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@). |
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854 |
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855 \section{Maintenance Algorithms} |
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856 |
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857 The previous sections describes how to initialize and clear an mp\_int structure. To further support operations |
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858 that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be |
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859 able to augment the precision of an mp\_int and |
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860 initialize mp\_ints with differing initial conditions. |
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861 |
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862 These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level |
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863 algorithms such as addition, multiplication and modular exponentiation. |
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864 |
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865 \subsection{Augmenting an mp\_int's Precision} |
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866 When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire |
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867 result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member |
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868 is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it |
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869 must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. |
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870 |
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871 \newpage\begin{figure}[here] |
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872 \begin{center} |
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873 \begin{tabular}{l} |
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874 \hline Algorithm \textbf{mp\_grow}. \\ |
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875 \textbf{Input}. An mp\_int $a$ and an integer $b$. \\ |
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876 \textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ |
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877 \hline \\ |
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878 1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ |
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879 2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ |
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880 3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ |
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881 4. Re-allocate the array of digits $a$ to size $v$ \\ |
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882 5. If the allocation failed then return(\textit{MP\_MEM}). \\ |
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883 6. for n from a.alloc to $v - 1$ do \\ |
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884 \hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ |
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885 7. $a.alloc \leftarrow v$ \\ |
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886 8. Return(\textit{MP\_OKAY}) \\ |
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887 \hline |
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888 \end{tabular} |
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889 \end{center} |
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890 \caption{Algorithm mp\_grow} |
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891 \end{figure} |
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892 |
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893 \textbf{Algorithm mp\_grow.} |
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894 It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to |
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895 prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. |
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896 |
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897 The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). |
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898 This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. |
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899 |
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900 It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much |
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901 akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are |
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902 assumed to contain undefined values they are initially set to zero. |
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903 |
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904 EXAM,bn_mp_grow.c |
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905 |
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906 A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line @24,alloc@) checks |
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907 if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} |
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908 the function skips the re-allocation part thus saving time. |
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909 |
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910 When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is |
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911 padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line @25, size@). The XREALLOC function is used |
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912 to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc |
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913 function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before |
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914 the re-allocation. All that is left is to clear the newly allocated digits and return. |
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915 |
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916 Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return |
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917 an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would |
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918 result in a memory leak if XREALLOC ever failed. |
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919 |
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920 \subsection{Initializing Variable Precision mp\_ints} |
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921 Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size |
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922 of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it |
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923 will allocate \textit{at least} a specified number of digits. |
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924 |
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925 \begin{figure}[here] |
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926 \begin{small} |
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927 \begin{center} |
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928 \begin{tabular}{l} |
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929 \hline Algorithm \textbf{mp\_init\_size}. \\ |
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930 \textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ |
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931 \textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ |
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932 \hline \\ |
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933 1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ |
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934 2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ |
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935 3. Allocate $v$ digits. \\ |
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936 4. for $n$ from $0$ to $v - 1$ do \\ |
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937 \hspace{3mm}4.1 $a_n \leftarrow 0$ \\ |
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938 5. $a.sign \leftarrow MP\_ZPOS$\\ |
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939 6. $a.used \leftarrow 0$\\ |
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940 7. $a.alloc \leftarrow v$\\ |
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941 8. Return(\textit{MP\_OKAY})\\ |
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942 \hline |
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943 \end{tabular} |
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944 \end{center} |
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945 \end{small} |
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946 \caption{Algorithm mp\_init\_size} |
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947 \end{figure} |
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948 |
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949 \textbf{Algorithm mp\_init\_size.} |
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950 This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of |
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951 digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a |
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952 multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial |
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953 allocations from becoming a bottleneck in the rest of the algorithms. |
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954 |
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955 Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This |
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956 particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is |
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957 correct no further memory re-allocations are required to work with the mp\_int. |
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958 |
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959 EXAM,bn_mp_init_size.c |
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960 |
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961 The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of |
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962 \textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the |
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963 mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be |
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964 returned (line @27,return@). |
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965 |
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966 The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The |
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967 \textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set |
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968 to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@). If the function |
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969 returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the |
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970 functions to work with. |
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971 |
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972 \subsection{Multiple Integer Initializations and Clearings} |
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973 Occasionally a function will require a series of mp\_int data types to be made available simultaneously. |
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974 The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single |
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975 statement. It is essentially a shortcut to multiple initializations. |
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976 |
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977 \newpage\begin{figure}[here] |
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978 \begin{center} |
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979 \begin{tabular}{l} |
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980 \hline Algorithm \textbf{mp\_init\_multi}. \\ |
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981 \textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ |
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982 \textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ |
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983 \hline \\ |
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984 1. for $n$ from 0 to $k - 1$ do \\ |
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985 \hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ |
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986 \hspace{+3mm}1.2. If initialization failed then do \\ |
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987 \hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ |
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988 \hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ |
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989 \hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ |
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990 2. Return(\textit{MP\_OKAY}) \\ |
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991 \hline |
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992 \end{tabular} |
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993 \end{center} |
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994 \caption{Algorithm mp\_init\_multi} |
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995 \end{figure} |
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996 |
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997 \textbf{Algorithm mp\_init\_multi.} |
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998 The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected |
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999 (\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' |
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1000 initialization which allows for quick recovery from runtime errors. |
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1001 |
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1002 EXAM,bn_mp_init_multi.c |
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1003 |
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1004 This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int |
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1005 structures in an actual C array they are simply passed as arguments to the function. This function makes use of the |
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1006 ``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument |
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1007 appended on the right. |
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1008 |
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1009 The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count |
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1010 $n$ of succesfully initialized mp\_int structures is maintained (line @47,n++@) such that if a failure does occur, |
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1011 the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@). |
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1012 |
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1013 |
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1014 \subsection{Clamping Excess Digits} |
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1015 When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of |
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1016 the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a |
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1017 $j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ |
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1018 though, with no final carry into the last position. However, suppose the destination had to be first expanded |
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1019 (\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. |
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1020 That would be a considerable waste of time since heap operations are relatively slow. |
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1021 |
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1022 The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function |
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1023 terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked |
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1024 there would be an excess high order zero digit. |
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1025 |
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1026 For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit |
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1027 will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would |
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1028 accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very |
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1029 low the representation is excessively large. |
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1030 |
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1031 The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the |
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1032 \textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a |
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1033 positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to |
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1034 \textbf{MP\_ZPOS}. |
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1035 |
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1036 \begin{figure}[here] |
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1037 \begin{center} |
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1038 \begin{tabular}{l} |
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1039 \hline Algorithm \textbf{mp\_clamp}. \\ |
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1040 \textbf{Input}. An mp\_int $a$ \\ |
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1041 \textbf{Output}. Any excess leading zero digits of $a$ are removed \\ |
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1042 \hline \\ |
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1043 1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ |
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1044 \hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ |
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1045 2. if $a.used = 0$ then do \\ |
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1046 \hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ |
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1047 \hline \\ |
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1048 \end{tabular} |
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1049 \end{center} |
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1050 \caption{Algorithm mp\_clamp} |
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1051 \end{figure} |
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1052 |
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1053 \textbf{Algorithm mp\_clamp.} |
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1054 As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at |
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1055 the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for |
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1056 when all of the digits are zero to ensure that the mp\_int is valid at all times. |
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1057 |
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1058 EXAM,bn_mp_clamp.c |
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1059 |
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1060 Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming |
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1061 language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is |
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1062 important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously |
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1063 undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not |
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1064 the pointer ``a''. |
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1065 |
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1066 \section*{Exercises} |
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1067 \begin{tabular}{cl} |
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1068 $\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ |
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1069 & \\ |
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1070 $\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ |
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1071 & \\ |
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1072 $\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ |
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1073 & encryption when $\beta = 2^{28}$. \\ |
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1074 & \\ |
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1075 $\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ |
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1076 & \\ |
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1077 $\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ |
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1078 & \\ |
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1079 \end{tabular} |
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1080 |
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1081 |
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1082 %%% |
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1083 % CHAPTER FOUR |
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1084 %%% |
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1085 |
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1086 \chapter{Basic Operations} |
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1087 |
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1088 \section{Introduction} |
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1089 In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining |
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1090 mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low |
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1091 level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they |
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1092 work before proceeding since these algorithms will be used almost intrinsically in the following chapters. |
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1093 |
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1094 The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of |
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1095 mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures |
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1096 represent. |
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1097 |
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1098 \section{Assigning Values to mp\_int Structures} |
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1099 \subsection{Copying an mp\_int} |
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1100 Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making |
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1101 a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same |
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1102 value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. |
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1103 |
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1104 \newpage\begin{figure}[here] |
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1105 \begin{center} |
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1106 \begin{tabular}{l} |
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1107 \hline Algorithm \textbf{mp\_copy}. \\ |
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1108 \textbf{Input}. An mp\_int $a$ and $b$. \\ |
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1109 \textbf{Output}. Store a copy of $a$ in $b$. \\ |
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1110 \hline \\ |
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1111 1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ |
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1112 2. for $n$ from 0 to $a.used - 1$ do \\ |
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1113 \hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ |
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1114 3. for $n$ from $a.used$ to $b.used - 1$ do \\ |
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1115 \hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ |
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1116 4. $b.used \leftarrow a.used$ \\ |
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1117 5. $b.sign \leftarrow a.sign$ \\ |
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1118 6. return(\textit{MP\_OKAY}) \\ |
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1119 \hline |
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1120 \end{tabular} |
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1121 \end{center} |
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1122 \caption{Algorithm mp\_copy} |
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1123 \end{figure} |
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1124 |
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1125 \textbf{Algorithm mp\_copy.} |
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1126 This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will |
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1127 represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the |
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1128 mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. |
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1129 |
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1130 If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow |
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1131 algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two |
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1132 and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of |
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1133 $b$. |
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1134 |
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1135 \textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the |
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1136 text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in |
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1137 step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is |
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1138 limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return |
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1139 the error code itself. However, the C code presented will demonstrate all of the error handling logic required to |
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1140 implement the pseudo-code. |
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1141 |
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1142 EXAM,bn_mp_copy.c |
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1143 |
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1144 Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output |
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1145 mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without |
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1146 copying digits (line @24,a == b@). |
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1147 |
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1148 The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than |
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1149 $a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines @29,alloc@ to @33,}@). In order to |
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1150 simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits |
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1151 of the mp\_ints $a$ and $b$ respectively. These aliases (lines @42,tmpa@ and @45,tmpb@) allow the compiler to access the digits without first dereferencing the |
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1152 mp\_int pointers and then subsequently the pointer to the digits. |
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1153 |
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1154 After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess |
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1155 digits of $b$ are set to zero (lines @53,for@ to @55,}@). Both ``for'' loops make use of the pointer aliases and in |
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1156 fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization |
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1157 allows the alias to stay in a machine register fairly easy between the two loops. |
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1158 |
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1159 \textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will |
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1160 be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the |
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1161 number of pointer dereferencing operations required to access data. For example, a for loop may resemble |
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1162 |
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1163 \begin{alltt} |
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1164 for (x = 0; x < 100; x++) \{ |
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1165 a->num[4]->dp[x] = 0; |
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1166 \} |
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1167 \end{alltt} |
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1168 |
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1169 This could be re-written using aliases as |
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1170 |
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1171 \begin{alltt} |
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1172 mp_digit *tmpa; |
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1173 a = a->num[4]->dp; |
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1174 for (x = 0; x < 100; x++) \{ |
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1175 *a++ = 0; |
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1176 \} |
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1177 \end{alltt} |
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1178 |
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1179 In this case an alias is used to access the |
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1180 array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required |
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1181 as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. |
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1182 |
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1183 The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations |
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1184 may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may |
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1185 work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer |
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1186 aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code |
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1187 stands a better chance of being faster. |
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1188 |
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1189 The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' |
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1190 loop of the function mp\_copy() re-written to not use pointer aliases. |
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1191 |
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1192 \begin{alltt} |
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1193 /* copy all the digits */ |
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1194 for (n = 0; n < a->used; n++) \{ |
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1195 b->dp[n] = a->dp[n]; |
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1196 \} |
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1197 \end{alltt} |
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1198 |
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1199 Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more |
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1200 complicated as there are four variables within the statement instead of just two. |
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1201 |
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1202 \subsubsection{Nested Statements} |
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1203 Another commonly used technique in the source routines is that certain sections of code are nested. This is used in |
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1204 particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) |
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1205 will typically have three different phases. First the temporaries are initialized, then the columns calculated and |
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1206 finally the carries are propagated. In this example the middle column production phase will typically be nested as it |
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1207 uses temporary variables and aliases the most. |
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1208 |
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1209 The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result |
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1210 the various temporary variables required do not propagate into other sections of code. |
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1211 |
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1212 |
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1213 \subsection{Creating a Clone} |
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1214 Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int |
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1215 and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is |
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1216 useful within functions that need to modify an argument but do not wish to actually modify the original copy. The |
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1217 mp\_init\_copy algorithm has been designed to help perform this task. |
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1218 |
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1219 \begin{figure}[here] |
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1220 \begin{center} |
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1221 \begin{tabular}{l} |
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1222 \hline Algorithm \textbf{mp\_init\_copy}. \\ |
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1223 \textbf{Input}. An mp\_int $a$ and $b$\\ |
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1224 \textbf{Output}. $a$ is initialized to be a copy of $b$. \\ |
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1225 \hline \\ |
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1226 1. Init $a$. (\textit{mp\_init}) \\ |
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1227 2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ |
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1228 3. Return the status of the copy operation. \\ |
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1229 \hline |
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1230 \end{tabular} |
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1231 \end{center} |
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1232 \caption{Algorithm mp\_init\_copy} |
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1233 \end{figure} |
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1234 |
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1235 \textbf{Algorithm mp\_init\_copy.} |
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1236 This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As |
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1237 such this algorithm will perform two operations in one step. |
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1238 |
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1239 EXAM,bn_mp_init_copy.c |
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1240 |
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1241 This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that |
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1242 \textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call |
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1243 and \textbf{a} will be left intact. |
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1244 |
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1245 \section{Zeroing an Integer} |
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1246 Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to |
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1247 perform this task. |
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1248 |
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1249 \begin{figure}[here] |
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1250 \begin{center} |
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1251 \begin{tabular}{l} |
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1252 \hline Algorithm \textbf{mp\_zero}. \\ |
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1253 \textbf{Input}. An mp\_int $a$ \\ |
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1254 \textbf{Output}. Zero the contents of $a$ \\ |
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1255 \hline \\ |
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1256 1. $a.used \leftarrow 0$ \\ |
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1257 2. $a.sign \leftarrow$ MP\_ZPOS \\ |
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1258 3. for $n$ from 0 to $a.alloc - 1$ do \\ |
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1259 \hspace{3mm}3.1 $a_n \leftarrow 0$ \\ |
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1260 \hline |
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1261 \end{tabular} |
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1262 \end{center} |
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1263 \caption{Algorithm mp\_zero} |
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1264 \end{figure} |
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1265 |
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1266 \textbf{Algorithm mp\_zero.} |
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1267 This algorithm simply resets a mp\_int to the default state. |
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1268 |
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1269 EXAM,bn_mp_zero.c |
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1270 |
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1271 After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the |
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1272 \textbf{sign} variable is set to \textbf{MP\_ZPOS}. |
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1273 |
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1274 \section{Sign Manipulation} |
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1275 \subsection{Absolute Value} |
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1276 With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute |
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1277 the absolute value of an mp\_int. |
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1278 |
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1279 \begin{figure}[here] |
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1280 \begin{center} |
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1281 \begin{tabular}{l} |
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1282 \hline Algorithm \textbf{mp\_abs}. \\ |
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1283 \textbf{Input}. An mp\_int $a$ \\ |
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1284 \textbf{Output}. Computes $b = \vert a \vert$ \\ |
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1285 \hline \\ |
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1286 1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ |
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1287 2. If the copy failed return(\textit{MP\_MEM}). \\ |
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1288 3. $b.sign \leftarrow MP\_ZPOS$ \\ |
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1289 4. Return(\textit{MP\_OKAY}) \\ |
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1290 \hline |
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1291 \end{tabular} |
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1292 \end{center} |
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1293 \caption{Algorithm mp\_abs} |
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1294 \end{figure} |
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1295 |
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1296 \textbf{Algorithm mp\_abs.} |
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1297 This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an |
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1298 algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, |
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1299 for instance, the developer to pass the same mp\_int as the source and destination to this function without addition |
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1300 logic to handle it. |
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1301 |
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1302 EXAM,bn_mp_abs.c |
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1303 |
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1304 This fairly trivial algorithm first eliminates non--required duplications (line @27,a != b@) and then sets the |
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1305 \textbf{sign} flag to \textbf{MP\_ZPOS}. |
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1306 |
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1307 \subsection{Integer Negation} |
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1308 With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute |
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1309 the negative of an mp\_int input. |
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1310 |
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1311 \begin{figure}[here] |
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1312 \begin{center} |
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1313 \begin{tabular}{l} |
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1314 \hline Algorithm \textbf{mp\_neg}. \\ |
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1315 \textbf{Input}. An mp\_int $a$ \\ |
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1316 \textbf{Output}. Computes $b = -a$ \\ |
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1317 \hline \\ |
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1318 1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ |
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1319 2. If the copy failed return(\textit{MP\_MEM}). \\ |
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1320 3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ |
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1321 4. If $a.sign = MP\_ZPOS$ then do \\ |
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1322 \hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ |
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1323 5. else do \\ |
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1324 \hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ |
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1325 6. Return(\textit{MP\_OKAY}) \\ |
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1326 \hline |
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1327 \end{tabular} |
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1328 \end{center} |
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1329 \caption{Algorithm mp\_neg} |
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1330 \end{figure} |
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1331 |
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1332 \textbf{Algorithm mp\_neg.} |
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1333 This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then |
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1334 the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if |
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1335 $a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return |
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1336 zero as negative. |
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1337 |
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1338 EXAM,bn_mp_neg.c |
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1339 |
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1340 Like mp\_abs() this function avoids non--required duplications (line @21,a != b@) and then sets the sign. We |
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1341 have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero |
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1342 than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. |
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1343 |
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1344 \section{Small Constants} |
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1345 \subsection{Setting Small Constants} |
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1346 Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. |
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1347 |
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1348 \newpage\begin{figure}[here] |
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1349 \begin{center} |
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1350 \begin{tabular}{l} |
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1351 \hline Algorithm \textbf{mp\_set}. \\ |
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1352 \textbf{Input}. An mp\_int $a$ and a digit $b$ \\ |
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1353 \textbf{Output}. Make $a$ equivalent to $b$ \\ |
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1354 \hline \\ |
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1355 1. Zero $a$ (\textit{mp\_zero}). \\ |
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1356 2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ |
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1357 3. $a.used \leftarrow \left \lbrace \begin{array}{ll} |
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1358 1 & \mbox{if }a_0 > 0 \\ |
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1359 0 & \mbox{if }a_0 = 0 |
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1360 \end{array} \right .$ \\ |
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1361 \hline |
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1362 \end{tabular} |
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1363 \end{center} |
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1364 \caption{Algorithm mp\_set} |
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1365 \end{figure} |
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1366 |
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1367 \textbf{Algorithm mp\_set.} |
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1368 This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The |
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1369 single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. |
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1370 |
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1371 EXAM,bn_mp_set.c |
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1372 |
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1373 First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a |
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1374 small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count |
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1375 is zero. Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@). After this step we have to |
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1376 check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise |
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1377 to zero. |
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1378 |
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1379 We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with |
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1380 $2^k - 1$ will perform the same operation. |
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1381 |
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1382 One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses |
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1383 this function should take that into account. Only trivially small constants can be set using this function. |
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1384 |
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1385 \subsection{Setting Large Constants} |
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1386 To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' |
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1387 data type as input and will always treat it as a 32-bit integer. |
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1388 |
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1389 \begin{figure}[here] |
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1390 \begin{center} |
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1391 \begin{tabular}{l} |
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1392 \hline Algorithm \textbf{mp\_set\_int}. \\ |
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1393 \textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ |
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1394 \textbf{Output}. Make $a$ equivalent to $b$ \\ |
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1395 \hline \\ |
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1396 1. Zero $a$ (\textit{mp\_zero}) \\ |
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1397 2. for $n$ from 0 to 7 do \\ |
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1398 \hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ |
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1399 \hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ |
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1400 \hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ |
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1401 \hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ |
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1402 3. Clamp excess used digits (\textit{mp\_clamp}) \\ |
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1403 \hline |
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1404 \end{tabular} |
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1405 \end{center} |
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1406 \caption{Algorithm mp\_set\_int} |
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1407 \end{figure} |
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1408 |
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1409 \textbf{Algorithm mp\_set\_int.} |
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1410 The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the |
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1411 mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the |
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1412 next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is |
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1413 incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have |
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1414 zero digits used and the newly added four bits would be ignored. |
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1415 |
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1416 Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. |
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1417 |
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1418 EXAM,bn_mp_set_int.c |
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1419 |
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1420 This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird |
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1421 addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not |
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1422 seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ |
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1423 as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps |
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1424 the number of used digits low. |
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1425 |
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1426 \section{Comparisons} |
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1427 \subsection{Unsigned Comparisions} |
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1428 Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, |
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1429 to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ |
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1430 to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude |
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1431 positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. |
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1432 |
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1433 The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two |
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1434 mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the |
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1435 signs are known to agree in advance. |
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1436 |
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1437 To facilitate working with the results of the comparison functions three constants are required. |
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1438 |
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1439 \begin{figure}[here] |
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1440 \begin{center} |
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1441 \begin{tabular}{|r|l|} |
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1442 \hline \textbf{Constant} & \textbf{Meaning} \\ |
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1443 \hline \textbf{MP\_GT} & Greater Than \\ |
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1444 \hline \textbf{MP\_EQ} & Equal To \\ |
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1445 \hline \textbf{MP\_LT} & Less Than \\ |
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1446 \hline |
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1447 \end{tabular} |
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1448 \end{center} |
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1449 \caption{Comparison Return Codes} |
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1450 \end{figure} |
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1451 |
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1452 \begin{figure}[here] |
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1453 \begin{center} |
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1454 \begin{tabular}{l} |
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1455 \hline Algorithm \textbf{mp\_cmp\_mag}. \\ |
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1456 \textbf{Input}. Two mp\_ints $a$ and $b$. \\ |
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1457 \textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ |
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1458 \hline \\ |
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1459 1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ |
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1460 2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ |
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1461 3. for n from $a.used - 1$ to 0 do \\ |
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1462 \hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ |
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1463 \hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ |
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1464 4. Return(\textit{MP\_EQ}) \\ |
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1465 \hline |
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1466 \end{tabular} |
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1467 \end{center} |
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1468 \caption{Algorithm mp\_cmp\_mag} |
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1469 \end{figure} |
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1470 |
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1471 \textbf{Algorithm mp\_cmp\_mag.} |
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1472 By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return |
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1473 \textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. |
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1474 Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. |
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1475 If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. |
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1476 |
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1477 By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to |
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1478 the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. |
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1479 |
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1480 EXAM,bn_mp_cmp_mag.c |
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1481 |
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1482 The two if statements (lines @24,if@ and @28,if@) compare the number of digits in the two inputs. These two are |
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1483 performed before all of the digits are compared since it is a very cheap test to perform and can potentially save |
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1484 considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be |
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1485 smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. |
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1486 |
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1487 |
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1488 |
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1489 \subsection{Signed Comparisons} |
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1490 Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude |
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1491 comparison a trivial signed comparison algorithm can be written. |
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1492 |
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1493 \begin{figure}[here] |
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1494 \begin{center} |
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1495 \begin{tabular}{l} |
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1496 \hline Algorithm \textbf{mp\_cmp}. \\ |
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1497 \textbf{Input}. Two mp\_ints $a$ and $b$ \\ |
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1498 \textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ |
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1499 \hline \\ |
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1500 1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ |
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1501 2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ |
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1502 3. if $a.sign = MP\_NEG$ then \\ |
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1503 \hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ |
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1504 4 Otherwise \\ |
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1505 \hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ |
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1506 \hline |
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1507 \end{tabular} |
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1508 \end{center} |
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1509 \caption{Algorithm mp\_cmp} |
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1510 \end{figure} |
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1511 |
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1512 \textbf{Algorithm mp\_cmp.} |
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1513 The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate |
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1514 comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step |
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1515 three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then |
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1516 $\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. |
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1517 |
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1518 EXAM,bn_mp_cmp.c |
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1519 |
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1520 The two if statements (lines @22,if@ and @26,if@) perform the initial sign comparison. If the signs are not the equal then which ever |
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1521 has the positive sign is larger. The inputs are compared (line @30,if@) based on magnitudes. If the signs were both |
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1522 negative then the unsigned comparison is performed in the opposite direction (line @31,mp_cmp_mag@). Otherwise, the signs are assumed to |
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1523 be both positive and a forward direction unsigned comparison is performed. |
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1524 |
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1525 \section*{Exercises} |
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1526 \begin{tabular}{cl} |
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1527 $\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ |
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1528 & \\ |
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1529 $\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ |
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1530 & of two random digits (of equal magnitude) before a difference is found. \\ |
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1531 & \\ |
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1532 $\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ |
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1533 & on the observations made in the previous problem. \\ |
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1534 & |
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1535 \end{tabular} |
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1536 |
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1537 \chapter{Basic Arithmetic} |
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1538 \section{Introduction} |
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1539 At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been |
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1540 established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These |
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1541 algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important |
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1542 that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms |
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1543 which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. |
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1544 |
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1545 MARK,SHIFTS |
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1546 All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right |
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1547 logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real |
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1548 number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). |
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1549 Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. |
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1550 For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. |
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1551 |
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1552 One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed |
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1553 from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the |
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1554 result is $110_2$. |
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1555 |
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1556 \section{Addition and Subtraction} |
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1557 In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers |
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1558 $a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. |
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1559 As a result subtraction can be performed with a trivial series of logical operations and an addition. |
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1560 |
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1561 However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the |
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1562 sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or |
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1563 subtraction algorithms with the sign fixed up appropriately. |
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1564 |
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1565 The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of |
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1566 the integers respectively. |
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1567 |
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1568 \subsection{Low Level Addition} |
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1569 An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the |
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1570 trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. |
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1571 Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. |
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1572 |
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1573 \newpage |
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1574 \begin{figure}[!here] |
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1575 \begin{center} |
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1576 \begin{small} |
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1577 \begin{tabular}{l} |
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1578 \hline Algorithm \textbf{s\_mp\_add}. \\ |
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1579 \textbf{Input}. Two mp\_ints $a$ and $b$ \\ |
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1580 \textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ |
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1581 \hline \\ |
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1582 1. if $a.used > b.used$ then \\ |
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1583 \hspace{+3mm}1.1 $min \leftarrow b.used$ \\ |
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1584 \hspace{+3mm}1.2 $max \leftarrow a.used$ \\ |
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1585 \hspace{+3mm}1.3 $x \leftarrow a$ \\ |
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1586 2. else \\ |
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1587 \hspace{+3mm}2.1 $min \leftarrow a.used$ \\ |
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1588 \hspace{+3mm}2.2 $max \leftarrow b.used$ \\ |
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1589 \hspace{+3mm}2.3 $x \leftarrow b$ \\ |
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1590 3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ |
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1591 4. $oldused \leftarrow c.used$ \\ |
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1592 5. $c.used \leftarrow max + 1$ \\ |
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1593 6. $u \leftarrow 0$ \\ |
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1594 7. for $n$ from $0$ to $min - 1$ do \\ |
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1595 \hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ |
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1596 \hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ |
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1597 \hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ |
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1598 8. if $min \ne max$ then do \\ |
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1599 \hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ |
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1600 \hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ |
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1601 \hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ |
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1602 \hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ |
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1603 9. $c_{max} \leftarrow u$ \\ |
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1604 10. if $olduse > max$ then \\ |
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1605 \hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ |
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1606 \hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ |
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1607 11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ |
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1608 12. Return(\textit{MP\_OKAY}) \\ |
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1609 \hline |
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1610 \end{tabular} |
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1611 \end{small} |
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1612 \end{center} |
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1613 \caption{Algorithm s\_mp\_add} |
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1614 \end{figure} |
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1615 |
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1616 \textbf{Algorithm s\_mp\_add.} |
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1617 This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. |
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1618 Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the |
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1619 MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. |
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1620 |
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1621 The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic |
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1622 will simply add all of the smallest input to the largest input and store that first part of the result in the |
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1623 destination. Then it will apply a simpler addition loop to excess digits of the larger input. |
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1624 |
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1625 The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two |
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1626 inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the |
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1627 same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum |
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1628 of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. |
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1629 |
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1630 At this point the first addition loop will go through as many digit positions that both inputs have. The carry |
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1631 variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce |
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1632 one digit of the summand. First |
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1633 two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored |
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1634 in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. |
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1635 |
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1636 Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias |
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1637 for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits |
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1638 and the carry to the destination. |
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1639 |
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1640 The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. |
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1641 |
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1642 |
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1643 EXAM,bn_s_mp_add.c |
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1644 |
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1645 We first sort (lines @27,if@ to @35,}@) the inputs based on magnitude and determine the $min$ and $max$ variables. |
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1646 Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we |
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1647 grow the destination (@37,init@ to @42,}@) ensure that it can accomodate the result of the addition. |
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1648 |
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1649 Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on |
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1650 lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the |
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1651 compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. |
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1652 |
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1653 The initial carry $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type |
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1654 compatibility within the implementation. The initial addition (line @66,for@ to @75,}@) adds digits from |
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1655 both inputs until the smallest input runs out of digits. Similarly the conditional addition loop |
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1656 (line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs. The addition is finished |
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1657 with the final carry being stored in $tmpc$ (line @94,tmpc++@). Note the ``++'' operator within the same expression. |
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1658 After line @94,tmpc++@, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful |
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1659 for the next loop (line @97,for@ to @99,}@) which set any old upper digits to zero. |
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1660 |
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1661 \subsection{Low Level Subtraction} |
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1662 The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the |
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1663 unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must |
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1664 be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. |
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1665 This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. |
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1666 |
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1667 MARK,GAMMA |
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1668 |
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1669 For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent |
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1670 the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For |
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1671 this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a |
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1672 mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). |
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1673 |
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1674 For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' |
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1675 data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. |
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1676 |
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1677 \newpage\begin{figure}[!here] |
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1678 \begin{center} |
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1679 \begin{small} |
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1680 \begin{tabular}{l} |
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1681 \hline Algorithm \textbf{s\_mp\_sub}. \\ |
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1682 \textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ |
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1683 \textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ |
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1684 \hline \\ |
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1685 1. $min \leftarrow b.used$ \\ |
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1686 2. $max \leftarrow a.used$ \\ |
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1687 3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ |
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1688 4. $oldused \leftarrow c.used$ \\ |
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1689 5. $c.used \leftarrow max$ \\ |
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1690 6. $u \leftarrow 0$ \\ |
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1691 7. for $n$ from $0$ to $min - 1$ do \\ |
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1692 \hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ |
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1693 \hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ |
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1694 \hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ |
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1695 8. if $min < max$ then do \\ |
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1696 \hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ |
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1697 \hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ |
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1698 \hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ |
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1699 \hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ |
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1700 9. if $oldused > max$ then do \\ |
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1701 \hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ |
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1702 \hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ |
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1703 10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ |
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1704 11. Return(\textit{MP\_OKAY}). \\ |
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1705 \hline |
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1706 \end{tabular} |
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1707 \end{small} |
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1708 \end{center} |
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1709 \caption{Algorithm s\_mp\_sub} |
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1710 \end{figure} |
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1711 |
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1712 \textbf{Algorithm s\_mp\_sub.} |
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1713 This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when |
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1714 passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This |
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1715 algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case |
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1716 of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. |
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1717 |
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1718 The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 |
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1719 set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at |
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1720 most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and |
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1721 set to the maximal count for the operation. |
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1722 |
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1723 The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision |
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1724 subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction |
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1725 loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. |
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1726 |
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1727 For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to |
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1728 the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the |
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1729 third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the |
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1730 way to the most significant bit. |
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1731 |
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1732 Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most |
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1733 significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that |
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1734 is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the |
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1735 carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. |
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1736 |
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1737 If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step |
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1738 10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. |
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1739 |
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1740 EXAM,bn_s_mp_sub.c |
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1741 |
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1742 Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded |
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1743 (lines @24,min@ and @25,max@). In reality the $min$ and $max$ variables are only aliases and are only |
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1744 used to make the source code easier to read. Again the pointer alias optimization is used |
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1745 within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized |
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1746 (lines @42,tmpa@, @43,tmpb@ and @44,tmpc@) for $a$, $b$ and $c$ respectively. |
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1747 |
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1748 The first subtraction loop (lines @47,u = 0@ through @61,}@) subtract digits from both inputs until the smaller of |
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1749 the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' |
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1750 method of extracting the carry (line @57, >>@). The traditional method for extracting the carry would be to shift |
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1751 by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of |
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1752 the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry |
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1753 extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the |
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1754 most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This |
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1755 optimization only works on twos compliment machines which is a safe assumption to make. |
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1756 |
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1757 If $a$ has a larger magnitude than $b$ an additional loop (lines @64,for@ through @73,}@) is required to propagate |
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1758 the carry through $a$ and copy the result to $c$. |
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1759 |
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1760 \subsection{High Level Addition} |
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1761 Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be |
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1762 established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data |
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1763 types. |
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1764 |
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1765 Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} |
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1766 flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. |
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1767 |
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1768 \begin{figure}[!here] |
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1769 \begin{center} |
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1770 \begin{tabular}{l} |
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1771 \hline Algorithm \textbf{mp\_add}. \\ |
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1772 \textbf{Input}. Two mp\_ints $a$ and $b$ \\ |
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1773 \textbf{Output}. The signed addition $c = a + b$. \\ |
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1774 \hline \\ |
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1775 1. if $a.sign = b.sign$ then do \\ |
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1776 \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ |
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1777 \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ |
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1778 2. else do \\ |
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1779 \hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ |
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1780 \hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ |
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1781 \hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ |
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1782 \hspace{3mm}2.2 else do \\ |
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1783 \hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ |
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1784 \hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ |
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1785 3. Return(\textit{MP\_OKAY}). \\ |
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1786 \hline |
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1787 \end{tabular} |
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1788 \end{center} |
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1789 \caption{Algorithm mp\_add} |
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1790 \end{figure} |
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1791 |
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1792 \textbf{Algorithm mp\_add.} |
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1793 This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from |
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1794 either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly |
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1795 straightforward but restricted since subtraction can only produce positive results. |
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1796 |
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1797 \begin{figure}[here] |
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1798 \begin{small} |
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1799 \begin{center} |
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1800 \begin{tabular}{|c|c|c|c|c|} |
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1801 \hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ |
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1802 \hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ |
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1803 \hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ |
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1804 \hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ |
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1805 \hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ |
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1806 \hline &&&&\\ |
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1807 |
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1808 \hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ |
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1809 \hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ |
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1810 |
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1811 \hline &&&&\\ |
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1812 |
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1813 \hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ |
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1814 \hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ |
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1815 |
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1816 \hline |
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1817 \end{tabular} |
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1818 \end{center} |
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1819 \end{small} |
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1820 \caption{Addition Guide Chart} |
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1821 \label{fig:AddChart} |
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1822 \end{figure} |
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1823 |
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1824 Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three |
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1825 specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are |
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1826 forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best |
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1827 follows how the implementation actually was achieved. |
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1828 |
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1829 Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms |
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1830 s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} |
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1831 to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. |
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1832 |
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1833 For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would |
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1834 produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp |
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1835 within algorithm s\_mp\_add will force $-0$ to become $0$. |
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1836 |
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1837 EXAM,bn_mp_add.c |
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1838 |
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1839 The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which |
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1840 is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without |
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1841 explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower |
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1842 level functions do so. Returning their return code is sufficient. |
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1843 |
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1844 \subsection{High Level Subtraction} |
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1845 The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. |
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1846 |
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1847 \newpage\begin{figure}[!here] |
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1848 \begin{center} |
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1849 \begin{tabular}{l} |
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1850 \hline Algorithm \textbf{mp\_sub}. \\ |
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1851 \textbf{Input}. Two mp\_ints $a$ and $b$ \\ |
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1852 \textbf{Output}. The signed subtraction $c = a - b$. \\ |
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1853 \hline \\ |
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1854 1. if $a.sign \ne b.sign$ then do \\ |
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1855 \hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ |
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1856 \hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ |
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1857 2. else do \\ |
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1858 \hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ |
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1859 \hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ |
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1860 \hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ |
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1861 \hspace{3mm}2.2 else do \\ |
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1862 \hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} |
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1863 MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ |
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1864 MP\_NEG & \mbox{otherwise} \\ |
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1865 \end{array} \right .$ \\ |
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1866 \hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ |
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1867 3. Return(\textit{MP\_OKAY}). \\ |
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1868 \hline |
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1869 \end{tabular} |
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1870 \end{center} |
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1871 \caption{Algorithm mp\_sub} |
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1872 \end{figure} |
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1873 |
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1874 \textbf{Algorithm mp\_sub.} |
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1875 This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or |
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1876 \cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and |
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1877 the operations required. |
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1878 |
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1879 \begin{figure}[!here] |
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1880 \begin{small} |
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1881 \begin{center} |
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1882 \begin{tabular}{|c|c|c|c|c|} |
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1883 \hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ |
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1884 \hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ |
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1885 \hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ |
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1886 \hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ |
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1887 \hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ |
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1888 \hline &&&& \\ |
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1889 \hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ |
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1890 \hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ |
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1891 \hline &&&& \\ |
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1892 \hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ |
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1893 \hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ |
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1894 \hline |
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1895 \end{tabular} |
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1896 \end{center} |
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1897 \end{small} |
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1898 \caption{Subtraction Guide Chart} |
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1899 \label{fig:SubChart} |
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1900 \end{figure} |
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1901 |
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1902 Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the |
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1903 algorithm from producing $-a - -a = -0$ as a result. |
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1904 |
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1905 EXAM,bn_mp_sub.c |
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1906 |
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1907 Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations |
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1908 and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a |
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1909 ``greater than or equal to'' comparison. |
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1910 |
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1911 \section{Bit and Digit Shifting} |
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1912 MARK,POLY |
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1913 It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. |
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1914 This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. |
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1915 |
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1916 In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift |
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1917 the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations |
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1918 are on radix-$\beta$ digits. |
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1919 |
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1920 \subsection{Multiplication by Two} |
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1921 |
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1922 In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient |
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1923 operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. |
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1924 |
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1925 \newpage\begin{figure}[!here] |
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1926 \begin{small} |
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1927 \begin{center} |
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1928 \begin{tabular}{l} |
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1929 \hline Algorithm \textbf{mp\_mul\_2}. \\ |
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1930 \textbf{Input}. One mp\_int $a$ \\ |
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1931 \textbf{Output}. $b = 2a$. \\ |
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1932 \hline \\ |
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1933 1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ |
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1934 2. $oldused \leftarrow b.used$ \\ |
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1935 3. $b.used \leftarrow a.used$ \\ |
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1936 4. $r \leftarrow 0$ \\ |
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1937 5. for $n$ from 0 to $a.used - 1$ do \\ |
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1938 \hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ |
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1939 \hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ |
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1940 \hspace{3mm}5.3 $r \leftarrow rr$ \\ |
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1941 6. If $r \ne 0$ then do \\ |
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1942 \hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ |
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1943 \hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ |
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1944 7. If $b.used < oldused - 1$ then do \\ |
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1945 \hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ |
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1946 \hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ |
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1947 8. $b.sign \leftarrow a.sign$ \\ |
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1948 9. Return(\textit{MP\_OKAY}).\\ |
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1949 \hline |
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1950 \end{tabular} |
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1951 \end{center} |
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1952 \end{small} |
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1953 \caption{Algorithm mp\_mul\_2} |
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1954 \end{figure} |
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1955 |
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1956 \textbf{Algorithm mp\_mul\_2.} |
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1957 This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such |
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1958 an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since |
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1959 it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. |
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1960 |
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1961 Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count |
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1962 is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. |
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1963 |
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1964 Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together |
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1965 are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to |
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1966 obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus |
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1967 the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with |
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1968 forwarding the carry to the next iteration. |
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1969 |
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1970 Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. |
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1971 Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. |
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1972 |
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1973 EXAM,bn_mp_mul_2.c |
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1974 |
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1975 This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference |
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1976 is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. |
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1977 |
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1978 \subsection{Division by Two} |
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1979 A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. |
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1980 |
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1981 \newpage\begin{figure}[!here] |
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1982 \begin{small} |
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1983 \begin{center} |
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1984 \begin{tabular}{l} |
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1985 \hline Algorithm \textbf{mp\_div\_2}. \\ |
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1986 \textbf{Input}. One mp\_int $a$ \\ |
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1987 \textbf{Output}. $b = a/2$. \\ |
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1988 \hline \\ |
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1989 1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ |
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1990 2. If the reallocation failed return(\textit{MP\_MEM}). \\ |
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1991 3. $oldused \leftarrow b.used$ \\ |
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1992 4. $b.used \leftarrow a.used$ \\ |
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1993 5. $r \leftarrow 0$ \\ |
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1994 6. for $n$ from $b.used - 1$ to $0$ do \\ |
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1995 \hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ |
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1996 \hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ |
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1997 \hspace{3mm}6.3 $r \leftarrow rr$ \\ |
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1998 7. If $b.used < oldused - 1$ then do \\ |
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1999 \hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ |
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2000 \hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ |
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2001 8. $b.sign \leftarrow a.sign$ \\ |
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2002 9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ |
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2003 10. Return(\textit{MP\_OKAY}).\\ |
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2004 \hline |
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2005 \end{tabular} |
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2006 \end{center} |
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2007 \end{small} |
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2008 \caption{Algorithm mp\_div\_2} |
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2009 \end{figure} |
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2010 |
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2011 \textbf{Algorithm mp\_div\_2.} |
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2012 This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition |
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2013 core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm |
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2014 could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent |
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2015 reading past the end of the array of digits. |
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2016 |
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2017 Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the |
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2018 least significant bit not the most significant bit. |
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2019 |
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2020 EXAM,bn_mp_div_2.c |
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2021 |
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2022 \section{Polynomial Basis Operations} |
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2023 Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as |
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2024 the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single |
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2025 place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer |
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2026 division and Karatsuba multiplication. |
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2027 |
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2028 Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that |
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2029 $y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the |
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2030 polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. |
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2031 |
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2032 \subsection{Multiplication by $x$} |
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2033 |
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2034 Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one |
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2035 degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to |
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2036 multiplying by the integer $\beta$. |
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2037 |
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2038 \newpage\begin{figure}[!here] |
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2039 \begin{small} |
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2040 \begin{center} |
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2041 \begin{tabular}{l} |
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2042 \hline Algorithm \textbf{mp\_lshd}. \\ |
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2043 \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ |
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2044 \textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ |
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2045 \hline \\ |
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2046 1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ |
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2047 2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ |
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2048 3. If the reallocation failed return(\textit{MP\_MEM}). \\ |
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2049 4. $a.used \leftarrow a.used + b$ \\ |
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2050 5. $i \leftarrow a.used - 1$ \\ |
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2051 6. $j \leftarrow a.used - 1 - b$ \\ |
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2052 7. for $n$ from $a.used - 1$ to $b$ do \\ |
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2053 \hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ |
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2054 \hspace{3mm}7.2 $i \leftarrow i - 1$ \\ |
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2055 \hspace{3mm}7.3 $j \leftarrow j - 1$ \\ |
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2056 8. for $n$ from 0 to $b - 1$ do \\ |
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2057 \hspace{3mm}8.1 $a_n \leftarrow 0$ \\ |
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2058 9. Return(\textit{MP\_OKAY}). \\ |
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2059 \hline |
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2060 \end{tabular} |
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2061 \end{center} |
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2062 \end{small} |
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2063 \caption{Algorithm mp\_lshd} |
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2064 \end{figure} |
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2065 |
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2066 \textbf{Algorithm mp\_lshd.} |
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2067 This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs |
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2068 from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The |
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2069 motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally |
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2070 different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is |
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2071 typically used on values where the original value is no longer required. The algorithm will return success immediately if |
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2072 $b \le 0$ since the rest of algorithm is only valid when $b > 0$. |
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2073 |
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2074 First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over |
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2075 the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). |
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2076 The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on |
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2077 step 8 sets the lower $b$ digits to zero. |
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2078 |
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2079 \newpage |
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2080 FIGU,sliding_window,Sliding Window Movement |
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2081 |
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2082 EXAM,bn_mp_lshd.c |
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2083 |
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2084 The if statement (line @24,if@) ensures that the $b$ variable is greater than zero since we do not interpret negative |
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2085 shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates |
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2086 the need for an additional variable in the for loop. The variable $top$ (line @42,top@) is an alias |
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2087 for the leading digit while $bottom$ (line @45,bottom@) is an alias for the trailing edge. The aliases form a |
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2088 window of exactly $b$ digits over the input. |
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2089 |
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2090 \subsection{Division by $x$} |
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2091 |
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2092 Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. |
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2093 |
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2094 \newpage\begin{figure}[!here] |
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2095 \begin{small} |
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2096 \begin{center} |
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2097 \begin{tabular}{l} |
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2098 \hline Algorithm \textbf{mp\_rshd}. \\ |
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2099 \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ |
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2100 \textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ |
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2101 \hline \\ |
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2102 1. If $b \le 0$ then return. \\ |
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2103 2. If $a.used \le b$ then do \\ |
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2104 \hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ |
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2105 \hspace{3mm}2.2 Return. \\ |
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2106 3. $i \leftarrow 0$ \\ |
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2107 4. $j \leftarrow b$ \\ |
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2108 5. for $n$ from 0 to $a.used - b - 1$ do \\ |
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2109 \hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ |
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2110 \hspace{3mm}5.2 $i \leftarrow i + 1$ \\ |
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2111 \hspace{3mm}5.3 $j \leftarrow j + 1$ \\ |
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2112 6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ |
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2113 \hspace{3mm}6.1 $a_n \leftarrow 0$ \\ |
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2114 7. $a.used \leftarrow a.used - b$ \\ |
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2115 8. Return. \\ |
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2116 \hline |
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2117 \end{tabular} |
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2118 \end{center} |
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2119 \end{small} |
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2120 \caption{Algorithm mp\_rshd} |
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2121 \end{figure} |
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2122 |
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2123 \textbf{Algorithm mp\_rshd.} |
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2124 This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since |
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2125 it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. |
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2126 |
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2127 If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal |
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2128 to the shift count $b$ then it will simply zero the input and return. |
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2129 |
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2130 After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that |
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2131 is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. |
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2132 Also the digits are copied from the leading to the trailing edge. |
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2133 |
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2134 Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. |
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2135 |
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2136 EXAM,bn_mp_rshd.c |
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2137 |
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2138 The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we |
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2139 form a sliding window except we copy in the other direction. After the window (line @59,for (;@) we then zero |
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2140 the upper digits of the input to make sure the result is correct. |
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2141 |
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2142 \section{Powers of Two} |
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2143 |
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2144 Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For |
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2145 example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single |
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2146 shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. |
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2147 |
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2148 \subsection{Multiplication by Power of Two} |
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2149 |
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2150 \newpage\begin{figure}[!here] |
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2151 \begin{small} |
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2152 \begin{center} |
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2153 \begin{tabular}{l} |
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2154 \hline Algorithm \textbf{mp\_mul\_2d}. \\ |
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2155 \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ |
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2156 \textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ |
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2157 \hline \\ |
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2158 1. $c \leftarrow a$. (\textit{mp\_copy}) \\ |
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2159 2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ |
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2160 3. If the reallocation failed return(\textit{MP\_MEM}). \\ |
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2161 4. If $b \ge lg(\beta)$ then \\ |
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2162 \hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ |
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2163 \hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ |
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2164 5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ |
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2165 6. If $d \ne 0$ then do \\ |
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2166 \hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ |
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2167 \hspace{3mm}6.2 $r \leftarrow 0$ \\ |
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2168 \hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ |
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2169 \hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ |
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2170 \hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ |
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2171 \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ |
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2172 \hspace{3mm}6.4 If $r > 0$ then do \\ |
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2173 \hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ |
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2174 \hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ |
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2175 7. Return(\textit{MP\_OKAY}). \\ |
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2176 \hline |
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2177 \end{tabular} |
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2178 \end{center} |
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2179 \end{small} |
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2180 \caption{Algorithm mp\_mul\_2d} |
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2181 \end{figure} |
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2182 |
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2183 \textbf{Algorithm mp\_mul\_2d.} |
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2184 This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to |
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2185 quickly compute the product. |
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2186 |
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2187 First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than |
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2188 $\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ |
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2189 left. |
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2190 |
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2191 After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts |
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2192 required. If it is non-zero a modified shift loop is used to calculate the remaining product. |
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2193 Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ |
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2194 variable is used to extract the upper $d$ bits to form the carry for the next iteration. |
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2195 |
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2196 This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to |
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2197 complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. |
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2198 |
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2199 EXAM,bn_mp_mul_2d.c |
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2200 |
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2201 The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the |
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2202 destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then |
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2203 has to be grown (line @31,grow@) to accomodate the result. |
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2204 |
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2205 If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples |
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2206 of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift |
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2207 loop (lines @45,if@ to @76,}@) we make use of pre--computed values $shift$ and $mask$. These are used to |
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2208 extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a |
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2209 chain between consecutive iterations to propagate the carry. |
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2210 |
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2211 \subsection{Division by Power of Two} |
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2212 |
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2213 \newpage\begin{figure}[!here] |
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2214 \begin{small} |
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2215 \begin{center} |
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2216 \begin{tabular}{l} |
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2217 \hline Algorithm \textbf{mp\_div\_2d}. \\ |
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2218 \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ |
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2219 \textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ |
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2220 \hline \\ |
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2221 1. If $b \le 0$ then do \\ |
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2222 \hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ |
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2223 \hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ |
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2224 \hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ |
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2225 2. $c \leftarrow a$ \\ |
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2226 3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ |
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2227 4. If $b \ge lg(\beta)$ then do \\ |
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2228 \hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ |
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2229 5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ |
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2230 6. If $k \ne 0$ then do \\ |
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2231 \hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ |
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2232 \hspace{3mm}6.2 $r \leftarrow 0$ \\ |
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2233 \hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ |
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2234 \hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ |
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2235 \hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ |
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2236 \hspace{6mm}6.3.3 $r \leftarrow rr$ \\ |
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2237 7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ |
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2238 8. Return(\textit{MP\_OKAY}). \\ |
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2239 \hline |
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2240 \end{tabular} |
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2241 \end{center} |
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2242 \end{small} |
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2243 \caption{Algorithm mp\_div\_2d} |
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2244 \end{figure} |
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2245 |
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2246 \textbf{Algorithm mp\_div\_2d.} |
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2247 This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm |
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2248 mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division |
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2249 by using algorithm mp\_mod\_2d. |
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2250 |
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2251 EXAM,bn_mp_div_2d.c |
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2252 |
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2253 The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally |
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2254 ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the |
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2255 result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before |
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2256 the quotient is obtained. |
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2257 |
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2258 The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is |
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2259 the direction of the shifts. |
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2260 |
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2261 \subsection{Remainder of Division by Power of Two} |
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2262 |
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2263 The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This |
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2264 algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. |
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2265 |
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2266 \begin{figure}[!here] |
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2267 \begin{small} |
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2268 \begin{center} |
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2269 \begin{tabular}{l} |
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2270 \hline Algorithm \textbf{mp\_mod\_2d}. \\ |
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2271 \textbf{Input}. One mp\_int $a$ and an integer $b$ \\ |
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2272 \textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ |
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2273 \hline \\ |
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2274 1. If $b \le 0$ then do \\ |
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2275 \hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ |
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2276 \hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ |
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2277 2. If $b > a.used \cdot lg(\beta)$ then do \\ |
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2278 \hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ |
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2279 \hspace{3mm}2.2 Return the result of step 2.1. \\ |
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2280 3. $c \leftarrow a$ \\ |
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2281 4. If step 3 failed return(\textit{MP\_MEM}). \\ |
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2282 5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ |
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2283 \hspace{3mm}5.1 $c_n \leftarrow 0$ \\ |
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2284 6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ |
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2285 7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ |
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2286 8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ |
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2287 9. Return(\textit{MP\_OKAY}). \\ |
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2288 \hline |
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2289 \end{tabular} |
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2290 \end{center} |
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2291 \end{small} |
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2292 \caption{Algorithm mp\_mod\_2d} |
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2293 \end{figure} |
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2294 |
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2295 \textbf{Algorithm mp\_mod\_2d.} |
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2296 This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the |
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2297 result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ |
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2298 is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. |
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2299 |
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2300 EXAM,bn_mp_mod_2d.c |
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2301 |
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2302 We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger |
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2303 than the input we just mp\_copy() the input and return right away. After this point we know we must actually |
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2304 perform some work to produce the remainder. |
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2305 |
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2306 Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce |
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2307 the number. First we zero any digits above the last digit in $2^b$ (line @41,for@). Next we reduce the |
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2308 leading digit of both (line @45,&=@) and then mp\_clamp(). |
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2309 |
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2310 \section*{Exercises} |
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2311 \begin{tabular}{cl} |
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2312 $\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ |
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2313 & in $O(n)$ time. \\ |
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2314 &\\ |
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2315 $\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ |
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2316 & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ |
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2317 & upto $64$ with a hamming weight less than three. \\ |
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2318 &\\ |
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2319 $\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ |
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2320 & $2^k - 1$ as well. \\ |
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2321 &\\ |
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2322 $\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ |
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2323 & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ |
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2324 & any $n$-bit input. Note that the time of addition is ignored in the \\ |
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2325 & calculation. \\ |
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2326 & \\ |
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2327 $\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ |
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2328 & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ |
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2329 & the cost of addition. \\ |
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2330 & \\ |
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2331 $\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ |
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2332 & for $n = 64 \ldots 1024$ in steps of $64$. \\ |
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2333 & \\ |
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2334 $\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ |
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2335 & calculating the result of a signed comparison. \\ |
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2336 & |
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2337 \end{tabular} |
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2338 |
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2339 \chapter{Multiplication and Squaring} |
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2340 \section{The Multipliers} |
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2341 For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of |
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2342 algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction |
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2343 where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication |
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2344 and squaring, leaving modular reductions for the subsequent chapter. |
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2345 |
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2346 The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular |
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2347 exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular |
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2348 exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, |
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2349 35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision |
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2350 multiplications. |
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2351 |
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2352 For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied |
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2353 against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the |
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2354 overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in |
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2355 1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. |
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2356 This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. |
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2357 |
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2358 \section{Multiplication} |
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2359 \subsection{The Baseline Multiplication} |
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2360 \label{sec:basemult} |
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2361 \index{baseline multiplication} |
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2362 Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication |
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2363 algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision |
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2364 multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To |
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2365 simplify most discussions, it will be assumed that the inputs have comparable number of digits. |
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2366 |
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2367 The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be |
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2368 used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important |
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2369 facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this |
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2370 modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product |
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2371 will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. |
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2372 |
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2373 Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to |
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2374 include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The |
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2375 constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}). |
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2376 |
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2377 \newpage\begin{figure}[!here] |
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2378 \begin{small} |
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2379 \begin{center} |
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2380 \begin{tabular}{l} |
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2381 \hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ |
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2382 \textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ |
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2383 \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ |
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2384 \hline \\ |
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2385 1. If min$(a.used, b.used) < \delta$ then do \\ |
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2386 \hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ |
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2387 \hspace{3mm}1.2 Return the result of step 1.1 \\ |
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2388 \\ |
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2389 Allocate and initialize a temporary mp\_int. \\ |
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2390 2. Init $t$ to be of size $digs$ \\ |
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2391 3. If step 2 failed return(\textit{MP\_MEM}). \\ |
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2392 4. $t.used \leftarrow digs$ \\ |
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2393 \\ |
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2394 Compute the product. \\ |
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2395 5. for $ix$ from $0$ to $a.used - 1$ do \\ |
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2396 \hspace{3mm}5.1 $u \leftarrow 0$ \\ |
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2397 \hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ |
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2398 \hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ |
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2399 \hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ |
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2400 \hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ |
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2401 \hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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2402 \hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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2403 \hspace{3mm}5.5 if $ix + pb < digs$ then do \\ |
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2404 \hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ |
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2405 6. Clamp excess digits of $t$. \\ |
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2406 7. Swap $c$ with $t$ \\ |
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2407 8. Clear $t$ \\ |
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2408 9. Return(\textit{MP\_OKAY}). \\ |
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2409 \hline |
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2410 \end{tabular} |
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2411 \end{center} |
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2412 \end{small} |
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2413 \caption{Algorithm s\_mp\_mul\_digs} |
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2414 \end{figure} |
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2415 |
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2416 \textbf{Algorithm s\_mp\_mul\_digs.} |
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2417 This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem |
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2418 a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent |
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2419 algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. |
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2420 Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the |
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2421 inputs. |
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2422 |
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2423 The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either |
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2424 input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A |
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2425 temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to |
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2426 compute products when either $a = c$ or $b = c$ without overwriting the inputs. |
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2427 |
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2428 All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable |
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2429 is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm |
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2430 will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the |
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2431 innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. |
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2432 |
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2433 For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best |
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2434 visualized in the following table. |
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2435 |
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2436 \begin{figure}[here] |
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2437 \begin{center} |
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2438 \begin{tabular}{|c|c|c|c|c|c|l|} |
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2439 \hline && & 5 & 7 & 6 & \\ |
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2440 \hline $\times$&& & 2 & 4 & 1 & \\ |
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2441 \hline &&&&&&\\ |
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2442 && & 5 & 7 & 6 & $10^0(1)(576)$ \\ |
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2443 &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ |
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2444 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ |
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2445 \hline |
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2446 \end{tabular} |
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2447 \end{center} |
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2448 \caption{Long-Hand Multiplication Diagram} |
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2449 \end{figure} |
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2450 |
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2451 Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate |
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2452 count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. |
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2453 |
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2454 Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step |
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2455 is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a |
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2456 double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step |
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2457 5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit |
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2458 $t_{ix+iy}$ and the result would be lost. |
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2459 |
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2460 At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th |
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2461 digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result |
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2462 exceed the precision requested. |
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2463 |
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2464 EXAM,bn_s_mp_mul_digs.c |
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2465 |
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2466 First we determine (line @30,if@) if the Comba method can be used first since it's faster. The conditions for |
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2467 sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than |
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2468 \textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is |
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2469 set to $\delta$ but can be reduced when memory is at a premium. |
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2470 |
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2471 If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int |
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2472 $t$ (line @36,init@) to the exact size of the output to avoid further re--allocations. At this point we now |
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2473 begin the $O(n^2)$ loop. |
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2474 |
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2475 This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of |
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2476 digits as output. In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum |
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2477 number of inner loop iterations. |
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2478 |
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2479 Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the |
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2480 carry from the previous iteration. A particularly important observation is that most modern optimizing |
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2481 C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that |
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2482 is required for the product. In x86 terms for example, this means using the MUL instruction. |
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2483 |
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2484 Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the |
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2485 next iteration. |
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2486 |
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2487 \subsection{Faster Multiplication by the ``Comba'' Method} |
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2488 MARK,COMBA |
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2489 |
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2490 One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be |
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2491 computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement |
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2492 in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. |
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2493 Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an |
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2494 interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written |
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2495 five years before. |
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2496 |
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2497 At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight |
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2498 twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products |
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2499 are produced then added together to form the final result. In the baseline algorithm the columns are added together |
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2500 after each iteration to get the result instantaneously. |
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2501 |
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2502 In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at |
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2503 the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated |
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2504 after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute |
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2505 the product vector $\vec x$ as follows. |
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2506 |
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2507 \begin{equation} |
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2508 \vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace |
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2509 \end{equation} |
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2510 |
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2511 Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication |
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2512 of $576$ and $241$. |
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2513 |
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2514 \newpage\begin{figure}[here] |
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2515 \begin{small} |
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2516 \begin{center} |
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2517 \begin{tabular}{|c|c|c|c|c|c|} |
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2518 \hline & & 5 & 7 & 6 & First Input\\ |
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2519 \hline $\times$ & & 2 & 4 & 1 & Second Input\\ |
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2520 \hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ |
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2521 & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ |
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2522 $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ |
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2523 \hline 10 & 34 & 45 & 31 & 6 & Final Result \\ |
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2524 \hline |
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2525 \end{tabular} |
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2526 \end{center} |
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2527 \end{small} |
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2528 \caption{Comba Multiplication Diagram} |
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2529 \end{figure} |
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2530 |
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2531 At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. |
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2532 Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is |
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2533 congruent to adding a leading zero digit. |
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2534 |
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2535 \begin{figure}[!here] |
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2536 \begin{small} |
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2537 \begin{center} |
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2538 \begin{tabular}{l} |
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2539 \hline Algorithm \textbf{Comba Fixup}. \\ |
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2540 \textbf{Input}. Vector $\vec x$ of dimension $k$ \\ |
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2541 \textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ |
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2542 \hline \\ |
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2543 1. for $n$ from $0$ to $k - 1$ do \\ |
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2544 \hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ |
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2545 \hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ |
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2546 2. Return($\vec x$). \\ |
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2547 \hline |
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2548 \end{tabular} |
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2549 \end{center} |
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2550 \end{small} |
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2551 \caption{Algorithm Comba Fixup} |
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2552 \end{figure} |
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2553 |
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2554 With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case |
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2555 $241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more |
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2556 efficient than the baseline algorithm why not simply always use this algorithm? |
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2557 |
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2558 \subsubsection{Column Weight.} |
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2559 At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output |
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2560 independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix |
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2561 the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of |
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2562 three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then |
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2563 an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is |
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2564 min$(m, n)$ which is fairly obvious. |
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2565 |
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2566 The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall |
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2567 from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these |
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2568 two quantities we must not violate the following |
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2569 |
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2570 \begin{equation} |
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2571 k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} |
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2572 \end{equation} |
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2573 |
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2574 Which reduces to |
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2575 |
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2576 \begin{equation} |
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2577 k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} |
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2578 \end{equation} |
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2579 |
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2580 Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is |
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2581 found. |
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2582 |
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2583 \begin{equation} |
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2584 k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} |
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2585 \end{equation} |
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2586 |
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2587 The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration |
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2588 the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since |
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2589 $256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. |
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2590 |
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2591 \newpage\begin{figure}[!here] |
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2592 \begin{small} |
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2593 \begin{center} |
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2594 \begin{tabular}{l} |
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2595 \hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ |
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2596 \textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ |
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2597 \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ |
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2598 \hline \\ |
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2599 Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ |
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2600 1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ |
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2601 2. If step 1 failed return(\textit{MP\_MEM}).\\ |
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2602 \\ |
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2603 3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ |
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2604 \\ |
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2605 4. $\_ \hat W \leftarrow 0$ \\ |
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2606 5. for $ix$ from 0 to $pa - 1$ do \\ |
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2607 \hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ |
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2608 \hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ |
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2609 \hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ |
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2610 \hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ |
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2611 \hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ |
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2612 \hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ |
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2613 \hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ |
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2614 6. $W_{pa} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ |
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2615 \\ |
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2616 7. $oldused \leftarrow c.used$ \\ |
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2617 8. $c.used \leftarrow digs$ \\ |
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2618 9. for $ix$ from $0$ to $pa$ do \\ |
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2619 \hspace{3mm}9.1 $c_{ix} \leftarrow W_{ix}$ \\ |
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2620 10. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ |
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2621 \hspace{3mm}10.1 $c_{ix} \leftarrow 0$ \\ |
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2622 \\ |
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2623 11. Clamp $c$. \\ |
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2624 12. Return MP\_OKAY. \\ |
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2625 \hline |
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2626 \end{tabular} |
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2627 \end{center} |
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2628 \end{small} |
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2629 \caption{Algorithm fast\_s\_mp\_mul\_digs} |
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2630 \label{fig:COMBAMULT} |
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2631 \end{figure} |
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2632 |
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2633 \textbf{Algorithm fast\_s\_mp\_mul\_digs.} |
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2634 This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. |
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2635 |
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2636 The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the |
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2637 loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and |
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2638 reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. |
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2639 |
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2640 The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than |
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2641 $b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable |
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2642 $ix$ is. This is used for the immediately subsequent statement where we find $iy$. |
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2643 |
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2644 The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time |
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2645 means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each |
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2646 pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to |
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2647 move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until |
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2648 $tx \ge a.used$ or $ty < 0$ occurs. |
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2649 |
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2650 After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator |
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2651 into the next round by dividing $\_ \hat W$ by $\beta$. |
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2652 |
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2653 To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the |
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2654 cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require |
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2655 $O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, |
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2656 the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply |
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2657 and addition operations in the nested loop in parallel. |
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2658 |
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2659 EXAM,bn_fast_s_mp_mul_digs.c |
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2660 |
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2661 As per the pseudo--code we first calculate $pa$ (line @47,MIN@) as the number of digits to output. Next we begin the outer loop |
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2662 to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines @61,tmpx@, @62,tmpy@) to point |
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2663 inside the two multiplicands quickly. |
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2664 |
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2665 The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play. Originally this comba |
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2666 implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix |
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2667 the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write |
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2668 one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth |
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2669 is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often |
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2670 slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the |
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2671 compiler has aliased $\_ \hat W$ to a CPU register. |
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2672 |
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2673 After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as |
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2674 a carry for the next pass. After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product. |
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2675 |
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2676 \subsection{Polynomial Basis Multiplication} |
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2677 To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms |
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2678 the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and |
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2679 $g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. |
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2680 |
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2681 The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will |
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2682 directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients |
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2683 requires $O(n^2)$ time and would in practice be slower than the Comba technique. |
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2684 |
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2685 However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown |
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2686 coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with |
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2687 Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in |
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2688 effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. |
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2689 |
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2690 The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since |
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2691 $W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the |
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2692 fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required |
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2693 by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. |
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2694 |
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2695 When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term |
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2696 is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product |
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2697 $W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather |
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2698 simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. |
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2699 The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the |
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2700 points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. |
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2701 |
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2702 If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} |
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2703 $\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that |
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2704 $\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For |
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2705 example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. |
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2706 |
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2707 \begin{eqnarray} |
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2708 \zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ |
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2709 16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) |
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2710 \end{eqnarray} |
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2711 |
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2712 Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the |
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2713 polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. |
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2714 |
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2715 As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of |
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2716 multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is |
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2717 $O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} |
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2718 summarizes the exponents for various values of $n$. |
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2719 |
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2720 \begin{figure} |
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2721 \begin{center} |
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2722 \begin{tabular}{|c|c|c|} |
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2723 \hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ |
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2724 \hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ |
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2725 \hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ |
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2726 \hline $4$ & $1.403677461$ &\\ |
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2727 \hline $5$ & $1.365212389$ &\\ |
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2728 \hline $10$ & $1.278753601$ &\\ |
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2729 \hline $100$ & $1.149426538$ &\\ |
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2730 \hline $1000$ & $1.100270931$ &\\ |
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2731 \hline $10000$ & $1.075252070$ &\\ |
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2732 \hline |
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2733 \end{tabular} |
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2734 \end{center} |
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2735 \caption{Asymptotic Running Time of Polynomial Basis Multiplication} |
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2736 \label{fig:exponent} |
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2737 \end{figure} |
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2738 |
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2739 At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead |
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2740 of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large |
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2741 numbers. |
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2742 |
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2743 \subsubsection{Cutoff Point} |
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2744 The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, |
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2745 the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the |
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2746 polynomial basis approach more costly to use with small inputs. |
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2747 |
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2748 Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a |
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2749 point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and |
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2750 when $m > y$ the Comba methods are slower than the polynomial basis algorithms. |
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2751 |
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2752 The exact location of $y$ depends on several key architectural elements of the computer platform in question. |
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2753 |
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2754 \begin{enumerate} |
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2755 \item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example |
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2756 on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower |
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2757 the cutoff point $y$ will be. |
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2758 |
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2759 \item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits |
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2760 grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This |
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2761 directly reflects on the ratio previous mentioned. |
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2762 |
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2763 \item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an |
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2764 influence over the cutoff point. |
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2765 |
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2766 \end{enumerate} |
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2767 |
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2768 A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point |
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2769 is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when |
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2770 a high resolution timer is available. |
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2771 |
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2772 \subsection{Karatsuba Multiplication} |
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2773 Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for |
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2774 general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with |
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2775 light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. |
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2776 |
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2777 \begin{equation} |
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2778 f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) - (ac + bd))x + bd |
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2779 \end{equation} |
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2780 |
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2781 Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying |
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2782 this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns |
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2783 out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points |
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2784 $\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations. |
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2785 |
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2786 \begin{center} |
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2787 \begin{tabular}{rcrcrcrc} |
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2788 $\zeta_{0}$ & $=$ & & & & & $w_0$ \\ |
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2789 $-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\ |
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2790 $\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ |
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2791 \end{tabular} |
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2792 \end{center} |
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2793 |
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2794 By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity |
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2795 of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} |
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2796 making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point |
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2797 $\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row. |
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2798 |
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2799 \newpage\begin{figure}[!here] |
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2800 \begin{small} |
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2801 \begin{center} |
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2802 \begin{tabular}{l} |
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2803 \hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ |
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2804 \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ |
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2805 \textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ |
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2806 \hline \\ |
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2807 1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ |
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2808 2. If step 2 failed then return(\textit{MP\_MEM}). \\ |
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2809 \\ |
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2810 Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ |
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2811 3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ |
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2812 4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ |
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2813 5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ |
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2814 6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ |
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2815 7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ |
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2816 \\ |
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2817 Calculate the three products. \\ |
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2818 8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ |
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2819 9. $x1y1 \leftarrow x1 \cdot y1$ \\ |
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2820 10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ |
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2821 11. $x0 \leftarrow y1 - y0$ \\ |
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2822 12. $t1 \leftarrow t1 \cdot x0$ \\ |
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2823 \\ |
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2824 Calculate the middle term. \\ |
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2825 13. $x0 \leftarrow x0y0 + x1y1$ \\ |
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2826 14. $t1 \leftarrow x0 - t1$ \\ |
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2827 \\ |
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2828 Calculate the final product. \\ |
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2829 15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ |
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2830 16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ |
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2831 17. $t1 \leftarrow x0y0 + t1$ \\ |
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2832 18. $c \leftarrow t1 + x1y1$ \\ |
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2833 19. Clear all of the temporary variables. \\ |
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2834 20. Return(\textit{MP\_OKAY}).\\ |
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2835 \hline |
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2836 \end{tabular} |
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2837 \end{center} |
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2838 \end{small} |
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2839 \caption{Algorithm mp\_karatsuba\_mul} |
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2840 \end{figure} |
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2841 |
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2842 \textbf{Algorithm mp\_karatsuba\_mul.} |
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2843 This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description |
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2844 from Knuth \cite[pp. 294-295]{TAOCPV2}. |
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2845 |
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2846 \index{radix point} |
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2847 In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must |
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2848 be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the |
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2849 smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 |
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2850 compute the lower halves. Step 6 and 7 computer the upper halves. |
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2851 |
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2852 After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products |
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2853 $x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead |
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2854 of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. |
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2855 |
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2856 The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. |
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2857 |
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2858 EXAM,bn_mp_karatsuba_mul.c |
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2859 |
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2860 The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional |
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2861 wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense |
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2862 to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables |
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2863 required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only |
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2864 the temporaries that have been successfully allocated so far. |
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2865 |
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2866 The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the |
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2867 additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective |
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2868 number of digits for the next section of code. |
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2869 |
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2870 The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd |
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2871 to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and |
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2872 \textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it |
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2873 is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and |
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2874 $y1$ respectively. |
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2875 |
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2876 By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. |
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2877 |
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2878 When line @152,err@ is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that |
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2879 the same code that handles errors can be used to clear the temporary variables and return. |
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2880 |
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2881 \subsection{Toom-Cook $3$-Way Multiplication} |
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2882 Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are |
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2883 chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, |
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2884 $16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients |
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2885 of the $W(x)$. |
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2886 |
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2887 With the five relations that Toom-Cook specifies, the following system of equations is formed. |
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2888 |
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2889 \begin{center} |
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2890 \begin{tabular}{rcrcrcrcrcr} |
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2891 $\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ |
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2892 $16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ |
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2893 $\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ |
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2894 $\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ |
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2895 $\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ |
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2896 \end{tabular} |
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2897 \end{center} |
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2898 |
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2899 A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power |
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2900 of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that |
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2901 the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point |
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2902 (\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. |
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2903 |
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2904 \begin{figure}[!here] |
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2905 \begin{small} |
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2906 \begin{center} |
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2907 \begin{tabular}{l} |
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2908 \hline Algorithm \textbf{mp\_toom\_mul}. \\ |
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2909 \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ |
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2910 \textbf{Output}. $c \leftarrow a \cdot b $ \\ |
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2911 \hline \\ |
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2912 Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ |
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2913 1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ |
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2914 2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ |
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2915 3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ |
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2916 4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ |
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2917 5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ |
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2918 6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ |
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2919 7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ |
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2920 \\ |
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2921 Find the five equations for $w_0, w_1, ..., w_4$. \\ |
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2922 8. $w_0 \leftarrow a_0 \cdot b_0$ \\ |
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2923 9. $w_4 \leftarrow a_2 \cdot b_2$ \\ |
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2924 10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ |
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2925 11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ |
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2926 12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ |
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2927 13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ |
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2928 14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ |
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2929 15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ |
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2930 16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ |
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2931 17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ |
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2932 \\ |
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2933 Continued on the next page.\\ |
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2934 \hline |
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2935 \end{tabular} |
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2936 \end{center} |
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2937 \end{small} |
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2938 \caption{Algorithm mp\_toom\_mul} |
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2939 \end{figure} |
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2940 |
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2941 \newpage\begin{figure}[!here] |
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2942 \begin{small} |
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2943 \begin{center} |
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2944 \begin{tabular}{l} |
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2945 \hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ |
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2946 \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ |
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2947 \textbf{Output}. $c \leftarrow a \cdot b $ \\ |
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2948 \hline \\ |
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2949 Now solve the system of equations. \\ |
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2950 18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ |
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2951 19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ |
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2952 20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ |
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2953 21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ |
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2954 22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ |
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2955 23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ |
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2956 24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ |
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2957 25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ |
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2958 \\ |
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2959 Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ |
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2960 26. for $n$ from $1$ to $4$ do \\ |
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2961 \hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ |
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2962 27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ |
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2963 28. Return(\textit{MP\_OKAY}) \\ |
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2964 \hline |
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|
2965 \end{tabular} |
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2966 \end{center} |
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2967 \end{small} |
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|
2968 \caption{Algorithm mp\_toom\_mul (continued)} |
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2969 \end{figure} |
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2970 |
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|
2971 \textbf{Algorithm mp\_toom\_mul.} |
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2972 This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this |
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2973 algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this |
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2974 description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across |
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2975 any given step. |
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2976 |
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2977 The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller |
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2978 integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. |
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2979 |
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2980 The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond |
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2981 to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find |
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2982 $f(y)$ and $g(y)$ which significantly speeds up the algorithm. |
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2983 |
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2984 After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients |
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2985 $w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of |
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2986 the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates |
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2987 that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. |
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2988 |
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2989 Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer |
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2990 result $a \cdot b$ is produced. |
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2991 |
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2992 EXAM,bn_mp_toom_mul.c |
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2993 |
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2994 The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very |
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2995 large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with |
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2996 Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this |
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2997 algorithm is not practical as Karatsuba has a much lower cutoff point. |
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2998 |
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2999 First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines @40,mod@ to @69,rshd@) with |
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3000 combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly |
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3001 for $b$. |
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3002 |
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3003 Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so |
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3004 we get those out of the way first (lines @72,mul@ and @77,mul@). Next we compute $w1, w2$ and $w3$ using Horners method. |
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3005 |
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3006 After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively |
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3007 straight forward. |
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3008 |
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3009 \subsection{Signed Multiplication} |
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3010 Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all |
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3011 of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. |
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3012 |
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3013 \begin{figure}[!here] |
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3014 \begin{small} |
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3015 \begin{center} |
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3016 \begin{tabular}{l} |
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3017 \hline Algorithm \textbf{mp\_mul}. \\ |
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3018 \textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ |
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3019 \textbf{Output}. $c \leftarrow a \cdot b$ \\ |
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3020 \hline \\ |
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3021 1. If $a.sign = b.sign$ then \\ |
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3022 \hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ |
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3023 2. else \\ |
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3024 \hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ |
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3025 3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ |
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3026 \hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ |
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3027 4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ |
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3028 \hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ |
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3029 5. else \\ |
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3030 \hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ |
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3031 \hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ |
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3032 \hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ |
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3033 \hspace{3mm}5.3 else \\ |
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3034 \hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ |
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3035 6. $c.sign \leftarrow sign$ \\ |
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3036 7. Return the result of the unsigned multiplication performed. \\ |
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3037 \hline |
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3038 \end{tabular} |
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3039 \end{center} |
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3040 \end{small} |
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3041 \caption{Algorithm mp\_mul} |
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3042 \end{figure} |
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3043 |
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3044 \textbf{Algorithm mp\_mul.} |
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3045 This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms |
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3046 available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm |
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3047 s\_mp\_mul\_digs will clear it. |
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3048 |
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3049 EXAM,bn_mp_mul.c |
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3050 |
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3051 The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?'' |
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3052 operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. |
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3053 |
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3054 \section{Squaring} |
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3055 \label{sec:basesquare} |
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3056 |
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3057 Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization |
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3058 available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications |
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3059 performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider |
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3060 the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, |
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3061 $1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ |
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3062 and $3 \cdot 1 = 1 \cdot 3$. |
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3063 |
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3064 For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ |
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3065 required for multiplication. The following diagram gives an example of the operations required. |
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3066 |
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3067 \begin{figure}[here] |
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3068 \begin{center} |
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3069 \begin{tabular}{ccccc|c} |
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3070 &&1&2&3&\\ |
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3071 $\times$ &&1&2&3&\\ |
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3072 \hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ |
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3073 & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ |
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3074 $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ |
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3075 \end{tabular} |
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3076 \end{center} |
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3077 \caption{Squaring Optimization Diagram} |
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3078 \end{figure} |
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3079 |
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3080 MARK,SQUARE |
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3081 Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ |
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3082 represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. |
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3083 |
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3084 The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will |
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3085 appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double |
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3086 products and at most one square (\textit{see the exercise section}). |
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3087 |
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3088 The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, |
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3089 occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. |
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3090 Column two of row one is a square and column three is the first unique column. |
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3091 |
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3092 \subsection{The Baseline Squaring Algorithm} |
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3093 The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines |
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3094 will not handle. |
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3095 |
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3096 \begin{figure}[!here] |
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3097 \begin{small} |
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3098 \begin{center} |
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3099 \begin{tabular}{l} |
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3100 \hline Algorithm \textbf{s\_mp\_sqr}. \\ |
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3101 \textbf{Input}. mp\_int $a$ \\ |
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3102 \textbf{Output}. $b \leftarrow a^2$ \\ |
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3103 \hline \\ |
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3104 1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ |
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3105 2. If step 1 failed return(\textit{MP\_MEM}) \\ |
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3106 3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ |
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3107 4. For $ix$ from 0 to $a.used - 1$ do \\ |
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3108 \hspace{3mm}Calculate the square. \\ |
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3109 \hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ |
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3110 \hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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3111 \hspace{3mm}Calculate the double products after the square. \\ |
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3112 \hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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3113 \hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ |
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3114 \hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ |
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3115 \hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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3116 \hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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3117 \hspace{3mm}Set the last carry. \\ |
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3118 \hspace{3mm}4.5 While $u > 0$ do \\ |
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3119 \hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ |
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3120 \hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ |
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3121 \hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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3122 \hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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3123 5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ |
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3124 6. Exchange $b$ and $t$. \\ |
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3125 7. Clear $t$ (\textit{mp\_clear}) \\ |
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3126 8. Return(\textit{MP\_OKAY}) \\ |
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3127 \hline |
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3128 \end{tabular} |
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3129 \end{center} |
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3130 \end{small} |
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3131 \caption{Algorithm s\_mp\_sqr} |
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3132 \end{figure} |
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3133 |
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3134 \textbf{Algorithm s\_mp\_sqr.} |
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3135 This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC |
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3136 \cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the |
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3137 destination mp\_int to be the same as the source mp\_int. |
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3138 |
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3139 The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while |
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3140 the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate |
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3141 the carry and compute the double products. |
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3142 |
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3143 The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this |
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3144 very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that |
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3145 when it is multiplied by two, it can be properly represented by a mp\_word. |
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3146 |
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3147 Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial |
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3148 results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. |
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3149 |
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3150 EXAM,bn_s_mp_sqr.c |
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3151 |
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3152 Inside the outer loop (line @32,for@) the square term is calculated on line @35,r =@. The carry (line @42,>>@) has been |
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3153 extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized |
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3154 (lines @45,tmpx@ and @48,tmpt@) to simplify the inner loop. The doubling is performed using two |
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3155 additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast. |
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3156 |
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3157 The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops |
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3158 get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to |
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Matt Johnston <matt@ucc.asn.au>
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3159 square a number. |
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3160 |
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3161 \subsection{Faster Squaring by the ``Comba'' Method} |
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3162 A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional |
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3163 drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these |
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3164 performance hazards. |
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3165 |
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3166 The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry |
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3167 propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact |
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3168 that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, |
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3169 $ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. |
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3170 |
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3171 However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two |
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3172 mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and |
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Matt Johnston <matt@ucc.asn.au>
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3173 carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. |
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3174 |
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3175 \newpage\begin{figure}[!here] |
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3176 \begin{small} |
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3177 \begin{center} |
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3178 \begin{tabular}{l} |
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3179 \hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ |
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3180 \textbf{Input}. mp\_int $a$ \\ |
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3181 \textbf{Output}. $b \leftarrow a^2$ \\ |
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3182 \hline \\ |
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3183 Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ |
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3184 1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ |
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3185 2. If step 1 failed return(\textit{MP\_MEM}). \\ |
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3186 \\ |
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3187 3. $pa \leftarrow 2 \cdot a.used$ \\ |
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3188 4. $\hat W1 \leftarrow 0$ \\ |
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3189 5. for $ix$ from $0$ to $pa - 1$ do \\ |
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3190 \hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ |
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3191 \hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ |
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3192 \hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ |
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3193 \hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ |
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3194 \hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ |
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3195 \hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ |
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3196 \hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ |
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3197 \hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ |
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3198 \hspace{3mm}5.8 if $ix$ is even then \\ |
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3199 \hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ |
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3200 \hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ |
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3201 \hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ |
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3202 \\ |
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3203 6. $oldused \leftarrow b.used$ \\ |
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3204 7. $b.used \leftarrow 2 \cdot a.used$ \\ |
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3205 8. for $ix$ from $0$ to $pa - 1$ do \\ |
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3206 \hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ |
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3207 9. for $ix$ from $pa$ to $oldused - 1$ do \\ |
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3208 \hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ |
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3209 10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ |
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3210 11. Return(\textit{MP\_OKAY}). \\ |
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3211 \hline |
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3212 \end{tabular} |
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3213 \end{center} |
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3214 \end{small} |
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3215 \caption{Algorithm fast\_s\_mp\_sqr} |
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3216 \end{figure} |
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3217 |
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3218 \textbf{Algorithm fast\_s\_mp\_sqr.} |
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3219 This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm |
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3220 s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. |
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3221 This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. |
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3222 |
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3223 First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop |
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3224 products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an |
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3225 addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal |
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3226 $a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum |
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3227 of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform |
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3228 fewer multiplications and the routine ends up being faster. |
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3229 |
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3230 Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square |
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3231 only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. |
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3232 |
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3233 EXAM,bn_fast_s_mp_sqr.c |
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3234 |
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3235 This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for |
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3236 the special case of squaring. |
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3237 |
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3238 \subsection{Polynomial Basis Squaring} |
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3239 The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception |
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3240 is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ |
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3241 multiplications to find the $\zeta$ relations, squaring operations are performed instead. |
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3242 |
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3243 \subsection{Karatsuba Squaring} |
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3244 Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. |
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3245 Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a |
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3246 number with the following equation. |
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3247 |
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3248 \begin{equation} |
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3249 h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2 |
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3250 \end{equation} |
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3251 |
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3252 Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in |
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3253 Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of |
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3254 $O \left ( n^{lg(3)} \right )$. |
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3255 |
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3256 If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm |
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3257 instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the |
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3258 time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff |
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3259 point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. |
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3260 |
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3261 Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. |
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3262 The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication |
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3263 were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. |
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3264 |
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3265 \newpage\begin{figure}[!here] |
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3266 \begin{small} |
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3267 \begin{center} |
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3268 \begin{tabular}{l} |
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3269 \hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ |
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3270 \textbf{Input}. mp\_int $a$ \\ |
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3271 \textbf{Output}. $b \leftarrow a^2$ \\ |
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3272 \hline \\ |
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3273 1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ |
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3274 2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ |
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3275 \\ |
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3276 Split the input. e.g. $a = x1\beta^B + x0$ \\ |
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3277 3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ |
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3278 4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ |
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3279 5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ |
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3280 \\ |
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3281 Calculate the three squares. \\ |
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3282 6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ |
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3283 7. $x1x1 \leftarrow x1^2$ \\ |
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3284 8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\ |
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3285 9. $t1 \leftarrow t1^2$ \\ |
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3286 \\ |
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3287 Compute the middle term. \\ |
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3288 10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ |
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3289 11. $t1 \leftarrow t2 - t1$ \\ |
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3290 \\ |
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3291 Compute final product. \\ |
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3292 12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ |
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3293 13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ |
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3294 14. $t1 \leftarrow t1 + x0x0$ \\ |
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3295 15. $b \leftarrow t1 + x1x1$ \\ |
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3296 16. Return(\textit{MP\_OKAY}). \\ |
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3297 \hline |
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3298 \end{tabular} |
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3299 \end{center} |
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3300 \end{small} |
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3301 \caption{Algorithm mp\_karatsuba\_sqr} |
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3302 \end{figure} |
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3303 |
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3304 \textbf{Algorithm mp\_karatsuba\_sqr.} |
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3305 This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based |
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3306 multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. |
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3307 |
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3308 The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is |
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3309 placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ |
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3310 as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. |
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3311 |
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3312 By expanding $\left (x1 - x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$. |
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3313 Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then |
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3314 this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. |
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3315 |
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3316 Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or |
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3317 machine clock cycles.}. |
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3318 |
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3319 \begin{equation} |
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3320 5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 |
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3321 \end{equation} |
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3322 |
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3323 For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. |
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3324 \begin{center} |
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3325 \begin{tabular}{rcl} |
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3326 ${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ |
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3327 ${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ |
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3328 ${13 \over 9}$ & $<$ & $n$ \\ |
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3329 \end{tabular} |
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3330 \end{center} |
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3331 |
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3332 This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors |
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3333 where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On |
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3334 the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a |
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3335 ratio of 1:7. } than simpler operations such as addition. |
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3336 |
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3337 EXAM,bn_mp_karatsuba_sqr.c |
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3338 |
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3339 This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and |
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3340 shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used} |
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3341 count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents |
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3342 to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. |
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3343 |
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3344 By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point |
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3345 is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 |
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3346 it is actually below the Comba limit (\textit{at 110 digits}). |
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3347 |
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3348 This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are |
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3349 redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and |
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3350 mp\_clears are executed normally. |
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3351 |
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3352 \subsection{Toom-Cook Squaring} |
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3353 The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used |
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3354 instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to |
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3355 derive their own Toom-Cook squaring algorithm. |
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3356 |
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3357 \subsection{High Level Squaring} |
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3358 \newpage\begin{figure}[!here] |
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3359 \begin{small} |
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3360 \begin{center} |
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3361 \begin{tabular}{l} |
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3362 \hline Algorithm \textbf{mp\_sqr}. \\ |
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3363 \textbf{Input}. mp\_int $a$ \\ |
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3364 \textbf{Output}. $b \leftarrow a^2$ \\ |
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3365 \hline \\ |
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3366 1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ |
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3367 \hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ |
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3368 2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ |
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3369 \hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ |
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3370 3. else \\ |
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3371 \hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ |
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3372 \hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ |
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3373 \hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ |
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3374 \hspace{3mm}3.3 else \\ |
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3375 \hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ |
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3376 4. $b.sign \leftarrow MP\_ZPOS$ \\ |
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3377 5. Return the result of the unsigned squaring performed. \\ |
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3378 \hline |
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3379 \end{tabular} |
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3380 \end{center} |
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3381 \end{small} |
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3382 \caption{Algorithm mp\_sqr} |
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3383 \end{figure} |
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3384 |
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3385 \textbf{Algorithm mp\_sqr.} |
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3386 This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least |
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3387 \textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If |
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3388 neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. |
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3389 |
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3390 EXAM,bn_mp_sqr.c |
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3391 |
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3392 \section*{Exercises} |
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3393 \begin{tabular}{cl} |
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3394 $\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ |
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3395 & that have different number of digits in Karatsuba multiplication. \\ |
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3396 & \\ |
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3397 $\left [ 2 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\ |
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3398 & of double products and at most one square is stated. Prove this statement. \\ |
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3399 & \\ |
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3400 $\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ |
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3401 & \\ |
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3402 $\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ |
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3403 & \\ |
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3404 $\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ |
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3405 & required for equation $6.7$ to be true. \\ |
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3406 & \\ |
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3407 $\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ |
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3408 & compute subsets of the columns in each thread. Determine a cutoff point where \\ |
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3409 & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ |
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3410 &\\ |
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3411 $\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ |
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3412 & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ |
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3413 & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ |
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3414 & \\ |
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3415 \end{tabular} |
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3416 |
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3417 \chapter{Modular Reduction} |
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3418 MARK,REDUCTION |
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3419 \section{Basics of Modular Reduction} |
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3420 \index{modular residue} |
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3421 Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, |
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3422 such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} |
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3423 modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered |
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3424 in~\ref{sec:division}. |
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3425 |
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3426 Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result |
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3427 $r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the |
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3428 ``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and |
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3429 other forms of residues. |
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3430 |
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3431 Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions |
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3432 is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the |
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3433 RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in |
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3434 elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular |
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3435 exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the |
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3436 range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check |
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3437 algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. |
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3438 |
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|
3439 \section{The Barrett Reduction} |
|
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3440 The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate |
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3441 division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to |
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3442 |
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3443 \begin{equation} |
|
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3444 c = a - b \cdot \lfloor a/b \rfloor |
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|
3445 \end{equation} |
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3446 |
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3447 Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper |
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3448 targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, |
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3449 DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. |
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3450 It would take another common optimization to optimize the algorithm. |
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3451 |
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3452 \subsection{Fixed Point Arithmetic} |
|
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3453 The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed |
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3454 point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were |
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3455 fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit |
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3456 integer and a $q$-bit fraction part (\textit{where $p+q = k$}). |
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3457 |
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3458 In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the |
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3459 value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by |
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3460 moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted |
|
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3461 to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the |
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3462 fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. |
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3463 |
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3464 This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication |
|
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3465 of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is |
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3466 equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer |
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3467 $a$ by another integer $b$ can be achieved with the following expression. |
|
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3468 |
|
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|
3469 \begin{equation} |
|
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3470 \lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor |
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3471 \end{equation} |
|
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3472 |
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3473 The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with |
|
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3474 modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations |
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|
3475 are considerably faster than division on most processors. |
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3476 |
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3477 Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which |
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3478 leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and |
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3479 the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally |
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3480 larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach |
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3481 to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. |
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3482 |
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3483 \begin{equation} |
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3484 c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor |
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3485 \end{equation} |
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3486 |
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3487 Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ |
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3488 variable also helps re-inforce the idea that it is meant to be computed once and re-used. |
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3489 |
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|
3490 \begin{equation} |
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3491 c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor |
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3492 \end{equation} |
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3493 |
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3494 Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett |
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3495 reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough |
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3496 precision. |
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3497 |
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3498 Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and |
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3499 another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to |
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3500 reduce the number. |
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3501 |
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3502 For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing |
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3503 $a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. |
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3504 By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. |
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3505 |
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3506 \subsection{Choosing a Radix Point} |
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3507 Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best |
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3508 that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. |
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3509 See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of |
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|
3510 the initial multiplication that finds the quotient. |
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3511 |
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|
3512 Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent |
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3513 the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if |
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3514 two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the |
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3515 $m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to |
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3516 express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then |
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3517 ${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient |
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3518 is bound by $0 \le {a' \over b} < 1$. |
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3519 |
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3520 Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits |
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3521 ``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input |
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Matt Johnston <matt@ucc.asn.au>
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3522 with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation |
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3523 |
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Matt Johnston <matt@ucc.asn.au>
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|
3524 \begin{equation} |
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3525 c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor |
|
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3526 \end{equation} |
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3527 |
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3528 Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the |
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3529 exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor |
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Matt Johnston <matt@ucc.asn.au>
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|
3530 would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient |
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Matt Johnston <matt@ucc.asn.au>
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3531 $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off |
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Matt Johnston <matt@ucc.asn.au>
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|
3532 by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient |
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3533 can be off by an additional value of one for a total of at most two. This implies that |
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Matt Johnston <matt@ucc.asn.au>
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3534 $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting |
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3535 $b$ once or twice the residue is found. |
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3536 |
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3537 The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single |
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3538 precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. |
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|
3539 This is considerably faster than the original attempt. |
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3540 |
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3541 For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ |
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3542 represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. |
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3543 With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then |
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Matt Johnston <matt@ucc.asn.au>
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3544 $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ |
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Matt Johnston <matt@ucc.asn.au>
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|
3545 is found. |
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3546 |
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|
3547 \subsection{Trimming the Quotient} |
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3548 So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As |
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|
3549 it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for |
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|
3550 optimization. |
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3551 |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3552 After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower |
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Matt Johnston <matt@ucc.asn.au>
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|
3553 half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision |
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3554 multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. |
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Matt Johnston <matt@ucc.asn.au>
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3555 In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. |
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3556 |
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Matt Johnston <matt@ucc.asn.au>
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3557 The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision |
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Matt Johnston <matt@ucc.asn.au>
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3558 multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number |
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|
3559 of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. |
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3560 |
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|
3561 \subsection{Trimming the Residue} |
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3562 After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small |
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3563 multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the |
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3564 result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are |
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|
3565 implicitly zero. |
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3566 |
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3567 The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full |
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3568 $O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can |
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3569 be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces |
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3570 only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. |
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3571 |
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3572 With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which |
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|
3573 is considerably faster than the straightforward $3m^2$ method. |
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3574 |
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|
3575 \subsection{The Barrett Algorithm} |
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|
3576 \newpage\begin{figure}[!here] |
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3577 \begin{small} |
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3578 \begin{center} |
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3579 \begin{tabular}{l} |
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3580 \hline Algorithm \textbf{mp\_reduce}. \\ |
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3581 \textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ |
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3582 \textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ |
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3583 \hline \\ |
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3584 Let $m$ represent the number of digits in $b$. \\ |
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3585 1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ |
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3586 2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ |
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3587 \\ |
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3588 Produce the quotient. \\ |
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3589 3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ |
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3590 4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ |
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3591 \\ |
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3592 Subtract the multiple of modulus from the input. \\ |
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3593 5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ |
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3594 6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ |
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3595 7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ |
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3596 \\ |
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3597 Add $\beta^{m+1}$ if a carry occured. \\ |
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3598 8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ |
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3599 \hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ |
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3600 \hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ |
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3601 \hspace{3mm}8.3 $a \leftarrow a + q$ \\ |
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3602 \\ |
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3603 Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ |
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3604 9. While $a \ge b$ do (\textit{mp\_cmp}) \\ |
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3605 \hspace{3mm}9.1 $c \leftarrow a - b$ \\ |
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3606 10. Clear $q$. \\ |
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3607 11. Return(\textit{MP\_OKAY}) \\ |
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3608 \hline |
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3609 \end{tabular} |
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3610 \end{center} |
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3611 \end{small} |
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3612 \caption{Algorithm mp\_reduce} |
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3613 \end{figure} |
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3614 |
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3615 \textbf{Algorithm mp\_reduce.} |
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3616 This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC |
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3617 \cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must |
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3618 be adhered to for the algorithm to work. |
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3619 |
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3620 First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting |
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3621 a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order |
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3622 for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. |
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3623 Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this |
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3624 algorithm and is assumed to be calculated and stored before the algorithm is used. |
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3625 |
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3626 Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called |
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3627 $s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that |
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3628 instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number |
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3629 of digits in $b$ is very much smaller than $\beta$. |
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3630 |
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3631 While it is known that |
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3632 $a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied |
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3633 ``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be |
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3634 fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. |
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3635 |
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3636 The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is |
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3637 performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. |
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3638 |
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3639 EXAM,bn_mp_reduce.c |
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3640 |
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3641 The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves |
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3642 the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits |
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3643 in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is |
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3644 safe to do so. |
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3645 |
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3646 \subsection{The Barrett Setup Algorithm} |
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3647 In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for |
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3648 future use so that the Barrett algorithm can be used without delay. |
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3649 |
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3650 \newpage\begin{figure}[!here] |
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3651 \begin{small} |
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3652 \begin{center} |
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3653 \begin{tabular}{l} |
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3654 \hline Algorithm \textbf{mp\_reduce\_setup}. \\ |
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3655 \textbf{Input}. mp\_int $a$ ($a > 1$) \\ |
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3656 \textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ |
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3657 \hline \\ |
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3658 1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ |
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3659 2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ |
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3660 3. Return(\textit{MP\_OKAY}) \\ |
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3661 \hline |
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3662 \end{tabular} |
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3663 \end{center} |
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3664 \end{small} |
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3665 \caption{Algorithm mp\_reduce\_setup} |
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3666 \end{figure} |
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3667 |
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3668 \textbf{Algorithm mp\_reduce\_setup.} |
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3669 This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which |
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3670 is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. |
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3671 |
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3672 EXAM,bn_mp_reduce_setup.c |
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3673 |
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3674 This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable |
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3675 which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the |
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3676 remainder to be passed as NULL meaning to ignore the value. |
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3677 |
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3678 \section{The Montgomery Reduction} |
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3679 Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting |
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3680 form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a |
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3681 residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. |
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3682 |
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3683 Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of |
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3684 $n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input |
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3685 is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. |
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3686 |
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3687 \textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way |
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3688 to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. |
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3689 |
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3690 \textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually |
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3691 this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to |
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3692 multiplication by $k^{-1}$ modulo $n$. |
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3693 |
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3694 From these two simple facts the following simple algorithm can be derived. |
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3695 |
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3696 \newpage\begin{figure}[!here] |
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3697 \begin{small} |
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3698 \begin{center} |
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3699 \begin{tabular}{l} |
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3700 \hline Algorithm \textbf{Montgomery Reduction}. \\ |
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3701 \textbf{Input}. Integer $x$, $n$ and $k$ \\ |
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3702 \textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ |
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3703 \hline \\ |
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3704 1. for $t$ from $1$ to $k$ do \\ |
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3705 \hspace{3mm}1.1 If $x$ is odd then \\ |
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3706 \hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ |
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3707 \hspace{3mm}1.2 $x \leftarrow x/2$ \\ |
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3708 2. Return $x$. \\ |
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3709 \hline |
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3710 \end{tabular} |
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3711 \end{center} |
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3712 \end{small} |
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3713 \caption{Algorithm Montgomery Reduction} |
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3714 \end{figure} |
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3715 |
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3716 The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is |
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3717 added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since |
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3718 $x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the |
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3719 final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to |
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3720 $0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. |
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3721 |
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3722 \begin{figure}[here] |
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3723 \begin{small} |
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3724 \begin{center} |
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3725 \begin{tabular}{|c|l|} |
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3726 \hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ |
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3727 \hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ |
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3728 \hline $2$ & $x/2 = 1453$ \\ |
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3729 \hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ |
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3730 \hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ |
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3731 \hline $5$ & $x/2 = 278$ \\ |
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3732 \hline $6$ & $x/2 = 139$ \\ |
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3733 \hline $7$ & $x + n = 396$, $x/2 = 198$ \\ |
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3734 \hline $8$ & $x/2 = 99$ \\ |
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3735 \hline |
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3736 \end{tabular} |
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3737 \end{center} |
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3738 \end{small} |
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3739 \caption{Example of Montgomery Reduction (I)} |
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3740 \label{fig:MONT1} |
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3741 \end{figure} |
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3742 |
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3743 Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 8$. The result of the algorithm $r = 99$ is |
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3744 congruent to the value of $2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^8$ modulo $257$ the correct residue |
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3745 $r \equiv 158$ is produced. |
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3746 |
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3747 Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts |
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3748 and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. |
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3749 Fortunately there exists an alternative representation of the algorithm. |
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3750 |
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3751 \begin{figure}[!here] |
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3752 \begin{small} |
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3753 \begin{center} |
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3754 \begin{tabular}{l} |
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3755 \hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ |
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3756 \textbf{Input}. Integer $x$, $n$ and $k$ \\ |
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3757 \textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ |
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3758 \hline \\ |
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3759 1. for $t$ from $0$ to $k - 1$ do \\ |
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3760 \hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ |
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3761 \hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ |
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3762 2. Return $x/2^k$. \\ |
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3763 \hline |
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3764 \end{tabular} |
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3765 \end{center} |
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3766 \end{small} |
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3767 \caption{Algorithm Montgomery Reduction (modified I)} |
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3768 \end{figure} |
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3769 |
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3770 This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single |
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3771 precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. |
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3772 |
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3773 \begin{figure}[here] |
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3774 \begin{small} |
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3775 \begin{center} |
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3776 \begin{tabular}{|c|l|r|} |
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3777 \hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ |
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3778 \hline -- & $5555$ & $1010110110011$ \\ |
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3779 \hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ |
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3780 \hline $2$ & $5812$ & $1011010110100$ \\ |
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3781 \hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ |
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3782 \hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ |
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3783 \hline $5$ & $8896$ & $10001011000000$ \\ |
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3784 \hline $6$ & $8896$ & $10001011000000$ \\ |
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3785 \hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ |
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3786 \hline $8$ & $25344$ & $110001100000000$ \\ |
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3787 \hline -- & $x/2^k = 99$ & \\ |
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3788 \hline |
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|
3789 \end{tabular} |
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3790 \end{center} |
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3791 \end{small} |
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3792 \caption{Example of Montgomery Reduction (II)} |
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3793 \label{fig:MONT2} |
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3794 \end{figure} |
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3795 |
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3796 Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 8$. |
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3797 With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the |
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3798 loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is |
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3799 zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. |
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3800 |
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3801 \subsection{Digit Based Montgomery Reduction} |
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3802 Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the |
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|
3803 previous algorithm re-written to compute the Montgomery reduction in this new fashion. |
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3804 |
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3805 \begin{figure}[!here] |
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3806 \begin{small} |
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3807 \begin{center} |
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3808 \begin{tabular}{l} |
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3809 \hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ |
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3810 \textbf{Input}. Integer $x$, $n$ and $k$ \\ |
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3811 \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ |
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3812 \hline \\ |
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3813 1. for $t$ from $0$ to $k - 1$ do \\ |
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3814 \hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ |
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3815 2. Return $x/\beta^k$. \\ |
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3816 \hline |
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3817 \end{tabular} |
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3818 \end{center} |
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3819 \end{small} |
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3820 \caption{Algorithm Montgomery Reduction (modified II)} |
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3821 \end{figure} |
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3822 |
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3823 The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of |
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3824 the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This |
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3825 problem breaks down to solving the following congruency. |
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3826 |
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3827 \begin{center} |
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3828 \begin{tabular}{rcl} |
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3829 $x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ |
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3830 $\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ |
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3831 $\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ |
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3832 \end{tabular} |
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3833 \end{center} |
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3834 |
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3835 In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used |
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3836 extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. |
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3837 |
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3838 For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ |
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3839 represent the value to reduce. |
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3840 |
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3841 \newpage\begin{figure} |
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3842 \begin{center} |
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3843 \begin{tabular}{|c|c|c|} |
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3844 \hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ |
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3845 \hline -- & $33$ & --\\ |
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3846 \hline $0$ & $33 + \mu n = 50$ & $1$ \\ |
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3847 \hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ |
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3848 \hline |
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3849 \end{tabular} |
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3850 \end{center} |
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3851 \caption{Example of Montgomery Reduction} |
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3852 \end{figure} |
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3853 |
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3854 The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ |
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3855 which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in |
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3856 the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and |
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3857 the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. |
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3858 |
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|
3859 \subsection{Baseline Montgomery Reduction} |
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3860 The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for |
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|
3861 Montgomery reductions. |
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3862 |
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3863 \newpage\begin{figure}[!here] |
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3864 \begin{small} |
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3865 \begin{center} |
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3866 \begin{tabular}{l} |
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3867 \hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ |
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3868 \textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ |
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3869 \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ |
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3870 \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ |
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3871 \hline \\ |
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3872 1. $digs \leftarrow 2n.used + 1$ \\ |
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3873 2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ |
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3874 \hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ |
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3875 \\ |
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3876 Setup $x$ for the reduction. \\ |
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3877 3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ |
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|
3878 4. $x.used \leftarrow digs$ \\ |
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3879 \\ |
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3880 Eliminate the lower $k$ digits. \\ |
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3881 5. For $ix$ from $0$ to $k - 1$ do \\ |
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3882 \hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ |
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3883 \hspace{3mm}5.2 $u \leftarrow 0$ \\ |
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3884 \hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ |
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3885 \hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ |
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3886 \hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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3887 \hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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3888 \hspace{3mm}5.4 While $u > 0$ do \\ |
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3889 \hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ |
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3890 \hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ |
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3891 \hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ |
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3892 \hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ |
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3893 \\ |
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3894 Divide by $\beta^k$ and fix up as required. \\ |
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3895 6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ |
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3896 7. If $x \ge n$ then \\ |
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3897 \hspace{3mm}7.1 $x \leftarrow x - n$ \\ |
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3898 8. Return(\textit{MP\_OKAY}). \\ |
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3899 \hline |
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3900 \end{tabular} |
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3901 \end{center} |
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3902 \end{small} |
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3903 \caption{Algorithm mp\_montgomery\_reduce} |
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3904 \end{figure} |
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3905 |
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3906 \textbf{Algorithm mp\_montgomery\_reduce.} |
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3907 This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based |
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3908 on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The |
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3909 restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as |
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3910 for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in |
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3911 advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. |
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3912 |
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3913 Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on |
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3914 the size of the input. This algorithm is discussed in ~COMBARED~. |
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3915 |
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3916 Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop |
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3917 calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and |
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3918 multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. |
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3919 |
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3920 Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications |
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3921 in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision |
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3922 multiplications. |
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3923 |
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3924 EXAM,bn_mp_montgomery_reduce.c |
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3925 |
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3926 This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based |
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3927 routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop. |
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3928 |
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3929 The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and |
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3930 the alias $tmpn$ refers to the modulus $n$. |
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3931 |
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3932 \subsection{Faster ``Comba'' Montgomery Reduction} |
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3933 MARK,COMBARED |
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3934 |
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3935 The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial |
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3936 nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba |
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3937 technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates |
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3938 a $k \times 1$ product $k$ times. |
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3939 |
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3940 The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the |
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3941 carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. |
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3942 Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. |
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3943 |
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3944 With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases |
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3945 the speed of the algorithm. |
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3946 |
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3947 \newpage\begin{figure}[!here] |
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3948 \begin{small} |
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3949 \begin{center} |
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3950 \begin{tabular}{l} |
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3951 \hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
3952 \textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3953 \hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3954 \textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3955 \hline \\ |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
3956 Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ |
|
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|
3957 1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ |
|
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|
3958 Copy the digits of $x$ into the array $\hat W$ \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3959 2. For $ix$ from $0$ to $x.used - 1$ do \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3960 \hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3961 3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ |
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Matt Johnston <matt@ucc.asn.au>
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3962 \hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ |
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|
3963 Elimiate the lower $k$ digits. \\ |
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|
3964 4. for $ix$ from $0$ to $n.used - 1$ do \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3965 \hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3966 \hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3967 \hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3968 \hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3969 Propagate carries upwards. \\ |
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3970 5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ |
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3971 \hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ |
|
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|
3972 Shift right and reduce modulo $\beta$ simultaneously. \\ |
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|
3973 6. for $ix$ from $0$ to $n.used + 1$ do \\ |
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|
3974 \hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ |
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|
3975 Zero excess digits and fixup $x$. \\ |
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|
3976 7. if $x.used > n.used + 1$ then do \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3977 \hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
3978 \hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ |
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|
3979 8. $x.used \leftarrow n.used + 1$ \\ |
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|
3980 9. Clamp excessive digits of $x$. \\ |
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|
3981 10. If $x \ge n$ then \\ |
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|
3982 \hspace{3mm}10.1 $x \leftarrow x - n$ \\ |
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|
3983 11. Return(\textit{MP\_OKAY}). \\ |
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|
3984 \hline |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3985 \end{tabular} |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3986 \end{center} |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3987 \end{small} |
|
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Matt Johnston <matt@ucc.asn.au>
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|
3988 \caption{Algorithm fast\_mp\_montgomery\_reduce} |
|
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|
3989 \end{figure} |
|
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|
3990 |
|
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|
3991 \textbf{Algorithm fast\_mp\_montgomery\_reduce.} |
|
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3992 This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly |
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3993 faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions |
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3994 on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the |
|
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3995 the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo |
|
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3996 a modulus of at most $3,556$ bits in length. |
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|
3997 |
|
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|
3998 As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the |
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|
3999 contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step |
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4000 4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such |
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4001 as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing |
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|
4002 a single precision multiplication instead half the amount of time is spent. |
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|
4003 |
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4004 Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step |
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4005 4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note |
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4006 how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no |
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|
4007 point. |
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4008 |
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4009 Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are |
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|
4010 stored in the destination $x$. |
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4011 |
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|
4012 EXAM,bn_fast_mp_montgomery_reduce.c |
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4013 |
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4014 The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share |
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4015 the same alias variables to make the code easier to read. |
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4016 |
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4017 The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This |
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4018 forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry |
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4019 for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. |
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4020 |
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4021 The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns |
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4022 modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th |
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4023 digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. |
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4024 |
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4025 \subsection{Montgomery Setup} |
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4026 To calculate the variable $\rho$ a relatively simple algorithm will be required. |
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4027 |
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4028 \begin{figure}[!here] |
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4029 \begin{small} |
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4030 \begin{center} |
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4031 \begin{tabular}{l} |
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4032 \hline Algorithm \textbf{mp\_montgomery\_setup}. \\ |
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4033 \textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ |
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4034 \textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ |
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4035 \hline \\ |
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4036 1. $b \leftarrow n_0$ \\ |
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4037 2. If $b$ is even return(\textit{MP\_VAL}) \\ |
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4038 3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\ |
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4039 4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ |
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4040 \hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ |
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4041 5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ |
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4042 6. Return(\textit{MP\_OKAY}). \\ |
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4043 \hline |
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4044 \end{tabular} |
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4045 \end{center} |
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4046 \end{small} |
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4047 \caption{Algorithm mp\_montgomery\_setup} |
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4048 \end{figure} |
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4049 |
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4050 \textbf{Algorithm mp\_montgomery\_setup.} |
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4051 This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick |
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4052 to calculate $1/n_0$ when $\beta$ is a power of two. |
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4053 |
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4054 EXAM,bn_mp_montgomery_setup.c |
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4055 |
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4056 This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess |
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4057 multiplications when $\beta$ is not the default 28-bits. |
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4058 |
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4059 \section{The Diminished Radix Algorithm} |
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4060 The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett |
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4061 or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. |
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4062 |
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4063 \begin{equation} |
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4064 (x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} |
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4065 \end{equation} |
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4066 |
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4067 This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that |
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4068 then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof |
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4069 of the above equation is very simple. First write $x$ in the product form. |
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4070 |
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4071 \begin{equation} |
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4072 x = qn + r |
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4073 \end{equation} |
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4074 |
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4075 Now reduce both sides modulo $(n - k)$. |
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4076 |
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4077 \begin{equation} |
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4078 x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} |
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4079 \end{equation} |
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4080 |
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4081 The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ |
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4082 into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. |
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4083 |
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4084 \begin{figure}[!here] |
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4085 \begin{small} |
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4086 \begin{center} |
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4087 \begin{tabular}{l} |
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4088 \hline Algorithm \textbf{Diminished Radix Reduction}. \\ |
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4089 \textbf{Input}. Integer $x$, $n$, $k$ \\ |
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4090 \textbf{Output}. $x \mbox{ mod } (n - k)$ \\ |
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4091 \hline \\ |
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4092 1. $q \leftarrow \lfloor x / n \rfloor$ \\ |
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4093 2. $q \leftarrow k \cdot q$ \\ |
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4094 3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ |
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4095 4. $x \leftarrow x + q$ \\ |
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4096 5. If $x \ge (n - k)$ then \\ |
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4097 \hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ |
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4098 \hspace{3mm}5.2 Goto step 1. \\ |
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4099 6. Return $x$ \\ |
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4100 \hline |
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4101 \end{tabular} |
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4102 \end{center} |
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4103 \end{small} |
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4104 \caption{Algorithm Diminished Radix Reduction} |
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4105 \label{fig:DR} |
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4106 \end{figure} |
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4107 |
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4108 This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always |
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4109 once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. |
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4110 |
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4111 \begin{equation} |
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4112 0 \le x < n^2 + k^2 - 2nk |
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4113 \end{equation} |
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4114 |
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4115 The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. |
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4116 |
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4117 \begin{equation} |
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4118 q < n - 2k - k^2/n |
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4119 \end{equation} |
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4120 |
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4121 Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as |
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4122 $0 \le x < n$. By step four the sum $x + q$ is bounded by |
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4123 |
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4124 \begin{equation} |
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4125 0 \le q + x < (k + 1)n - 2k^2 - 1 |
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4126 \end{equation} |
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4127 |
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4128 With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the |
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4129 sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the |
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4130 range $0 \le x < (n - k - 1)^2$. |
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4131 |
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4132 \begin{figure} |
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4133 \begin{small} |
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4134 \begin{center} |
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4135 \begin{tabular}{|l|} |
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4136 \hline |
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4137 $x = 123456789, n = 256, k = 3$ \\ |
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4138 \hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ |
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4139 $q \leftarrow q*k = 1446759$ \\ |
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4140 $x \leftarrow x \mbox{ mod } n = 21$ \\ |
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4141 $x \leftarrow x + q = 1446780$ \\ |
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4142 $x \leftarrow x - (n - k) = 1446527$ \\ |
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4143 \hline |
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4144 $q \leftarrow \lfloor x/n \rfloor = 5650$ \\ |
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4145 $q \leftarrow q*k = 16950$ \\ |
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4146 $x \leftarrow x \mbox{ mod } n = 127$ \\ |
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4147 $x \leftarrow x + q = 17077$ \\ |
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4148 $x \leftarrow x - (n - k) = 16824$ \\ |
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4149 \hline |
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4150 $q \leftarrow \lfloor x/n \rfloor = 65$ \\ |
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4151 $q \leftarrow q*k = 195$ \\ |
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4152 $x \leftarrow x \mbox{ mod } n = 184$ \\ |
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4153 $x \leftarrow x + q = 379$ \\ |
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4154 $x \leftarrow x - (n - k) = 126$ \\ |
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4155 \hline |
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4156 \end{tabular} |
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4157 \end{center} |
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4158 \end{small} |
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4159 \caption{Example Diminished Radix Reduction} |
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4160 \label{fig:EXDR} |
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4161 \end{figure} |
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4162 |
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4163 Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ |
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4164 is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only |
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4165 three passes were required to find the residue $x \equiv 126$. |
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4166 |
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4167 |
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4168 \subsection{Choice of Moduli} |
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4169 On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other |
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4170 modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. |
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4171 |
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4172 Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. |
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4173 Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division |
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4174 by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ |
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4175 which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. |
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4176 |
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4177 However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be |
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4178 performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. |
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4179 Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. |
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4180 |
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4181 Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted |
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4182 modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the |
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4183 $2^p$ logic except $p$ must be a multiple of $lg(\beta)$. |
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4184 |
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4185 \subsection{Choice of $k$} |
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4186 Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ |
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4187 in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might |
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4188 as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. |
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4189 |
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4190 \subsection{Restricted Diminished Radix Reduction} |
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4191 The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce |
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4192 an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation |
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4193 of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition |
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4194 of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular |
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4195 exponentiations are performed. |
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4196 |
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4197 \newpage\begin{figure}[!here] |
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4198 \begin{small} |
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4199 \begin{center} |
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4200 \begin{tabular}{l} |
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4201 \hline Algorithm \textbf{mp\_dr\_reduce}. \\ |
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4202 \textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ |
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4203 \hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ |
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4204 \textbf{Output}. $x \mbox{ mod } n$ \\ |
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4205 \hline \\ |
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4206 1. $m \leftarrow n.used$ \\ |
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4207 2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ |
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4208 3. $\mu \leftarrow 0$ \\ |
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4209 4. for $i$ from $0$ to $m - 1$ do \\ |
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4210 \hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ |
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4211 \hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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4212 \hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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4213 5. $x_{m} \leftarrow \mu$ \\ |
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4214 6. for $i$ from $m + 1$ to $x.used - 1$ do \\ |
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4215 \hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ |
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4216 7. Clamp excess digits of $x$. \\ |
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4217 8. If $x \ge n$ then \\ |
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4218 \hspace{3mm}8.1 $x \leftarrow x - n$ \\ |
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4219 \hspace{3mm}8.2 Goto step 3. \\ |
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4220 9. Return(\textit{MP\_OKAY}). \\ |
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4221 \hline |
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4222 \end{tabular} |
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4223 \end{center} |
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4224 \end{small} |
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4225 \caption{Algorithm mp\_dr\_reduce} |
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4226 \end{figure} |
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4227 |
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4228 \textbf{Algorithm mp\_dr\_reduce.} |
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4229 This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction |
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4230 with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. |
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4231 |
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4232 This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ |
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4233 and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing |
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4234 the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th |
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4235 digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to |
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4236 $x$ before the addition of the multiple of the upper half. |
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4237 |
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4238 At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes |
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4239 at step 3. |
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4240 |
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4241 EXAM,bn_mp_dr_reduce.c |
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4242 |
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4243 The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where |
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4244 the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of |
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4245 the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. |
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4246 |
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4247 The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits |
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4248 a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) |
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4249 in this algorithm. |
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4250 |
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4251 By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the |
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4252 same pointer will point to the $m+1$'th digit where the zeroes will be placed. |
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4253 |
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4254 Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. |
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4255 With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used |
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4256 as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code |
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4257 does not need to be checked. |
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4258 |
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4259 \subsubsection{Setup} |
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4260 To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for |
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4261 completeness. |
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4262 |
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4263 \begin{figure}[!here] |
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4264 \begin{small} |
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4265 \begin{center} |
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4266 \begin{tabular}{l} |
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4267 \hline Algorithm \textbf{mp\_dr\_setup}. \\ |
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4268 \textbf{Input}. mp\_int $n$ \\ |
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4269 \textbf{Output}. $k = \beta - n_0$ \\ |
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4270 \hline \\ |
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4271 1. $k \leftarrow \beta - n_0$ \\ |
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4272 \hline |
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4273 \end{tabular} |
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4274 \end{center} |
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4275 \end{small} |
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4276 \caption{Algorithm mp\_dr\_setup} |
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4277 \end{figure} |
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4278 |
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4279 EXAM,bn_mp_dr_setup.c |
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4280 |
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4281 \subsubsection{Modulus Detection} |
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4282 Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be |
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4283 of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. |
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4284 |
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4285 \begin{figure}[!here] |
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4286 \begin{small} |
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4287 \begin{center} |
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4288 \begin{tabular}{l} |
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4289 \hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ |
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4290 \textbf{Input}. mp\_int $n$ \\ |
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4291 \textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ |
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4292 \hline |
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4293 1. If $n.used < 2$ then return($0$). \\ |
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4294 2. for $ix$ from $1$ to $n.used - 1$ do \\ |
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4295 \hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ |
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4296 3. Return($1$). \\ |
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4297 \hline |
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4298 \end{tabular} |
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4299 \end{center} |
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4300 \end{small} |
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4301 \caption{Algorithm mp\_dr\_is\_modulus} |
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4302 \end{figure} |
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4303 |
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4304 \textbf{Algorithm mp\_dr\_is\_modulus.} |
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4305 This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are |
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4306 in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to |
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4307 step 3 then $n$ must be of Diminished Radix form. |
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4308 |
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4309 EXAM,bn_mp_dr_is_modulus.c |
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4310 |
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4311 \subsection{Unrestricted Diminished Radix Reduction} |
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4312 The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm |
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4313 is a straightforward adaptation of algorithm~\ref{fig:DR}. |
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4314 |
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4315 In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new |
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4316 algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. |
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4317 |
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4318 \begin{figure}[!here] |
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4319 \begin{small} |
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4320 \begin{center} |
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4321 \begin{tabular}{l} |
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4322 \hline Algorithm \textbf{mp\_reduce\_2k}. \\ |
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4323 \textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ |
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4324 \hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ |
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4325 \textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ |
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4326 \hline |
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4327 1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ |
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4328 2. While $a \ge n$ do \\ |
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4329 \hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ |
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4330 \hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ |
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4331 \hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ |
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4332 \hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ |
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4333 \hspace{3mm}2.5 If $a \ge n$ then do \\ |
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4334 \hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ |
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4335 3. Return(\textit{MP\_OKAY}). \\ |
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4336 \hline |
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4337 \end{tabular} |
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4338 \end{center} |
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4339 \end{small} |
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4340 \caption{Algorithm mp\_reduce\_2k} |
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4341 \end{figure} |
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4342 |
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4343 \textbf{Algorithm mp\_reduce\_2k.} |
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4344 This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right |
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4345 shift which makes the algorithm fairly inexpensive to use. |
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4346 |
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4347 EXAM,bn_mp_reduce_2k.c |
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4348 |
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4349 The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d |
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4350 on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size |
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4351 is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without |
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4352 any multiplications. |
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4353 |
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4354 The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are |
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4355 positive. By using the unsigned versions the overhead is kept to a minimum. |
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4356 |
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4357 \subsubsection{Unrestricted Setup} |
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4358 To setup this reduction algorithm the value of $k = 2^p - n$ is required. |
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4359 |
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4360 \begin{figure}[!here] |
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4361 \begin{small} |
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4362 \begin{center} |
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4363 \begin{tabular}{l} |
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4364 \hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ |
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4365 \textbf{Input}. mp\_int $n$ \\ |
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4366 \textbf{Output}. $k = 2^p - n$ \\ |
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4367 \hline |
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4368 1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ |
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4369 2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ |
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4370 3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ |
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4371 4. $k \leftarrow x_0$ \\ |
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4372 5. Return(\textit{MP\_OKAY}). \\ |
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4373 \hline |
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4374 \end{tabular} |
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4375 \end{center} |
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4376 \end{small} |
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4377 \caption{Algorithm mp\_reduce\_2k\_setup} |
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4378 \end{figure} |
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4379 |
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4380 \textbf{Algorithm mp\_reduce\_2k\_setup.} |
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4381 This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction |
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4382 is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. |
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4383 |
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4384 EXAM,bn_mp_reduce_2k_setup.c |
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4385 |
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4386 \subsubsection{Unrestricted Detection} |
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4387 An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. |
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4388 |
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4389 \begin{enumerate} |
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4390 \item The number has only one digit. |
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4391 \item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. |
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4392 \end{enumerate} |
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4393 |
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4394 If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only |
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4395 one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact |
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4396 that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most |
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4397 significant bit. The resulting sum will be a power of two. |
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4398 |
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4399 \begin{figure}[!here] |
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4400 \begin{small} |
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4401 \begin{center} |
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4402 \begin{tabular}{l} |
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4403 \hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ |
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4404 \textbf{Input}. mp\_int $n$ \\ |
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4405 \textbf{Output}. $1$ if of proper form, $0$ otherwise \\ |
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4406 \hline |
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4407 1. If $n.used = 0$ then return($0$). \\ |
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4408 2. If $n.used = 1$ then return($1$). \\ |
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4409 3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ |
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4410 4. for $x$ from $lg(\beta)$ to $p$ do \\ |
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4411 \hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ |
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4412 5. Return($1$). \\ |
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4413 \hline |
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4414 \end{tabular} |
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4415 \end{center} |
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4416 \end{small} |
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4417 \caption{Algorithm mp\_reduce\_is\_2k} |
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4418 \end{figure} |
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4419 |
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4420 \textbf{Algorithm mp\_reduce\_is\_2k.} |
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4421 This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. |
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4422 |
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4423 EXAM,bn_mp_reduce_is_2k.c |
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4424 |
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4425 |
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4426 |
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4427 \section{Algorithm Comparison} |
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4428 So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses |
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4429 that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since |
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4430 all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. |
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4431 |
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4432 \begin{center} |
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4433 \begin{small} |
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4434 \begin{tabular}{|c|c|c|c|c|c|} |
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4435 \hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ |
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4436 \hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ |
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4437 \hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ |
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4438 \hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ |
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4439 \hline |
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4440 \end{tabular} |
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4441 \end{small} |
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4442 \end{center} |
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4443 |
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4444 In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery |
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4445 reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of |
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4446 calling the half precision multipliers, addition and division by $\beta$ algorithms. |
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4447 |
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4448 For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly |
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4449 shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms |
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4450 primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in |
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4451 modular exponentiation to greatly speed up the operation. |
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4452 |
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4453 |
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4454 |
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4455 \section*{Exercises} |
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4456 \begin{tabular}{cl} |
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4457 $\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ |
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4458 & calculates the correct value of $\rho$. \\ |
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4459 & \\ |
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4460 $\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ |
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4461 & \\ |
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4462 $\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ |
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4463 & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ |
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4464 & terminate within $1 \le k \le 10$ iterations. \\ |
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4465 & \\ |
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4466 \end{tabular} |
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4467 |
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4468 |
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4469 \chapter{Exponentiation} |
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4470 Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed |
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4471 in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key |
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4472 cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any |
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4473 such cryptosystem and many methods have been sought to speed it up. |
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4474 |
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4475 \section{Exponentiation Basics} |
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4476 A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size |
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4477 the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature |
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4478 with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. |
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4479 |
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4480 Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which |
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4481 are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least |
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4482 significant bit. If $b$ is a $k$-bit integer than the following equation is true. |
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4483 |
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4484 \begin{equation} |
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4485 a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} |
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4486 \end{equation} |
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4487 |
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4488 By taking the base $a$ logarithm of both sides of the equation the following equation is the result. |
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4489 |
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4490 \begin{equation} |
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4491 b = \sum_{i=0}^{k-1}2^i \cdot b_i |
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4492 \end{equation} |
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4493 |
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4494 The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to |
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4495 $a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average |
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4496 $k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. |
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4497 |
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4498 While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to |
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4499 be computed in an auxilary variable. Consider the following equivalent algorithm. |
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4500 |
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4501 \begin{figure}[!here] |
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4502 \begin{small} |
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4503 \begin{center} |
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4504 \begin{tabular}{l} |
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4505 \hline Algorithm \textbf{Left to Right Exponentiation}. \\ |
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4506 \textbf{Input}. Integer $a$, $b$ and $k$ \\ |
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4507 \textbf{Output}. $c = a^b$ \\ |
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4508 \hline \\ |
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4509 1. $c \leftarrow 1$ \\ |
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4510 2. for $i$ from $k - 1$ to $0$ do \\ |
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4511 \hspace{3mm}2.1 $c \leftarrow c^2$ \\ |
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4512 \hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ |
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4513 3. Return $c$. \\ |
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4514 \hline |
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4515 \end{tabular} |
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4516 \end{center} |
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4517 \end{small} |
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4518 \caption{Left to Right Exponentiation} |
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4519 \label{fig:LTOR} |
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4520 \end{figure} |
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4521 |
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4522 This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is |
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4523 multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the |
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4524 product. |
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4525 |
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4526 For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. |
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4527 |
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4528 \newpage\begin{figure} |
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4529 \begin{center} |
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4530 \begin{tabular}{|c|c|} |
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4531 \hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ |
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4532 \hline - & $1$ \\ |
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4533 \hline $5$ & $a$ \\ |
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4534 \hline $4$ & $a^2$ \\ |
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4535 \hline $3$ & $a^4 \cdot a$ \\ |
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4536 \hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ |
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4537 \hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ |
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4538 \hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ |
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4539 \hline |
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4540 \end{tabular} |
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4541 \end{center} |
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4542 \caption{Example of Left to Right Exponentiation} |
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4543 \end{figure} |
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4544 |
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4545 When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is |
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4546 called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. |
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4547 |
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4548 \subsection{Single Digit Exponentiation} |
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4549 The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended |
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4550 to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of |
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4551 $b$ that are greater than three. |
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4552 |
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4553 \newpage\begin{figure}[!here] |
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4554 \begin{small} |
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4555 \begin{center} |
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4556 \begin{tabular}{l} |
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4557 \hline Algorithm \textbf{mp\_expt\_d}. \\ |
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4558 \textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ |
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4559 \textbf{Output}. $c = a^b$ \\ |
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4560 \hline \\ |
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4561 1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ |
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4562 2. $c \leftarrow 1$ (\textit{mp\_set}) \\ |
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4563 3. for $x$ from 1 to $lg(\beta)$ do \\ |
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4564 \hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ |
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4565 \hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ |
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4566 \hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ |
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4567 \hspace{3mm}3.3 $b \leftarrow b << 1$ \\ |
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4568 4. Clear $g$. \\ |
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4569 5. Return(\textit{MP\_OKAY}). \\ |
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4570 \hline |
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4571 \end{tabular} |
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4572 \end{center} |
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4573 \end{small} |
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4574 \caption{Algorithm mp\_expt\_d} |
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4575 \end{figure} |
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4576 |
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4577 \textbf{Algorithm mp\_expt\_d.} |
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4578 This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to |
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4579 quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the |
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4580 exponent is a fixed width. |
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4581 |
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4582 A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of |
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4583 $1$ in the subsequent step. |
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4584 |
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4585 Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared |
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4586 on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value |
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4587 of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each |
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4588 iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. |
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4589 |
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4590 EXAM,bn_mp_expt_d.c |
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4591 |
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4592 Line @29,mp_set@ sets the initial value of the result to $1$. Next the loop on line @31,for@ steps through each bit of the exponent starting from |
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4593 the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first. After |
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4594 the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line |
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4595 @47,<<@ moves all of the bits of the exponent upwards towards the most significant location. |
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4596 |
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4597 \section{$k$-ary Exponentiation} |
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4598 When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor |
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4599 slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to |
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4600 the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} |
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4601 computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a |
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4602 portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. |
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4603 |
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4604 \begin{figure}[!here] |
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4605 \begin{small} |
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4606 \begin{center} |
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4607 \begin{tabular}{l} |
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4608 \hline Algorithm \textbf{$k$-ary Exponentiation}. \\ |
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4609 \textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ |
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4610 \textbf{Output}. $c = a^b$ \\ |
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4611 \hline \\ |
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4612 1. $c \leftarrow 1$ \\ |
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4613 2. for $i$ from $t - 1$ to $0$ do \\ |
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4614 \hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ |
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4615 \hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ |
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4616 \hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ |
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4617 3. Return $c$. \\ |
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4618 \hline |
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4619 \end{tabular} |
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4620 \end{center} |
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4621 \end{small} |
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4622 \caption{$k$-ary Exponentiation} |
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4623 \label{fig:KARY} |
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4624 \end{figure} |
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4625 |
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4626 The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been |
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4627 precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and |
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4628 $2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. |
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4629 However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. |
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4630 |
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4631 Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The |
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4632 original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings |
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4633 has increased slightly but the number of multiplications has nearly halved. |
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4634 |
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4635 \subsection{Optimal Values of $k$} |
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4636 An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest |
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4637 approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ |
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4638 for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. |
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4639 |
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4640 \begin{figure}[here] |
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4641 \begin{center} |
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4642 \begin{small} |
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4643 \begin{tabular}{|c|c|c|c|c|c|} |
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4644 \hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ |
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4645 \hline $16$ & $2$ & $27$ & $24$ \\ |
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4646 \hline $32$ & $3$ & $49$ & $48$ \\ |
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4647 \hline $64$ & $3$ & $92$ & $96$ \\ |
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4648 \hline $128$ & $4$ & $175$ & $192$ \\ |
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4649 \hline $256$ & $4$ & $335$ & $384$ \\ |
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4650 \hline $512$ & $5$ & $645$ & $768$ \\ |
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4651 \hline $1024$ & $6$ & $1257$ & $1536$ \\ |
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4652 \hline $2048$ & $6$ & $2452$ & $3072$ \\ |
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4653 \hline $4096$ & $7$ & $4808$ & $6144$ \\ |
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4654 \hline |
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4655 \end{tabular} |
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4656 \end{small} |
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4657 \end{center} |
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4658 \caption{Optimal Values of $k$ for $k$-ary Exponentiation} |
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4659 \label{fig:OPTK} |
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4660 \end{figure} |
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4661 |
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4662 \subsection{Sliding-Window Exponentiation} |
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4663 A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially |
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4664 this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the |
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4665 algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. |
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4666 |
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4667 Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. |
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4668 |
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4669 \begin{figure}[here] |
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4670 \begin{center} |
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4671 \begin{small} |
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4672 \begin{tabular}{|c|c|c|c|c|c|} |
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4673 \hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ |
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4674 \hline $16$ & $3$ & $24$ & $27$ \\ |
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4675 \hline $32$ & $3$ & $45$ & $49$ \\ |
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4676 \hline $64$ & $4$ & $87$ & $92$ \\ |
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4677 \hline $128$ & $4$ & $167$ & $175$ \\ |
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4678 \hline $256$ & $5$ & $322$ & $335$ \\ |
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4679 \hline $512$ & $6$ & $628$ & $645$ \\ |
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4680 \hline $1024$ & $6$ & $1225$ & $1257$ \\ |
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4681 \hline $2048$ & $7$ & $2403$ & $2452$ \\ |
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4682 \hline $4096$ & $8$ & $4735$ & $4808$ \\ |
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4683 \hline |
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4684 \end{tabular} |
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4685 \end{small} |
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4686 \end{center} |
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4687 \caption{Optimal Values of $k$ for Sliding Window Exponentiation} |
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4688 \label{fig:OPTK2} |
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4689 \end{figure} |
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4690 |
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4691 \newpage\begin{figure}[!here] |
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4692 \begin{small} |
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4693 \begin{center} |
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4694 \begin{tabular}{l} |
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4695 \hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ |
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4696 \textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ |
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4697 \textbf{Output}. $c = a^b$ \\ |
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4698 \hline \\ |
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4699 1. $c \leftarrow 1$ \\ |
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4700 2. for $i$ from $t - 1$ to $0$ do \\ |
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4701 \hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ |
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4702 \hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ |
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4703 \hspace{3mm}2.2 else do \\ |
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4704 \hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ |
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4705 \hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ |
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4706 \hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ |
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4707 \hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ |
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4708 3. Return $c$. \\ |
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4709 \hline |
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4710 \end{tabular} |
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4711 \end{center} |
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4712 \end{small} |
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4713 \caption{Sliding Window $k$-ary Exponentiation} |
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4714 \end{figure} |
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4715 |
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4716 Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this |
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4717 algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half |
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4718 the size as the previous table. |
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4719 |
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4720 Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as |
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4721 the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the |
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4722 exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where |
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4723 a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ |
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4724 squarings. The second method requires $8$ multiplications and $18$ squarings. |
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4725 |
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4726 In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. |
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4727 |
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4728 \section{Modular Exponentiation} |
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4729 |
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4730 Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing |
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4731 $d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it |
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4732 modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. |
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4733 |
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4734 This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using |
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4735 one of the algorithms presented in ~REDUCTION~. |
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4736 |
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4737 Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm |
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4738 will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The |
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4739 value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm |
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4740 terminates with an error. |
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4741 |
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4742 \begin{figure}[!here] |
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4743 \begin{small} |
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4744 \begin{center} |
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4745 \begin{tabular}{l} |
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4746 \hline Algorithm \textbf{mp\_exptmod}. \\ |
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4747 \textbf{Input}. mp\_int $a$, $b$ and $c$ \\ |
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4748 \textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ |
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4749 \hline \\ |
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4750 1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ |
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4751 2. If $b.sign = MP\_NEG$ then \\ |
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4752 \hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ |
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4753 \hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ |
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4754 \hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ |
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4755 3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ |
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4756 \hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ |
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4757 4. else \\ |
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4758 \hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ |
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4759 \hline |
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4760 \end{tabular} |
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4761 \end{center} |
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4762 \end{small} |
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4763 \caption{Algorithm mp\_exptmod} |
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4764 \end{figure} |
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4765 |
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4766 \textbf{Algorithm mp\_exptmod.} |
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4767 The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm |
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4768 which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation |
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4769 except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation |
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4770 algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). |
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4771 |
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4772 EXAM,bn_mp_exptmod.c |
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4773 |
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4774 In order to keep the algorithms in a known state the first step on line @29,if@ is to reject any negative modulus as input. If the exponent is |
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4775 negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned |
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4776 the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive |
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4777 exponent. |
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4778 |
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|
4779 If the exponent is positive the algorithm resumes the exponentiation. Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix |
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4780 form. If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one |
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4781 of three values. |
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4782 |
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4783 \begin{enumerate} |
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4784 \item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. |
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4785 \item $dr = 1$ means that the modulus is of restricted Diminished Radix form. |
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4786 \item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. |
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4787 \end{enumerate} |
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4788 |
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4789 Line @69,if@ determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, |
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4790 the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. |
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4791 |
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4792 \subsection{Barrett Modular Exponentiation} |
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4793 |
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4794 \newpage\begin{figure}[!here] |
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4795 \begin{small} |
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4796 \begin{center} |
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4797 \begin{tabular}{l} |
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4798 \hline Algorithm \textbf{s\_mp\_exptmod}. \\ |
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4799 \textbf{Input}. mp\_int $a$, $b$ and $c$ \\ |
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4800 \textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ |
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4801 \hline \\ |
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4802 1. $k \leftarrow lg(x)$ \\ |
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4803 2. $winsize \leftarrow \left \lbrace \begin{array}{ll} |
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4804 2 & \mbox{if }k \le 7 \\ |
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4805 3 & \mbox{if }7 < k \le 36 \\ |
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4806 4 & \mbox{if }36 < k \le 140 \\ |
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4807 5 & \mbox{if }140 < k \le 450 \\ |
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4808 6 & \mbox{if }450 < k \le 1303 \\ |
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4809 7 & \mbox{if }1303 < k \le 3529 \\ |
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4810 8 & \mbox{if }3529 < k \\ |
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4811 \end{array} \right .$ \\ |
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4812 3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ |
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4813 4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ |
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4814 5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ |
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4815 \\ |
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4816 Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ |
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4817 6. $k \leftarrow 2^{winsize - 1}$ \\ |
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4818 7. $M_{k} \leftarrow M_1$ \\ |
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4819 8. for $ix$ from 0 to $winsize - 2$ do \\ |
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4820 \hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ |
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4821 \hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ |
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4822 9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ |
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4823 \hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ |
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4824 \hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ |
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4825 10. $res \leftarrow 1$ \\ |
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4826 \\ |
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4827 Start Sliding Window. \\ |
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4828 11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ |
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4829 12. Loop \\ |
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4830 \hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ |
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4831 \hspace{3mm}12.2 If $bitcnt = 0$ then do \\ |
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4832 \hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ |
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4833 \hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ |
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4834 \hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ |
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4835 \hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ |
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4836 Continued on next page. \\ |
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4837 \hline |
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4838 \end{tabular} |
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4839 \end{center} |
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4840 \end{small} |
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4841 \caption{Algorithm s\_mp\_exptmod} |
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4842 \end{figure} |
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4843 |
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4844 \newpage\begin{figure}[!here] |
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4845 \begin{small} |
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4846 \begin{center} |
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4847 \begin{tabular}{l} |
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4848 \hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ |
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4849 \textbf{Input}. mp\_int $a$, $b$ and $c$ \\ |
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4850 \textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ |
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4851 \hline \\ |
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4852 \hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ |
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Matt Johnston <matt@ucc.asn.au>
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|
4853 \hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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diff
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|
4854 \hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4855 \hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
4856 \hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4857 \hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ |
|
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|
4858 \hspace{6mm}12.6.3 Goto step 12. \\ |
|
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|
4859 \hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4860 \hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4861 \hspace{3mm}12.9 $mode \leftarrow 2$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4862 \hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4863 \hspace{6mm}Window is full so perform the squarings and single multiplication. \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4864 \hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ |
|
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|
4865 \hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4866 \hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4867 \hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4868 \hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4869 \hspace{6mm}Reset the window. \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4870 \hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4871 \\ |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
4872 No more windows left. Check for residual bits of exponent. \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4873 13. If $mode = 2$ and $bitcpy > 0$ then do \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4874 \hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4875 \hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4876 \hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4877 \hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4878 \hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4879 \hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4880 \hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4881 14. $y \leftarrow res$ \\ |
|
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|
4882 15. Clear $res$, $mu$ and the $M$ array. \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4883 16. Return(\textit{MP\_OKAY}). \\ |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4884 \hline |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4885 \end{tabular} |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
4886 \end{center} |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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diff
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|
4887 \end{small} |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
4888 \caption{Algorithm s\_mp\_exptmod (continued)} |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
4889 \end{figure} |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4890 |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
4891 \textbf{Algorithm s\_mp\_exptmod.} |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
4892 This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4893 algorithm to keep the product small throughout the algorithm. |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4894 |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
4895 The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
4896 larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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diff
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|
4897 table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
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changeset
|
4898 |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
4899 After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
4900 the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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diff
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|
4901 times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. |
|
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
4902 |
|
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
4903 Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4904 \begin{enumerate} |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4905 \item The variable $mode$ dictates how the bits of the exponent are interpreted. |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4906 \begin{enumerate} |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4907 \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4908 $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. |
|
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Matt Johnston <matt@ucc.asn.au>
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|
4909 \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits |
|
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|
4910 are read and a single squaring is performed. If a non-zero bit is read a new window is created. |
|
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|
4911 \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit |
|
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|
4912 downwards. |
|
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|
4913 \end{enumerate} |
|
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|
4914 \item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit |
|
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|
4915 is fetched from the exponent. |
|
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|
4916 \item The variable $buf$ holds the currently read digit of the exponent. |
|
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|
4917 \item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. |
|
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|
4918 \item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and |
|
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|
4919 the appropriate operations performed. |
|
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|
4920 \item The variable $bitbuf$ holds the current bits of the window being formed. |
|
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|
4921 \end{enumerate} |
|
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|
4922 |
|
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|
4923 All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step |
|
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|
4924 inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is |
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4925 read and if there are no digits left than the loop terminates. |
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4926 |
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4927 After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit |
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4928 upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to |
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4929 trailing edges the entire exponent is read from most significant bit to least significant bit. |
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4930 |
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4931 At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the |
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4932 algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle |
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4933 the two cases of $mode = 1$ and $mode = 2$ respectively. |
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4934 |
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4935 FIGU,expt_state,Sliding Window State Diagram |
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4936 |
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4937 By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then |
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4938 a Left-to-Right algorithm is used to process the remaining few bits. |
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4939 |
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4940 EXAM,bn_s_mp_exptmod.c |
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4941 |
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4942 Lines @26,if@ through @40,}@ determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted |
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4943 from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement |
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4944 on line @32,if@ the value of $x$ is already known to be greater than $140$. |
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4945 |
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4946 The conditional piece of code beginning on line @42,ifdef@ allows the window size to be restricted to five bits. This logic is used to ensure |
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4947 the table of precomputed powers of $G$ remains relatively small. |
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4948 |
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4949 The for loop on line @49,for@ initializes the $M$ array while lines @59,mp_init@ and @62,mp_reduce@ compute the value of $\mu$ required for |
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4950 Barrett reduction. |
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4951 |
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4952 -- More later. |
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4953 |
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4954 \section{Quick Power of Two} |
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4955 Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is |
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4956 equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. |
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4957 |
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4958 \begin{figure}[!here] |
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4959 \begin{small} |
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4960 \begin{center} |
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4961 \begin{tabular}{l} |
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4962 \hline Algorithm \textbf{mp\_2expt}. \\ |
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4963 \textbf{Input}. integer $b$ \\ |
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4964 \textbf{Output}. $a \leftarrow 2^b$ \\ |
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4965 \hline \\ |
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4966 1. $a \leftarrow 0$ \\ |
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4967 2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ |
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4968 3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ |
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4969 4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ |
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4970 5. Return(\textit{MP\_OKAY}). \\ |
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4971 \hline |
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4972 \end{tabular} |
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4973 \end{center} |
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4974 \end{small} |
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4975 \caption{Algorithm mp\_2expt} |
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4976 \end{figure} |
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4977 |
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4978 \textbf{Algorithm mp\_2expt.} |
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4979 |
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4980 EXAM,bn_mp_2expt.c |
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4981 |
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4982 \chapter{Higher Level Algorithms} |
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4983 |
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4984 This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These |
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4985 routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. |
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4986 |
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4987 The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic |
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4988 for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. |
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4989 These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate |
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4990 various representations of integers. For example, converting from an mp\_int to a string of character. |
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4991 |
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4992 \section{Integer Division with Remainder} |
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4993 \label{sec:division} |
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4994 |
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4995 Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication |
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4996 the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables |
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4997 will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and |
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4998 let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. |
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4999 |
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5000 \newpage\begin{figure}[!here] |
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5001 \begin{small} |
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5002 \begin{center} |
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5003 \begin{tabular}{l} |
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5004 \hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ |
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5005 \textbf{Input}. integer $x$ and $y$ \\ |
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5006 \textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ |
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5007 \hline \\ |
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5008 1. $q \leftarrow 0$ \\ |
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5009 2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ |
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5010 3. for $t$ from $n$ down to $0$ do \\ |
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5011 \hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ |
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5012 \hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ |
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5013 \hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ |
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5014 4. $r \leftarrow y$ \\ |
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5015 5. Return($q, r$) \\ |
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5016 \hline |
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5017 \end{tabular} |
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5018 \end{center} |
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5019 \end{small} |
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5020 \caption{Algorithm Radix-$\beta$ Integer Division} |
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5021 \label{fig:raddiv} |
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5022 \end{figure} |
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5023 |
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5024 As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which |
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5025 their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. |
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5026 |
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5027 To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and |
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5028 simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method |
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5029 used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading |
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5030 digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly |
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5031 arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. |
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5032 As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. |
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5033 |
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5034 Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder |
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5035 $y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the |
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5036 remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since |
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5037 $237 \cdot 23 + 20 = 5471$ is true. |
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5038 |
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5039 \subsection{Quotient Estimation} |
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5040 \label{sec:divest} |
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5041 As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading |
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5042 digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically |
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5043 speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the |
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5044 dividend and divisor are zero. |
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5045 |
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5046 The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} |
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5047 of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate |
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5048 using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ |
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5049 represent the most significant digits of the dividend and divisor respectively. |
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5050 |
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5051 \textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to |
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5052 $k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } |
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5053 The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other |
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5054 cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility |
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5055 $-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of |
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5056 inequalities will prove the hypothesis. |
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5057 |
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5058 \begin{equation} |
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5059 y - \hat k x \le y - \hat k x_s\beta^s |
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5060 \end{equation} |
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5061 |
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5062 This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. |
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5063 |
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5064 \begin{equation} |
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5065 y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) |
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5066 \end{equation} |
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5067 |
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5068 By simplifying the previous inequality the following inequality is formed. |
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5069 |
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5070 \begin{equation} |
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5071 y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s |
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5072 \end{equation} |
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5073 |
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5074 Subsequently, |
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5075 |
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5076 \begin{equation} |
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5077 y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x |
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5078 \end{equation} |
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5079 |
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5080 Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} |
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5081 |
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5082 |
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5083 \subsection{Normalized Integers} |
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5084 For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both |
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5085 $x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original |
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5086 remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will |
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5087 lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. |
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5088 |
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5089 \begin{equation} |
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5090 {{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} |
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5091 \end{equation} |
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5092 |
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5093 At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. |
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5094 |
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5095 \subsection{Radix-$\beta$ Division with Remainder} |
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5096 \newpage\begin{figure}[!here] |
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5097 \begin{small} |
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5098 \begin{center} |
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5099 \begin{tabular}{l} |
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5100 \hline Algorithm \textbf{mp\_div}. \\ |
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5101 \textbf{Input}. mp\_int $a, b$ \\ |
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5102 \textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ |
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5103 \hline \\ |
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5104 1. If $b = 0$ return(\textit{MP\_VAL}). \\ |
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5105 2. If $\vert a \vert < \vert b \vert$ then do \\ |
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5106 \hspace{3mm}2.1 $d \leftarrow a$ \\ |
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5107 \hspace{3mm}2.2 $c \leftarrow 0$ \\ |
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5108 \hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ |
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5109 \\ |
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5110 Setup the quotient to receive the digits. \\ |
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5111 3. Grow $q$ to $a.used + 2$ digits. \\ |
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5112 4. $q \leftarrow 0$ \\ |
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5113 5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ |
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5114 6. $sign \leftarrow \left \lbrace \begin{array}{ll} |
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5115 MP\_ZPOS & \mbox{if }a.sign = b.sign \\ |
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5116 MP\_NEG & \mbox{otherwise} \\ |
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5117 \end{array} \right .$ \\ |
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5118 \\ |
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5119 Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ |
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5120 7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ |
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5121 8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ |
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5122 \\ |
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5123 Find the leading digit of the quotient. \\ |
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5124 9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ |
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5125 10. $y \leftarrow y \cdot \beta^{n - t}$ \\ |
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5126 11. While ($x \ge y$) do \\ |
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5127 \hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ |
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5128 \hspace{3mm}11.2 $x \leftarrow x - y$ \\ |
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5129 12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ |
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5130 \\ |
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5131 Continued on the next page. \\ |
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5132 \hline |
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5133 \end{tabular} |
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5134 \end{center} |
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5135 \end{small} |
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5136 \caption{Algorithm mp\_div} |
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5137 \end{figure} |
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5138 |
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5139 \newpage\begin{figure}[!here] |
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5140 \begin{small} |
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5141 \begin{center} |
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5142 \begin{tabular}{l} |
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5143 \hline Algorithm \textbf{mp\_div} (continued). \\ |
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5144 \textbf{Input}. mp\_int $a, b$ \\ |
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5145 \textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ |
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5146 \hline \\ |
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5147 Now find the remainder fo the digits. \\ |
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5148 13. for $i$ from $n$ down to $(t + 1)$ do \\ |
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5149 \hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ |
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5150 \hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ |
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5151 \hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ |
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5152 \hspace{3mm}13.3 else \\ |
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5153 \hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ |
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5154 \hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ |
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5155 \hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ |
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5156 \hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ |
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5157 \\ |
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5158 Fixup quotient estimation. \\ |
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5159 \hspace{3mm}13.5 Loop \\ |
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5160 \hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ |
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5161 \hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ |
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5162 \hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ |
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5163 \hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ |
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5164 \hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ |
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5165 \hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ |
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5166 \hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ |
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5167 \hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ |
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5168 \hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ |
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5169 \hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ |
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5170 \hspace{6mm}13.10 t$1 \leftarrow y$ \\ |
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5171 \hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ |
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5172 \hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ |
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5173 \hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ |
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5174 \\ |
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5175 Finalize the result. \\ |
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5176 14. Clamp excess digits of $q$ \\ |
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5177 15. $c \leftarrow q, c.sign \leftarrow sign$ \\ |
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5178 16. $x.sign \leftarrow a.sign$ \\ |
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5179 17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ |
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5180 18. Return(\textit{MP\_OKAY}). \\ |
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5181 \hline |
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5182 \end{tabular} |
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|
5183 \end{center} |
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|
5184 \end{small} |
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|
5185 \caption{Algorithm mp\_div (continued)} |
|
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|
5186 \end{figure} |
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|
5187 \textbf{Algorithm mp\_div.} |
|
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5188 This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed |
|
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5189 division and will produce a fully qualified quotient and remainder. |
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5190 |
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5191 First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly |
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5192 zero and the remainder is the dividend. |
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5193 |
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5194 After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the |
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5195 divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are |
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5196 positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. |
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5197 This is performed by shifting both to the left by enough bits to get the desired normalization. |
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5198 |
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5199 At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is |
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5200 $2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted |
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5201 to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the |
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5202 shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two |
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5203 times to produce the desired leading digit of the quotient. |
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5204 |
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5205 Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly |
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5206 accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by |
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5207 induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. |
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5208 |
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5209 Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is |
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5210 to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher |
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5211 order approximation to adjust the quotient digit. |
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5212 |
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5213 After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced |
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5214 by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of |
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5215 algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. |
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5216 |
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5217 Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the |
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5218 remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} |
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5219 is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie |
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5220 outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should |
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5221 respectively be replaced with a zero. |
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5222 |
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5223 EXAM,bn_mp_div.c |
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5224 |
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5225 The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or |
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5226 remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division |
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5227 algorithm with only the quotient is |
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5228 |
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5229 \begin{verbatim} |
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5230 mp_div(&a, &b, &c, NULL); /* c = [a/b] */ |
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5231 \end{verbatim} |
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5232 |
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5233 Lines @37,if@ and @42,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor |
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5234 respectively. After the two trivial cases all of the temporary variables are initialized. Line @76,neg@ determines the sign of |
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5235 the quotient and line @77,sign@ ensures that both $x$ and $y$ are positive. |
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5236 |
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5237 The number of bits in the leading digit is calculated on line @80,norm@. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits |
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5238 of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is |
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5239 exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting |
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5240 them to the left by $lg(\beta) - 1 - k$ bits. |
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5241 |
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5242 Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the |
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5243 leading digit of the quotient. The loop beginning on line @113,for@ will produce the remainder of the quotient digits. |
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5244 |
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5245 The conditional ``continue'' on line @114,if@ is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the |
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5246 algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits |
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5247 above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. |
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5248 |
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5249 Lines @142,t1@, @143,t1@ and @150,t2@ through @152,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int |
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5250 variables directly. |
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5251 |
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5252 \section{Single Digit Helpers} |
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5253 |
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5254 This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of |
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5255 the helper functions assume the single digit input is positive and will treat them as such. |
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5256 |
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5257 \subsection{Single Digit Addition and Subtraction} |
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5258 |
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5259 Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction |
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5260 algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. |
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5261 |
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5262 \newpage\begin{figure}[!here] |
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5263 \begin{small} |
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5264 \begin{center} |
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5265 \begin{tabular}{l} |
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5266 \hline Algorithm \textbf{mp\_add\_d}. \\ |
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5267 \textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ |
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5268 \textbf{Output}. $c = a + b$ \\ |
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5269 \hline \\ |
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5270 1. $t \leftarrow b$ (\textit{mp\_set}) \\ |
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5271 2. $c \leftarrow a + t$ \\ |
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5272 3. Return(\textit{MP\_OKAY}) \\ |
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5273 \hline |
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5274 \end{tabular} |
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5275 \end{center} |
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5276 \end{small} |
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5277 \caption{Algorithm mp\_add\_d} |
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5278 \end{figure} |
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5279 |
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5280 \textbf{Algorithm mp\_add\_d.} |
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5281 This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. |
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5282 |
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5283 EXAM,bn_mp_add_d.c |
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5284 |
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5285 Clever use of the letter 't'. |
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5286 |
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5287 \subsubsection{Subtraction} |
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5288 The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. |
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5289 |
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5290 \subsection{Single Digit Multiplication} |
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5291 Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline |
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5292 multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands |
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5293 only has one digit. |
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5294 |
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5295 \begin{figure}[!here] |
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|
5296 \begin{small} |
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|
5297 \begin{center} |
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5298 \begin{tabular}{l} |
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5299 \hline Algorithm \textbf{mp\_mul\_d}. \\ |
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|
5300 \textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ |
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5301 \textbf{Output}. $c = ab$ \\ |
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5302 \hline \\ |
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5303 1. $pa \leftarrow a.used$ \\ |
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5304 2. Grow $c$ to at least $pa + 1$ digits. \\ |
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5305 3. $oldused \leftarrow c.used$ \\ |
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5306 4. $c.used \leftarrow pa + 1$ \\ |
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5307 5. $c.sign \leftarrow a.sign$ \\ |
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5308 6. $\mu \leftarrow 0$ \\ |
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5309 7. for $ix$ from $0$ to $pa - 1$ do \\ |
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5310 \hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ |
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5311 \hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ |
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5312 \hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ |
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5313 8. $c_{pa} \leftarrow \mu$ \\ |
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5314 9. for $ix$ from $pa + 1$ to $oldused$ do \\ |
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5315 \hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ |
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5316 10. Clamp excess digits of $c$. \\ |
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5317 11. Return(\textit{MP\_OKAY}). \\ |
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5318 \hline |
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5319 \end{tabular} |
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5320 \end{center} |
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5321 \end{small} |
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5322 \caption{Algorithm mp\_mul\_d} |
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5323 \end{figure} |
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|
5324 \textbf{Algorithm mp\_mul\_d.} |
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5325 This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. |
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5326 Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. |
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5327 |
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5328 EXAM,bn_mp_mul_d.c |
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5329 |
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5330 In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is |
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5331 read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. |
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5332 |
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5333 \subsection{Single Digit Division} |
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5334 Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the |
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5335 divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. |
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5336 |
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5337 \newpage\begin{figure}[!here] |
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5338 \begin{small} |
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5339 \begin{center} |
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5340 \begin{tabular}{l} |
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5341 \hline Algorithm \textbf{mp\_div\_d}. \\ |
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5342 \textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ |
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5343 \textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ |
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5344 \hline \\ |
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5345 1. If $b = 0$ then return(\textit{MP\_VAL}).\\ |
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5346 2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ |
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5347 3. Init $q$ to $a.used$ digits. \\ |
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5348 4. $q.used \leftarrow a.used$ \\ |
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5349 5. $q.sign \leftarrow a.sign$ \\ |
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5350 6. $\hat w \leftarrow 0$ \\ |
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5351 7. for $ix$ from $a.used - 1$ down to $0$ do \\ |
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5352 \hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ |
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5353 \hspace{3mm}7.2 If $\hat w \ge b$ then \\ |
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5354 \hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ |
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5355 \hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ |
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5356 \hspace{3mm}7.3 else\\ |
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5357 \hspace{6mm}7.3.1 $t \leftarrow 0$ \\ |
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5358 \hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ |
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5359 8. $d \leftarrow \hat w$ \\ |
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5360 9. Clamp excess digits of $q$. \\ |
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5361 10. $c \leftarrow q$ \\ |
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5362 11. Return(\textit{MP\_OKAY}). \\ |
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5363 \hline |
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5364 \end{tabular} |
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5365 \end{center} |
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5366 \end{small} |
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5367 \caption{Algorithm mp\_div\_d} |
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5368 \end{figure} |
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5369 \textbf{Algorithm mp\_div\_d.} |
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5370 This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the |
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5371 algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ |
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5372 after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. |
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5373 |
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5374 If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with |
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5375 a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction |
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|
5376 from chapter seven. |
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5377 |
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5378 EXAM,bn_mp_div_d.c |
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5379 |
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5380 Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to |
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5381 indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. |
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5382 |
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5383 The division and remainder on lines @44,/@ and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based |
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5384 processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC |
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5385 compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. |
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5386 |
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5387 \subsection{Single Digit Root Extraction} |
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5388 |
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5389 Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation |
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5390 (\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. |
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5391 |
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5392 \begin{equation} |
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5393 x_{i+1} = x_i - {f(x_i) \over f'(x_i)} |
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5394 \label{eqn:newton} |
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5395 \end{equation} |
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5396 |
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5397 In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is |
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5398 simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain |
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5399 such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the |
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5400 algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. |
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5401 |
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5402 \newpage\begin{figure}[!here] |
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5403 \begin{small} |
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5404 \begin{center} |
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5405 \begin{tabular}{l} |
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5406 \hline Algorithm \textbf{mp\_n\_root}. \\ |
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5407 \textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ |
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5408 \textbf{Output}. $c^b \le a$ \\ |
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5409 \hline \\ |
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5410 1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ |
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5411 2. $sign \leftarrow a.sign$ \\ |
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5412 3. $a.sign \leftarrow MP\_ZPOS$ \\ |
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5413 4. t$2 \leftarrow 2$ \\ |
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5414 5. Loop \\ |
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5415 \hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ |
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5416 \hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ |
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5417 \hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ |
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5418 \hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ |
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5419 \hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ |
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5420 \hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ |
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5421 \hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ |
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5422 \hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ |
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5423 6. Loop \\ |
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5424 \hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ |
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5425 \hspace{3mm}6.2 If t$2 > a$ then \\ |
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5426 \hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ |
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5427 \hspace{6mm}6.2.2 Goto step 6. \\ |
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5428 7. $a.sign \leftarrow sign$ \\ |
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5429 8. $c \leftarrow $ t$1$ \\ |
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5430 9. $c.sign \leftarrow sign$ \\ |
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5431 10. Return(\textit{MP\_OKAY}). \\ |
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5432 \hline |
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5433 \end{tabular} |
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5434 \end{center} |
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5435 \end{small} |
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5436 \caption{Algorithm mp\_n\_root} |
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5437 \end{figure} |
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5438 \textbf{Algorithm mp\_n\_root.} |
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5439 This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation |
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5440 that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding |
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5441 $x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ |
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5442 multiplications by t$1$ inside the loop. |
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5443 |
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5444 The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the |
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5445 root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. |
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5446 |
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5447 EXAM,bn_mp_n_root.c |
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5448 |
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5449 \section{Random Number Generation} |
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5450 |
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5451 Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho |
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5452 factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented |
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5453 is solely for simulations and not intended for cryptographic use. |
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5454 |
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5455 \newpage\begin{figure}[!here] |
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5456 \begin{small} |
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5457 \begin{center} |
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5458 \begin{tabular}{l} |
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5459 \hline Algorithm \textbf{mp\_rand}. \\ |
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5460 \textbf{Input}. An integer $b$ \\ |
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5461 \textbf{Output}. A pseudo-random number of $b$ digits \\ |
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5462 \hline \\ |
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5463 1. $a \leftarrow 0$ \\ |
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5464 2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ |
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5465 3. Pick a non-zero random digit $d$. \\ |
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5466 4. $a \leftarrow a + d$ \\ |
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5467 5. for $ix$ from 1 to $d - 1$ do \\ |
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5468 \hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ |
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5469 \hspace{3mm}5.2 Pick a random digit $d$. \\ |
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5470 \hspace{3mm}5.3 $a \leftarrow a + d$ \\ |
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5471 6. Return(\textit{MP\_OKAY}). \\ |
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5472 \hline |
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5473 \end{tabular} |
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5474 \end{center} |
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5475 \end{small} |
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5476 \caption{Algorithm mp\_rand} |
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5477 \end{figure} |
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5478 \textbf{Algorithm mp\_rand.} |
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5479 This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the |
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5480 final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of |
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5481 the integers from $0$ to $\beta - 1$. |
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5482 |
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5483 EXAM,bn_mp_rand.c |
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5484 |
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5485 \section{Formatted Representations} |
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5486 The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to |
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5487 be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers |
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5488 into a program. |
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5489 |
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5490 \subsection{Reading Radix-n Input} |
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5491 For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to |
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5492 printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the |
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5493 map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen |
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5494 such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary |
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5495 mediums. |
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5496 |
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5497 \newpage\begin{figure}[here] |
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5498 \begin{center} |
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5499 \begin{tabular}{cc|cc|cc|cc} |
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5500 \hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ |
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5501 \hline |
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5502 0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ |
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5503 4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ |
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5504 8 & 8 & 9 & 9 & 10 & A & 11 & B \\ |
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5505 12 & C & 13 & D & 14 & E & 15 & F \\ |
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5506 16 & G & 17 & H & 18 & I & 19 & J \\ |
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5507 20 & K & 21 & L & 22 & M & 23 & N \\ |
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5508 24 & O & 25 & P & 26 & Q & 27 & R \\ |
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5509 28 & S & 29 & T & 30 & U & 31 & V \\ |
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5510 32 & W & 33 & X & 34 & Y & 35 & Z \\ |
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5511 36 & a & 37 & b & 38 & c & 39 & d \\ |
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5512 40 & e & 41 & f & 42 & g & 43 & h \\ |
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5513 44 & i & 45 & j & 46 & k & 47 & l \\ |
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5514 48 & m & 49 & n & 50 & o & 51 & p \\ |
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5515 52 & q & 53 & r & 54 & s & 55 & t \\ |
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5516 56 & u & 57 & v & 58 & w & 59 & x \\ |
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5517 60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ |
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5518 \hline |
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5519 \end{tabular} |
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5520 \end{center} |
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5521 \caption{Lower ASCII Map} |
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|
5522 \label{fig:ASC} |
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5523 \end{figure} |
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5524 |
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|
5525 \newpage\begin{figure}[!here] |
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5526 \begin{small} |
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5527 \begin{center} |
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5528 \begin{tabular}{l} |
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5529 \hline Algorithm \textbf{mp\_read\_radix}. \\ |
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5530 \textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ |
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5531 \textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ |
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5532 \hline \\ |
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5533 1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ |
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5534 2. $ix \leftarrow 0$ \\ |
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5535 3. If $str_0 =$ ``-'' then do \\ |
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5536 \hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ |
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5537 \hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ |
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5538 4. else \\ |
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5539 \hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ |
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5540 5. $a \leftarrow 0$ \\ |
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5541 6. for $iy$ from $ix$ to $sn - 1$ do \\ |
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5542 \hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ |
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5543 \hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ |
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5544 \hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ |
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5545 \hspace{3mm}6.4 $a \leftarrow a + y$ \\ |
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5546 7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ |
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5547 8. Return(\textit{MP\_OKAY}). \\ |
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5548 \hline |
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5549 \end{tabular} |
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5550 \end{center} |
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5551 \end{small} |
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5552 \caption{Algorithm mp\_read\_radix} |
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5553 \end{figure} |
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5554 \textbf{Algorithm mp\_read\_radix.} |
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5555 This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the |
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5556 string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input |
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5557 and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded |
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5558 as part of larger input without any significant problem. |
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5559 |
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5560 EXAM,bn_mp_read_radix.c |
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5561 |
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5562 \subsection{Generating Radix-$n$ Output} |
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5563 Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. |
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5564 |
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5565 \newpage\begin{figure}[!here] |
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5566 \begin{small} |
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5567 \begin{center} |
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5568 \begin{tabular}{l} |
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5569 \hline Algorithm \textbf{mp\_toradix}. \\ |
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5570 \textbf{Input}. A mp\_int $a$ and an integer $r$\\ |
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5571 \textbf{Output}. The radix-$r$ representation of $a$ \\ |
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5572 \hline \\ |
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5573 1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ |
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5574 2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ |
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5575 3. $t \leftarrow a$ \\ |
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5576 4. $str \leftarrow$ ``'' \\ |
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5577 5. if $t.sign = MP\_NEG$ then \\ |
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5578 \hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ |
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5579 \hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ |
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5580 6. While ($t \ne 0$) do \\ |
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5581 \hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ |
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5582 \hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ |
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5583 \hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ |
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5584 \hspace{3mm}6.4 $str \leftarrow str + y$ \\ |
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5585 7. If $str_0 = $``$-$'' then \\ |
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5586 \hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ |
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5587 8. Otherwise \\ |
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5588 \hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ |
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5589 9. Return(\textit{MP\_OKAY}).\\ |
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5590 \hline |
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5591 \end{tabular} |
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5592 \end{center} |
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5593 \end{small} |
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5594 \caption{Algorithm mp\_toradix} |
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5595 \end{figure} |
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5596 \textbf{Algorithm mp\_toradix.} |
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5597 This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing |
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5598 successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in |
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5599 each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions |
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5600 are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order |
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5601 (see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. |
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5602 |
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5603 \begin{figure} |
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5604 \begin{center} |
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5605 \begin{tabular}{|c|c|c|} |
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5606 \hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ |
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5607 \hline $1234$ & -- & -- \\ |
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5608 \hline $123$ & $4$ & ``4'' \\ |
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5609 \hline $12$ & $3$ & ``43'' \\ |
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5610 \hline $1$ & $2$ & ``432'' \\ |
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5611 \hline $0$ & $1$ & ``4321'' \\ |
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5612 \hline |
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5613 \end{tabular} |
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5614 \end{center} |
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5615 \caption{Example of Algorithm mp\_toradix.} |
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5616 \label{fig:mpradix} |
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5617 \end{figure} |
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5618 |
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5619 EXAM,bn_mp_toradix.c |
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5620 |
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5621 \chapter{Number Theoretic Algorithms} |
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5622 This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi |
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5623 symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and |
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5624 various Sieve based factoring algorithms. |
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5625 |
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5626 \section{Greatest Common Divisor} |
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5627 The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of |
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5628 both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur |
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5629 simultaneously. |
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5630 |
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5631 The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then |
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5632 $r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. |
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5633 |
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5634 \newpage\begin{figure}[!here] |
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5635 \begin{small} |
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5636 \begin{center} |
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5637 \begin{tabular}{l} |
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5638 \hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ |
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5639 \textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ |
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5640 \textbf{Output}. The greatest common divisor $(a, b)$. \\ |
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5641 \hline \\ |
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5642 1. While ($b > 0$) do \\ |
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5643 \hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ |
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5644 \hspace{3mm}1.2 $a \leftarrow b$ \\ |
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5645 \hspace{3mm}1.3 $b \leftarrow r$ \\ |
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5646 2. Return($a$). \\ |
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5647 \hline |
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5648 \end{tabular} |
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5649 \end{center} |
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5650 \end{small} |
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5651 \caption{Algorithm Greatest Common Divisor (I)} |
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5652 \label{fig:gcd1} |
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5653 \end{figure} |
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5654 |
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5655 This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are |
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5656 relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of |
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5657 greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. |
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5658 In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. |
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5659 |
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5660 \begin{figure}[!here] |
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5661 \begin{small} |
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5662 \begin{center} |
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5663 \begin{tabular}{l} |
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5664 \hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ |
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5665 \textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ |
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5666 \textbf{Output}. The greatest common divisor $(a, b)$. \\ |
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5667 \hline \\ |
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5668 1. While ($b > 0$) do \\ |
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5669 \hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ |
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5670 \hspace{3mm}1.2 $b \leftarrow b - a$ \\ |
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5671 2. Return($a$). \\ |
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5672 \hline |
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5673 \end{tabular} |
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5674 \end{center} |
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5675 \end{small} |
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5676 \caption{Algorithm Greatest Common Divisor (II)} |
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5677 \label{fig:gcd2} |
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5678 \end{figure} |
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5679 |
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5680 \textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} |
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5681 The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other |
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5682 words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always |
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5683 divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the |
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5684 second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. |
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5685 |
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5686 As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that |
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5687 $b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does |
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5688 not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by |
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5689 the greatest common divisor. |
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5690 |
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5691 However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. |
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5692 Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. |
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5693 |
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5694 \begin{figure}[!here] |
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5695 \begin{small} |
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5696 \begin{center} |
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5697 \begin{tabular}{l} |
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5698 \hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ |
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5699 \textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ |
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5700 \textbf{Output}. The greatest common divisor $(a, b)$. \\ |
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5701 \hline \\ |
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5702 1. $k \leftarrow 0$ \\ |
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5703 2. While $a$ and $b$ are both divisible by $p$ do \\ |
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5704 \hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ |
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5705 \hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ |
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5706 \hspace{3mm}2.3 $k \leftarrow k + 1$ \\ |
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5707 3. While $a$ is divisible by $p$ do \\ |
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5708 \hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ |
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5709 4. While $b$ is divisible by $p$ do \\ |
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5710 \hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ |
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5711 5. While ($b > 0$) do \\ |
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5712 \hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ |
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5713 \hspace{3mm}5.2 $b \leftarrow b - a$ \\ |
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5714 \hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ |
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5715 \hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ |
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5716 6. Return($a \cdot p^k$). \\ |
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5717 \hline |
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5718 \end{tabular} |
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5719 \end{center} |
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5720 \end{small} |
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5721 \caption{Algorithm Greatest Common Divisor (III)} |
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5722 \label{fig:gcd3} |
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5723 \end{figure} |
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5724 |
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5725 This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ |
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5726 decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common |
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5727 divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely |
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5728 divided out of the difference $b - a$ so long as the division leaves no remainder. |
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5729 |
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5730 In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy |
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5731 to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by |
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5732 step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the |
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5733 largest of the pair. |
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5734 |
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5735 \subsection{Complete Greatest Common Divisor} |
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5736 The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly |
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5737 and will produce the greatest common divisor. |
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5738 |
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5739 \newpage\begin{figure}[!here] |
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5740 \begin{small} |
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5741 \begin{center} |
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5742 \begin{tabular}{l} |
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5743 \hline Algorithm \textbf{mp\_gcd}. \\ |
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5744 \textbf{Input}. mp\_int $a$ and $b$ \\ |
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5745 \textbf{Output}. The greatest common divisor $c = (a, b)$. \\ |
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5746 \hline \\ |
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5747 1. If $a = 0$ and $b \ne 0$ then \\ |
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|
5748 \hspace{3mm}1.1 $c \leftarrow b$ \\ |
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|
5749 \hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ |
|
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|
5750 2. If $a \ne 0$ and $b = 0$ then \\ |
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|
5751 \hspace{3mm}2.1 $c \leftarrow a$ \\ |
|
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|
5752 \hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ |
|
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|
5753 3. If $a = b = 0$ then \\ |
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|
5754 \hspace{3mm}3.1 $c \leftarrow 1$ \\ |
|
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|
5755 \hspace{3mm}3.2 Return(\textit{MP\_OKAY}). \\ |
|
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|
5756 4. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ |
|
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|
5757 5. $k \leftarrow 0$ \\ |
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5758 6. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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5759 \hspace{3mm}6.1 $k \leftarrow k + 1$ \\ |
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|
5760 \hspace{3mm}6.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ |
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5761 \hspace{3mm}6.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ |
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5762 7. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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|
5763 \hspace{3mm}7.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ |
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5764 8. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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5765 \hspace{3mm}8.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ |
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5766 9. While $v.used > 0$ \\ |
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5767 \hspace{3mm}9.1 If $\vert u \vert > \vert v \vert$ then \\ |
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5768 \hspace{6mm}9.1.1 Swap $u$ and $v$. \\ |
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5769 \hspace{3mm}9.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ |
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5770 \hspace{3mm}9.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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5771 \hspace{6mm}9.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ |
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|
5772 10. $c \leftarrow u \cdot 2^k$ \\ |
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5773 11. Return(\textit{MP\_OKAY}). \\ |
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|
5774 \hline |
|
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|
5775 \end{tabular} |
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|
5776 \end{center} |
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|
5777 \end{small} |
|
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|
5778 \caption{Algorithm mp\_gcd} |
|
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|
5779 \end{figure} |
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|
5780 \textbf{Algorithm mp\_gcd.} |
|
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5781 This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of |
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|
5782 Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as |
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|
5783 Algorithm B and in practice this appears to be true. |
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5784 |
|
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5785 The first three steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the |
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5786 largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of |
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5787 $a$ and $b$ respectively and the algorithm will proceed to reduce the pair. |
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5788 |
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5789 Step six will divide out any common factors of two and keep track of the count in the variable $k$. After this step two is no longer a |
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|
5790 factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step |
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|
5791 seven and eight ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while loops will iterate since |
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|
5792 they cannot both be even. |
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|
5793 |
|
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5794 By step nine both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to |
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|
5795 or greater than $u$. This ensures that the subtraction on step 9.2 will always produce a positive and even result. Step 9.3 removes any |
|
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|
5796 factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. |
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|
5797 |
|
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5798 After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result |
|
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|
5799 must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. |
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|
5800 |
|
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|
5801 EXAM,bn_mp_gcd.c |
|
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5802 |
|
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5803 This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the |
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|
5804 integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise |
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|
5805 it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three |
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|
5806 trivial cases of inputs are handled on lines @25,zero@ through @34,}@. After those lines the inputs are assumed to be non-zero. |
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|
5807 |
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5808 Lines @36,if@ and @40,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two |
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5809 must be divided out of the two inputs. The while loop on line @49,while@ iterates so long as both are even. The local integer $k$ is used to |
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5810 keep track of how many factors of $2$ are pulled out of both values. It is assumed that the number of factors will not exceed the maximum |
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5811 value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than entries than are accessible by an ``int'' so this is not |
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5812 a limitation.}. |
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5813 |
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5814 At this point there are no more common factors of two in the two values. The while loops on lines @60,while@ and @65,while@ remove any independent |
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5815 factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop |
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5816 on line @71, while@ performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in |
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5817 place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. |
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|
5818 |
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|
5819 \section{Least Common Multiple} |
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5820 The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the |
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5821 least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ |
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5822 and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. |
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5823 |
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5824 The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will |
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5825 collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on |
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5826 Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). |
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5827 Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. |
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5828 |
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5829 \begin{figure}[!here] |
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5830 \begin{small} |
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5831 \begin{center} |
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5832 \begin{tabular}{l} |
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5833 \hline Algorithm \textbf{mp\_lcm}. \\ |
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5834 \textbf{Input}. mp\_int $a$ and $b$ \\ |
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5835 \textbf{Output}. The least common multiple $c = [a, b]$. \\ |
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5836 \hline \\ |
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5837 1. $c \leftarrow (a, b)$ \\ |
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5838 2. $t \leftarrow a \cdot b$ \\ |
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5839 3. $c \leftarrow \lfloor t / c \rfloor$ \\ |
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5840 4. Return(\textit{MP\_OKAY}). \\ |
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5841 \hline |
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5842 \end{tabular} |
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5843 \end{center} |
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5844 \end{small} |
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5845 \caption{Algorithm mp\_lcm} |
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5846 \end{figure} |
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5847 \textbf{Algorithm mp\_lcm.} |
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5848 This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by |
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5849 dividing the product of the two inputs by their greatest common divisor. |
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5850 |
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5851 EXAM,bn_mp_lcm.c |
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5852 |
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5853 \section{Jacobi Symbol Computation} |
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5854 To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is |
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5855 defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is |
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5856 equivalent to equation \ref{eqn:legendre}. |
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5857 |
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5858 \textit{-- Tom, don't be an ass, cite your source here...!} |
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5859 |
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5860 \begin{equation} |
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5861 a^{(p-1)/2} \equiv \begin{array}{rl} |
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5862 -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ |
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5863 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ |
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5864 1 & \mbox{if }a\mbox{ is a quadratic residue}. |
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5865 \end{array} \mbox{ (mod }p\mbox{)} |
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5866 \label{eqn:legendre} |
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5867 \end{equation} |
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5868 |
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5869 \textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} |
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5870 An integer $a$ is a quadratic residue if the following equation has a solution. |
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5871 |
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5872 \begin{equation} |
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5873 x^2 \equiv a \mbox{ (mod }p\mbox{)} |
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5874 \label{eqn:root} |
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5875 \end{equation} |
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5876 |
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5877 Consider the following equation. |
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5878 |
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5879 \begin{equation} |
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5880 0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} |
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5881 \label{eqn:rooti} |
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5882 \end{equation} |
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5883 |
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5884 Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ |
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5885 then the quantity in the braces must be zero. By reduction, |
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5886 |
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5887 \begin{eqnarray} |
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5888 \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ |
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5889 \left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ |
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5890 x^2 \equiv a \mbox{ (mod }p\mbox{)} |
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5891 \end{eqnarray} |
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5892 |
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5893 As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ |
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5894 is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since |
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|
5895 \begin{equation} |
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5896 0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} |
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|
5897 \end{equation} |
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5898 One of the terms on the right hand side must be zero. \textbf{QED} |
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5899 |
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|
5900 \subsection{Jacobi Symbol} |
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5901 The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then |
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5902 the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. |
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5903 |
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5904 \begin{equation} |
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5905 \left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) |
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5906 \end{equation} |
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5907 |
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5908 By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for |
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5909 further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the |
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5910 following are true. |
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5911 |
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5912 \begin{enumerate} |
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5913 \item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. |
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5914 \item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. |
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5915 \item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. |
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5916 \item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. |
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5917 \item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically |
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5918 $\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. |
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5919 \end{enumerate} |
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5920 |
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5921 Using these facts if $a = 2^k \cdot a'$ then |
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5922 |
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5923 \begin{eqnarray} |
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5924 \left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ |
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5925 = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) |
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|
5926 \label{eqn:jacobi} |
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5927 \end{eqnarray} |
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5928 |
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5929 By fact five, |
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5930 |
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5931 \begin{equation} |
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5932 \left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} |
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5933 \end{equation} |
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5934 |
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5935 Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then |
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5936 |
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5937 \begin{equation} |
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5938 \left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} |
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5939 \end{equation} |
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5940 |
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5941 By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. |
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5942 |
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|
5943 \begin{equation} |
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5944 \left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} |
|
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|
5945 \end{equation} |
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5946 |
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5947 The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of |
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5948 $\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the |
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5949 factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the |
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|
5950 Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. |
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5951 |
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|
5952 \newpage\begin{figure}[!here] |
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5953 \begin{small} |
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5954 \begin{center} |
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5955 \begin{tabular}{l} |
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|
5956 \hline Algorithm \textbf{mp\_jacobi}. \\ |
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5957 \textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ |
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5958 \textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ |
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5959 \hline \\ |
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5960 1. If $a = 0$ then \\ |
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|
5961 \hspace{3mm}1.1 $c \leftarrow 0$ \\ |
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5962 \hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ |
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5963 2. If $a = 1$ then \\ |
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5964 \hspace{3mm}2.1 $c \leftarrow 1$ \\ |
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5965 \hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ |
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5966 3. $a' \leftarrow a$ \\ |
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5967 4. $k \leftarrow 0$ \\ |
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5968 5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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5969 \hspace{3mm}5.1 $k \leftarrow k + 1$ \\ |
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5970 \hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ |
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5971 6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ |
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5972 \hspace{3mm}6.1 $s \leftarrow 1$ \\ |
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5973 7. else \\ |
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5974 \hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ |
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5975 \hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ |
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5976 \hspace{6mm}7.2.1 $s \leftarrow 1$ \\ |
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5977 \hspace{3mm}7.3 else \\ |
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5978 \hspace{6mm}7.3.1 $s \leftarrow -1$ \\ |
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5979 8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ |
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5980 \hspace{3mm}8.1 $s \leftarrow -s$ \\ |
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5981 9. If $a' \ne 1$ then \\ |
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5982 \hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ |
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5983 \hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ |
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5984 10. $c \leftarrow s$ \\ |
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5985 11. Return(\textit{MP\_OKAY}). \\ |
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5986 \hline |
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5987 \end{tabular} |
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5988 \end{center} |
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5989 \end{small} |
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5990 \caption{Algorithm mp\_jacobi} |
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5991 \end{figure} |
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5992 \textbf{Algorithm mp\_jacobi.} |
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5993 This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm |
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5994 is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. |
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5995 |
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5996 Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the |
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5997 input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one |
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5998 if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled |
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5999 the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ |
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6000 are congruent to one modulo four, otherwise it evaluates to negative one. |
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6001 |
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6002 By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute |
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6003 $\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. |
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6004 |
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6005 EXAM,bn_mp_jacobi.c |
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6006 |
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6007 As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C |
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6008 variable name character. |
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6009 |
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6010 The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm |
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6011 has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since |
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6012 the values it may obtain are merely $-1$, $0$ and $1$. |
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6013 |
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6014 After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant |
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6015 bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same |
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6016 processor requirements and neither is faster than the other. |
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6017 |
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6018 Line @59, if@ through @70, }@ determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than |
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6019 $k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of |
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6020 $(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@. |
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6021 |
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6022 Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. |
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6023 |
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6024 \textit{-- Comment about default $s$ and such...} |
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6025 |
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6026 \section{Modular Inverse} |
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6027 \label{sec:modinv} |
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6028 The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there |
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6029 exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is |
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6030 denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and |
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6031 fields of integers. However, the former will be the matter of discussion. |
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6032 |
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6033 The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the |
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6034 order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. |
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6035 |
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6036 \begin{equation} |
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6037 ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} |
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6038 \end{equation} |
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6039 |
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6040 However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite |
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6041 requires all of the prime factors. This approach also is very slow as the size of $p$ grows. |
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6042 |
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6043 A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear |
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6044 Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. |
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6045 |
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6046 \begin{equation} |
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6047 ab + pq = 1 |
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6048 \end{equation} |
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6049 |
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6050 Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of |
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6051 $a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. |
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6052 However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The |
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6053 binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine |
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6054 equation. |
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6055 |
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6056 \subsection{General Case} |
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6057 \newpage\begin{figure}[!here] |
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6058 \begin{small} |
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6059 \begin{center} |
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6060 \begin{tabular}{l} |
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6061 \hline Algorithm \textbf{mp\_invmod}. \\ |
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6062 \textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ |
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6063 \textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ |
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6064 \hline \\ |
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6065 1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ |
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6066 2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ |
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6067 3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ |
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6068 4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ |
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6069 5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ |
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6070 6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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6071 \hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ |
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6072 \hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ |
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6073 \hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ |
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6074 \hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ |
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6075 \hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ |
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6076 \hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ |
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6077 7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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6078 \hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ |
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6079 \hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ |
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6080 \hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ |
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6081 \hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ |
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6082 \hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ |
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6083 \hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ |
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6084 8. If $u \ge v$ then \\ |
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6085 \hspace{3mm}8.1 $u \leftarrow u - v$ \\ |
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6086 \hspace{3mm}8.2 $A \leftarrow A - C$ \\ |
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6087 \hspace{3mm}8.3 $B \leftarrow B - D$ \\ |
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6088 9. else \\ |
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6089 \hspace{3mm}9.1 $v \leftarrow v - u$ \\ |
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6090 \hspace{3mm}9.2 $C \leftarrow C - A$ \\ |
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6091 \hspace{3mm}9.3 $D \leftarrow D - B$ \\ |
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6092 10. If $u \ne 0$ goto step 6. \\ |
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6093 11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ |
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6094 12. While $C \le 0$ do \\ |
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6095 \hspace{3mm}12.1 $C \leftarrow C + b$ \\ |
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6096 13. While $C \ge b$ do \\ |
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6097 \hspace{3mm}13.1 $C \leftarrow C - b$ \\ |
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6098 14. $c \leftarrow C$ \\ |
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6099 15. Return(\textit{MP\_OKAY}). \\ |
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6100 \hline |
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6101 \end{tabular} |
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6102 \end{center} |
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6103 \end{small} |
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6104 \end{figure} |
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6105 \textbf{Algorithm mp\_invmod.} |
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6106 This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the |
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6107 extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete |
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6108 Diophantine solution. |
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6109 |
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6110 If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative |
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6111 inverse for $a$ and the error is reported. |
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6112 |
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6113 The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case |
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6114 the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is |
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6115 |
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6116 \begin{equation} |
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6117 Ca + Db = v |
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6118 \end{equation} |
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6119 |
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6120 If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ |
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6121 is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie |
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6122 within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ |
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6123 then only a couple of additions or subtractions will be required to adjust the inverse. |
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6124 |
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6125 EXAM,bn_mp_invmod.c |
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6126 |
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6127 \subsubsection{Odd Moduli} |
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6128 |
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6129 When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve |
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6130 the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. |
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6131 |
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6132 The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This |
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6133 optimization will halve the time required to compute the modular inverse. |
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6134 |
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6135 \section{Primality Tests} |
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6136 |
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6137 A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime |
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6138 since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. |
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6139 |
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6140 Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or |
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6141 not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all |
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6142 probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is |
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6143 prime the algorithm may be incorrect. |
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6144 |
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6145 As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as |
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6146 well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. |
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6147 |
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6148 \subsection{Trial Division} |
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6149 |
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6150 Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously |
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6151 cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test |
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6152 would require a prohibitive amount of time as $n$ grows. |
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6153 |
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6154 Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset |
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6155 of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. |
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6156 |
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6157 The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be |
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6158 discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by |
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6159 $1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range |
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6160 $3 \le q \le 100$. |
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6161 |
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6162 At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to |
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6163 be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate |
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6164 approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The |
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6165 array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. |
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6166 |
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6167 \begin{figure}[!here] |
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6168 \begin{small} |
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6169 \begin{center} |
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6170 \begin{tabular}{l} |
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6171 \hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ |
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6172 \textbf{Input}. mp\_int $a$ \\ |
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6173 \textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ |
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6174 \hline \\ |
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6175 1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ |
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6176 \hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ |
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6177 \hspace{3mm}1.2 If $d = 0$ then \\ |
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6178 \hspace{6mm}1.2.1 $c \leftarrow 1$ \\ |
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6179 \hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ |
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6180 2. $c \leftarrow 0$ \\ |
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6181 3. Return(\textit{MP\_OKAY}). \\ |
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6182 \hline |
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6183 \end{tabular} |
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6184 \end{center} |
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6185 \end{small} |
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6186 \caption{Algorithm mp\_prime\_is\_divisible} |
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6187 \end{figure} |
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6188 \textbf{Algorithm mp\_prime\_is\_divisible.} |
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6189 This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. |
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6190 |
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6191 EXAM,bn_mp_prime_is_divisible.c |
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6192 |
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6193 The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a |
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6194 mp\_digit. The table \_\_prime\_tab is defined in the following file. |
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6195 |
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6196 EXAM,bn_prime_tab.c |
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6197 |
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6198 Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes |
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6199 upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. |
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6200 |
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6201 \subsection{The Fermat Test} |
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6202 The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in |
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6203 fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of |
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6204 the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to |
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6205 $a^1 = a$. |
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6206 |
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6207 If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case |
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6208 it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order |
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6209 of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several |
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6210 integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows |
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6211 in size. |
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6212 |
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6213 \begin{figure}[!here] |
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6214 \begin{small} |
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6215 \begin{center} |
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6216 \begin{tabular}{l} |
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6217 \hline Algorithm \textbf{mp\_prime\_fermat}. \\ |
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6218 \textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ |
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6219 \textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ |
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6220 \hline \\ |
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6221 1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ |
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6222 2. If $t = b$ then \\ |
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6223 \hspace{3mm}2.1 $c = 1$ \\ |
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6224 3. else \\ |
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6225 \hspace{3mm}3.1 $c = 0$ \\ |
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6226 4. Return(\textit{MP\_OKAY}). \\ |
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6227 \hline |
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6228 \end{tabular} |
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6229 \end{center} |
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6230 \end{small} |
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6231 \caption{Algorithm mp\_prime\_fermat} |
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6232 \end{figure} |
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6233 \textbf{Algorithm mp\_prime\_fermat.} |
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6234 This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to |
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6235 determine the result. |
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6236 |
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6237 EXAM,bn_mp_prime_fermat.c |
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6238 |
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6239 \subsection{The Miller-Rabin Test} |
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6240 The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen |
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6241 candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the |
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6242 value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that |
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6243 some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. |
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6244 |
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6245 \begin{figure}[!here] |
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6246 \begin{small} |
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6247 \begin{center} |
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6248 \begin{tabular}{l} |
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6249 \hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ |
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6250 \textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ |
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6251 \textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ |
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6252 \hline |
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6253 1. $a' \leftarrow a - 1$ \\ |
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6254 2. $r \leftarrow n1$ \\ |
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6255 3. $c \leftarrow 0, s \leftarrow 0$ \\ |
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6256 4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ |
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6257 \hspace{3mm}4.1 $s \leftarrow s + 1$ \\ |
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6258 \hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ |
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6259 5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ |
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6260 6. If $y \nequiv \pm 1$ then \\ |
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6261 \hspace{3mm}6.1 $j \leftarrow 1$ \\ |
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6262 \hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ |
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6263 \hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ |
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6264 \hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ |
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6265 \hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ |
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6266 \hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ |
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6267 7. $c \leftarrow 1$\\ |
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6268 8. Return(\textit{MP\_OKAY}). \\ |
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6269 \hline |
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6270 \end{tabular} |
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6271 \end{center} |
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6272 \end{small} |
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6273 \caption{Algorithm mp\_prime\_miller\_rabin} |
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6274 \end{figure} |
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6275 \textbf{Algorithm mp\_prime\_miller\_rabin.} |
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6276 This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine |
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6277 if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. |
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6278 |
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6279 If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will |
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6280 square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ |
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6281 is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably |
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6282 composite then it is \textit{probably} prime. |
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6283 |
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6284 EXAM,bn_mp_prime_miller_rabin.c |
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6285 |
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6286 |
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6287 |
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6288 |
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6289 \backmatter |
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6290 \appendix |
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6291 \begin{thebibliography}{ABCDEF} |
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6292 \bibitem[1]{TAOCPV2} |
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6293 Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 |
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6294 |
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6295 \bibitem[2]{HAC} |
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6296 A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 |
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6297 |
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6298 \bibitem[3]{ROSE} |
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6299 Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 |
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6300 |
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6301 \bibitem[4]{COMBA} |
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6302 Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) |
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6303 |
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6304 \bibitem[5]{KARA} |
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6305 A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 |
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6306 |
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6307 \bibitem[6]{KARAP} |
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6308 Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 |
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6309 |
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6310 \bibitem[7]{BARRETT} |
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6311 Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. |
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6312 |
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6313 \bibitem[8]{MONT} |
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6314 P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. |
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6315 |
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6316 \bibitem[9]{DRMET} |
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6317 Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories |
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6318 |
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6319 \bibitem[10]{MMB} |
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6320 J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 |
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6321 |
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6322 \bibitem[11]{RSAREF} |
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6323 R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} |
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6324 |
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6325 \bibitem[12]{DHREF} |
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6326 Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 |
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6327 |
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6328 \bibitem[13]{IEEE} |
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6329 IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) |
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6330 |
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6331 \bibitem[14]{GMP} |
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6332 GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} |
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6333 |
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6334 \bibitem[15]{MPI} |
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6335 Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} |
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6336 |
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6337 \bibitem[16]{OPENSSL} |
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6338 OpenSSL Cryptographic Toolkit, \url{http://openssl.org} |
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6339 |
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6340 \bibitem[17]{LIP} |
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6341 Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} |
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6342 |
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6343 \bibitem[18]{ISOC} |
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6344 JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' |
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6345 |
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6346 \bibitem[19]{JAVA} |
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6347 The Sun Java Website, \url{http://java.sun.com/} |
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6348 |
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6349 \end{thebibliography} |
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6350 |
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6351 \input{tommath.ind} |
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6352 |
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6353 \end{document} |