Mercurial > dropbear
annotate libtommath/bn.tex @ 994:5c5ade336926
Prefer stronger algorithms in algorithm negotiation.
Prefer diffie-hellman-group14-sha1 (2048 bit) over
diffie-hellman-group1-sha1 (1024 bit).
Due to meet-in-the-middle attacks the effective key length of
three key 3DES is 112 bits. AES is stronger and faster then 3DES.
Prefer to delay the start of compression until after authentication
has completed. This avoids exposing compression code to attacks
from unauthenticated users.
(github pull request #9)
author | Fedor Brunner <fedor.brunner@azet.sk> |
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date | Fri, 23 Jan 2015 23:00:25 +0800 |
parents | 5ff8218bcee9 |
children | 60fc6476e044 |
rev | line source |
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1 \documentclass[synpaper]{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{LibTomMath User Manual \\ v0.40} |
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53 \author{Tom St Denis \\ [email protected]} |
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54 \maketitle |
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55 This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been |
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56 formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package. |
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57 |
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58 \vspace{10cm} |
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59 |
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60 \begin{flushright}Open Source. Open Academia. Open Minds. |
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61 |
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62 \mbox{ } |
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63 |
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64 Tom St Denis, |
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65 |
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66 Ontario, Canada |
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67 \end{flushright} |
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68 |
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69 \tableofcontents |
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70 \listoffigures |
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71 \mainmatter |
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72 \pagestyle{headings} |
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73 \chapter{Introduction} |
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74 \section{What is LibTomMath?} |
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75 LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating |
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76 large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming |
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77 C compiler. |
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78 |
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79 In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how |
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80 to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous |
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81 universities, commercial and open source software developers. It has been used on a variety of platforms ranging from |
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82 Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines. |
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83 |
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84 \section{License} |
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85 As of the v0.25 the library source code has been placed in the public domain with every new release. As of the v0.28 |
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86 release the textbook ``Implementing Multiple Precision Arithmetic'' has been placed in the public domain with every new |
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87 release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development |
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88 algorithms used in the library. |
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89 |
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90 Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the |
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91 public domain everyone is entitled to do with them as they see fit. |
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92 |
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93 \section{Building LibTomMath} |
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94 |
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95 LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will |
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96 also build in MSVC, Borland C out of the box. For any other ISO C compiler a makefile will have to be made by the end |
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97 developer. |
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98 |
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99 \subsection{Static Libraries} |
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100 To build as a static library for GCC issue the following |
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101 \begin{alltt} |
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102 make |
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103 \end{alltt} |
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104 |
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105 command. This will build the library and archive the object files in ``libtommath.a''. Now you link against |
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106 that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following |
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107 \begin{alltt} |
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108 nmake -f makefile.msvc |
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109 \end{alltt} |
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110 |
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111 This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC |
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112 version 6.00 with service pack 5. |
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113 |
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114 \subsection{Shared Libraries} |
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115 To build as a shared library for GCC issue the following |
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116 \begin{alltt} |
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117 make -f makefile.shared |
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118 \end{alltt} |
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119 This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared |
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120 and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared |
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121 library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally |
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122 you use libtool to link your application against the shared object. |
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123 |
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124 There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires |
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125 Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library |
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126 ``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin. |
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127 |
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128 \subsection{Testing} |
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129 To build the library and the test harness type |
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130 |
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131 \begin{alltt} |
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132 make test |
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133 \end{alltt} |
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134 |
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135 This will build the library, ``test'' and ``mtest/mtest''. The ``test'' program will accept test vectors and verify the |
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136 results. ``mtest/mtest'' will generate test vectors using the MPI library by Michael Fromberger\footnote{A copy of MPI |
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137 is included in the package}. Simply pipe mtest into test using |
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138 |
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139 \begin{alltt} |
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140 mtest/mtest | test |
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141 \end{alltt} |
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142 |
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143 If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into |
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144 mtest. For example, if your PRNG program is called ``myprng'' simply invoke |
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145 |
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146 \begin{alltt} |
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147 myprng | mtest/mtest | test |
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148 \end{alltt} |
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149 |
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150 This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc) |
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151 that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program |
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152 will exit with a dump of the relevent numbers it was working with. |
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153 |
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154 \section{Build Configuration} |
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155 LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''. |
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156 Each phase changes how the library is built and they are applied one after another respectively. |
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157 |
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158 To make the system more powerful you can tweak the build process. Classes are defined in the file |
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159 ``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply |
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160 instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you |
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161 access to every function LibTomMath offers. |
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162 |
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163 However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You |
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164 don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is |
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165 another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional |
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166 classes can be defined base on the need of the user. |
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167 |
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168 \subsection{Build Depends} |
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169 In the file tommath\_class.h you will see a large list of C ``defines'' followed by a series of ``ifdefs'' |
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170 which further define symbols. All of the symbols (technically they're macros $\ldots$) represent a given C source |
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171 file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When a define has been enabled the |
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172 function in the respective file will be compiled and linked into the library. Accordingly when the define |
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173 is absent the file will not be compiled and not contribute any size to the library. |
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174 |
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175 You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice). |
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176 This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined. |
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177 This is useful for ``trims''. |
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178 |
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179 \subsection{Build Tweaks} |
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180 A tweak is an algorithm ``alternative''. For example, to provide tradeoffs (usually between size and space). |
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181 They can be enabled at any pass of the configuration phase. |
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182 |
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183 \begin{small} |
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184 \begin{center} |
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185 \begin{tabular}{|l|l|} |
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186 \hline \textbf{Define} & \textbf{Purpose} \\ |
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187 \hline BN\_MP\_DIV\_SMALL & Enables a slower, smaller and equally \\ |
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188 & functional mp\_div() function \\ |
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189 \hline |
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190 \end{tabular} |
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191 \end{center} |
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192 \end{small} |
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193 |
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194 \subsection{Build Trims} |
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195 A trim is a manner of removing functionality from a function that is not required. For instance, to perform |
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196 RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed. |
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197 Build trims are meant to be defined on the last pass of the configuration which means they are to be defined |
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198 only if LTM\_LAST has been defined. |
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199 |
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200 \subsubsection{Moduli Related} |
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201 \begin{small} |
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202 \begin{center} |
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203 \begin{tabular}{|l|l|} |
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204 \hline \textbf{Restriction} & \textbf{Undefine} \\ |
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205 \hline Exponentiation with odd moduli only & BN\_S\_MP\_EXPTMOD\_C \\ |
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206 & BN\_MP\_REDUCE\_C \\ |
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207 & BN\_MP\_REDUCE\_SETUP\_C \\ |
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208 & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ |
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209 & BN\_FAST\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ |
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210 \hline Exponentiation with random odd moduli & (The above plus the following) \\ |
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211 & BN\_MP\_REDUCE\_2K\_C \\ |
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212 & BN\_MP\_REDUCE\_2K\_SETUP\_C \\ |
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213 & BN\_MP\_REDUCE\_IS\_2K\_C \\ |
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214 & BN\_MP\_DR\_IS\_MODULUS\_C \\ |
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215 & BN\_MP\_DR\_REDUCE\_C \\ |
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216 & BN\_MP\_DR\_SETUP\_C \\ |
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217 \hline Modular inverse odd moduli only & BN\_MP\_INVMOD\_SLOW\_C \\ |
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218 \hline Modular inverse (both, smaller/slower) & BN\_FAST\_MP\_INVMOD\_C \\ |
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219 \hline |
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220 \end{tabular} |
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221 \end{center} |
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222 \end{small} |
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223 |
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224 \subsubsection{Operand Size Related} |
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225 \begin{small} |
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226 \begin{center} |
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227 \begin{tabular}{|l|l|} |
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228 \hline \textbf{Restriction} & \textbf{Undefine} \\ |
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229 \hline Moduli $\le 2560$ bits & BN\_MP\_MONTGOMERY\_REDUCE\_C \\ |
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230 & BN\_S\_MP\_MUL\_DIGS\_C \\ |
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231 & BN\_S\_MP\_MUL\_HIGH\_DIGS\_C \\ |
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232 & BN\_S\_MP\_SQR\_C \\ |
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233 \hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\ |
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234 & BN\_MP\_KARATSUBA\_SQR\_C \\ |
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235 & BN\_MP\_TOOM\_MUL\_C \\ |
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236 & BN\_MP\_TOOM\_SQR\_C \\ |
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237 |
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238 \hline |
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239 \end{tabular} |
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240 \end{center} |
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241 \end{small} |
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242 |
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243 |
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244 \section{Purpose of LibTomMath} |
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245 Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with |
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246 bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the |
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247 source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the |
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248 source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision |
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249 arithmetic techniques. |
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250 |
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251 LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one |
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252 function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed |
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253 increase. |
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254 |
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255 Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies |
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256 the library (beat that!). |
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257 |
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258 So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think |
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259 are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}. |
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260 |
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261 \newpage\begin{figure}[here] |
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262 \begin{small} |
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263 \begin{center} |
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264 \begin{tabular}{|l|c|c|l|} |
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265 \hline \textbf{Criteria} & \textbf{Pro} & \textbf{Con} & \textbf{Notes} \\ |
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266 \hline Few lines of code per file & X & & GnuPG $ = 300.9$, LibTomMath $ = 71.97$ \\ |
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267 \hline Commented function prototypes & X && GnuPG function names are cryptic. \\ |
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268 \hline Speed && X & LibTomMath is slower. \\ |
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269 \hline Totally free & X & & GPL has unfavourable restrictions.\\ |
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270 \hline Large function base & X & & GnuPG is barebones. \\ |
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271 \hline Five modular reduction algorithms & X & & Faster modular exponentiation for a variety of moduli. \\ |
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272 \hline Portable & X & & GnuPG requires configuration to build. \\ |
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273 \hline |
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274 \end{tabular} |
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275 \end{center} |
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276 \end{small} |
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277 \caption{LibTomMath Valuation} |
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278 \end{figure} |
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279 |
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280 It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application. |
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281 However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem |
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282 would require when working with large integers. |
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283 |
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284 So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your |
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285 own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is |
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286 not normally significantly slower. On x86 machines the difference is normally a factor of two when performing modular |
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287 exponentiations. It depends largely on the processor, compiler and the moduli being used. |
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288 |
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289 Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. However, |
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290 on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library |
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291 that is very flexible, complete and performs well in resource contrained environments. Fast RSA for example can |
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292 be performed with as little as 8KB of ram for data (again depending on build options). |
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293 |
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294 \chapter{Getting Started with LibTomMath} |
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295 \section{Building Programs} |
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296 In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically |
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297 libtommath.a). There is no library initialization required and the entire library is thread safe. |
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298 |
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299 \section{Return Codes} |
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300 There are three possible return codes a function may return. |
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301 |
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302 \index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM} |
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303 \begin{figure}[here!] |
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304 \begin{center} |
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305 \begin{small} |
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306 \begin{tabular}{|l|l|} |
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307 \hline \textbf{Code} & \textbf{Meaning} \\ |
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308 \hline MP\_OKAY & The function succeeded. \\ |
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309 \hline MP\_VAL & The function input was invalid. \\ |
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310 \hline MP\_MEM & Heap memory exhausted. \\ |
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311 \hline &\\ |
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312 \hline MP\_YES & Response is yes. \\ |
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313 \hline MP\_NO & Response is no. \\ |
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314 \hline |
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315 \end{tabular} |
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316 \end{small} |
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317 \end{center} |
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318 \caption{Return Codes} |
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319 \end{figure} |
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320 |
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321 The last two codes listed are not actually ``return'ed'' by a function. They are placed in an integer (the caller must |
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322 provide the address of an integer it can store to) which the caller can access. To convert one of the three return codes |
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323 to a string use the following function. |
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324 |
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325 \index{mp\_error\_to\_string} |
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326 \begin{alltt} |
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327 char *mp_error_to_string(int code); |
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328 \end{alltt} |
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329 |
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330 This will return a pointer to a string which describes the given error code. It will not work for the return codes |
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331 MP\_YES and MP\_NO. |
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332 |
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333 \section{Data Types} |
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334 The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to |
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335 organize all of the data required to manipulate the integer it represents. Within LibTomMath it has been prototyped |
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336 as the following. |
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337 |
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338 \index{mp\_int} |
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339 \begin{alltt} |
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340 typedef struct \{ |
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341 int used, alloc, sign; |
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342 mp_digit *dp; |
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343 \} mp_int; |
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344 \end{alltt} |
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345 |
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346 Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the |
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347 ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other |
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348 platforms by defining the appropriate macros. |
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349 |
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350 All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to |
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351 hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be |
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352 done to use an mp\_int is that it must be initialized. |
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353 |
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354 \section{Function Organization} |
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355 |
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356 The arithmetic functions of the library are all organized to have the same style prototype. That is source operands |
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357 are passed on the left and the destination is on the right. For instance, |
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358 |
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359 \begin{alltt} |
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360 mp_add(&a, &b, &c); /* c = a + b */ |
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361 mp_mul(&a, &a, &c); /* c = a * a */ |
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362 mp_div(&a, &b, &c, &d); /* c = [a/b], d = a mod b */ |
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363 \end{alltt} |
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364 |
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365 Another feature of the way the functions have been implemented is that source operands can be destination operands as well. |
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366 For instance, |
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367 |
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368 \begin{alltt} |
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369 mp_add(&a, &b, &b); /* b = a + b */ |
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370 mp_div(&a, &b, &a, &c); /* a = [a/b], c = a mod b */ |
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371 \end{alltt} |
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372 |
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373 This allows operands to be re-used which can make programming simpler. |
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374 |
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375 \section{Initialization} |
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376 \subsection{Single Initialization} |
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377 A single mp\_int can be initialized with the ``mp\_init'' function. |
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378 |
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379 \index{mp\_init} |
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380 \begin{alltt} |
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381 int mp_init (mp_int * a); |
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382 \end{alltt} |
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383 |
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384 This function expects a pointer to an mp\_int structure and will initialize the members of the structure so the mp\_int |
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385 represents the default integer which is zero. If the functions returns MP\_OKAY then the mp\_int is ready to be used |
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386 by the other LibTomMath functions. |
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387 |
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388 \begin{small} \begin{alltt} |
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389 int main(void) |
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390 \{ |
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391 mp_int number; |
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392 int result; |
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393 |
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394 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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395 printf("Error initializing the number. \%s", |
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396 mp_error_to_string(result)); |
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397 return EXIT_FAILURE; |
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398 \} |
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399 |
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400 /* use the number */ |
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401 |
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402 return EXIT_SUCCESS; |
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403 \} |
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404 \end{alltt} \end{small} |
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405 |
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406 \subsection{Single Free} |
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407 When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function |
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408 provides this functionality. |
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409 |
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410 \index{mp\_clear} |
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411 \begin{alltt} |
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412 void mp_clear (mp_int * a); |
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413 \end{alltt} |
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414 |
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415 The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the |
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416 pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations. |
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417 Is is legal to call mp\_clear() twice on the same mp\_int in a row. |
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418 |
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419 \begin{small} \begin{alltt} |
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420 int main(void) |
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421 \{ |
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422 mp_int number; |
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423 int result; |
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424 |
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425 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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426 printf("Error initializing the number. \%s", |
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427 mp_error_to_string(result)); |
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428 return EXIT_FAILURE; |
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429 \} |
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430 |
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431 /* use the number */ |
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432 |
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433 /* We're done with it. */ |
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434 mp_clear(&number); |
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435 |
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436 return EXIT_SUCCESS; |
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437 \} |
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438 \end{alltt} \end{small} |
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439 |
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440 \subsection{Multiple Initializations} |
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441 Certain algorithms require more than one large integer. In these instances it is ideal to initialize all of the mp\_int |
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442 variables in an ``all or nothing'' fashion. That is, they are either all initialized successfully or they are all |
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443 not initialized. |
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444 |
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445 The mp\_init\_multi() function provides this functionality. |
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446 |
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447 \index{mp\_init\_multi} \index{mp\_clear\_multi} |
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448 \begin{alltt} |
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449 int mp_init_multi(mp_int *mp, ...); |
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450 \end{alltt} |
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451 |
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452 It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all |
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453 at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them |
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454 are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd |
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455 from the heap at the same time. |
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456 |
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457 \begin{small} \begin{alltt} |
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458 int main(void) |
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459 \{ |
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460 mp_int num1, num2, num3; |
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461 int result; |
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462 |
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463 if ((result = mp_init_multi(&num1, |
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464 &num2, |
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465 &num3, NULL)) != MP\_OKAY) \{ |
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466 printf("Error initializing the numbers. \%s", |
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467 mp_error_to_string(result)); |
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468 return EXIT_FAILURE; |
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469 \} |
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470 |
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471 /* use the numbers */ |
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472 |
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473 /* We're done with them. */ |
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474 mp_clear_multi(&num1, &num2, &num3, NULL); |
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475 |
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476 return EXIT_SUCCESS; |
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477 \} |
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478 \end{alltt} \end{small} |
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479 |
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480 \subsection{Other Initializers} |
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481 To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided. |
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482 |
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483 \index{mp\_init\_copy} |
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484 \begin{alltt} |
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485 int mp_init_copy (mp_int * a, mp_int * b); |
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486 \end{alltt} |
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487 |
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488 This function will initialize $a$ and make it a copy of $b$ if all goes well. |
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489 |
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490 \begin{small} \begin{alltt} |
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491 int main(void) |
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492 \{ |
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493 mp_int num1, num2; |
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494 int result; |
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495 |
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496 /* initialize and do work on num1 ... */ |
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497 |
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498 /* We want a copy of num1 in num2 now */ |
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499 if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{ |
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500 printf("Error initializing the copy. \%s", |
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501 mp_error_to_string(result)); |
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502 return EXIT_FAILURE; |
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503 \} |
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504 |
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505 /* now num2 is ready and contains a copy of num1 */ |
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506 |
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507 /* We're done with them. */ |
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508 mp_clear_multi(&num1, &num2, NULL); |
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509 |
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510 return EXIT_SUCCESS; |
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511 \} |
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512 \end{alltt} \end{small} |
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513 |
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514 Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given |
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515 default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets |
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516 you override this behaviour. |
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517 |
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518 \index{mp\_init\_size} |
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519 \begin{alltt} |
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520 int mp_init_size (mp_int * a, int size); |
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521 \end{alltt} |
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522 |
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523 The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized |
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524 to have $size$ digits (which are all initially zero). |
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525 |
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526 \begin{small} \begin{alltt} |
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527 int main(void) |
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528 \{ |
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529 mp_int number; |
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530 int result; |
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531 |
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532 /* we need a 60-digit number */ |
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533 if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{ |
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534 printf("Error initializing the number. \%s", |
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535 mp_error_to_string(result)); |
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536 return EXIT_FAILURE; |
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537 \} |
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538 |
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539 /* use the number */ |
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540 |
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541 return EXIT_SUCCESS; |
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542 \} |
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543 \end{alltt} \end{small} |
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544 |
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545 \section{Maintenance Functions} |
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546 |
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547 \subsection{Reducing Memory Usage} |
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548 When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess |
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549 digits can be removed to return memory to the heap with the mp\_shrink() function. |
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550 |
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551 \index{mp\_shrink} |
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552 \begin{alltt} |
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553 int mp_shrink (mp_int * a); |
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554 \end{alltt} |
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555 |
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556 This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the |
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557 excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations |
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558 will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further |
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559 modify in the system (unless you are seriously low on memory). |
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560 |
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561 \begin{small} \begin{alltt} |
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562 int main(void) |
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563 \{ |
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564 mp_int number; |
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565 int result; |
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566 |
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567 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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568 printf("Error initializing the number. \%s", |
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569 mp_error_to_string(result)); |
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570 return EXIT_FAILURE; |
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571 \} |
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572 |
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573 /* use the number [e.g. pre-computation] */ |
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574 |
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575 /* We're done with it for now. */ |
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576 if ((result = mp_shrink(&number)) != MP_OKAY) \{ |
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577 printf("Error shrinking the number. \%s", |
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578 mp_error_to_string(result)); |
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579 return EXIT_FAILURE; |
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580 \} |
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581 |
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582 /* use it .... */ |
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583 |
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584 |
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585 /* we're done with it. */ |
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586 mp_clear(&number); |
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587 |
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588 return EXIT_SUCCESS; |
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589 \} |
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590 \end{alltt} \end{small} |
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591 |
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592 \subsection{Adding additional digits} |
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593 |
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594 Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent |
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595 the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is, |
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596 contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in |
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597 the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to |
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598 your desired size. |
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599 |
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600 \index{mp\_grow} |
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601 \begin{alltt} |
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602 int mp_grow (mp_int * a, int size); |
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603 \end{alltt} |
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604 |
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605 This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than |
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606 $size$ the function will not do anything. |
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607 |
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608 \begin{small} \begin{alltt} |
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609 int main(void) |
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610 \{ |
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611 mp_int number; |
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612 int result; |
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613 |
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614 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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615 printf("Error initializing the number. \%s", |
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616 mp_error_to_string(result)); |
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617 return EXIT_FAILURE; |
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618 \} |
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619 |
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620 /* use the number */ |
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621 |
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622 /* We need to add 20 digits to the number */ |
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623 if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{ |
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624 printf("Error growing the number. \%s", |
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625 mp_error_to_string(result)); |
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626 return EXIT_FAILURE; |
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627 \} |
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628 |
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629 |
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630 /* use the number */ |
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631 |
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632 /* we're done with it. */ |
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633 mp_clear(&number); |
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634 |
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635 return EXIT_SUCCESS; |
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636 \} |
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637 \end{alltt} \end{small} |
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638 |
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639 \chapter{Basic Operations} |
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640 \section{Small Constants} |
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641 Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two |
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642 small constant assignment functions. The first function is used to set a single digit constant while the second sets |
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643 an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the |
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644 domain of a digit can change (it's always at least $0 \ldots 127$). |
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645 |
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646 \subsection{Single Digit} |
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647 |
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648 Setting a single digit can be accomplished with the following function. |
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649 |
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650 \index{mp\_set} |
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651 \begin{alltt} |
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652 void mp_set (mp_int * a, mp_digit b); |
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653 \end{alltt} |
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654 |
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655 This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this |
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656 function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function |
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657 succeeded. |
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658 |
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659 \begin{small} \begin{alltt} |
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660 int main(void) |
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661 \{ |
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662 mp_int number; |
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663 int result; |
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664 |
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665 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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666 printf("Error initializing the number. \%s", |
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667 mp_error_to_string(result)); |
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668 return EXIT_FAILURE; |
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669 \} |
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670 |
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671 /* set the number to 5 */ |
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672 mp_set(&number, 5); |
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673 |
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674 /* we're done with it. */ |
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675 mp_clear(&number); |
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676 |
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677 return EXIT_SUCCESS; |
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678 \} |
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679 \end{alltt} \end{small} |
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680 |
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681 \subsection{Long Constants} |
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682 |
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683 To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function |
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684 can be used. |
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685 |
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686 \index{mp\_set\_int} |
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687 \begin{alltt} |
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688 int mp_set_int (mp_int * a, unsigned long b); |
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689 \end{alltt} |
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690 |
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691 This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always |
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692 accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits |
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693 this function can fail if it runs out of heap memory. |
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694 |
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695 To get the ``unsigned long'' copy of an mp\_int the following function can be used. |
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696 |
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697 \index{mp\_get\_int} |
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698 \begin{alltt} |
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699 unsigned long mp_get_int (mp_int * a); |
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700 \end{alltt} |
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701 |
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702 This will return the 32 least significant bits of the mp\_int $a$. |
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703 |
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704 \begin{small} \begin{alltt} |
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705 int main(void) |
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706 \{ |
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707 mp_int number; |
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708 int result; |
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709 |
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710 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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711 printf("Error initializing the number. \%s", |
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712 mp_error_to_string(result)); |
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713 return EXIT_FAILURE; |
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714 \} |
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715 |
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716 /* set the number to 654321 (note this is bigger than 127) */ |
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717 if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{ |
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718 printf("Error setting the value of the number. \%s", |
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719 mp_error_to_string(result)); |
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720 return EXIT_FAILURE; |
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721 \} |
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722 |
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723 printf("number == \%lu", mp_get_int(&number)); |
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724 |
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725 /* we're done with it. */ |
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726 mp_clear(&number); |
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727 |
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728 return EXIT_SUCCESS; |
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729 \} |
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730 \end{alltt} \end{small} |
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731 |
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732 This should output the following if the program succeeds. |
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733 |
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734 \begin{alltt} |
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735 number == 654321 |
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736 \end{alltt} |
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737 |
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738 \subsection{Initialize and Setting Constants} |
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739 To both initialize and set small constants the following two functions are available. |
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740 \index{mp\_init\_set} \index{mp\_init\_set\_int} |
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741 \begin{alltt} |
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742 int mp_init_set (mp_int * a, mp_digit b); |
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743 int mp_init_set_int (mp_int * a, unsigned long b); |
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744 \end{alltt} |
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745 |
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746 Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values. |
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747 |
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748 \begin{alltt} |
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749 int main(void) |
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750 \{ |
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751 mp_int number1, number2; |
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752 int result; |
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753 |
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754 /* initialize and set a single digit */ |
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755 if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{ |
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756 printf("Error setting number1: \%s", |
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757 mp_error_to_string(result)); |
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758 return EXIT_FAILURE; |
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759 \} |
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760 |
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761 /* initialize and set a long */ |
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762 if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{ |
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763 printf("Error setting number2: \%s", |
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764 mp_error_to_string(result)); |
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765 return EXIT_FAILURE; |
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766 \} |
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767 |
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768 /* display */ |
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769 printf("Number1, Number2 == \%lu, \%lu", |
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770 mp_get_int(&number1), mp_get_int(&number2)); |
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771 |
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772 /* clear */ |
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773 mp_clear_multi(&number1, &number2, NULL); |
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774 |
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775 return EXIT_SUCCESS; |
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776 \} |
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777 \end{alltt} |
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778 |
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779 If this program succeeds it shall output. |
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780 \begin{alltt} |
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781 Number1, Number2 == 100, 1023 |
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782 \end{alltt} |
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783 |
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784 \section{Comparisons} |
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785 |
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786 Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes |
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787 for any comparison. |
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788 |
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789 \index{MP\_GT} \index{MP\_EQ} \index{MP\_LT} |
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790 \begin{figure}[here] |
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791 \begin{center} |
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792 \begin{tabular}{|c|c|} |
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793 \hline \textbf{Result Code} & \textbf{Meaning} \\ |
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794 \hline MP\_GT & $a > b$ \\ |
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795 \hline MP\_EQ & $a = b$ \\ |
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796 \hline MP\_LT & $a < b$ \\ |
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797 \hline |
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798 \end{tabular} |
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799 \end{center} |
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800 \caption{Comparison Codes for $a, b$} |
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801 \label{fig:CMP} |
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802 \end{figure} |
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803 |
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804 In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of |
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805 $b$. |
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806 |
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807 \subsection{Unsigned comparison} |
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808 |
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809 An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the |
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810 mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two |
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811 mp\_int variables based on their digits only. |
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812 |
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813 \index{mp\_cmp\_mag} |
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814 \begin{alltt} |
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815 int mp_cmp_mag(mp_int * a, mp_int * b); |
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816 \end{alltt} |
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817 This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the |
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818 three compare codes listed in figure \ref{fig:CMP}. |
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819 |
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820 \begin{small} \begin{alltt} |
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821 int main(void) |
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822 \{ |
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823 mp_int number1, number2; |
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824 int result; |
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825 |
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826 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ |
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827 printf("Error initializing the numbers. \%s", |
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828 mp_error_to_string(result)); |
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829 return EXIT_FAILURE; |
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830 \} |
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831 |
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832 /* set the number1 to 5 */ |
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833 mp_set(&number1, 5); |
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834 |
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835 /* set the number2 to -6 */ |
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836 mp_set(&number2, 6); |
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837 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ |
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838 printf("Error negating number2. \%s", |
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839 mp_error_to_string(result)); |
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840 return EXIT_FAILURE; |
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841 \} |
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842 |
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843 switch(mp_cmp_mag(&number1, &number2)) \{ |
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844 case MP_GT: printf("|number1| > |number2|"); break; |
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845 case MP_EQ: printf("|number1| = |number2|"); break; |
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846 case MP_LT: printf("|number1| < |number2|"); break; |
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847 \} |
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848 |
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849 /* we're done with it. */ |
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850 mp_clear_multi(&number1, &number2, NULL); |
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851 |
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852 return EXIT_SUCCESS; |
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853 \} |
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854 \end{alltt} \end{small} |
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855 |
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856 If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes |
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857 successfully it should print the following. |
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858 |
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859 \begin{alltt} |
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860 |number1| < |number2| |
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861 \end{alltt} |
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862 |
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863 This is because $\vert -6 \vert = 6$ and obviously $5 < 6$. |
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864 |
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865 \subsection{Signed comparison} |
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866 |
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867 To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided. |
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868 |
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869 \index{mp\_cmp} |
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870 \begin{alltt} |
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871 int mp_cmp(mp_int * a, mp_int * b); |
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872 \end{alltt} |
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873 |
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874 This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they |
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875 differ it will return immediately based on their signs. If the signs are equal then it will compare the digits |
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876 individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}. |
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877 |
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878 \begin{small} \begin{alltt} |
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879 int main(void) |
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880 \{ |
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881 mp_int number1, number2; |
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882 int result; |
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883 |
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884 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ |
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885 printf("Error initializing the numbers. \%s", |
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886 mp_error_to_string(result)); |
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887 return EXIT_FAILURE; |
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888 \} |
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889 |
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890 /* set the number1 to 5 */ |
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891 mp_set(&number1, 5); |
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892 |
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893 /* set the number2 to -6 */ |
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894 mp_set(&number2, 6); |
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895 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ |
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896 printf("Error negating number2. \%s", |
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897 mp_error_to_string(result)); |
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898 return EXIT_FAILURE; |
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899 \} |
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900 |
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901 switch(mp_cmp(&number1, &number2)) \{ |
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902 case MP_GT: printf("number1 > number2"); break; |
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903 case MP_EQ: printf("number1 = number2"); break; |
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904 case MP_LT: printf("number1 < number2"); break; |
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905 \} |
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906 |
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907 /* we're done with it. */ |
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908 mp_clear_multi(&number1, &number2, NULL); |
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909 |
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910 return EXIT_SUCCESS; |
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911 \} |
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912 \end{alltt} \end{small} |
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913 |
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914 If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes |
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915 successfully it should print the following. |
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916 |
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917 \begin{alltt} |
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918 number1 > number2 |
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919 \end{alltt} |
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920 |
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921 \subsection{Single Digit} |
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922 |
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923 To compare a single digit against an mp\_int the following function has been provided. |
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924 |
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925 \index{mp\_cmp\_d} |
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926 \begin{alltt} |
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927 int mp_cmp_d(mp_int * a, mp_digit b); |
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928 \end{alltt} |
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929 |
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930 This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as |
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931 positive. This function is rather handy when you have to compare against small values such as $1$ (which often |
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932 comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes |
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933 listed in figure \ref{fig:CMP}. |
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934 |
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935 |
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936 \begin{small} \begin{alltt} |
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937 int main(void) |
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938 \{ |
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939 mp_int number; |
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940 int result; |
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941 |
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942 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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943 printf("Error initializing the number. \%s", |
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944 mp_error_to_string(result)); |
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945 return EXIT_FAILURE; |
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946 \} |
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947 |
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948 /* set the number to 5 */ |
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949 mp_set(&number, 5); |
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950 |
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951 switch(mp_cmp_d(&number, 7)) \{ |
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952 case MP_GT: printf("number > 7"); break; |
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953 case MP_EQ: printf("number = 7"); break; |
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954 case MP_LT: printf("number < 7"); break; |
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955 \} |
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956 |
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957 /* we're done with it. */ |
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958 mp_clear(&number); |
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959 |
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960 return EXIT_SUCCESS; |
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961 \} |
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962 \end{alltt} \end{small} |
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963 |
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964 If this program functions properly it will print out the following. |
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965 |
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966 \begin{alltt} |
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967 number < 7 |
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968 \end{alltt} |
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969 |
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970 \section{Logical Operations} |
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971 |
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972 Logical operations are operations that can be performed either with simple shifts or boolean operators such as |
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973 AND, XOR and OR directly. These operations are very quick. |
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974 |
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975 \subsection{Multiplication by two} |
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976 |
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977 Multiplications and divisions by any power of two can be performed with quick logical shifts either left or |
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978 right depending on the operation. |
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979 |
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980 When multiplying or dividing by two a special case routine can be used which are as follows. |
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981 \index{mp\_mul\_2} \index{mp\_div\_2} |
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982 \begin{alltt} |
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983 int mp_mul_2(mp_int * a, mp_int * b); |
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984 int mp_div_2(mp_int * a, mp_int * b); |
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985 \end{alltt} |
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986 |
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987 The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast |
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988 since the shift counts and maskes are hardcoded into the routines. |
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989 |
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990 \begin{small} \begin{alltt} |
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991 int main(void) |
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992 \{ |
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993 mp_int number; |
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994 int result; |
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995 |
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996 if ((result = mp_init(&number)) != MP_OKAY) \{ |
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997 printf("Error initializing the number. \%s", |
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998 mp_error_to_string(result)); |
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999 return EXIT_FAILURE; |
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1000 \} |
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1001 |
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1002 /* set the number to 5 */ |
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1003 mp_set(&number, 5); |
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1004 |
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1005 /* multiply by two */ |
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1006 if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{ |
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1007 printf("Error multiplying the number. \%s", |
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1008 mp_error_to_string(result)); |
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1009 return EXIT_FAILURE; |
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1010 \} |
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1011 switch(mp_cmp_d(&number, 7)) \{ |
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1012 case MP_GT: printf("2*number > 7"); break; |
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1013 case MP_EQ: printf("2*number = 7"); break; |
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1014 case MP_LT: printf("2*number < 7"); break; |
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1015 \} |
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1016 |
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1017 /* now divide by two */ |
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1018 if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{ |
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1019 printf("Error dividing the number. \%s", |
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1020 mp_error_to_string(result)); |
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1021 return EXIT_FAILURE; |
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1022 \} |
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1023 switch(mp_cmp_d(&number, 7)) \{ |
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1024 case MP_GT: printf("2*number/2 > 7"); break; |
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1025 case MP_EQ: printf("2*number/2 = 7"); break; |
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1026 case MP_LT: printf("2*number/2 < 7"); break; |
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1027 \} |
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1028 |
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1029 /* we're done with it. */ |
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1030 mp_clear(&number); |
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1031 |
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1032 return EXIT_SUCCESS; |
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1033 \} |
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1034 \end{alltt} \end{small} |
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1035 |
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1036 If this program is successful it will print out the following text. |
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1037 |
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1038 \begin{alltt} |
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1039 2*number > 7 |
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1040 2*number/2 < 7 |
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1041 \end{alltt} |
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1042 |
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1043 Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used. |
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1044 |
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1045 \index{mp\_mul\_2d} |
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1046 \begin{alltt} |
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1047 int mp_mul_2d(mp_int * a, int b, mp_int * c); |
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1048 \end{alltt} |
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1049 |
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1050 This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to |
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1051 zero the function will copy $a$ to ``c'' without performing any further actions. |
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1052 |
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1053 To divide by a power of two use the following. |
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1054 |
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1055 \index{mp\_div\_2d} |
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1056 \begin{alltt} |
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1057 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d); |
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1058 \end{alltt} |
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1059 Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the |
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1060 function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL} |
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1061 value to signal that the remainder is not desired. |
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1062 |
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1063 \subsection{Polynomial Basis Operations} |
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1064 |
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1065 Strictly speaking the organization of the integers within the mp\_int structures is what is known as a |
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1066 ``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if |
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1067 $f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be |
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1068 the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$. |
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1069 |
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1070 To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The |
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1071 following function provides this operation. |
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1072 |
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1073 \index{mp\_lshd} |
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1074 \begin{alltt} |
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1075 int mp_lshd (mp_int * a, int b); |
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1076 \end{alltt} |
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1077 |
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1078 This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes |
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1079 in the least significant digits. Similarly to divide by a power of $x$ the following function is provided. |
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1080 |
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1081 \index{mp\_rshd} |
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1082 \begin{alltt} |
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1083 void mp_rshd (mp_int * a, int b) |
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1084 \end{alltt} |
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1085 This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations |
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1086 in place and no new digits are required to complete it. |
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1087 |
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1088 \subsection{AND, OR and XOR Operations} |
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1089 |
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1090 While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The |
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1091 three functions are prototyped as follows. |
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1092 |
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1093 \index{mp\_or} \index{mp\_and} \index{mp\_xor} |
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1094 \begin{alltt} |
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1095 int mp_or (mp_int * a, mp_int * b, mp_int * c); |
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1096 int mp_and (mp_int * a, mp_int * b, mp_int * c); |
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1097 int mp_xor (mp_int * a, mp_int * b, mp_int * c); |
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1098 \end{alltt} |
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1099 |
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1100 Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR. |
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1101 |
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1102 \section{Addition and Subtraction} |
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1103 |
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1104 To compute an addition or subtraction the following two functions can be used. |
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1105 |
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1106 \index{mp\_add} \index{mp\_sub} |
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1107 \begin{alltt} |
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1108 int mp_add (mp_int * a, mp_int * b, mp_int * c); |
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1109 int mp_sub (mp_int * a, mp_int * b, mp_int * c) |
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1110 \end{alltt} |
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1111 |
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1112 Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign |
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1113 aware. |
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1114 |
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1115 \section{Sign Manipulation} |
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1116 \subsection{Negation} |
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1117 \label{sec:NEG} |
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1118 Simple integer negation can be performed with the following. |
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1119 |
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1120 \index{mp\_neg} |
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1121 \begin{alltt} |
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1122 int mp_neg (mp_int * a, mp_int * b); |
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1123 \end{alltt} |
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1124 |
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1125 Which assigns $-a$ to $b$. |
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1126 |
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1127 \subsection{Absolute} |
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1128 Simple integer absolutes can be performed with the following. |
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1129 |
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1130 \index{mp\_neg} |
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1131 \begin{alltt} |
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1132 int mp_abs (mp_int * a, mp_int * b); |
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1133 \end{alltt} |
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1134 |
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1135 Which assigns $\vert a \vert$ to $b$. |
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1136 |
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1137 \section{Integer Division and Remainder} |
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1138 To perform a complete and general integer division with remainder use the following function. |
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1139 |
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1140 \index{mp\_div} |
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1141 \begin{alltt} |
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1142 int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d); |
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1143 \end{alltt} |
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1144 |
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1145 This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that |
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1146 $bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If |
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1147 $b$ is zero the function returns \textbf{MP\_VAL}. |
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1148 |
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1149 |
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1150 \chapter{Multiplication and Squaring} |
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1151 \section{Multiplication} |
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1152 A full signed integer multiplication can be performed with the following. |
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1153 \index{mp\_mul} |
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1154 \begin{alltt} |
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1155 int mp_mul (mp_int * a, mp_int * b, mp_int * c); |
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1156 \end{alltt} |
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1157 Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are |
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1158 specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which |
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1159 should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate |
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1160 sized inputs. Then followed by the Comba and baseline multipliers. |
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1161 |
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1162 Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul() |
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1163 will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called. |
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1164 |
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1165 \begin{alltt} |
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1166 int main(void) |
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1167 \{ |
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1168 mp_int number1, number2; |
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1169 int result; |
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1170 |
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1171 /* Initialize the numbers */ |
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1172 if ((result = mp_init_multi(&number1, |
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1173 &number2, NULL)) != MP_OKAY) \{ |
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1174 printf("Error initializing the numbers. \%s", |
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1175 mp_error_to_string(result)); |
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1176 return EXIT_FAILURE; |
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1177 \} |
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1178 |
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1179 /* set the terms */ |
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1180 if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{ |
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1181 printf("Error setting number1. \%s", |
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1182 mp_error_to_string(result)); |
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1183 return EXIT_FAILURE; |
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1184 \} |
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1185 |
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1186 if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{ |
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1187 printf("Error setting number2. \%s", |
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1188 mp_error_to_string(result)); |
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1189 return EXIT_FAILURE; |
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1190 \} |
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1191 |
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1192 /* multiply them */ |
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1193 if ((result = mp_mul(&number1, &number2, |
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1194 &number1)) != MP_OKAY) \{ |
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1195 printf("Error multiplying terms. \%s", |
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1196 mp_error_to_string(result)); |
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1197 return EXIT_FAILURE; |
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1198 \} |
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1199 |
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1200 /* display */ |
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1201 printf("number1 * number2 == \%lu", mp_get_int(&number1)); |
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1202 |
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1203 /* free terms and return */ |
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1204 mp_clear_multi(&number1, &number2, NULL); |
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1205 |
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1206 return EXIT_SUCCESS; |
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1207 \} |
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1208 \end{alltt} |
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1209 |
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1210 If this program succeeds it shall output the following. |
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1211 |
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1212 \begin{alltt} |
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1213 number1 * number2 == 262911 |
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1214 \end{alltt} |
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1215 |
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1216 \section{Squaring} |
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1217 Since squaring can be performed faster than multiplication it is performed it's own function instead of just using |
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1218 mp\_mul(). |
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1219 |
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1220 \index{mp\_sqr} |
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1221 \begin{alltt} |
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1222 int mp_sqr (mp_int * a, mp_int * b); |
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1223 \end{alltt} |
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1224 |
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1225 Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring |
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1226 algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms because |
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1227 of the speed difference. |
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1228 |
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1229 \section{Tuning Polynomial Basis Routines} |
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1230 |
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1231 Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that |
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1232 the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require |
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1233 considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision |
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1234 multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor |
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1235 of 138). |
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1236 |
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1237 So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not |
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1238 actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration, |
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1239 GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at |
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1240 110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster. |
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1241 |
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1242 Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points |
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1243 exist and for the most part I just set the cutoff points very high to make sure they're not called. |
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1244 |
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1245 A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This |
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1246 can be built with GCC as follows |
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1247 |
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1248 \begin{alltt} |
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1249 make XXX |
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1250 \end{alltt} |
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1251 Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}. |
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1252 |
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1253 \begin{figure}[here] |
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1254 \begin{center} |
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1255 \begin{small} |
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1256 \begin{tabular}{|l|l|} |
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1257 \hline \textbf{Value of XXX} & \textbf{Meaning} \\ |
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1258 \hline tune & Builds portable tuning application \\ |
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1259 \hline tune86 & Builds x86 (pentium and up) program for COFF \\ |
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1260 \hline tune86c & Builds x86 program for Cygwin \\ |
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1261 \hline tune86l & Builds x86 program for Linux (ELF format) \\ |
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1262 \hline |
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1263 \end{tabular} |
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1264 \end{small} |
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1265 \end{center} |
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1266 \caption{Build Names for Tuning Programs} |
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1267 \label{fig:tuning} |
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1268 \end{figure} |
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1269 |
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1270 When the program is running it will output a series of measurements for different cutoff points. It will first find |
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1271 good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook |
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1272 tuning takes a very long time as the cutoff points are likely to be very high. |
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1273 |
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1274 \chapter{Modular Reduction} |
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1275 |
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1276 Modular reduction is process of taking the remainder of one quantity divided by another. Expressed |
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1277 as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$. |
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1278 |
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1279 \begin{equation} |
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1280 a \equiv b \mbox{ (mod }c\mbox{)} |
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1281 \label{eqn:mod} |
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1282 \end{equation} |
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1283 |
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1284 Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly |
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1285 fast reduction algorithms can be written for the limited range. |
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1286 |
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1287 Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation |
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1288 algorithm mp\_exptmod when an appropriate modulus is detected. |
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1289 |
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1290 \section{Straight Division} |
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1291 In order to effect an arbitrary modular reduction the following algorithm is provided. |
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1292 |
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1293 \index{mp\_mod} |
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1294 \begin{alltt} |
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1295 int mp_mod(mp_int *a, mp_int *b, mp_int *c); |
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1296 \end{alltt} |
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1297 |
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1298 This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign |
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1299 of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$. |
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1300 |
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1301 \section{Barrett Reduction} |
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1302 |
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1303 Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve |
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1304 a decent speedup over straight division. First a $\mu$ value must be precomputed with the following function. |
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1305 |
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1306 \index{mp\_reduce\_setup} |
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Matt Johnston <matt@ucc.asn.au>
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|
1307 \begin{alltt} |
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|
1308 int mp_reduce_setup(mp_int *a, mp_int *b); |
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|
1309 \end{alltt} |
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|
1310 |
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|
1311 Given a modulus in $b$ this produces the required $\mu$ value in $a$. For any given modulus this only has to |
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|
1312 be computed once. Modular reduction can now be performed with the following. |
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1313 |
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Matt Johnston <matt@ucc.asn.au>
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|
1314 \index{mp\_reduce} |
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Matt Johnston <matt@ucc.asn.au>
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|
1315 \begin{alltt} |
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Matt Johnston <matt@ucc.asn.au>
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|
1316 int mp_reduce(mp_int *a, mp_int *b, mp_int *c); |
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|
1317 \end{alltt} |
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1318 |
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Matt Johnston <matt@ucc.asn.au>
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|
1319 This will reduce $a$ in place modulo $b$ with the precomputed $\mu$ value in $c$. $a$ must be in the range |
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|
1320 $0 \le a < b^2$. |
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1321 |
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Matt Johnston <matt@ucc.asn.au>
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|
1322 \begin{alltt} |
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|
1323 int main(void) |
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1324 \{ |
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|
1325 mp_int a, b, c, mu; |
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|
1326 int result; |
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|
1327 |
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Matt Johnston <matt@ucc.asn.au>
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|
1328 /* initialize a,b to desired values, mp_init mu, |
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|
1329 * c and set c to 1...we want to compute a^3 mod b |
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|
1330 */ |
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1331 |
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Matt Johnston <matt@ucc.asn.au>
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|
1332 /* get mu value */ |
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1333 if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{ |
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Matt Johnston <matt@ucc.asn.au>
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|
1334 printf("Error getting mu. \%s", |
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|
1335 mp_error_to_string(result)); |
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|
1336 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
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|
1337 \} |
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|
1338 |
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Matt Johnston <matt@ucc.asn.au>
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|
1339 /* square a to get c = a^2 */ |
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Matt Johnston <matt@ucc.asn.au>
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1340 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ |
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Matt Johnston <matt@ucc.asn.au>
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|
1341 printf("Error squaring. \%s", |
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|
1342 mp_error_to_string(result)); |
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|
1343 return EXIT_FAILURE; |
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|
1344 \} |
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|
1345 |
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Matt Johnston <matt@ucc.asn.au>
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|
1346 /* now reduce `c' modulo b */ |
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1347 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ |
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Matt Johnston <matt@ucc.asn.au>
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|
1348 printf("Error reducing. \%s", |
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1349 mp_error_to_string(result)); |
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|
1350 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
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|
1351 \} |
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Matt Johnston <matt@ucc.asn.au>
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|
1352 |
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Matt Johnston <matt@ucc.asn.au>
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|
1353 /* multiply a to get c = a^3 */ |
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1354 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ |
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Matt Johnston <matt@ucc.asn.au>
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|
1355 printf("Error reducing. \%s", |
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|
1356 mp_error_to_string(result)); |
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|
1357 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
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|
1358 \} |
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Matt Johnston <matt@ucc.asn.au>
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|
1359 |
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Matt Johnston <matt@ucc.asn.au>
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|
1360 /* now reduce `c' modulo b */ |
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Matt Johnston <matt@ucc.asn.au>
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1361 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ |
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Matt Johnston <matt@ucc.asn.au>
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|
1362 printf("Error reducing. \%s", |
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|
1363 mp_error_to_string(result)); |
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|
1364 return EXIT_FAILURE; |
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|
1365 \} |
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|
1366 |
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Matt Johnston <matt@ucc.asn.au>
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|
1367 /* c now equals a^3 mod b */ |
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|
1368 |
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|
1369 return EXIT_SUCCESS; |
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1370 \} |
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|
1371 \end{alltt} |
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1372 |
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|
1373 This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed. |
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1374 |
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|
1375 \section{Montgomery Reduction} |
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1376 |
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|
1377 Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation |
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|
1378 step is required. This is accomplished with the following. |
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1379 |
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|
1380 \index{mp\_montgomery\_setup} |
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|
1381 \begin{alltt} |
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|
1382 int mp_montgomery_setup(mp_int *a, mp_digit *mp); |
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|
1383 \end{alltt} |
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1384 |
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|
1385 For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the |
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|
1386 following. |
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1387 |
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|
1388 \index{mp\_montgomery\_reduce} |
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|
1389 \begin{alltt} |
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Matt Johnston <matt@ucc.asn.au>
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|
1390 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); |
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|
1391 \end{alltt} |
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Matt Johnston <matt@ucc.asn.au>
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|
1392 This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range |
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|
1393 $0 \le a < b^2$. |
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|
1394 |
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Matt Johnston <matt@ucc.asn.au>
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|
1395 Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default |
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Matt Johnston <matt@ucc.asn.au>
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|
1396 setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to |
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Matt Johnston <matt@ucc.asn.au>
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|
1397 $127$ digits just that it falls back to a baseline algorithm after that point. |
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Matt Johnston <matt@ucc.asn.au>
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|
1398 |
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Matt Johnston <matt@ucc.asn.au>
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|
1399 An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$ |
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Matt Johnston <matt@ucc.asn.au>
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1400 where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$). |
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|
1401 |
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Matt Johnston <matt@ucc.asn.au>
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|
1402 To quickly calculate $R$ the following function was provided. |
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Matt Johnston <matt@ucc.asn.au>
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|
1403 |
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Matt Johnston <matt@ucc.asn.au>
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|
1404 \index{mp\_montgomery\_calc\_normalization} |
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Matt Johnston <matt@ucc.asn.au>
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|
1405 \begin{alltt} |
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Matt Johnston <matt@ucc.asn.au>
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|
1406 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); |
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Matt Johnston <matt@ucc.asn.au>
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|
1407 \end{alltt} |
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Matt Johnston <matt@ucc.asn.au>
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|
1408 Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division. |
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Matt Johnston <matt@ucc.asn.au>
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|
1409 |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1410 The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1411 example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1412 multiplying it by $R$. Consider the following code snippet. |
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Matt Johnston <matt@ucc.asn.au>
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|
1413 |
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Matt Johnston <matt@ucc.asn.au>
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|
1414 \begin{alltt} |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1415 int main(void) |
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Matt Johnston <matt@ucc.asn.au>
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|
1416 \{ |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1417 mp_int a, b, c, R; |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1418 mp_digit mp; |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1419 int result; |
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Matt Johnston <matt@ucc.asn.au>
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|
1420 |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1421 /* initialize a,b to desired values, |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1422 * mp_init R, c and set c to 1.... |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
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|
1423 */ |
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Matt Johnston <matt@ucc.asn.au>
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|
1424 |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1425 /* get normalization */ |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1426 if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{ |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1427 printf("Error getting norm. \%s", |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1428 mp_error_to_string(result)); |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
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|
1429 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1430 \} |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1431 |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1432 /* get mp value */ |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1433 if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{ |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1434 printf("Error setting up montgomery. \%s", |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1435 mp_error_to_string(result)); |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1436 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1437 \} |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1438 |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1439 /* normalize `a' so now a is equal to aR */ |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1440 if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{ |
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propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1441 printf("Error computing aR. \%s", |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1442 mp_error_to_string(result)); |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1443 return EXIT_FAILURE; |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
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changeset
|
1444 \} |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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changeset
|
1445 |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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changeset
|
1446 /* square a to get c = a^2R^2 */ |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1447 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1448 printf("Error squaring. \%s", |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
1449 mp_error_to_string(result)); |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
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|
1450 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
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changeset
|
1451 \} |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
1452 |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1453 /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */ |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1454 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1455 printf("Error reducing. \%s", |
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Matt Johnston <matt@ucc.asn.au>
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|
1456 mp_error_to_string(result)); |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
1457 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1458 \} |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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|
1459 |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
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changeset
|
1460 /* multiply a to get c = a^3R^2 */ |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1461 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1462 printf("Error reducing. \%s", |
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Matt Johnston <matt@ucc.asn.au>
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|
1463 mp_error_to_string(result)); |
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Matt Johnston <matt@ucc.asn.au>
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|
1464 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1465 \} |
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Matt Johnston <matt@ucc.asn.au>
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|
1466 |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
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changeset
|
1467 /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */ |
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1468 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ |
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1469 printf("Error reducing. \%s", |
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1470 mp_error_to_string(result)); |
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1471 return EXIT_FAILURE; |
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1472 \} |
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1473 |
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1474 /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */ |
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1475 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ |
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1476 printf("Error reducing. \%s", |
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1477 mp_error_to_string(result)); |
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1478 return EXIT_FAILURE; |
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1479 \} |
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1480 |
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1481 /* c now equals a^3 mod b */ |
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1482 |
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1483 return EXIT_SUCCESS; |
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1484 \} |
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1485 \end{alltt} |
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1486 |
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1487 This particular example does not look too efficient but it demonstrates the point of the algorithm. By |
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1488 normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows |
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1489 a single final reduction to correct for the normalization and the fast reduction used within the algorithm. |
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1490 |
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1491 For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}. |
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1492 |
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1493 \section{Restricted Dimminished Radix} |
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1494 |
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1495 ``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple |
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1496 digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the |
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1497 form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$). |
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1498 |
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1499 As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus. |
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1500 |
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1501 \index{mp\_dr\_setup} |
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1502 \begin{alltt} |
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1503 void mp_dr_setup(mp_int *a, mp_digit *d); |
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1504 \end{alltt} |
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1505 |
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1506 This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail |
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1507 and does not return any error codes. After the pre--computation a reduction can be performed with the |
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1508 following. |
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1509 |
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1510 \index{mp\_dr\_reduce} |
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1511 \begin{alltt} |
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1512 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); |
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1513 \end{alltt} |
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1514 |
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1515 This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted |
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1516 dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are |
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1517 much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time. |
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1518 |
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1519 Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or |
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1520 BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed |
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1521 primes are acceptable. |
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1522 |
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1523 Note that unlike Montgomery reduction there is no normalization process. The result of this function is |
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1524 equal to the correct residue. |
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1525 |
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1526 \section{Unrestricted Dimminshed Radix} |
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1527 |
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1528 Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the |
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1529 form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they |
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1530 can be applied to a wider range of numbers. |
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1531 |
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1532 \index{mp\_reduce\_2k\_setup} |
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1533 \begin{alltt} |
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1534 int mp_reduce_2k_setup(mp_int *a, mp_digit *d); |
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1535 \end{alltt} |
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1536 |
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1537 This will compute the required $d$ value for the given moduli $a$. |
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1538 |
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1539 \index{mp\_reduce\_2k} |
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1540 \begin{alltt} |
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1541 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); |
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1542 \end{alltt} |
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1543 |
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1544 This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is |
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1545 slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction. |
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1546 |
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1547 \chapter{Exponentiation} |
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1548 \section{Single Digit Exponentiation} |
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1549 \index{mp\_expt\_d} |
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1550 \begin{alltt} |
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1551 int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) |
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1552 \end{alltt} |
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1553 This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by |
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1554 $a$ for all values of $b$ greater than three. |
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1555 |
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1556 \section{Modular Exponentiation} |
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1557 \index{mp\_exptmod} |
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1558 \begin{alltt} |
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1559 int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) |
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1560 \end{alltt} |
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1561 This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function |
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1562 will automatically detect the fastest modular reduction technique to use during the operation. For negative values of |
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1563 $X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that |
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1564 $gcd(G, P) = 1$. |
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1565 |
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1566 This function is actually a shell around the two internal exponentiation functions. This routine will automatically |
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1567 detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally |
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1568 moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery |
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1569 and the other two algorithms. |
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1570 |
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1571 \section{Root Finding} |
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1572 \index{mp\_n\_root} |
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1573 \begin{alltt} |
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1574 int mp_n_root (mp_int * a, mp_digit b, mp_int * c) |
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1575 \end{alltt} |
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1576 This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. The implementation of this function is not |
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1577 ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small |
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1578 numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return |
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1579 a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$ |
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1580 will return $-2$. |
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1581 |
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1582 This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since |
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1583 the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large |
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1584 values of $b$. If particularly large roots are required then a factor method could be used instead. For example, |
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1585 $a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply |
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1586 $\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$ |
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1587 |
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1588 \chapter{Prime Numbers} |
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1589 \section{Trial Division} |
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1590 \index{mp\_prime\_is\_divisible} |
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1591 \begin{alltt} |
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1592 int mp_prime_is_divisible (mp_int * a, int *result) |
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1593 \end{alltt} |
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1594 This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the |
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1595 outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that |
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1596 if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently |
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1597 the default is to set it to zero first.}. |
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1598 |
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1599 \section{Fermat Test} |
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1600 \index{mp\_prime\_fermat} |
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1601 \begin{alltt} |
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1602 int mp_prime_fermat (mp_int * a, mp_int * b, int *result) |
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1603 \end{alltt} |
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1604 Performs a Fermat primality test to the base $b$. That is it computes $b^a \mbox{ mod }a$ and tests whether the value is |
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1605 equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$ |
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1606 is set to zero. |
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1607 |
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1608 \section{Miller-Rabin Test} |
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1609 \index{mp\_prime\_miller\_rabin} |
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1610 \begin{alltt} |
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1611 int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) |
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1612 \end{alltt} |
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1613 Performs a Miller-Rabin test to the base $b$ of $a$. This test is much stronger than the Fermat test and is very hard to |
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1614 fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one. |
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1615 Otherwise $result$ is set to zero. |
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1616 |
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1617 Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of |
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1618 Miller-Rabin are a subset of the failures of the Fermat test. |
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1619 |
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1620 \subsection{Required Number of Tests} |
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1621 Generally to ensure a number is very likely to be prime you have to perform the Miller-Rabin with at least a half-dozen |
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1622 or so unique bases. However, it has been proven that the probability of failure goes down as the size of the input goes up. |
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1623 This is why a simple function has been provided to help out. |
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1624 |
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1625 \index{mp\_prime\_rabin\_miller\_trials} |
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1626 \begin{alltt} |
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1627 int mp_prime_rabin_miller_trials(int size) |
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1628 \end{alltt} |
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1629 This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed |
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1630 in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would |
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1631 require ten tests whereas a 1024-bit number would only require four tests. |
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1632 |
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1633 You should always still perform a trial division before a Miller-Rabin test though. |
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1634 |
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1635 \section{Primality Testing} |
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1636 \index{mp\_prime\_is\_prime} |
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1637 \begin{alltt} |
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1638 int mp_prime_is_prime (mp_int * a, int t, int *result) |
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1639 \end{alltt} |
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1640 This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$. |
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1641 If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by |
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1642 $1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$). |
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1643 |
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1644 \section{Next Prime} |
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1645 \index{mp\_prime\_next\_prime} |
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1646 \begin{alltt} |
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1647 int mp_prime_next_prime(mp_int *a, int t, int bbs_style) |
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1648 \end{alltt} |
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1649 This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you |
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1650 want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime. |
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1651 |
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1652 \section{Random Primes} |
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1653 \index{mp\_prime\_random} |
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1654 \begin{alltt} |
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1655 int mp_prime_random(mp_int *a, int t, int size, int bbs, |
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1656 ltm_prime_callback cb, void *dat) |
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1657 \end{alltt} |
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1658 This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass |
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1659 $t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for |
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1660 |
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1661 \begin{alltt} |
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1662 typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); |
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1663 \end{alltt} |
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1664 |
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1665 Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply |
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1666 copied from the original input. It can be used to pass RNG context data to the callback. The function |
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1667 mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there |
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1668 is no skew on the least significant bits. |
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1669 |
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1670 \textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available |
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1671 but users are encouraged to use the new mp\_prime\_random\_ex() function instead. |
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1672 |
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1673 \subsection{Extended Generation} |
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1674 \index{mp\_prime\_random\_ex} |
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1675 \begin{alltt} |
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1676 int mp_prime_random_ex(mp_int *a, int t, |
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1677 int size, int flags, |
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1678 ltm_prime_callback cb, void *dat); |
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1679 \end{alltt} |
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1680 This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$ |
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1681 specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available |
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1682 (see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in |
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1683 mp\_prime\_random(). |
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1684 |
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1685 \begin{figure}[here] |
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1686 \begin{center} |
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1687 \begin{small} |
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1688 \begin{tabular}{|r|l|} |
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1689 \hline \textbf{Flag} & \textbf{Meaning} \\ |
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1690 \hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\ |
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1691 \hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\ |
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1692 & This option implies LTM\_PRIME\_BBS as well. \\ |
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1693 \hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\ |
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1694 & Is forced to zero. \\ |
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1695 \hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\ |
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1696 & Is forced to one. \\ |
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1697 \hline |
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1698 \end{tabular} |
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1699 \end{small} |
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1700 \end{center} |
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1701 \caption{Primality Generation Options} |
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1702 \label{fig:primeopts} |
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1703 \end{figure} |
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1704 |
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1705 \chapter{Input and Output} |
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1706 \section{ASCII Conversions} |
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1707 \subsection{To ASCII} |
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1708 \index{mp\_toradix} |
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1709 \begin{alltt} |
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1710 int mp_toradix (mp_int * a, char *str, int radix); |
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1711 \end{alltt} |
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1712 This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character |
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1713 to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required |
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1714 by the conversion before storing any data use the following function. |
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1715 |
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1716 \index{mp\_radix\_size} |
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1717 \begin{alltt} |
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1718 int mp_radix_size (mp_int * a, int radix, int *size) |
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1719 \end{alltt} |
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1720 This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this |
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1721 function returns an error code and ``size'' will be zero. |
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1722 |
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1723 \subsection{From ASCII} |
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1724 \index{mp\_read\_radix} |
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1725 \begin{alltt} |
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1726 int mp_read_radix (mp_int * a, char *str, int radix); |
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1727 \end{alltt} |
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1728 This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a |
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1729 character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign |
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1730 can be used to denote a negative number. |
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1731 |
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1732 \section{Binary Conversions} |
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1733 |
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1734 Converting an mp\_int to and from binary is another keen idea. |
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1735 |
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1736 \index{mp\_unsigned\_bin\_size} |
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1737 \begin{alltt} |
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1738 int mp_unsigned_bin_size(mp_int *a); |
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1739 \end{alltt} |
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1740 |
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1741 This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$. |
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1742 |
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1743 \index{mp\_to\_unsigned\_bin} |
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1744 \begin{alltt} |
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1745 int mp_to_unsigned_bin(mp_int *a, unsigned char *b); |
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1746 \end{alltt} |
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1747 This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?) |
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1748 requires. It does not store the sign of the integer. |
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1749 |
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1750 \index{mp\_read\_unsigned\_bin} |
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1751 \begin{alltt} |
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1752 int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c); |
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1753 \end{alltt} |
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1754 This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting |
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1755 integer $a$ will always be positive. |
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1756 |
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1757 For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the |
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1758 previous functions. |
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1759 |
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1760 \begin{alltt} |
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1761 int mp_signed_bin_size(mp_int *a); |
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1762 int mp_read_signed_bin(mp_int *a, unsigned char *b, int c); |
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1763 int mp_to_signed_bin(mp_int *a, unsigned char *b); |
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1764 \end{alltt} |
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1765 They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero |
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1766 byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix |
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1767 is non--zero. |
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1768 |
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1769 \chapter{Algebraic Functions} |
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1770 \section{Extended Euclidean Algorithm} |
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1771 \index{mp\_exteuclid} |
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1772 \begin{alltt} |
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1773 int mp_exteuclid(mp_int *a, mp_int *b, |
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1774 mp_int *U1, mp_int *U2, mp_int *U3); |
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1775 \end{alltt} |
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1776 |
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1777 This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds. |
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1778 |
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1779 \begin{equation} |
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1780 a \cdot U1 + b \cdot U2 = U3 |
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1781 \end{equation} |
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1782 |
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1783 Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired. |
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1784 |
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1785 \section{Greatest Common Divisor} |
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1786 \index{mp\_gcd} |
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1787 \begin{alltt} |
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1788 int mp_gcd (mp_int * a, mp_int * b, mp_int * c) |
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1789 \end{alltt} |
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1790 This will compute the greatest common divisor of $a$ and $b$ and store it in $c$. |
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1791 |
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1792 \section{Least Common Multiple} |
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1793 \index{mp\_lcm} |
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1794 \begin{alltt} |
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1795 int mp_lcm (mp_int * a, mp_int * b, mp_int * c) |
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1796 \end{alltt} |
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1797 This will compute the least common multiple of $a$ and $b$ and store it in $c$. |
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1798 |
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1799 \section{Jacobi Symbol} |
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1800 \index{mp\_jacobi} |
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1801 \begin{alltt} |
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1802 int mp_jacobi (mp_int * a, mp_int * p, int *c) |
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1803 \end{alltt} |
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1804 This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre |
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1805 symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime |
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1806 then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$ |
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1807 and the result will be $1$ if $a$ is a quadratic residue modulo $p$. |
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1808 |
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1809 \section{Modular Inverse} |
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1810 \index{mp\_invmod} |
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1811 \begin{alltt} |
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1812 int mp_invmod (mp_int * a, mp_int * b, mp_int * c) |
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1813 \end{alltt} |
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1814 Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$. |
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1815 |
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1816 \section{Single Digit Functions} |
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1817 |
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1818 For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions |
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1819 |
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1820 \index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d} |
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1821 \begin{alltt} |
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1822 int mp_add_d(mp_int *a, mp_digit b, mp_int *c); |
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1823 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); |
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1824 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); |
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1825 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); |
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1826 int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); |
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1827 \end{alltt} |
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1828 |
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1829 These work like the full mp\_int capable variants except the second parameter $b$ is a mp\_digit. These |
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1830 functions fairly handy if you have to work with relatively small numbers since you will not have to allocate |
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1831 an entire mp\_int to store a number like $1$ or $2$. |
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1832 |
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1833 \input{bn.ind} |
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1834 |
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1835 \end{document} |