Mercurial > dropbear
annotate libtommath/bn.tex @ 708:b207d5183bb7
document a few more changes
author | Matt Johnston <matt@ucc.asn.au> |
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date | Wed, 20 Mar 2013 00:05:19 +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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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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Matt Johnston <matt@ucc.asn.au>
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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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|
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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|
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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Matt Johnston <matt@ucc.asn.au>
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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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|
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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Matt Johnston <matt@ucc.asn.au>
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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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Matt Johnston <matt@ucc.asn.au>
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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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Matt Johnston <matt@ucc.asn.au>
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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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|
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:
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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:
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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>
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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>
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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:
diff
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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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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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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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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1427 printf("Error getting norm. \%s", |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1428 mp_error_to_string(result)); |
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Matt Johnston <matt@ucc.asn.au>
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|
1429 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
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|
1430 \} |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1431 |
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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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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1434 printf("Error setting up montgomery. \%s", |
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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:
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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
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|
1443 return EXIT_FAILURE; |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1444 \} |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1445 |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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:
diff
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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
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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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>
parents:
diff
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|
1450 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1451 \} |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
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diff
changeset
|
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:
diff
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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", |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
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|
1456 mp_error_to_string(result)); |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
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|
1457 return EXIT_FAILURE; |
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1458 \} |
eed26cff980b
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Matt Johnston <matt@ucc.asn.au>
parents:
diff
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|
1459 |
eed26cff980b
propagate from branch 'au.asn.ucc.matt.ltm.dropbear' (head 6c790cad5a7fa866ad062cb3a0c279f7ba788583)
Matt Johnston <matt@ucc.asn.au>
parents:
diff
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>
parents:
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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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Matt Johnston <matt@ucc.asn.au>
parents:
diff
changeset
|
1469 printf("Error reducing. \%s", |
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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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|
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} |