annotate bn.tex @ 282:91fbc376f010 libtommath-orig libtommath-0.35

Import of libtommath 0.35 From ltm-0.35.tar.bz2 SHA1 of 3f193dbae9351e92d02530994fa18236f7fde01c
author Matt Johnston <matt@ucc.asn.au>
date Wed, 08 Mar 2006 13:16:18 +0000
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1 \documentclass[b5paper]{book}
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2 \usepackage{hyperref}
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3 \usepackage{makeidx}
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4 \usepackage{amssymb}
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5 \usepackage{color}
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6 \usepackage{alltt}
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7 \usepackage{graphicx}
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8 \usepackage{layout}
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9 \def\union{\cup}
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10 \def\intersect{\cap}
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11 \def\getsrandom{\stackrel{\rm R}{\gets}}
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12 \def\cross{\times}
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13 \def\cat{\hspace{0.5em} \| \hspace{0.5em}}
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14 \def\catn{$\|$}
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15 \def\divides{\hspace{0.3em} | \hspace{0.3em}}
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16 \def\nequiv{\not\equiv}
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17 \def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
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18 \def\lcm{{\rm lcm}}
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19 \def\gcd{{\rm gcd}}
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20 \def\log{{\rm log}}
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21 \def\ord{{\rm ord}}
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22 \def\abs{{\mathit abs}}
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23 \def\rep{{\mathit rep}}
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24 \def\mod{{\mathit\ mod\ }}
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25 \renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
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26 \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
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27 \newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
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28 \def\Or{{\rm\ or\ }}
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29 \def\And{{\rm\ and\ }}
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30 \def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
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31 \def\implies{\Rightarrow}
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32 \def\undefined{{\rm ``undefined"}}
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33 \def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
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34 \let\oldphi\phi
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35 \def\phi{\varphi}
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36 \def\Pr{{\rm Pr}}
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37 \newcommand{\str}[1]{{\mathbf{#1}}}
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38 \def\F{{\mathbb F}}
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39 \def\N{{\mathbb N}}
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40 \def\Z{{\mathbb Z}}
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41 \def\R{{\mathbb R}}
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42 \def\C{{\mathbb C}}
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43 \def\Q{{\mathbb Q}}
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44 \definecolor{DGray}{gray}{0.5}
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45 \newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
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46 \def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
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47 \def\gap{\vspace{0.5ex}}
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48 \makeindex
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49 \begin{document}
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50 \frontmatter
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51 \pagestyle{empty}
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52 \title{LibTomMath User Manual \\ v0.35}
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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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parents:
diff changeset
453 at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
454 are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
455 from the heap at the same time.
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
456
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
457 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
458 int main(void)
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
459 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
460 mp_int num1, num2, num3;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
461 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
462
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
463 if ((result = mp_init_multi(&num1,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
464 &num2,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
465 &num3, NULL)) != MP\_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
466 printf("Error initializing the numbers. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
467 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
468 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
469 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
470
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
471 /* use the numbers */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
472
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
473 /* We're done with them. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
474 mp_clear_multi(&num1, &num2, &num3, NULL);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
475
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
476 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
477 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
478 \end{alltt} \end{small}
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
479
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
480 \subsection{Other Initializers}
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
481 To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided.
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
482
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
483 \index{mp\_init\_copy}
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
484 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
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diff changeset
485 int mp_init_copy (mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
486 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
487
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
488 This function will initialize $a$ and make it a copy of $b$ if all goes well.
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
489
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
490 \begin{small} \begin{alltt}
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
491 int main(void)
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parents:
diff changeset
492 \{
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
493 mp_int num1, num2;
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
494 int result;
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Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
495
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
496 /* initialize and do work on num1 ... */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
497
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
498 /* We want a copy of num1 in num2 now */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
499 if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
500 printf("Error initializing the copy. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
501 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
502 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
503 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
504
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
505 /* now num2 is ready and contains a copy of num1 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
506
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
507 /* We're done with them. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
508 mp_clear_multi(&num1, &num2, NULL);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
509
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
510 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
511 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
512 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
513
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
514 Another less common initializer is mp\_init\_size() which allows the user to initialize an mp\_int with a given
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
515 default number of digits. By default, all initializers allocate \textbf{MP\_PREC} digits. This function lets
91fbc376f010 Import of libtommath 0.35
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parents:
diff changeset
516 you override this behaviour.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
517
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
518 \index{mp\_init\_size}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
519 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
520 int mp_init_size (mp_int * a, int size);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
521 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
522
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
523 The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
524 to have $size$ digits (which are all initially zero).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
525
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
526 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
527 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
528 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
529 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
530 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
531
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
532 /* we need a 60-digit number */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
533 if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
534 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
535 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
536 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
537 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
538
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
539 /* use the number */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
540
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
541 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
542 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
543 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
544
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
545 \section{Maintenance Functions}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
546
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
547 \subsection{Reducing Memory Usage}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
548 When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
549 digits can be removed to return memory to the heap with the mp\_shrink() function.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
550
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
551 \index{mp\_shrink}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
552 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
553 int mp_shrink (mp_int * a);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
554 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
555
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
556 This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
557 excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
558 will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
559 modify in the system (unless you are seriously low on memory).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
560
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
561 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
562 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
563 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
564 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
565 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
566
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
567 if ((result = mp_init(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
568 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
569 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
570 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
571 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
572
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
573 /* use the number [e.g. pre-computation] */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
574
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
575 /* We're done with it for now. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
576 if ((result = mp_shrink(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
577 printf("Error shrinking the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
578 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
579 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
580 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
581
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
582 /* use it .... */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
583
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
584
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
585 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
586 mp_clear(&number);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
587
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
588 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
589 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
590 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
591
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
592 \subsection{Adding additional digits}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
593
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
594 Within the mp\_int structure are two parameters which control the limitations of the array of digits that represent
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
595 the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
596 contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
597 the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
598 your desired size.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
599
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
600 \index{mp\_grow}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
601 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
602 int mp_grow (mp_int * a, int size);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
603 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
604
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
605 This will grow the array of digits of $a$ to $size$. If the \textit{alloc} parameter is already bigger than
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
606 $size$ the function will not do anything.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
607
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
608 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
609 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
610 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
611 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
612 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
613
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
614 if ((result = mp_init(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
615 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
616 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
617 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
618 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
619
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
620 /* use the number */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
621
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
622 /* We need to add 20 digits to the number */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
623 if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
624 printf("Error growing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
625 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
626 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
627 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
628
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
629
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
630 /* use the number */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
631
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
632 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
633 mp_clear(&number);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
634
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
635 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
636 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
637 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
638
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
639 \chapter{Basic Operations}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
640 \section{Small Constants}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
641 Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
642 small constant assignment functions. The first function is used to set a single digit constant while the second sets
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
643 an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
644 domain of a digit can change (it's always at least $0 \ldots 127$).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
645
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
646 \subsection{Single Digit}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
647
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
648 Setting a single digit can be accomplished with the following function.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
649
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
650 \index{mp\_set}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
651 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
652 void mp_set (mp_int * a, mp_digit b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
653 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
654
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
655 This will zero the contents of $a$ and make it represent an integer equal to the value of $b$. Note that this
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
656 function has a return type of \textbf{void}. It cannot cause an error so it is safe to assume the function
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
657 succeeded.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
658
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
659 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
660 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
661 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
662 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
663 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
664
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
665 if ((result = mp_init(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
666 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
667 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
668 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
669 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
670
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
671 /* set the number to 5 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
672 mp_set(&number, 5);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
673
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
674 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
675 mp_clear(&number);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
676
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
677 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
678 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
679 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
680
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
681 \subsection{Long Constants}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
682
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
683 To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
684 can be used.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
685
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
686 \index{mp\_set\_int}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
687 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
688 int mp_set_int (mp_int * a, unsigned long b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
689 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
690
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
691 This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
692 accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
693 this function can fail if it runs out of heap memory.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
694
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
695 To get the ``unsigned long'' copy of an mp\_int the following function can be used.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
696
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
697 \index{mp\_get\_int}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
698 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
699 unsigned long mp_get_int (mp_int * a);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
700 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
701
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
702 This will return the 32 least significant bits of the mp\_int $a$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
703
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
704 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
705 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
706 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
707 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
708 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
709
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
710 if ((result = mp_init(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
711 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
712 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
713 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
714 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
715
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
716 /* set the number to 654321 (note this is bigger than 127) */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
717 if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
718 printf("Error setting the value of the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
719 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
720 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
721 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
722
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
723 printf("number == \%lu", mp_get_int(&number));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
724
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
725 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
726 mp_clear(&number);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
727
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
728 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
729 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
730 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
731
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
732 This should output the following if the program succeeds.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
733
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
734 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
735 number == 654321
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
736 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
737
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
738 \subsection{Initialize and Setting Constants}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
739 To both initialize and set small constants the following two functions are available.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
740 \index{mp\_init\_set} \index{mp\_init\_set\_int}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
741 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
742 int mp_init_set (mp_int * a, mp_digit b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
743 int mp_init_set_int (mp_int * a, unsigned long b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
744 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
745
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
746 Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
747
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
748 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
749 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
750 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
751 mp_int number1, number2;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
752 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
753
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
754 /* initialize and set a single digit */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
755 if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
756 printf("Error setting number1: \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
757 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
758 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
759 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
760
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
761 /* initialize and set a long */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
762 if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
763 printf("Error setting number2: \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
764 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
765 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
766 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
767
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
768 /* display */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
769 printf("Number1, Number2 == \%lu, \%lu",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
770 mp_get_int(&number1), mp_get_int(&number2));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
771
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
772 /* clear */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
773 mp_clear_multi(&number1, &number2, NULL);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
774
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
775 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
776 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
777 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
778
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
779 If this program succeeds it shall output.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
780 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
781 Number1, Number2 == 100, 1023
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
782 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
783
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
784 \section{Comparisons}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
785
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
786 Comparisons in LibTomMath are always performed in a ``left to right'' fashion. There are three possible return codes
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
787 for any comparison.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
788
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
789 \index{MP\_GT} \index{MP\_EQ} \index{MP\_LT}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
790 \begin{figure}[here]
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
791 \begin{center}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
792 \begin{tabular}{|c|c|}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
793 \hline \textbf{Result Code} & \textbf{Meaning} \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
794 \hline MP\_GT & $a > b$ \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
795 \hline MP\_EQ & $a = b$ \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
796 \hline MP\_LT & $a < b$ \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
797 \hline
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
798 \end{tabular}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
799 \end{center}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
800 \caption{Comparison Codes for $a, b$}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
801 \label{fig:CMP}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
802 \end{figure}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
803
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
805 $b$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
806
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
807 \subsection{Unsigned comparison}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
808
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
809 An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
810 mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
811 mp\_int variables based on their digits only.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
812
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
813 \index{mp\_cmp\_mag}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
814 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
815 int mp_cmp_mag(mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
816 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
817 This will compare $a$ to $b$ placing $a$ to the left of $b$. This function cannot fail and will return one of the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
818 three compare codes listed in figure \ref{fig:CMP}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
819
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
820 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
821 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
822 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
823 mp_int number1, number2;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
824 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
825
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
826 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
827 printf("Error initializing the numbers. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
828 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
829 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
830 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
831
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
832 /* set the number1 to 5 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
833 mp_set(&number1, 5);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
834
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
835 /* set the number2 to -6 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
836 mp_set(&number2, 6);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
837 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
838 printf("Error negating number2. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
839 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
840 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
841 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
842
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
843 switch(mp_cmp_mag(&number1, &number2)) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
844 case MP_GT: printf("|number1| > |number2|"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
845 case MP_EQ: printf("|number1| = |number2|"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
846 case MP_LT: printf("|number1| < |number2|"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
847 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
848
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
849 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
850 mp_clear_multi(&number1, &number2, NULL);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
851
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
852 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
853 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
854 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
855
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
856 If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
857 successfully it should print the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
858
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
859 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
860 |number1| < |number2|
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
861 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
862
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
863 This is because $\vert -6 \vert = 6$ and obviously $5 < 6$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
864
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
865 \subsection{Signed comparison}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
866
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
867 To compare two mp\_int variables based on their signed value the mp\_cmp() function is provided.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
868
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
869 \index{mp\_cmp}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
870 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
871 int mp_cmp(mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
872 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
873
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
874 This will compare $a$ to the left of $b$. It will first compare the signs of the two mp\_int variables. If they
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
875 differ it will return immediately based on their signs. If the signs are equal then it will compare the digits
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
876 individually. This function will return one of the compare conditions codes listed in figure \ref{fig:CMP}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
877
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
878 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
879 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
880 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
881 mp_int number1, number2;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
882 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
883
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
884 if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
885 printf("Error initializing the numbers. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
886 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
887 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
888 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
889
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
890 /* set the number1 to 5 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
891 mp_set(&number1, 5);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
892
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
893 /* set the number2 to -6 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
894 mp_set(&number2, 6);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
895 if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
896 printf("Error negating number2. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
897 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
898 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
899 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
900
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
901 switch(mp_cmp(&number1, &number2)) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
902 case MP_GT: printf("number1 > number2"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
903 case MP_EQ: printf("number1 = number2"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
904 case MP_LT: printf("number1 < number2"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
905 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
906
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
907 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
908 mp_clear_multi(&number1, &number2, NULL);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
909
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
910 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
911 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
912 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
913
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
914 If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
915 successfully it should print the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
916
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
917 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
918 number1 > number2
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
919 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
920
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
921 \subsection{Single Digit}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
922
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
923 To compare a single digit against an mp\_int the following function has been provided.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
924
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
925 \index{mp\_cmp\_d}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
926 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
927 int mp_cmp_d(mp_int * a, mp_digit b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
928 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
929
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
930 This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
931 positive. This function is rather handy when you have to compare against small values such as $1$ (which often
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
932 comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
933 listed in figure \ref{fig:CMP}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
934
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
935
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
936 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
937 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
938 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
939 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
940 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
941
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
942 if ((result = mp_init(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
943 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
944 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
945 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
946 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
947
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
948 /* set the number to 5 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
949 mp_set(&number, 5);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
950
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
951 switch(mp_cmp_d(&number, 7)) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
952 case MP_GT: printf("number > 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
953 case MP_EQ: printf("number = 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
954 case MP_LT: printf("number < 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
955 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
956
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
957 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
958 mp_clear(&number);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
959
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
960 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
961 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
962 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
963
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
964 If this program functions properly it will print out the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
965
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
966 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
967 number < 7
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
968 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
969
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
970 \section{Logical Operations}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
971
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
972 Logical operations are operations that can be performed either with simple shifts or boolean operators such as
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
973 AND, XOR and OR directly. These operations are very quick.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
974
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
975 \subsection{Multiplication by two}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
976
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
977 Multiplications and divisions by any power of two can be performed with quick logical shifts either left or
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
978 right depending on the operation.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
979
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
980 When multiplying or dividing by two a special case routine can be used which are as follows.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
981 \index{mp\_mul\_2} \index{mp\_div\_2}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
982 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
983 int mp_mul_2(mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
984 int mp_div_2(mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
985 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
986
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
987 The former will assign twice $a$ to $b$ while the latter will assign half $a$ to $b$. These functions are fast
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
988 since the shift counts and maskes are hardcoded into the routines.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
989
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
990 \begin{small} \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
991 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
992 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
993 mp_int number;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
994 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
995
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
996 if ((result = mp_init(&number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
997 printf("Error initializing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
998 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
999 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1000 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1001
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1002 /* set the number to 5 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1003 mp_set(&number, 5);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1004
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1005 /* multiply by two */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1006 if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1007 printf("Error multiplying the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1008 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1009 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1010 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1011 switch(mp_cmp_d(&number, 7)) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1012 case MP_GT: printf("2*number > 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1013 case MP_EQ: printf("2*number = 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1014 case MP_LT: printf("2*number < 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1015 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1016
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1017 /* now divide by two */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1018 if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1019 printf("Error dividing the number. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1020 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1021 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1022 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1023 switch(mp_cmp_d(&number, 7)) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1024 case MP_GT: printf("2*number/2 > 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1025 case MP_EQ: printf("2*number/2 = 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1026 case MP_LT: printf("2*number/2 < 7"); break;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1027 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1028
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1029 /* we're done with it. */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1030 mp_clear(&number);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1031
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1032 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1033 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1034 \end{alltt} \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1035
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1036 If this program is successful it will print out the following text.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1037
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1038 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1039 2*number > 7
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1040 2*number/2 < 7
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1041 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1042
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1043 Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1044
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1045 \index{mp\_mul\_2d}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1046 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1047 int mp_mul_2d(mp_int * a, int b, mp_int * c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1048 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1049
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1051 zero the function will copy $a$ to ``c'' without performing any further actions.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1052
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1053 To divide by a power of two use the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1054
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1055 \index{mp\_div\_2d}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1056 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1057 int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1058 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1059 Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1060 function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1061 value to signal that the remainder is not desired.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1062
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1063 \subsection{Polynomial Basis Operations}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1064
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1065 Strictly speaking the organization of the integers within the mp\_int structures is what is known as a
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1066 ``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1068 the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1069
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1070 To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1071 following function provides this operation.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1072
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1073 \index{mp\_lshd}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1074 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1075 int mp_lshd (mp_int * a, int b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1076 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1077
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1078 This will multiply $a$ in place by $x^b$ which is equivalent to shifting the digits left $b$ places and inserting zeroes
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1079 in the least significant digits. Similarly to divide by a power of $x$ the following function is provided.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1080
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1081 \index{mp\_rshd}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1082 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1083 void mp_rshd (mp_int * a, int b)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1084 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1085 This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1086 in place and no new digits are required to complete it.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1087
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1088 \subsection{AND, OR and XOR Operations}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1089
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1090 While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1091 three functions are prototyped as follows.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1092
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1093 \index{mp\_or} \index{mp\_and} \index{mp\_xor}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1094 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1095 int mp_or (mp_int * a, mp_int * b, mp_int * c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1096 int mp_and (mp_int * a, mp_int * b, mp_int * c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1097 int mp_xor (mp_int * a, mp_int * b, mp_int * c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1098 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1099
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1100 Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1101
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1102 \section{Addition and Subtraction}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1103
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1104 To compute an addition or subtraction the following two functions can be used.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1105
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1106 \index{mp\_add} \index{mp\_sub}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1107 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1108 int mp_add (mp_int * a, mp_int * b, mp_int * c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1109 int mp_sub (mp_int * a, mp_int * b, mp_int * c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1110 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1111
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1112 Which perform $c = a \odot b$ where $\odot$ is one of signed addition or subtraction. The operations are fully sign
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1113 aware.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1114
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1115 \section{Sign Manipulation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1116 \subsection{Negation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1117 \label{sec:NEG}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1118 Simple integer negation can be performed with the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1119
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1120 \index{mp\_neg}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1121 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1122 int mp_neg (mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1123 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1124
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1125 Which assigns $-a$ to $b$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1126
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1127 \subsection{Absolute}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1128 Simple integer absolutes can be performed with the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1129
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1130 \index{mp\_neg}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1131 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1132 int mp_abs (mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1133 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1134
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1135 Which assigns $\vert a \vert$ to $b$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1136
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1137 \section{Integer Division and Remainder}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1138 To perform a complete and general integer division with remainder use the following function.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1139
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1140 \index{mp\_div}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1141 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1142 int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1143 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1144
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1145 This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1146 $bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1147 $b$ is zero the function returns \textbf{MP\_VAL}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1148
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1149
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1150 \chapter{Multiplication and Squaring}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1151 \section{Multiplication}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1152 A full signed integer multiplication can be performed with the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1153 \index{mp\_mul}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1154 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1155 int mp_mul (mp_int * a, mp_int * b, mp_int * c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1156 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1157 Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1158 specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1159 should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1160 sized inputs. Then followed by the Comba and baseline multipliers.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1161
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1162 Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul()
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1163 will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1164
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1165 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1166 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1167 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1168 mp_int number1, number2;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1169 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1170
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1171 /* Initialize the numbers */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1172 if ((result = mp_init_multi(&number1,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1173 &number2, NULL)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1174 printf("Error initializing the numbers. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1175 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1176 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1177 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1178
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1179 /* set the terms */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1180 if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1181 printf("Error setting number1. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1182 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1183 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1184 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1185
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1186 if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1187 printf("Error setting number2. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1188 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1189 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1190 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1191
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1192 /* multiply them */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1193 if ((result = mp_mul(&number1, &number2,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1194 &number1)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1195 printf("Error multiplying terms. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1196 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1197 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1198 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1199
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1200 /* display */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1201 printf("number1 * number2 == \%lu", mp_get_int(&number1));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1202
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1203 /* free terms and return */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1204 mp_clear_multi(&number1, &number2, NULL);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1205
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1206 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1207 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1208 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1209
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1210 If this program succeeds it shall output the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1211
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1212 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1213 number1 * number2 == 262911
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1214 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1215
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1216 \section{Squaring}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1217 Since squaring can be performed faster than multiplication it is performed it's own function instead of just using
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1218 mp\_mul().
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1219
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1220 \index{mp\_sqr}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1221 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1222 int mp_sqr (mp_int * a, mp_int * b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1223 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1224
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1225 Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1226 algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms because
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1227 of the speed difference.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1228
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1229 \section{Tuning Polynomial Basis Routines}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1230
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1231 Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1232 the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1233 considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1234 multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1235 of 138).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1236
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1237 So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1238 actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1240 110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1241
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1242 Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1243 exist and for the most part I just set the cutoff points very high to make sure they're not called.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1244
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1245 A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1246 can be built with GCC as follows
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1247
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1248 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1249 make XXX
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1250 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1251 Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1252
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1253 \begin{figure}[here]
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1254 \begin{center}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1255 \begin{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1256 \begin{tabular}{|l|l|}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1257 \hline \textbf{Value of XXX} & \textbf{Meaning} \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1258 \hline tune & Builds portable tuning application \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1259 \hline tune86 & Builds x86 (pentium and up) program for COFF \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1260 \hline tune86c & Builds x86 program for Cygwin \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1261 \hline tune86l & Builds x86 program for Linux (ELF format) \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1262 \hline
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1263 \end{tabular}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1264 \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1265 \end{center}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1266 \caption{Build Names for Tuning Programs}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1267 \label{fig:tuning}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1268 \end{figure}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1269
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1270 When the program is running it will output a series of measurements for different cutoff points. It will first find
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1271 good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1272 tuning takes a very long time as the cutoff points are likely to be very high.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1273
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1274 \chapter{Modular Reduction}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1275
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1276 Modular reduction is process of taking the remainder of one quantity divided by another. Expressed
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1277 as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1278
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1279 \begin{equation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1280 a \equiv b \mbox{ (mod }c\mbox{)}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1281 \label{eqn:mod}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1282 \end{equation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1283
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1284 Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1285 fast reduction algorithms can be written for the limited range.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1286
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1287 Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1288 algorithm mp\_exptmod when an appropriate modulus is detected.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1289
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1290 \section{Straight Division}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1291 In order to effect an arbitrary modular reduction the following algorithm is provided.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1292
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1293 \index{mp\_mod}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1294 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1295 int mp_mod(mp_int *a, mp_int *b, mp_int *c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1296 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1297
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1298 This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1299 of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1300
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1301 \section{Barrett Reduction}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1302
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1303 Barrett reduction is a generic optimized reduction algorithm that requires pre--computation to achieve
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1304 a decent speedup over straight division. First a $\mu$ value must be precomputed with the following function.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1305
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1306 \index{mp\_reduce\_setup}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1307 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1308 int mp_reduce_setup(mp_int *a, mp_int *b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1309 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1310
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1311 Given a modulus in $b$ this produces the required $\mu$ value in $a$. For any given modulus this only has to
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1312 be computed once. Modular reduction can now be performed with the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1313
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1314 \index{mp\_reduce}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1315 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1316 int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1317 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1318
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1319 This will reduce $a$ in place modulo $b$ with the precomputed $\mu$ value in $c$. $a$ must be in the range
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1320 $0 \le a < b^2$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1321
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1322 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1323 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1324 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1325 mp_int a, b, c, mu;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1326 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1327
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1328 /* initialize a,b to desired values, mp_init mu,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1329 * c and set c to 1...we want to compute a^3 mod b
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1330 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1331
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1332 /* get mu value */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1333 if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1334 printf("Error getting mu. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1335 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1336 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1337 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1338
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1339 /* square a to get c = a^2 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1340 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1341 printf("Error squaring. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1342 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1343 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1344 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1345
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1346 /* now reduce `c' modulo b */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1347 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1348 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1349 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1350 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1351 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1352
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1353 /* multiply a to get c = a^3 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1354 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1355 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1356 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1357 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1358 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1359
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1360 /* now reduce `c' modulo b */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1361 if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1362 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1363 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1364 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1365 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1366
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1367 /* c now equals a^3 mod b */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1368
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1369 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1370 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1371 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1372
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1373 This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1374
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1375 \section{Montgomery Reduction}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1376
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1377 Montgomery is a specialized reduction algorithm for any odd moduli. Like Barrett reduction a pre--computation
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1378 step is required. This is accomplished with the following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1379
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1380 \index{mp\_montgomery\_setup}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1381 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1382 int mp_montgomery_setup(mp_int *a, mp_digit *mp);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1383 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1384
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1385 For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1386 following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1387
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1388 \index{mp\_montgomery\_reduce}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1389 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1390 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1391 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1392 This reduces $a$ in place modulo $m$ with the pre--computed value $mp$. $a$ must be in the range
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1393 $0 \le a < b^2$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1394
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1395 Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1396 setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1397 $127$ digits just that it falls back to a baseline algorithm after that point.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1398
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1399 An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1400 where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1401
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1402 To quickly calculate $R$ the following function was provided.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1403
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1404 \index{mp\_montgomery\_calc\_normalization}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1405 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1406 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1407 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1408 Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1409
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1410 The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1411 example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1412 multiplying it by $R$. Consider the following code snippet.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1413
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1414 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1415 int main(void)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1416 \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1417 mp_int a, b, c, R;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1418 mp_digit mp;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1419 int result;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1420
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1421 /* initialize a,b to desired values,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1422 * mp_init R, c and set c to 1....
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1423 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1424
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1425 /* get normalization */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1426 if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1427 printf("Error getting norm. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1428 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1429 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1430 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1431
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1432 /* get mp value */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1433 if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1434 printf("Error setting up montgomery. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1435 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1436 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1437 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1438
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1439 /* normalize `a' so now a is equal to aR */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1440 if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1441 printf("Error computing aR. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1442 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1443 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1444 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1445
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1446 /* square a to get c = a^2R^2 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1447 if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1448 printf("Error squaring. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1449 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1450 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1451 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1452
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1453 /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1454 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1455 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1456 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1457 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1458 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1459
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1460 /* multiply a to get c = a^3R^2 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1461 if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1462 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1463 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1464 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1465 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1466
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1467 /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1468 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1469 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1470 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1471 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1472 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1473
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1474 /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1475 if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1476 printf("Error reducing. \%s",
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1477 mp_error_to_string(result));
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1478 return EXIT_FAILURE;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1479 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1480
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1481 /* c now equals a^3 mod b */
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1482
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1483 return EXIT_SUCCESS;
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1484 \}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1485 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1486
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1487 This particular example does not look too efficient but it demonstrates the point of the algorithm. By
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1488 normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1489 a single final reduction to correct for the normalization and the fast reduction used within the algorithm.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1490
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1491 For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1492
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1493 \section{Restricted Dimminished Radix}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1494
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1495 ``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1496 digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1497 form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1498
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1499 As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1500
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1501 \index{mp\_dr\_setup}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1502 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1503 void mp_dr_setup(mp_int *a, mp_digit *d);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1504 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1505
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1506 This computes the value required for the modulus $a$ and stores it in $d$. This function cannot fail
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1507 and does not return any error codes. After the pre--computation a reduction can be performed with the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1508 following.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1509
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1510 \index{mp\_dr\_reduce}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1511 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1512 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1513 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1514
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1515 This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1516 dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1517 much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1518
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1519 Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1520 BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1521 primes are acceptable.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1522
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1523 Note that unlike Montgomery reduction there is no normalization process. The result of this function is
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1524 equal to the correct residue.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1525
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1526 \section{Unrestricted Dimminshed Radix}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1527
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1528 Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1529 form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1530 can be applied to a wider range of numbers.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1531
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1532 \index{mp\_reduce\_2k\_setup}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1533 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1534 int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1535 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1536
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1537 This will compute the required $d$ value for the given moduli $a$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1538
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1539 \index{mp\_reduce\_2k}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1540 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1541 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1542 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1543
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1544 This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1545 slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1546
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1547 \chapter{Exponentiation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1548 \section{Single Digit Exponentiation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1549 \index{mp\_expt\_d}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1550 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1551 int mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1552 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1553 This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1554 $a$ for all values of $b$ greater than three.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1555
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1556 \section{Modular Exponentiation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1557 \index{mp\_exptmod}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1558 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1559 int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1560 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1561 This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1562 will automatically detect the fastest modular reduction technique to use during the operation. For negative values of
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1563 $X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1564 $gcd(G, P) = 1$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1565
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1566 This function is actually a shell around the two internal exponentiation functions. This routine will automatically
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1567 detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1568 moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1569 and the other two algorithms.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1570
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1571 \section{Root Finding}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1572 \index{mp\_n\_root}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1573 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1574 int mp_n_root (mp_int * a, mp_digit b, mp_int * c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1575 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1577 ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1578 numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1579 a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1580 will return $-2$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1581
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1582 This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1583 the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1584 values of $b$. If particularly large roots are required then a factor method could be used instead. For example,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1585 $a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1586 $\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1587
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1588 \chapter{Prime Numbers}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1589 \section{Trial Division}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1590 \index{mp\_prime\_is\_divisible}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1591 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1592 int mp_prime_is_divisible (mp_int * a, int *result)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1593 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1594 This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1595 outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1596 if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1597 the default is to set it to zero first.}.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1598
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1599 \section{Fermat Test}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1600 \index{mp\_prime\_fermat}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1601 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1602 int mp_prime_fermat (mp_int * a, mp_int * b, int *result)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1603 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1605 equal to $b$ or not. If the values are equal then $a$ is probably prime and $result$ is set to one. Otherwise $result$
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1606 is set to zero.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1607
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1608 \section{Miller-Rabin Test}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1609 \index{mp\_prime\_miller\_rabin}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1610 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1611 int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1612 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1614 fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1615 Otherwise $result$ is set to zero.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1616
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1617 Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1618 Miller-Rabin are a subset of the failures of the Fermat test.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1619
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1620 \subsection{Required Number of Tests}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1623 This is why a simple function has been provided to help out.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1624
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1625 \index{mp\_prime\_rabin\_miller\_trials}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1626 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1627 int mp_prime_rabin_miller_trials(int size)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1628 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1629 This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1630 in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1631 require ten tests whereas a 1024-bit number would only require four tests.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1632
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1633 You should always still perform a trial division before a Miller-Rabin test though.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1634
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1635 \section{Primality Testing}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1636 \index{mp\_prime\_is\_prime}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1637 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1638 int mp_prime_is_prime (mp_int * a, int t, int *result)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1639 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1640 This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1642 $1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1643
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1644 \section{Next Prime}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1645 \index{mp\_prime\_next\_prime}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1646 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1647 int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1648 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1649 This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1650 want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1651
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1652 \section{Random Primes}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1653 \index{mp\_prime\_random}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1654 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1655 int mp_prime_random(mp_int *a, int t, int size, int bbs,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1656 ltm_prime_callback cb, void *dat)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1657 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1658 This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1659 $t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1660
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1661 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1662 typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1663 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1664
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1665 Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1666 copied from the original input. It can be used to pass RNG context data to the callback. The function
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1667 mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1668 is no skew on the least significant bits.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1669
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1670 \textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1671 but users are encouraged to use the new mp\_prime\_random\_ex() function instead.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1672
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1673 \subsection{Extended Generation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1674 \index{mp\_prime\_random\_ex}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1675 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1676 int mp_prime_random_ex(mp_int *a, int t,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1677 int size, int flags,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1678 ltm_prime_callback cb, void *dat);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1679 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1680 This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1681 specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1682 (see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1683 mp\_prime\_random().
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1684
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1685 \begin{figure}[here]
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1686 \begin{center}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1687 \begin{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1688 \begin{tabular}{|r|l|}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1689 \hline \textbf{Flag} & \textbf{Meaning} \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1690 \hline LTM\_PRIME\_BBS & Make the prime congruent to $3$ modulo $4$ \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1691 \hline LTM\_PRIME\_SAFE & Make a prime $p$ such that $(p - 1)/2$ is also prime. \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1692 & This option implies LTM\_PRIME\_BBS as well. \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1693 \hline LTM\_PRIME\_2MSB\_OFF & Makes sure that the bit adjacent to the most significant bit \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1694 & Is forced to zero. \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1695 \hline LTM\_PRIME\_2MSB\_ON & Makes sure that the bit adjacent to the most significant bit \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1696 & Is forced to one. \\
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1697 \hline
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1698 \end{tabular}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1699 \end{small}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1700 \end{center}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1701 \caption{Primality Generation Options}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1702 \label{fig:primeopts}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1703 \end{figure}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1704
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1705 \chapter{Input and Output}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1706 \section{ASCII Conversions}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1707 \subsection{To ASCII}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1708 \index{mp\_toradix}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1709 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1710 int mp_toradix (mp_int * a, char *str, int radix);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1711 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1712 This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1713 to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1714 by the conversion before storing any data use the following function.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1715
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1716 \index{mp\_radix\_size}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1717 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1718 int mp_radix_size (mp_int * a, int radix, int *size)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1719 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1720 This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1721 function returns an error code and ``size'' will be zero.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1722
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1723 \subsection{From ASCII}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1724 \index{mp\_read\_radix}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1725 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1726 int mp_read_radix (mp_int * a, char *str, int radix);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1727 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1728 This will read the base-``radix'' NUL terminated string from ``str'' into $a$. It will stop reading when it reads a
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1729 character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1730 can be used to denote a negative number.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1731
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1732 \section{Binary Conversions}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1733
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1734 Converting an mp\_int to and from binary is another keen idea.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1735
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1736 \index{mp\_unsigned\_bin\_size}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1737 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1738 int mp_unsigned_bin_size(mp_int *a);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1739 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1740
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1741 This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1742
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1743 \index{mp\_to\_unsigned\_bin}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1744 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1745 int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1746 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1747 This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1748 requires. It does not store the sign of the integer.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1749
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1750 \index{mp\_read\_unsigned\_bin}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1751 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1752 int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1753 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1754 This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1755 integer $a$ will always be positive.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1756
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1757 For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1758 previous functions.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1759
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1760 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1761 int mp_signed_bin_size(mp_int *a);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1762 int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1763 int mp_to_signed_bin(mp_int *a, unsigned char *b);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1764 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1765 They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1766 byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1767 is non--zero.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1768
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1769 \chapter{Algebraic Functions}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1770 \section{Extended Euclidean Algorithm}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1771 \index{mp\_exteuclid}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1772 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1773 int mp_exteuclid(mp_int *a, mp_int *b,
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1774 mp_int *U1, mp_int *U2, mp_int *U3);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1775 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1776
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1777 This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that the following equation holds.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1778
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1779 \begin{equation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1780 a \cdot U1 + b \cdot U2 = U3
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1781 \end{equation}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1782
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1783 Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1784
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1785 \section{Greatest Common Divisor}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1786 \index{mp\_gcd}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1787 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1788 int mp_gcd (mp_int * a, mp_int * b, mp_int * c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1789 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1790 This will compute the greatest common divisor of $a$ and $b$ and store it in $c$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1791
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1792 \section{Least Common Multiple}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1793 \index{mp\_lcm}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1794 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1795 int mp_lcm (mp_int * a, mp_int * b, mp_int * c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1796 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1797 This will compute the least common multiple of $a$ and $b$ and store it in $c$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1798
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1799 \section{Jacobi Symbol}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1800 \index{mp\_jacobi}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1801 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1802 int mp_jacobi (mp_int * a, mp_int * p, int *c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1803 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1804 This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
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
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1806 then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1807 and the result will be $1$ if $a$ is a quadratic residue modulo $p$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1808
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1809 \section{Modular Inverse}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1810 \index{mp\_invmod}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1811 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1812 int mp_invmod (mp_int * a, mp_int * b, mp_int * c)
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1813 \end{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1814 Computes the multiplicative inverse of $a$ modulo $b$ and stores the result in $c$ such that $ac \equiv 1 \mbox{ (mod }b\mbox{)}$.
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1815
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1816 \section{Single Digit Functions}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1817
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1818 For those using small numbers (\textit{snicker snicker}) there are several ``helper'' functions
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1819
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1820 \index{mp\_add\_d} \index{mp\_sub\_d} \index{mp\_mul\_d} \index{mp\_div\_d} \index{mp\_mod\_d}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1821 \begin{alltt}
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1822 int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1823 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1824 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
91fbc376f010 Import of libtommath 0.35
Matt Johnston <matt@ucc.asn.au>
parents:
diff changeset
1825 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
91fbc376f010 Import of libtommath 0.35
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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}