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1 /* LibTomMath, multiple-precision integer library -- Tom St Denis |
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2 * |
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3 * LibTomMath is a library that provides multiple-precision |
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4 * integer arithmetic as well as number theoretic functionality. |
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5 * |
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6 * The library was designed directly after the MPI library by |
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7 * Michael Fromberger but has been written from scratch with |
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8 * additional optimizations in place. |
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9 * |
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10 * The library is free for all purposes without any express |
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11 * guarantee it works. |
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12 * |
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13 * Tom St Denis, [email protected], http://math.libtomcrypt.org |
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14 */ |
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15 #ifndef BN_H_ |
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16 #define BN_H_ |
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17 |
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18 #include <stdio.h> |
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19 #include <string.h> |
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20 #include <stdlib.h> |
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21 #include <ctype.h> |
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22 #include <limits.h> |
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23 |
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24 #undef MIN |
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25 #define MIN(x,y) ((x)<(y)?(x):(y)) |
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26 #undef MAX |
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27 #define MAX(x,y) ((x)>(y)?(x):(y)) |
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28 |
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29 #ifdef __cplusplus |
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30 extern "C" { |
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31 |
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32 /* C++ compilers don't like assigning void * to mp_digit * */ |
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33 #define OPT_CAST(x) (x *) |
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34 |
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35 #else |
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36 |
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37 /* C on the other hand doesn't care */ |
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38 #define OPT_CAST(x) |
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39 |
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40 #endif |
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41 |
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42 /* some default configurations. |
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43 * |
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44 * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits |
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45 * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits |
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46 * |
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47 * At the very least a mp_digit must be able to hold 7 bits |
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48 * [any size beyond that is ok provided it doesn't overflow the data type] |
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49 */ |
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50 #ifdef MP_8BIT |
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51 typedef unsigned char mp_digit; |
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52 typedef unsigned short mp_word; |
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53 #elif defined(MP_16BIT) |
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54 typedef unsigned short mp_digit; |
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55 typedef unsigned long mp_word; |
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56 #elif defined(MP_64BIT) |
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57 /* for GCC only on supported platforms */ |
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58 #ifndef CRYPT |
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59 typedef unsigned long long ulong64; |
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60 typedef signed long long long64; |
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61 #endif |
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62 |
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63 typedef ulong64 mp_digit; |
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64 typedef unsigned long mp_word __attribute__ ((mode(TI))); |
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65 |
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66 #define DIGIT_BIT 60 |
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67 #else |
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68 /* this is the default case, 28-bit digits */ |
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69 |
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70 /* this is to make porting into LibTomCrypt easier :-) */ |
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71 #ifndef CRYPT |
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72 #if defined(_MSC_VER) || defined(__BORLANDC__) |
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73 typedef unsigned __int64 ulong64; |
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74 typedef signed __int64 long64; |
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75 #else |
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76 typedef unsigned long long ulong64; |
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77 typedef signed long long long64; |
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78 #endif |
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79 #endif |
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80 |
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81 typedef unsigned long mp_digit; |
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82 typedef ulong64 mp_word; |
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83 |
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84 #ifdef MP_31BIT |
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85 /* this is an extension that uses 31-bit digits */ |
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86 #define DIGIT_BIT 31 |
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87 #else |
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88 /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */ |
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89 #define DIGIT_BIT 28 |
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90 #define MP_28BIT |
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91 #endif |
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92 #endif |
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93 |
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94 /* define heap macros */ |
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95 #ifndef CRYPT |
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96 /* default to libc stuff */ |
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97 #ifndef XMALLOC |
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98 #define XMALLOC malloc |
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99 #define XFREE free |
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100 #define XREALLOC realloc |
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101 #define XCALLOC calloc |
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102 #else |
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103 /* prototypes for our heap functions */ |
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104 extern void *XMALLOC(size_t n); |
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105 extern void *REALLOC(void *p, size_t n); |
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106 extern void *XCALLOC(size_t n, size_t s); |
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107 extern void XFREE(void *p); |
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108 #endif |
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109 #endif |
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110 |
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111 |
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112 /* otherwise the bits per digit is calculated automatically from the size of a mp_digit */ |
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113 #ifndef DIGIT_BIT |
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114 #define DIGIT_BIT ((int)((CHAR_BIT * sizeof(mp_digit) - 1))) /* bits per digit */ |
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115 #endif |
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116 |
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117 #define MP_DIGIT_BIT DIGIT_BIT |
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118 #define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1)) |
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119 #define MP_DIGIT_MAX MP_MASK |
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120 |
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121 /* equalities */ |
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122 #define MP_LT -1 /* less than */ |
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123 #define MP_EQ 0 /* equal to */ |
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124 #define MP_GT 1 /* greater than */ |
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125 |
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126 #define MP_ZPOS 0 /* positive integer */ |
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127 #define MP_NEG 1 /* negative */ |
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128 |
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129 #define MP_OKAY 0 /* ok result */ |
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130 #define MP_MEM -2 /* out of mem */ |
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131 #define MP_VAL -3 /* invalid input */ |
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132 #define MP_RANGE MP_VAL |
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133 |
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134 #define MP_YES 1 /* yes response */ |
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135 #define MP_NO 0 /* no response */ |
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136 |
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137 /* Primality generation flags */ |
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138 #define LTM_PRIME_BBS 0x0001 /* BBS style prime */ |
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139 #define LTM_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */ |
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140 #define LTM_PRIME_2MSB_OFF 0x0004 /* force 2nd MSB to 0 */ |
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141 #define LTM_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */ |
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142 |
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143 typedef int mp_err; |
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144 |
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145 /* you'll have to tune these... */ |
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146 extern int KARATSUBA_MUL_CUTOFF, |
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147 KARATSUBA_SQR_CUTOFF, |
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148 TOOM_MUL_CUTOFF, |
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149 TOOM_SQR_CUTOFF; |
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150 |
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151 /* define this to use lower memory usage routines (exptmods mostly) */ |
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152 /* #define MP_LOW_MEM */ |
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153 |
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154 /* default precision */ |
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155 #ifndef MP_PREC |
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156 #ifdef MP_LOW_MEM |
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157 #define MP_PREC 64 /* default digits of precision */ |
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158 #else |
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159 #define MP_PREC 8 /* default digits of precision */ |
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160 #endif |
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161 #endif |
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162 |
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163 /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */ |
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164 #define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1)) |
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165 |
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166 /* the infamous mp_int structure */ |
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167 typedef struct { |
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168 int used, alloc, sign; |
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169 mp_digit *dp; |
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170 } mp_int; |
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171 |
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172 /* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */ |
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173 typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); |
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174 |
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175 |
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176 #define USED(m) ((m)->used) |
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177 #define DIGIT(m,k) ((m)->dp[(k)]) |
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178 #define SIGN(m) ((m)->sign) |
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179 |
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180 /* error code to char* string */ |
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181 char *mp_error_to_string(int code); |
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182 |
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183 /* ---> init and deinit bignum functions <--- */ |
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184 /* init a bignum */ |
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185 int mp_init(mp_int *a); |
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186 |
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187 /* free a bignum */ |
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188 void mp_clear(mp_int *a); |
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189 |
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190 /* init a null terminated series of arguments */ |
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191 int mp_init_multi(mp_int *mp, ...); |
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192 |
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193 /* clear a null terminated series of arguments */ |
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194 void mp_clear_multi(mp_int *mp, ...); |
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195 |
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196 /* exchange two ints */ |
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197 void mp_exch(mp_int *a, mp_int *b); |
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198 |
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199 /* shrink ram required for a bignum */ |
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200 int mp_shrink(mp_int *a); |
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201 |
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202 /* grow an int to a given size */ |
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203 int mp_grow(mp_int *a, int size); |
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204 |
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205 /* init to a given number of digits */ |
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206 int mp_init_size(mp_int *a, int size); |
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207 |
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208 /* ---> Basic Manipulations <--- */ |
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209 #define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO) |
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210 #define mp_iseven(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO) |
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211 #define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO) |
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212 |
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213 /* set to zero */ |
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214 void mp_zero(mp_int *a); |
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215 |
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216 /* set to a digit */ |
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217 void mp_set(mp_int *a, mp_digit b); |
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218 |
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219 /* set a 32-bit const */ |
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220 int mp_set_int(mp_int *a, unsigned long b); |
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221 |
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222 /* get a 32-bit value */ |
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223 unsigned long mp_get_int(mp_int * a); |
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224 |
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225 /* initialize and set a digit */ |
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226 int mp_init_set (mp_int * a, mp_digit b); |
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227 |
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228 /* initialize and set 32-bit value */ |
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229 int mp_init_set_int (mp_int * a, unsigned long b); |
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230 |
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231 /* copy, b = a */ |
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232 int mp_copy(mp_int *a, mp_int *b); |
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233 |
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234 /* inits and copies, a = b */ |
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235 int mp_init_copy(mp_int *a, mp_int *b); |
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236 |
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237 /* trim unused digits */ |
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238 void mp_clamp(mp_int *a); |
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239 |
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240 /* ---> digit manipulation <--- */ |
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241 |
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242 /* right shift by "b" digits */ |
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243 void mp_rshd(mp_int *a, int b); |
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244 |
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245 /* left shift by "b" digits */ |
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246 int mp_lshd(mp_int *a, int b); |
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247 |
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248 /* c = a / 2**b */ |
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249 int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d); |
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250 |
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251 /* b = a/2 */ |
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252 int mp_div_2(mp_int *a, mp_int *b); |
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253 |
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254 /* c = a * 2**b */ |
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255 int mp_mul_2d(mp_int *a, int b, mp_int *c); |
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256 |
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257 /* b = a*2 */ |
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258 int mp_mul_2(mp_int *a, mp_int *b); |
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259 |
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260 /* c = a mod 2**d */ |
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261 int mp_mod_2d(mp_int *a, int b, mp_int *c); |
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262 |
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263 /* computes a = 2**b */ |
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264 int mp_2expt(mp_int *a, int b); |
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265 |
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266 /* Counts the number of lsbs which are zero before the first zero bit */ |
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267 int mp_cnt_lsb(mp_int *a); |
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268 |
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269 /* I Love Earth! */ |
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270 |
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271 /* makes a pseudo-random int of a given size */ |
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272 int mp_rand(mp_int *a, int digits); |
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273 |
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274 /* ---> binary operations <--- */ |
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275 /* c = a XOR b */ |
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276 int mp_xor(mp_int *a, mp_int *b, mp_int *c); |
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277 |
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278 /* c = a OR b */ |
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279 int mp_or(mp_int *a, mp_int *b, mp_int *c); |
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280 |
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281 /* c = a AND b */ |
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282 int mp_and(mp_int *a, mp_int *b, mp_int *c); |
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283 |
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284 /* ---> Basic arithmetic <--- */ |
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285 |
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286 /* b = -a */ |
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287 int mp_neg(mp_int *a, mp_int *b); |
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288 |
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289 /* b = |a| */ |
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290 int mp_abs(mp_int *a, mp_int *b); |
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291 |
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292 /* compare a to b */ |
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293 int mp_cmp(mp_int *a, mp_int *b); |
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294 |
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295 /* compare |a| to |b| */ |
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296 int mp_cmp_mag(mp_int *a, mp_int *b); |
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297 |
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298 /* c = a + b */ |
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299 int mp_add(mp_int *a, mp_int *b, mp_int *c); |
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300 |
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301 /* c = a - b */ |
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302 int mp_sub(mp_int *a, mp_int *b, mp_int *c); |
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303 |
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304 /* c = a * b */ |
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305 int mp_mul(mp_int *a, mp_int *b, mp_int *c); |
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306 |
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307 /* b = a*a */ |
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308 int mp_sqr(mp_int *a, mp_int *b); |
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309 |
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310 /* a/b => cb + d == a */ |
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311 int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d); |
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312 |
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313 /* c = a mod b, 0 <= c < b */ |
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314 int mp_mod(mp_int *a, mp_int *b, mp_int *c); |
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315 |
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316 /* ---> single digit functions <--- */ |
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317 |
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318 /* compare against a single digit */ |
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319 int mp_cmp_d(mp_int *a, mp_digit b); |
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320 |
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321 /* c = a + b */ |
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322 int mp_add_d(mp_int *a, mp_digit b, mp_int *c); |
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323 |
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324 /* c = a - b */ |
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325 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); |
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326 |
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327 /* c = a * b */ |
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328 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); |
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329 |
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330 /* a/b => cb + d == a */ |
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331 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); |
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332 |
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333 /* a/3 => 3c + d == a */ |
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334 int mp_div_3(mp_int *a, mp_int *c, mp_digit *d); |
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335 |
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336 /* c = a**b */ |
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337 int mp_expt_d(mp_int *a, mp_digit b, mp_int *c); |
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338 |
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339 /* c = a mod b, 0 <= c < b */ |
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340 int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); |
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341 |
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342 /* ---> number theory <--- */ |
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343 |
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344 /* d = a + b (mod c) */ |
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345 int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); |
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346 |
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347 /* d = a - b (mod c) */ |
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348 int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); |
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349 |
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350 /* d = a * b (mod c) */ |
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351 int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); |
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352 |
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353 /* c = a * a (mod b) */ |
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354 int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c); |
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355 |
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356 /* c = 1/a (mod b) */ |
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357 int mp_invmod(mp_int *a, mp_int *b, mp_int *c); |
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358 |
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359 /* c = (a, b) */ |
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360 int mp_gcd(mp_int *a, mp_int *b, mp_int *c); |
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361 |
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362 /* produces value such that U1*a + U2*b = U3 */ |
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363 int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3); |
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364 |
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365 /* c = [a, b] or (a*b)/(a, b) */ |
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366 int mp_lcm(mp_int *a, mp_int *b, mp_int *c); |
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367 |
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368 /* finds one of the b'th root of a, such that |c|**b <= |a| |
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369 * |
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370 * returns error if a < 0 and b is even |
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371 */ |
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372 int mp_n_root(mp_int *a, mp_digit b, mp_int *c); |
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373 |
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374 /* special sqrt algo */ |
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375 int mp_sqrt(mp_int *arg, mp_int *ret); |
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376 |
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377 /* is number a square? */ |
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378 int mp_is_square(mp_int *arg, int *ret); |
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379 |
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380 /* computes the jacobi c = (a | n) (or Legendre if b is prime) */ |
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381 int mp_jacobi(mp_int *a, mp_int *n, int *c); |
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382 |
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383 /* used to setup the Barrett reduction for a given modulus b */ |
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384 int mp_reduce_setup(mp_int *a, mp_int *b); |
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385 |
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386 /* Barrett Reduction, computes a (mod b) with a precomputed value c |
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387 * |
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388 * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely |
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389 * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code]. |
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390 */ |
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391 int mp_reduce(mp_int *a, mp_int *b, mp_int *c); |
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392 |
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393 /* setups the montgomery reduction */ |
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394 int mp_montgomery_setup(mp_int *a, mp_digit *mp); |
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395 |
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396 /* computes a = B**n mod b without division or multiplication useful for |
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397 * normalizing numbers in a Montgomery system. |
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398 */ |
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399 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); |
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400 |
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401 /* computes x/R == x (mod N) via Montgomery Reduction */ |
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402 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); |
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403 |
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404 /* returns 1 if a is a valid DR modulus */ |
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405 int mp_dr_is_modulus(mp_int *a); |
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406 |
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407 /* sets the value of "d" required for mp_dr_reduce */ |
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408 void mp_dr_setup(mp_int *a, mp_digit *d); |
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409 |
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410 /* reduces a modulo b using the Diminished Radix method */ |
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411 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); |
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412 |
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413 /* returns true if a can be reduced with mp_reduce_2k */ |
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414 int mp_reduce_is_2k(mp_int *a); |
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415 |
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416 /* determines k value for 2k reduction */ |
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417 int mp_reduce_2k_setup(mp_int *a, mp_digit *d); |
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418 |
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419 /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */ |
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420 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); |
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421 |
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422 /* d = a**b (mod c) */ |
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423 int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); |
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424 |
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425 /* ---> Primes <--- */ |
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426 |
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427 /* number of primes */ |
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428 #ifdef MP_8BIT |
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429 #define PRIME_SIZE 31 |
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430 #else |
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431 #define PRIME_SIZE 256 |
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432 #endif |
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433 |
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434 /* table of first PRIME_SIZE primes */ |
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435 extern const mp_digit __prime_tab[]; |
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436 |
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437 /* result=1 if a is divisible by one of the first PRIME_SIZE primes */ |
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438 int mp_prime_is_divisible(mp_int *a, int *result); |
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439 |
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440 /* performs one Fermat test of "a" using base "b". |
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441 * Sets result to 0 if composite or 1 if probable prime |
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442 */ |
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443 int mp_prime_fermat(mp_int *a, mp_int *b, int *result); |
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444 |
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445 /* performs one Miller-Rabin test of "a" using base "b". |
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446 * Sets result to 0 if composite or 1 if probable prime |
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447 */ |
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448 int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result); |
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449 |
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450 /* This gives [for a given bit size] the number of trials required |
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451 * such that Miller-Rabin gives a prob of failure lower than 2^-96 |
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452 */ |
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453 int mp_prime_rabin_miller_trials(int size); |
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454 |
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455 /* performs t rounds of Miller-Rabin on "a" using the first |
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456 * t prime bases. Also performs an initial sieve of trial |
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457 * division. Determines if "a" is prime with probability |
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458 * of error no more than (1/4)**t. |
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459 * |
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460 * Sets result to 1 if probably prime, 0 otherwise |
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461 */ |
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462 int mp_prime_is_prime(mp_int *a, int t, int *result); |
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463 |
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464 /* finds the next prime after the number "a" using "t" trials |
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465 * of Miller-Rabin. |
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466 * |
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467 * bbs_style = 1 means the prime must be congruent to 3 mod 4 |
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468 */ |
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469 int mp_prime_next_prime(mp_int *a, int t, int bbs_style); |
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470 |
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471 /* makes a truly random prime of a given size (bytes), |
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472 * call with bbs = 1 if you want it to be congruent to 3 mod 4 |
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473 * |
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474 * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can |
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475 * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself |
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476 * so it can be NULL |
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477 * |
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478 * The prime generated will be larger than 2^(8*size). |
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479 */ |
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480 #define mp_prime_random(a, t, size, bbs, cb, dat) mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?LTM_PRIME_BBS:0, cb, dat) |
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481 |
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482 /* makes a truly random prime of a given size (bits), |
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483 * |
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484 * Flags are as follows: |
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485 * |
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486 * LTM_PRIME_BBS - make prime congruent to 3 mod 4 |
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487 * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) |
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488 * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero |
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489 * LTM_PRIME_2MSB_ON - make the 2nd highest bit one |
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490 * |
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491 * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can |
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492 * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself |
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493 * so it can be NULL |
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494 * |
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495 */ |
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496 int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat); |
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497 |
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498 /* ---> radix conversion <--- */ |
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499 int mp_count_bits(mp_int *a); |
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500 |
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501 int mp_unsigned_bin_size(mp_int *a); |
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502 int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c); |
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503 int mp_to_unsigned_bin(mp_int *a, unsigned char *b); |
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504 |
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505 int mp_signed_bin_size(mp_int *a); |
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506 int mp_read_signed_bin(mp_int *a, unsigned char *b, int c); |
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507 int mp_to_signed_bin(mp_int *a, unsigned char *b); |
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508 |
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509 int mp_read_radix(mp_int *a, char *str, int radix); |
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510 int mp_toradix(mp_int *a, char *str, int radix); |
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511 int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen); |
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512 int mp_radix_size(mp_int *a, int radix, int *size); |
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513 |
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514 int mp_fread(mp_int *a, int radix, FILE *stream); |
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515 int mp_fwrite(mp_int *a, int radix, FILE *stream); |
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516 |
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517 #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) |
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518 #define mp_raw_size(mp) mp_signed_bin_size(mp) |
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519 #define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) |
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520 #define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) |
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521 #define mp_mag_size(mp) mp_unsigned_bin_size(mp) |
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522 #define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) |
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523 |
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524 #define mp_tobinary(M, S) mp_toradix((M), (S), 2) |
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525 #define mp_tooctal(M, S) mp_toradix((M), (S), 8) |
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526 #define mp_todecimal(M, S) mp_toradix((M), (S), 10) |
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527 #define mp_tohex(M, S) mp_toradix((M), (S), 16) |
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528 |
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529 /* lowlevel functions, do not call! */ |
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530 int s_mp_add(mp_int *a, mp_int *b, mp_int *c); |
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531 int s_mp_sub(mp_int *a, mp_int *b, mp_int *c); |
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532 #define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) |
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533 int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); |
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534 int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); |
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535 int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); |
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536 int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); |
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537 int fast_s_mp_sqr(mp_int *a, mp_int *b); |
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538 int s_mp_sqr(mp_int *a, mp_int *b); |
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539 int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c); |
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540 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c); |
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541 int mp_karatsuba_sqr(mp_int *a, mp_int *b); |
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542 int mp_toom_sqr(mp_int *a, mp_int *b); |
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543 int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c); |
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544 int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); |
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545 int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode); |
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546 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y); |
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547 void bn_reverse(unsigned char *s, int len); |
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548 |
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549 extern const char *mp_s_rmap; |
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550 |
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551 #ifdef __cplusplus |
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552 } |
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553 #endif |
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554 |
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555 #endif |
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556 |
