Mercurial > dropbear
annotate libtommath/bn_s_mp_karatsuba_mul.c @ 1729:f091f7536aa5
Refer to libtom license files
author | Matt Johnston <matt@ucc.asn.au> |
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date | Thu, 25 Jun 2020 23:18:16 +0800 |
parents | 1051e4eea25a |
children |
rev | line source |
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1692
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1 #include "tommath_private.h" |
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2 #ifdef BN_S_MP_KARATSUBA_MUL_C |
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3 /* LibTomMath, multiple-precision integer library -- Tom St Denis */ |
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4 /* SPDX-License-Identifier: Unlicense */ |
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5 |
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6 /* c = |a| * |b| using Karatsuba Multiplication using |
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7 * three half size multiplications |
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8 * |
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9 * Let B represent the radix [e.g. 2**MP_DIGIT_BIT] and |
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10 * let n represent half of the number of digits in |
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11 * the min(a,b) |
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12 * |
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13 * a = a1 * B**n + a0 |
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14 * b = b1 * B**n + b0 |
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15 * |
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16 * Then, a * b => |
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17 a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 |
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18 * |
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19 * Note that a1b1 and a0b0 are used twice and only need to be |
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20 * computed once. So in total three half size (half # of |
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21 * digit) multiplications are performed, a0b0, a1b1 and |
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22 * (a1+b1)(a0+b0) |
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23 * |
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24 * Note that a multiplication of half the digits requires |
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25 * 1/4th the number of single precision multiplications so in |
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26 * total after one call 25% of the single precision multiplications |
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27 * are saved. Note also that the call to mp_mul can end up back |
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28 * in this function if the a0, a1, b0, or b1 are above the threshold. |
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29 * This is known as divide-and-conquer and leads to the famous |
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30 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than |
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31 * the standard O(N**2) that the baseline/comba methods use. |
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32 * Generally though the overhead of this method doesn't pay off |
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33 * until a certain size (N ~ 80) is reached. |
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34 */ |
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35 mp_err s_mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c) |
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36 { |
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37 mp_int x0, x1, y0, y1, t1, x0y0, x1y1; |
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38 int B; |
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39 mp_err err = MP_MEM; /* default the return code to an error */ |
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40 |
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41 /* min # of digits */ |
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42 B = MP_MIN(a->used, b->used); |
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43 |
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44 /* now divide in two */ |
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45 B = B >> 1; |
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46 |
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47 /* init copy all the temps */ |
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48 if (mp_init_size(&x0, B) != MP_OKAY) { |
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49 goto LBL_ERR; |
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50 } |
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51 if (mp_init_size(&x1, a->used - B) != MP_OKAY) { |
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52 goto X0; |
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53 } |
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54 if (mp_init_size(&y0, B) != MP_OKAY) { |
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55 goto X1; |
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56 } |
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57 if (mp_init_size(&y1, b->used - B) != MP_OKAY) { |
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58 goto Y0; |
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59 } |
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60 |
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61 /* init temps */ |
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62 if (mp_init_size(&t1, B * 2) != MP_OKAY) { |
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63 goto Y1; |
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64 } |
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65 if (mp_init_size(&x0y0, B * 2) != MP_OKAY) { |
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66 goto T1; |
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67 } |
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68 if (mp_init_size(&x1y1, B * 2) != MP_OKAY) { |
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69 goto X0Y0; |
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70 } |
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71 |
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72 /* now shift the digits */ |
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73 x0.used = y0.used = B; |
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74 x1.used = a->used - B; |
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75 y1.used = b->used - B; |
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76 |
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77 { |
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78 int x; |
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79 mp_digit *tmpa, *tmpb, *tmpx, *tmpy; |
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80 |
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81 /* we copy the digits directly instead of using higher level functions |
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82 * since we also need to shift the digits |
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83 */ |
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84 tmpa = a->dp; |
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85 tmpb = b->dp; |
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86 |
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87 tmpx = x0.dp; |
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88 tmpy = y0.dp; |
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89 for (x = 0; x < B; x++) { |
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90 *tmpx++ = *tmpa++; |
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91 *tmpy++ = *tmpb++; |
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92 } |
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93 |
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94 tmpx = x1.dp; |
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95 for (x = B; x < a->used; x++) { |
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96 *tmpx++ = *tmpa++; |
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97 } |
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98 |
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99 tmpy = y1.dp; |
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100 for (x = B; x < b->used; x++) { |
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101 *tmpy++ = *tmpb++; |
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102 } |
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103 } |
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104 |
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105 /* only need to clamp the lower words since by definition the |
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106 * upper words x1/y1 must have a known number of digits |
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107 */ |
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108 mp_clamp(&x0); |
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109 mp_clamp(&y0); |
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110 |
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111 /* now calc the products x0y0 and x1y1 */ |
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112 /* after this x0 is no longer required, free temp [x0==t2]! */ |
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113 if (mp_mul(&x0, &y0, &x0y0) != MP_OKAY) { |
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114 goto X1Y1; /* x0y0 = x0*y0 */ |
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115 } |
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116 if (mp_mul(&x1, &y1, &x1y1) != MP_OKAY) { |
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117 goto X1Y1; /* x1y1 = x1*y1 */ |
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118 } |
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119 |
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120 /* now calc x1+x0 and y1+y0 */ |
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121 if (s_mp_add(&x1, &x0, &t1) != MP_OKAY) { |
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122 goto X1Y1; /* t1 = x1 - x0 */ |
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123 } |
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124 if (s_mp_add(&y1, &y0, &x0) != MP_OKAY) { |
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125 goto X1Y1; /* t2 = y1 - y0 */ |
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126 } |
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127 if (mp_mul(&t1, &x0, &t1) != MP_OKAY) { |
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128 goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */ |
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129 } |
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130 |
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131 /* add x0y0 */ |
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132 if (mp_add(&x0y0, &x1y1, &x0) != MP_OKAY) { |
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133 goto X1Y1; /* t2 = x0y0 + x1y1 */ |
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134 } |
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135 if (s_mp_sub(&t1, &x0, &t1) != MP_OKAY) { |
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136 goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ |
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137 } |
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138 |
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139 /* shift by B */ |
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140 if (mp_lshd(&t1, B) != MP_OKAY) { |
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141 goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */ |
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142 } |
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143 if (mp_lshd(&x1y1, B * 2) != MP_OKAY) { |
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144 goto X1Y1; /* x1y1 = x1y1 << 2*B */ |
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145 } |
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146 |
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147 if (mp_add(&x0y0, &t1, &t1) != MP_OKAY) { |
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148 goto X1Y1; /* t1 = x0y0 + t1 */ |
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149 } |
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150 if (mp_add(&t1, &x1y1, c) != MP_OKAY) { |
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151 goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */ |
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152 } |
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153 |
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154 /* Algorithm succeeded set the return code to MP_OKAY */ |
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155 err = MP_OKAY; |
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156 |
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157 X1Y1: |
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158 mp_clear(&x1y1); |
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159 X0Y0: |
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160 mp_clear(&x0y0); |
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161 T1: |
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162 mp_clear(&t1); |
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163 Y1: |
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164 mp_clear(&y1); |
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165 Y0: |
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166 mp_clear(&y0); |
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167 X1: |
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168 mp_clear(&x1); |
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169 X0: |
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170 mp_clear(&x0); |
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171 LBL_ERR: |
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172 return err; |
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173 } |
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174 #endif |