src/field_impl.h

   1 /**********************************************************************
   2  * Copyright (c) 2013, 2014 Pieter Wuille                             *
   3  * Distributed under the MIT software license, see the accompanying   *
   4  * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
   5  **********************************************************************/
   6
   7 #ifndef SECP256K1_FIELD_IMPL_H
   8 #define SECP256K1_FIELD_IMPL_H
   9
  10 #if defined HAVE_CONFIG_H
  11 #include "libsecp256k1-config.h"
  12 #endif
  13
  14 #include "util.h"
  15
  16 #if defined(USE_FIELD_10X26)
  17 #include "field_10x26_impl.h"
  18 #elif defined(USE_FIELD_5X52)
  19 #include "field_5x52_impl.h"
  20 #else
  21 #error "Please select field implementation"
  22 #endif
  23
  24 SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe *a, const secp256k1_fe *b) {
  25     secp256k1_fe na;
  26     secp256k1_fe_negate(&na, a, 1);
  27     secp256k1_fe_add(&na, b);
  28     return secp256k1_fe_normalizes_to_zero(&na);
  29 }
  30
  31 SECP256K1_INLINE static int secp256k1_fe_equal_var(const secp256k1_fe *a, const secp256k1_fe *b) {
  32     secp256k1_fe na;
  33     secp256k1_fe_negate(&na, a, 1);
  34     secp256k1_fe_add(&na, b);
  35     return secp256k1_fe_normalizes_to_zero_var(&na);
  36 }
  37
  38 static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a) {
  39     /** Given that p is congruent to 3 mod 4, we can compute the square root of
  40      *  a mod p as the (p+1)/4'th power of a.
  41      *
  42      *  As (p+1)/4 is an even number, it will have the same result for a and for
  43      *  (-a). Only one of these two numbers actually has a square root however,
  44      *  so we test at the end by squaring and comparing to the input.
  45      *  Also because (p+1)/4 is an even number, the computed square root is
  46      *  itself always a square (a ** ((p+1)/4) is the square of a ** ((p+1)/8)).
  47      */
  48     secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1;
  49     int j;
  50
  51     /** The binary representation of (p + 1)/4 has 3 blocks of 1s, with lengths in
  52      *  { 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block:
  53      *  1, [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
  54      */
  55
  56     secp256k1_fe_sqr(&x2, a);
  57     secp256k1_fe_mul(&x2, &x2, a);
  58
  59     secp256k1_fe_sqr(&x3, &x2);
  60     secp256k1_fe_mul(&x3, &x3, a);
  61
  62     x6 = x3;
  63     for (j=0; j<3; j++) {
  64         secp256k1_fe_sqr(&x6, &x6);
  65     }
  66     secp256k1_fe_mul(&x6, &x6, &x3);
  67
  68     x9 = x6;
  69     for (j=0; j<3; j++) {
  70         secp256k1_fe_sqr(&x9, &x9);
  71     }
  72     secp256k1_fe_mul(&x9, &x9, &x3);
  73
  74     x11 = x9;
  75     for (j=0; j<2; j++) {
  76         secp256k1_fe_sqr(&x11, &x11);
  77     }
  78     secp256k1_fe_mul(&x11, &x11, &x2);
  79
  80     x22 = x11;
  81     for (j=0; j<11; j++) {
  82         secp256k1_fe_sqr(&x22, &x22);
  83     }
  84     secp256k1_fe_mul(&x22, &x22, &x11);
  85
  86     x44 = x22;
  87     for (j=0; j<22; j++) {
  88         secp256k1_fe_sqr(&x44, &x44);
  89     }
  90     secp256k1_fe_mul(&x44, &x44, &x22);
  91
  92     x88 = x44;
  93     for (j=0; j<44; j++) {
  94         secp256k1_fe_sqr(&x88, &x88);
  95     }
  96     secp256k1_fe_mul(&x88, &x88, &x44);
  97
  98     x176 = x88;
  99     for (j=0; j<88; j++) {
 100         secp256k1_fe_sqr(&x176, &x176);
 101     }
 102     secp256k1_fe_mul(&x176, &x176, &x88);
 103
 104     x220 = x176;
 105     for (j=0; j<44; j++) {
 106         secp256k1_fe_sqr(&x220, &x220);
 107     }
 108     secp256k1_fe_mul(&x220, &x220, &x44);
 109
 110     x223 = x220;
 111     for (j=0; j<3; j++) {
 112         secp256k1_fe_sqr(&x223, &x223);
 113     }
 114     secp256k1_fe_mul(&x223, &x223, &x3);
 115
 116     /* The final result is then assembled using a sliding window over the blocks. */
 117
 118     t1 = x223;
 119     for (j=0; j<23; j++) {
 120         secp256k1_fe_sqr(&t1, &t1);
 121     }
 122     secp256k1_fe_mul(&t1, &t1, &x22);
 123     for (j=0; j<6; j++) {
 124         secp256k1_fe_sqr(&t1, &t1);
 125     }
 126     secp256k1_fe_mul(&t1, &t1, &x2);
 127     secp256k1_fe_sqr(&t1, &t1);
 128     secp256k1_fe_sqr(r, &t1);
 129
 130     /* Check that a square root was actually calculated */
 131
 132     secp256k1_fe_sqr(&t1, r);
 133     return secp256k1_fe_equal(&t1, a);
 134 }
 135
 136 static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a) {
 137     secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1;
 138     int j;
 139
 140     /** The binary representation of (p - 2) has 5 blocks of 1s, with lengths in
 141      *  { 1, 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block:
 142      *  [1], [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
 143      */
 144
 145     secp256k1_fe_sqr(&x2, a);
 146     secp256k1_fe_mul(&x2, &x2, a);
 147
 148     secp256k1_fe_sqr(&x3, &x2);
 149     secp256k1_fe_mul(&x3, &x3, a);
 150
 151     x6 = x3;
 152     for (j=0; j<3; j++) {
 153         secp256k1_fe_sqr(&x6, &x6);
 154     }
 155     secp256k1_fe_mul(&x6, &x6, &x3);
 156
 157     x9 = x6;
 158     for (j=0; j<3; j++) {
 159         secp256k1_fe_sqr(&x9, &x9);
 160     }
 161     secp256k1_fe_mul(&x9, &x9, &x3);
 162
 163     x11 = x9;
 164     for (j=0; j<2; j++) {
 165         secp256k1_fe_sqr(&x11, &x11);
 166     }
 167     secp256k1_fe_mul(&x11, &x11, &x2);
 168
 169     x22 = x11;
 170     for (j=0; j<11; j++) {
 171         secp256k1_fe_sqr(&x22, &x22);
 172     }
 173     secp256k1_fe_mul(&x22, &x22, &x11);
 174
 175     x44 = x22;
 176     for (j=0; j<22; j++) {
 177         secp256k1_fe_sqr(&x44, &x44);
 178     }
 179     secp256k1_fe_mul(&x44, &x44, &x22);
 180
 181     x88 = x44;
 182     for (j=0; j<44; j++) {
 183         secp256k1_fe_sqr(&x88, &x88);
 184     }
 185     secp256k1_fe_mul(&x88, &x88, &x44);
 186
 187     x176 = x88;
 188     for (j=0; j<88; j++) {
 189         secp256k1_fe_sqr(&x176, &x176);
 190     }
 191     secp256k1_fe_mul(&x176, &x176, &x88);
 192
 193     x220 = x176;
 194     for (j=0; j<44; j++) {
 195         secp256k1_fe_sqr(&x220, &x220);
 196     }
 197     secp256k1_fe_mul(&x220, &x220, &x44);
 198
 199     x223 = x220;
 200     for (j=0; j<3; j++) {
 201         secp256k1_fe_sqr(&x223, &x223);
 202     }
 203     secp256k1_fe_mul(&x223, &x223, &x3);
 204
 205     /* The final result is then assembled using a sliding window over the blocks. */
 206
 207     t1 = x223;
 208     for (j=0; j<23; j++) {
 209         secp256k1_fe_sqr(&t1, &t1);
 210     }
 211     secp256k1_fe_mul(&t1, &t1, &x22);
 212     for (j=0; j<5; j++) {
 213         secp256k1_fe_sqr(&t1, &t1);
 214     }
 215     secp256k1_fe_mul(&t1, &t1, a);
 216     for (j=0; j<3; j++) {
 217         secp256k1_fe_sqr(&t1, &t1);
 218     }
 219     secp256k1_fe_mul(&t1, &t1, &x2);
 220     for (j=0; j<2; j++) {
 221         secp256k1_fe_sqr(&t1, &t1);
 222     }
 223     secp256k1_fe_mul(r, a, &t1);
 224 }
 225
 226 static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *a) {
 227 #if defined(USE_FIELD_INV_BUILTIN)
 228     secp256k1_fe_inv(r, a);
 229 #elif defined(USE_FIELD_INV_NUM)
 230     secp256k1_num n, m;
 231     static const secp256k1_fe negone = SECP256K1_FE_CONST(
 232         0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL,
 233         0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL, 0xFFFFFC2EUL
 234     );
 235     /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */
 236     static const unsigned char prime[32] = {
 237         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
 238         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
 239         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
 240         0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
 241     };
 242     unsigned char b[32];
 243     int res;
 244     secp256k1_fe c = *a;
 245     secp256k1_fe_normalize_var(&c);
 246     secp256k1_fe_get_b32(b, &c);
 247     secp256k1_num_set_bin(&n, b, 32);
 248     secp256k1_num_set_bin(&m, prime, 32);
 249     secp256k1_num_mod_inverse(&n, &n, &m);
 250     secp256k1_num_get_bin(b, 32, &n);
 251     res = secp256k1_fe_set_b32(r, b);
 252     (void)res;
 253     VERIFY_CHECK(res);
 254     /* Verify the result is the (unique) valid inverse using non-GMP code. */
 255     secp256k1_fe_mul(&c, &c, r);
 256     secp256k1_fe_add(&c, &negone);
 257     CHECK(secp256k1_fe_normalizes_to_zero_var(&c));
 258 #else
 259 #error "Please select field inverse implementation"
 260 #endif
 261 }
 262
 263 static void secp256k1_fe_inv_all_var(secp256k1_fe *r, const secp256k1_fe *a, size_t len) {
 264     secp256k1_fe u;
 265     size_t i;
 266     if (len < 1) {
 267         return;
 268     }
 269
 270     VERIFY_CHECK((r + len <= a) || (a + len <= r));
 271
 272     r[0] = a[0];
 273
 274     i = 0;
 275     while (++i < len) {
 276         secp256k1_fe_mul(&r[i], &r[i - 1], &a[i]);
 277     }
 278
 279     secp256k1_fe_inv_var(&u, &r[--i]);
 280
 281     while (i > 0) {
 282         size_t j = i--;
 283         secp256k1_fe_mul(&r[j], &r[i], &u);
 284         secp256k1_fe_mul(&u, &u, &a[j]);
 285     }
 286
 287     r[0] = u;
 288 }
 289
 290 static int secp256k1_fe_is_quad_var(const secp256k1_fe *a) {
 291 #ifndef USE_NUM_NONE
 292     unsigned char b[32];
 293     secp256k1_num n;
 294     secp256k1_num m;
 295     /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */
 296     static const unsigned char prime[32] = {
 297         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
 298         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
 299         0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
 300         0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
 301     };
 302
 303     secp256k1_fe c = *a;
 304     secp256k1_fe_normalize_var(&c);
 305     secp256k1_fe_get_b32(b, &c);
 306     secp256k1_num_set_bin(&n, b, 32);
 307     secp256k1_num_set_bin(&m, prime, 32);
 308     return secp256k1_num_jacobi(&n, &m) >= 0;
 309 #else
 310     secp256k1_fe r;
 311     return secp256k1_fe_sqrt(&r, a);
 312 #endif
 313 }
 314
 315 #endif /* SECP256K1_FIELD_IMPL_H */