lib/libcrypto/ec/ecp_nistp256.c

   1 /* $OpenBSD: ecp_nistp256.c,v 1.18 2017/05/02 03:59:44 deraadt Exp $ */
   2 /*
   3  * Written by Adam Langley (Google) for the OpenSSL project
   4  */
   5 /*
   6  * Copyright (c) 2011 Google Inc.
   7  *
   8  * Permission to use, copy, modify, and distribute this software for any
   9  * purpose with or without fee is hereby granted, provided that the above
  10  * copyright notice and this permission notice appear in all copies.
  11  *
  12  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  13  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  14  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  15  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  16  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  17  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  18  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  19  */
  20
  21 /*
  22  * A 64-bit implementation of the NIST P-256 elliptic curve point multiplication
  23  *
  24  * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
  25  * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
  26  * work which got its smarts from Daniel J. Bernstein's work on the same.
  27  */
  28
  29 #include <stdint.h>
  30 #include <string.h>
  31
  32 #include <openssl/opensslconf.h>
  33
  34 #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128
  35
  36 #include <openssl/err.h>
  37 #include "ec_lcl.h"
  38
  39 #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
  40   /* even with gcc, the typedef won't work for 32-bit platforms */
  41   typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
  42   typedef __int128_t int128_t;
  43 #else
  44   #error "Need GCC 3.1 or later to define type uint128_t"
  45 #endif
  46
  47 typedef uint8_t u8;
  48 typedef uint32_t u32;
  49 typedef uint64_t u64;
  50 typedef int64_t s64;
  51
  52 /* The underlying field.
  53  *
  54  * P256 operates over GF(2^256-2^224+2^192+2^96-1). We can serialise an element
  55  * of this field into 32 bytes. We call this an felem_bytearray. */
  56
  57 typedef u8 felem_bytearray[32];
  58
  59 /* These are the parameters of P256, taken from FIPS 186-3, page 86. These
  60  * values are big-endian. */
  61 static const felem_bytearray nistp256_curve_params[5] = {
  62         {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,       /* p */
  63          0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  64          0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
  65          0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
  66         {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,       /* a = -3 */
  67          0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
  68          0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
  69          0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc},      /* b */
  70         {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
  71          0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
  72          0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
  73          0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
  74         {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47,       /* x */
  75          0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
  76          0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
  77          0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
  78         {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b,       /* y */
  79          0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
  80          0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
  81          0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
  82 };
  83
  84 /* The representation of field elements.
  85  * ------------------------------------
  86  *
  87  * We represent field elements with either four 128-bit values, eight 128-bit
  88  * values, or four 64-bit values. The field element represented is:
  89  *   v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192  (mod p)
  90  * or:
  91  *   v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512  (mod p)
  92  *
  93  * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
  94  * apart, but are 128-bits wide, the most significant bits of each limb overlap
  95  * with the least significant bits of the next.
  96  *
  97  * A field element with four limbs is an 'felem'. One with eight limbs is a
  98  * 'longfelem'
  99  *
 100  * A field element with four, 64-bit values is called a 'smallfelem'. Small
 101  * values are used as intermediate values before multiplication.
 102  */
 103
 104 #define NLIMBS 4
 105
 106 typedef uint128_t limb;
 107 typedef limb felem[NLIMBS];
 108 typedef limb longfelem[NLIMBS * 2];
 109 typedef u64 smallfelem[NLIMBS];
 110
 111 /* This is the value of the prime as four 64-bit words, little-endian. */
 112 static const u64 kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0, 0xffffffff00000001ul};
 113 static const limb bottom32bits = 0xffffffff;
 114 static const u64 bottom63bits = 0x7ffffffffffffffful;
 115
 116 /* bin32_to_felem takes a little-endian byte array and converts it into felem
 117  * form. This assumes that the CPU is little-endian. */
 118 static void
 119 bin32_to_felem(felem out, const u8 in[32])
 120 {
 121         out[0] = *((u64 *) & in[0]);
 122         out[1] = *((u64 *) & in[8]);
 123         out[2] = *((u64 *) & in[16]);
 124         out[3] = *((u64 *) & in[24]);
 125 }
 126
 127 /* smallfelem_to_bin32 takes a smallfelem and serialises into a little endian,
 128  * 32 byte array. This assumes that the CPU is little-endian. */
 129 static void
 130 smallfelem_to_bin32(u8 out[32], const smallfelem in)
 131 {
 132         *((u64 *) & out[0]) = in[0];
 133         *((u64 *) & out[8]) = in[1];
 134         *((u64 *) & out[16]) = in[2];
 135         *((u64 *) & out[24]) = in[3];
 136 }
 137
 138 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
 139 static void
 140 flip_endian(u8 * out, const u8 * in, unsigned len)
 141 {
 142         unsigned i;
 143         for (i = 0; i < len; ++i)
 144                 out[i] = in[len - 1 - i];
 145 }
 146
 147 /* BN_to_felem converts an OpenSSL BIGNUM into an felem */
 148 static int
 149 BN_to_felem(felem out, const BIGNUM * bn)
 150 {
 151         felem_bytearray b_in;
 152         felem_bytearray b_out;
 153         unsigned num_bytes;
 154
 155         /* BN_bn2bin eats leading zeroes */
 156         memset(b_out, 0, sizeof b_out);
 157         num_bytes = BN_num_bytes(bn);
 158         if (num_bytes > sizeof b_out) {
 159                 ECerror(EC_R_BIGNUM_OUT_OF_RANGE);
 160                 return 0;
 161         }
 162         if (BN_is_negative(bn)) {
 163                 ECerror(EC_R_BIGNUM_OUT_OF_RANGE);
 164                 return 0;
 165         }
 166         num_bytes = BN_bn2bin(bn, b_in);
 167         flip_endian(b_out, b_in, num_bytes);
 168         bin32_to_felem(out, b_out);
 169         return 1;
 170 }
 171
 172 /* felem_to_BN converts an felem into an OpenSSL BIGNUM */
 173 static BIGNUM *
 174 smallfelem_to_BN(BIGNUM * out, const smallfelem in)
 175 {
 176         felem_bytearray b_in, b_out;
 177         smallfelem_to_bin32(b_in, in);
 178         flip_endian(b_out, b_in, sizeof b_out);
 179         return BN_bin2bn(b_out, sizeof b_out, out);
 180 }
 181
 182
 183 /* Field operations
 184  * ---------------- */
 185
 186 static void
 187 smallfelem_one(smallfelem out)
 188 {
 189         out[0] = 1;
 190         out[1] = 0;
 191         out[2] = 0;
 192         out[3] = 0;
 193 }
 194
 195 static void
 196 smallfelem_assign(smallfelem out, const smallfelem in)
 197 {
 198         out[0] = in[0];
 199         out[1] = in[1];
 200         out[2] = in[2];
 201         out[3] = in[3];
 202 }
 203
 204 static void
 205 felem_assign(felem out, const felem in)
 206 {
 207         out[0] = in[0];
 208         out[1] = in[1];
 209         out[2] = in[2];
 210         out[3] = in[3];
 211 }
 212
 213 /* felem_sum sets out = out + in. */
 214 static void
 215 felem_sum(felem out, const felem in)
 216 {
 217         out[0] += in[0];
 218         out[1] += in[1];
 219         out[2] += in[2];
 220         out[3] += in[3];
 221 }
 222
 223 /* felem_small_sum sets out = out + in. */
 224 static void
 225 felem_small_sum(felem out, const smallfelem in)
 226 {
 227         out[0] += in[0];
 228         out[1] += in[1];
 229         out[2] += in[2];
 230         out[3] += in[3];
 231 }
 232
 233 /* felem_scalar sets out = out * scalar */
 234 static void
 235 felem_scalar(felem out, const u64 scalar)
 236 {
 237         out[0] *= scalar;
 238         out[1] *= scalar;
 239         out[2] *= scalar;
 240         out[3] *= scalar;
 241 }
 242
 243 /* longfelem_scalar sets out = out * scalar */
 244 static void
 245 longfelem_scalar(longfelem out, const u64 scalar)
 246 {
 247         out[0] *= scalar;
 248         out[1] *= scalar;
 249         out[2] *= scalar;
 250         out[3] *= scalar;
 251         out[4] *= scalar;
 252         out[5] *= scalar;
 253         out[6] *= scalar;
 254         out[7] *= scalar;
 255 }
 256
 257 #define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
 258 #define two105 (((limb)1) << 105)
 259 #define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)
 260
 261 /* zero105 is 0 mod p */
 262 static const felem zero105 = {two105m41m9, two105, two105m41p9, two105m41p9};
 263
 264 /* smallfelem_neg sets |out| to |-small|
 265  * On exit:
 266  *   out[i] < out[i] + 2^105
 267  */
 268 static void
 269 smallfelem_neg(felem out, const smallfelem small)
 270 {
 271         /* In order to prevent underflow, we subtract from 0 mod p. */
 272         out[0] = zero105[0] - small[0];
 273         out[1] = zero105[1] - small[1];
 274         out[2] = zero105[2] - small[2];
 275         out[3] = zero105[3] - small[3];
 276 }
 277
 278 /* felem_diff subtracts |in| from |out|
 279  * On entry:
 280  *   in[i] < 2^104
 281  * On exit:
 282  *   out[i] < out[i] + 2^105
 283  */
 284 static void
 285 felem_diff(felem out, const felem in)
 286 {
 287         /* In order to prevent underflow, we add 0 mod p before subtracting. */
 288         out[0] += zero105[0];
 289         out[1] += zero105[1];
 290         out[2] += zero105[2];
 291         out[3] += zero105[3];
 292
 293         out[0] -= in[0];
 294         out[1] -= in[1];
 295         out[2] -= in[2];
 296         out[3] -= in[3];
 297 }
 298
 299 #define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
 300 #define two107 (((limb)1) << 107)
 301 #define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)
 302
 303 /* zero107 is 0 mod p */
 304 static const felem zero107 = {two107m43m11, two107, two107m43p11, two107m43p11};
 305
 306 /* An alternative felem_diff for larger inputs |in|
 307  * felem_diff_zero107 subtracts |in| from |out|
 308  * On entry:
 309  *   in[i] < 2^106
 310  * On exit:
 311  *   out[i] < out[i] + 2^107
 312  */
 313 static void
 314 felem_diff_zero107(felem out, const felem in)
 315 {
 316         /* In order to prevent underflow, we add 0 mod p before subtracting. */
 317         out[0] += zero107[0];
 318         out[1] += zero107[1];
 319         out[2] += zero107[2];
 320         out[3] += zero107[3];
 321
 322         out[0] -= in[0];
 323         out[1] -= in[1];
 324         out[2] -= in[2];
 325         out[3] -= in[3];
 326 }
 327
 328 /* longfelem_diff subtracts |in| from |out|
 329  * On entry:
 330  *   in[i] < 7*2^67
 331  * On exit:
 332  *   out[i] < out[i] + 2^70 + 2^40
 333  */
 334 static void
 335 longfelem_diff(longfelem out, const longfelem in)
 336 {
 337         static const limb two70m8p6 = (((limb) 1) << 70) - (((limb) 1) << 8) + (((limb) 1) << 6);
 338         static const limb two70p40 = (((limb) 1) << 70) + (((limb) 1) << 40);
 339         static const limb two70 = (((limb) 1) << 70);
 340         static const limb two70m40m38p6 = (((limb) 1) << 70) - (((limb) 1) << 40) - (((limb) 1) << 38) + (((limb) 1) << 6);
 341         static const limb two70m6 = (((limb) 1) << 70) - (((limb) 1) << 6);
 342
 343         /* add 0 mod p to avoid underflow */
 344         out[0] += two70m8p6;
 345         out[1] += two70p40;
 346         out[2] += two70;
 347         out[3] += two70m40m38p6;
 348         out[4] += two70m6;
 349         out[5] += two70m6;
 350         out[6] += two70m6;
 351         out[7] += two70m6;
 352
 353         /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
 354         out[0] -= in[0];
 355         out[1] -= in[1];
 356         out[2] -= in[2];
 357         out[3] -= in[3];
 358         out[4] -= in[4];
 359         out[5] -= in[5];
 360         out[6] -= in[6];
 361         out[7] -= in[7];
 362 }
 363
 364 #define two64m0 (((limb)1) << 64) - 1
 365 #define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
 366 #define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
 367 #define two64m32 (((limb)1) << 64) - (((limb)1) << 32)
 368
 369 /* zero110 is 0 mod p */
 370 static const felem zero110 = {two64m0, two110p32m0, two64m46, two64m32};
 371
 372 /* felem_shrink converts an felem into a smallfelem. The result isn't quite
 373  * minimal as the value may be greater than p.
 374  *
 375  * On entry:
 376  *   in[i] < 2^109
 377  * On exit:
 378  *   out[i] < 2^64
 379  */
 380 static void
 381 felem_shrink(smallfelem out, const felem in)
 382 {
 383         felem tmp;
 384         u64 a, b, mask;
 385         s64 high, low;
 386         static const u64 kPrime3Test = 0x7fffffff00000001ul;    /* 2^63 - 2^32 + 1 */
 387
 388         /* Carry 2->3 */
 389         tmp[3] = zero110[3] + in[3] + ((u64) (in[2] >> 64));
 390         /* tmp[3] < 2^110 */
 391
 392         tmp[2] = zero110[2] + (u64) in[2];
 393         tmp[0] = zero110[0] + in[0];
 394         tmp[1] = zero110[1] + in[1];
 395         /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */
 396
 397         /*
 398          * We perform two partial reductions where we eliminate the high-word
 399          * of tmp[3]. We don't update the other words till the end.
 400          */
 401         a = tmp[3] >> 64;       /* a < 2^46 */
 402         tmp[3] = (u64) tmp[3];
 403         tmp[3] -= a;
 404         tmp[3] += ((limb) a) << 32;
 405         /* tmp[3] < 2^79 */
 406
 407         b = a;
 408         a = tmp[3] >> 64;       /* a < 2^15 */
 409         b += a;                 /* b < 2^46 + 2^15 < 2^47 */
 410         tmp[3] = (u64) tmp[3];
 411         tmp[3] -= a;
 412         tmp[3] += ((limb) a) << 32;
 413         /* tmp[3] < 2^64 + 2^47 */
 414
 415         /*
 416          * This adjusts the other two words to complete the two partial
 417          * reductions.
 418          */
 419         tmp[0] += b;
 420         tmp[1] -= (((limb) b) << 32);
 421
 422         /*
 423          * In order to make space in tmp[3] for the carry from 2 -> 3, we
 424          * conditionally subtract kPrime if tmp[3] is large enough.
 425          */
 426         high = tmp[3] >> 64;
 427         /* As tmp[3] < 2^65, high is either 1 or 0 */
 428         high <<= 63;
 429         high >>= 63;
 430         /*
 431          * high is: all ones   if the high word of tmp[3] is 1 all zeros  if
 432          * the high word of tmp[3] if 0
 433          */
 434         low = tmp[3];
 435         mask = low >> 63;
 436         /*
 437          * mask is: all ones   if the MSB of low is 1 all zeros  if the MSB
 438          * of low if 0
 439          */
 440         low &= bottom63bits;
 441         low -= kPrime3Test;
 442         /* if low was greater than kPrime3Test then the MSB is zero */
 443         low = ~low;
 444         low >>= 63;
 445         /*
 446          * low is: all ones   if low was > kPrime3Test all zeros  if low was
 447          * <= kPrime3Test
 448          */
 449         mask = (mask & low) | high;
 450         tmp[0] -= mask & kPrime[0];
 451         tmp[1] -= mask & kPrime[1];
 452         /* kPrime[2] is zero, so omitted */
 453         tmp[3] -= mask & kPrime[3];
 454         /* tmp[3] < 2**64 - 2**32 + 1 */
 455
 456         tmp[1] += ((u64) (tmp[0] >> 64));
 457         tmp[0] = (u64) tmp[0];
 458         tmp[2] += ((u64) (tmp[1] >> 64));
 459         tmp[1] = (u64) tmp[1];
 460         tmp[3] += ((u64) (tmp[2] >> 64));
 461         tmp[2] = (u64) tmp[2];
 462         /* tmp[i] < 2^64 */
 463
 464         out[0] = tmp[0];
 465         out[1] = tmp[1];
 466         out[2] = tmp[2];
 467         out[3] = tmp[3];
 468 }
 469
 470 /* smallfelem_expand converts a smallfelem to an felem */
 471 static void
 472 smallfelem_expand(felem out, const smallfelem in)
 473 {
 474         out[0] = in[0];
 475         out[1] = in[1];
 476         out[2] = in[2];
 477         out[3] = in[3];
 478 }
 479
 480 /* smallfelem_square sets |out| = |small|^2
 481  * On entry:
 482  *   small[i] < 2^64
 483  * On exit:
 484  *   out[i] < 7 * 2^64 < 2^67
 485  */
 486 static void
 487 smallfelem_square(longfelem out, const smallfelem small)
 488 {
 489         limb a;
 490         u64 high, low;
 491
 492         a = ((uint128_t) small[0]) * small[0];
 493         low = a;
 494         high = a >> 64;
 495         out[0] = low;
 496         out[1] = high;
 497
 498         a = ((uint128_t) small[0]) * small[1];
 499         low = a;
 500         high = a >> 64;
 501         out[1] += low;
 502         out[1] += low;
 503         out[2] = high;
 504
 505         a = ((uint128_t) small[0]) * small[2];
 506         low = a;
 507         high = a >> 64;
 508         out[2] += low;
 509         out[2] *= 2;
 510         out[3] = high;
 511
 512         a = ((uint128_t) small[0]) * small[3];
 513         low = a;
 514         high = a >> 64;
 515         out[3] += low;
 516         out[4] = high;
 517
 518         a = ((uint128_t) small[1]) * small[2];
 519         low = a;
 520         high = a >> 64;
 521         out[3] += low;
 522         out[3] *= 2;
 523         out[4] += high;
 524
 525         a = ((uint128_t) small[1]) * small[1];
 526         low = a;
 527         high = a >> 64;
 528         out[2] += low;
 529         out[3] += high;
 530
 531         a = ((uint128_t) small[1]) * small[3];
 532         low = a;
 533         high = a >> 64;
 534         out[4] += low;
 535         out[4] *= 2;
 536         out[5] = high;
 537
 538         a = ((uint128_t) small[2]) * small[3];
 539         low = a;
 540         high = a >> 64;
 541         out[5] += low;
 542         out[5] *= 2;
 543         out[6] = high;
 544         out[6] += high;
 545
 546         a = ((uint128_t) small[2]) * small[2];
 547         low = a;
 548         high = a >> 64;
 549         out[4] += low;
 550         out[5] += high;
 551
 552         a = ((uint128_t) small[3]) * small[3];
 553         low = a;
 554         high = a >> 64;
 555         out[6] += low;
 556         out[7] = high;
 557 }
 558
 559 /* felem_square sets |out| = |in|^2
 560  * On entry:
 561  *   in[i] < 2^109
 562  * On exit:
 563  *   out[i] < 7 * 2^64 < 2^67
 564  */
 565 static void
 566 felem_square(longfelem out, const felem in)
 567 {
 568         u64 small[4];
 569         felem_shrink(small, in);
 570         smallfelem_square(out, small);
 571 }
 572
 573 /* smallfelem_mul sets |out| = |small1| * |small2|
 574  * On entry:
 575  *   small1[i] < 2^64
 576  *   small2[i] < 2^64
 577  * On exit:
 578  *   out[i] < 7 * 2^64 < 2^67
 579  */
 580 static void
 581 smallfelem_mul(longfelem out, const smallfelem small1, const smallfelem small2)
 582 {
 583         limb a;
 584         u64 high, low;
 585
 586         a = ((uint128_t) small1[0]) * small2[0];
 587         low = a;
 588         high = a >> 64;
 589         out[0] = low;
 590         out[1] = high;
 591
 592
 593         a = ((uint128_t) small1[0]) * small2[1];
 594         low = a;
 595         high = a >> 64;
 596         out[1] += low;
 597         out[2] = high;
 598
 599         a = ((uint128_t) small1[1]) * small2[0];
 600         low = a;
 601         high = a >> 64;
 602         out[1] += low;
 603         out[2] += high;
 604
 605
 606         a = ((uint128_t) small1[0]) * small2[2];
 607         low = a;
 608         high = a >> 64;
 609         out[2] += low;
 610         out[3] = high;
 611
 612         a = ((uint128_t) small1[1]) * small2[1];
 613         low = a;
 614         high = a >> 64;
 615         out[2] += low;
 616         out[3] += high;
 617
 618         a = ((uint128_t) small1[2]) * small2[0];
 619         low = a;
 620         high = a >> 64;
 621         out[2] += low;
 622         out[3] += high;
 623
 624
 625         a = ((uint128_t) small1[0]) * small2[3];
 626         low = a;
 627         high = a >> 64;
 628         out[3] += low;
 629         out[4] = high;
 630
 631         a = ((uint128_t) small1[1]) * small2[2];
 632         low = a;
 633         high = a >> 64;
 634         out[3] += low;
 635         out[4] += high;
 636
 637         a = ((uint128_t) small1[2]) * small2[1];
 638         low = a;
 639         high = a >> 64;
 640         out[3] += low;
 641         out[4] += high;
 642
 643         a = ((uint128_t) small1[3]) * small2[0];
 644         low = a;
 645         high = a >> 64;
 646         out[3] += low;
 647         out[4] += high;
 648
 649
 650         a = ((uint128_t) small1[1]) * small2[3];
 651         low = a;
 652         high = a >> 64;
 653         out[4] += low;
 654         out[5] = high;
 655
 656         a = ((uint128_t) small1[2]) * small2[2];
 657         low = a;
 658         high = a >> 64;
 659         out[4] += low;
 660         out[5] += high;
 661
 662         a = ((uint128_t) small1[3]) * small2[1];
 663         low = a;
 664         high = a >> 64;
 665         out[4] += low;
 666         out[5] += high;
 667
 668
 669         a = ((uint128_t) small1[2]) * small2[3];
 670         low = a;
 671         high = a >> 64;
 672         out[5] += low;
 673         out[6] = high;
 674
 675         a = ((uint128_t) small1[3]) * small2[2];
 676         low = a;
 677         high = a >> 64;
 678         out[5] += low;
 679         out[6] += high;
 680
 681
 682         a = ((uint128_t) small1[3]) * small2[3];
 683         low = a;
 684         high = a >> 64;
 685         out[6] += low;
 686         out[7] = high;
 687 }
 688
 689 /* felem_mul sets |out| = |in1| * |in2|
 690  * On entry:
 691  *   in1[i] < 2^109
 692  *   in2[i] < 2^109
 693  * On exit:
 694  *   out[i] < 7 * 2^64 < 2^67
 695  */
 696 static void
 697 felem_mul(longfelem out, const felem in1, const felem in2)
 698 {
 699         smallfelem small1, small2;
 700         felem_shrink(small1, in1);
 701         felem_shrink(small2, in2);
 702         smallfelem_mul(out, small1, small2);
 703 }
 704
 705 /* felem_small_mul sets |out| = |small1| * |in2|
 706  * On entry:
 707  *   small1[i] < 2^64
 708  *   in2[i] < 2^109
 709  * On exit:
 710  *   out[i] < 7 * 2^64 < 2^67
 711  */
 712 static void
 713 felem_small_mul(longfelem out, const smallfelem small1, const felem in2)
 714 {
 715         smallfelem small2;
 716         felem_shrink(small2, in2);
 717         smallfelem_mul(out, small1, small2);
 718 }
 719
 720 #define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
 721 #define two100 (((limb)1) << 100)
 722 #define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
 723 /* zero100 is 0 mod p */
 724 static const felem zero100 = {two100m36m4, two100, two100m36p4, two100m36p4};
 725
 726 /* Internal function for the different flavours of felem_reduce.
 727  * felem_reduce_ reduces the higher coefficients in[4]-in[7].
 728  * On entry:
 729  *   out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
 730  *   out[1] >= in[7] + 2^32*in[4]
 731  *   out[2] >= in[5] + 2^32*in[5]
 732  *   out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
 733  * On exit:
 734  *   out[0] <= out[0] + in[4] + 2^32*in[5]
 735  *   out[1] <= out[1] + in[5] + 2^33*in[6]
 736  *   out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
 737  *   out[3] <= out[3] + 2^32*in[4] + 3*in[7]
 738  */
 739 static void
 740 felem_reduce_(felem out, const longfelem in)
 741 {
 742         int128_t c;
 743         /* combine common terms from below */
 744         c = in[4] + (in[5] << 32);
 745         out[0] += c;
 746         out[3] -= c;
 747
 748         c = in[5] - in[7];
 749         out[1] += c;
 750         out[2] -= c;
 751
 752         /* the remaining terms */
 753         /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
 754         out[1] -= (in[4] << 32);
 755         out[3] += (in[4] << 32);
 756
 757         /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
 758         out[2] -= (in[5] << 32);
 759
 760         /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
 761         out[0] -= in[6];
 762         out[0] -= (in[6] << 32);
 763         out[1] += (in[6] << 33);
 764         out[2] += (in[6] * 2);
 765         out[3] -= (in[6] << 32);
 766
 767         /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
 768         out[0] -= in[7];
 769         out[0] -= (in[7] << 32);
 770         out[2] += (in[7] << 33);
 771         out[3] += (in[7] * 3);
 772 }
 773
 774 /* felem_reduce converts a longfelem into an felem.
 775  * To be called directly after felem_square or felem_mul.
 776  * On entry:
 777  *   in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
 778  *   in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
 779  * On exit:
 780  *   out[i] < 2^101
 781  */
 782 static void
 783 felem_reduce(felem out, const longfelem in)
 784 {
 785         out[0] = zero100[0] + in[0];
 786         out[1] = zero100[1] + in[1];
 787         out[2] = zero100[2] + in[2];
 788         out[3] = zero100[3] + in[3];
 789
 790         felem_reduce_(out, in);
 791
 792         /*
 793          * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
 794          * out[1] > 2^100 - 2^64 - 7*2^96 > 0 out[2] > 2^100 - 2^36 + 2^4 -
 795          * 5*2^64 - 5*2^96 > 0 out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96
 796          * - 3*2^96 > 0
 797          *
 798          * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101 out[1] < 2^100 +
 799          * 3*2^64 + 5*2^64 + 3*2^97 < 2^101 out[2] < 2^100 + 5*2^64 + 2^64 +
 800          * 3*2^65 + 2^97 < 2^101 out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 <
 801          * 2^101
 802          */
 803 }
 804
 805 /* felem_reduce_zero105 converts a larger longfelem into an felem.
 806  * On entry:
 807  *   in[0] < 2^71
 808  * On exit:
 809  *   out[i] < 2^106
 810  */
 811 static void
 812 felem_reduce_zero105(felem out, const longfelem in)
 813 {
 814         out[0] = zero105[0] + in[0];
 815         out[1] = zero105[1] + in[1];
 816         out[2] = zero105[2] + in[2];
 817         out[3] = zero105[3] + in[3];
 818
 819         felem_reduce_(out, in);
 820
 821         /*
 822          * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
 823          * out[1] > 2^105 - 2^71 - 2^103 > 0 out[2] > 2^105 - 2^41 + 2^9 -
 824          * 2^71 - 2^103 > 0 out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 -
 825          * 2^103 > 0
 826          *
 827          * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 out[1] < 2^105 + 2^71 +
 828          * 2^71 + 2^103 < 2^106 out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 <
 829          * 2^106 out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
 830          */
 831 }
 832
 833 /* subtract_u64 sets *result = *result - v and *carry to one if the subtraction
 834  * underflowed. */
 835 static void
 836 subtract_u64(u64 * result, u64 * carry, u64 v)
 837 {
 838         uint128_t r = *result;
 839         r -= v;
 840         *carry = (r >> 64) & 1;
 841         *result = (u64) r;
 842 }
 843
 844 /* felem_contract converts |in| to its unique, minimal representation.
 845  * On entry:
 846  *   in[i] < 2^109
 847  */
 848 static void
 849 felem_contract(smallfelem out, const felem in)
 850 {
 851         unsigned i;
 852         u64 all_equal_so_far = 0, result = 0, carry;
 853
 854         felem_shrink(out, in);
 855         /* small is minimal except that the value might be > p */
 856
 857         all_equal_so_far--;
 858         /*
 859          * We are doing a constant time test if out >= kPrime. We need to
 860          * compare each u64, from most-significant to least significant. For
 861          * each one, if all words so far have been equal (m is all ones) then
 862          * a non-equal result is the answer. Otherwise we continue.
 863          */
 864         for (i = 3; i < 4; i--) {
 865                 u64 equal;
 866                 uint128_t a = ((uint128_t) kPrime[i]) - out[i];
 867                 /*
 868                  * if out[i] > kPrime[i] then a will underflow and the high
 869                  * 64-bits will all be set.
 870                  */
 871                 result |= all_equal_so_far & ((u64) (a >> 64));
 872
 873                 /*
 874                  * if kPrime[i] == out[i] then |equal| will be all zeros and
 875                  * the decrement will make it all ones.
 876                  */
 877                 equal = kPrime[i] ^ out[i];
 878                 equal--;
 879                 equal &= equal << 32;
 880                 equal &= equal << 16;
 881                 equal &= equal << 8;
 882                 equal &= equal << 4;
 883                 equal &= equal << 2;
 884                 equal &= equal << 1;
 885                 equal = ((s64) equal) >> 63;
 886
 887                 all_equal_so_far &= equal;
 888         }
 889
 890         /*
 891          * if all_equal_so_far is still all ones then the two values are
 892          * equal and so out >= kPrime is true.
 893          */
 894         result |= all_equal_so_far;
 895
 896         /* if out >= kPrime then we subtract kPrime. */
 897         subtract_u64(&out[0], &carry, result & kPrime[0]);
 898         subtract_u64(&out[1], &carry, carry);
 899         subtract_u64(&out[2], &carry, carry);
 900         subtract_u64(&out[3], &carry, carry);
 901
 902         subtract_u64(&out[1], &carry, result & kPrime[1]);
 903         subtract_u64(&out[2], &carry, carry);
 904         subtract_u64(&out[3], &carry, carry);
 905
 906         subtract_u64(&out[2], &carry, result & kPrime[2]);
 907         subtract_u64(&out[3], &carry, carry);
 908
 909         subtract_u64(&out[3], &carry, result & kPrime[3]);
 910 }
 911
 912 static void
 913 smallfelem_square_contract(smallfelem out, const smallfelem in)
 914 {
 915         longfelem longtmp;
 916         felem tmp;
 917
 918         smallfelem_square(longtmp, in);
 919         felem_reduce(tmp, longtmp);
 920         felem_contract(out, tmp);
 921 }
 922
 923 static void
 924 smallfelem_mul_contract(smallfelem out, const smallfelem in1, const smallfelem in2)
 925 {
 926         longfelem longtmp;
 927         felem tmp;
 928
 929         smallfelem_mul(longtmp, in1, in2);
 930         felem_reduce(tmp, longtmp);
 931         felem_contract(out, tmp);
 932 }
 933
 934 /* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
 935  * otherwise.
 936  * On entry:
 937  *   small[i] < 2^64
 938  */
 939 static limb
 940 smallfelem_is_zero(const smallfelem small)
 941 {
 942         limb result;
 943         u64 is_p;
 944
 945         u64 is_zero = small[0] | small[1] | small[2] | small[3];
 946         is_zero--;
 947         is_zero &= is_zero << 32;
 948         is_zero &= is_zero << 16;
 949         is_zero &= is_zero << 8;
 950         is_zero &= is_zero << 4;
 951         is_zero &= is_zero << 2;
 952         is_zero &= is_zero << 1;
 953         is_zero = ((s64) is_zero) >> 63;
 954
 955         is_p = (small[0] ^ kPrime[0]) |
 956             (small[1] ^ kPrime[1]) |
 957             (small[2] ^ kPrime[2]) |
 958             (small[3] ^ kPrime[3]);
 959         is_p--;
 960         is_p &= is_p << 32;
 961         is_p &= is_p << 16;
 962         is_p &= is_p << 8;
 963         is_p &= is_p << 4;
 964         is_p &= is_p << 2;
 965         is_p &= is_p << 1;
 966         is_p = ((s64) is_p) >> 63;
 967
 968         is_zero |= is_p;
 969
 970         result = is_zero;
 971         result |= ((limb) is_zero) << 64;
 972         return result;
 973 }
 974
 975 static int
 976 smallfelem_is_zero_int(const smallfelem small)
 977 {
 978         return (int) (smallfelem_is_zero(small) & ((limb) 1));
 979 }
 980
 981 /* felem_inv calculates |out| = |in|^{-1}
 982  *
 983  * Based on Fermat's Little Theorem:
 984  *   a^p = a (mod p)
 985  *   a^{p-1} = 1 (mod p)
 986  *   a^{p-2} = a^{-1} (mod p)
 987  */
 988 static void
 989 felem_inv(felem out, const felem in)
 990 {
 991         felem ftmp, ftmp2;
 992         /* each e_I will hold |in|^{2^I - 1} */
 993         felem e2, e4, e8, e16, e32, e64;
 994         longfelem tmp;
 995         unsigned i;
 996
 997         felem_square(tmp, in);
 998         felem_reduce(ftmp, tmp);/* 2^1 */
 999         felem_mul(tmp, in, ftmp);
1000         felem_reduce(ftmp, tmp);/* 2^2 - 2^0 */
1001         felem_assign(e2, ftmp);
1002         felem_square(tmp, ftmp);
1003         felem_reduce(ftmp, tmp);/* 2^3 - 2^1 */
1004         felem_square(tmp, ftmp);
1005         felem_reduce(ftmp, tmp);/* 2^4 - 2^2 */
1006         felem_mul(tmp, ftmp, e2);
1007         felem_reduce(ftmp, tmp);/* 2^4 - 2^0 */
1008         felem_assign(e4, ftmp);
1009         felem_square(tmp, ftmp);
1010         felem_reduce(ftmp, tmp);/* 2^5 - 2^1 */
1011         felem_square(tmp, ftmp);
1012         felem_reduce(ftmp, tmp);/* 2^6 - 2^2 */
1013         felem_square(tmp, ftmp);
1014         felem_reduce(ftmp, tmp);/* 2^7 - 2^3 */
1015         felem_square(tmp, ftmp);
1016         felem_reduce(ftmp, tmp);/* 2^8 - 2^4 */
1017         felem_mul(tmp, ftmp, e4);
1018         felem_reduce(ftmp, tmp);/* 2^8 - 2^0 */
1019         felem_assign(e8, ftmp);
1020         for (i = 0; i < 8; i++) {
1021                 felem_square(tmp, ftmp);
1022                 felem_reduce(ftmp, tmp);
1023         }                       /* 2^16 - 2^8 */
1024         felem_mul(tmp, ftmp, e8);
1025         felem_reduce(ftmp, tmp);/* 2^16 - 2^0 */
1026         felem_assign(e16, ftmp);
1027         for (i = 0; i < 16; i++) {
1028                 felem_square(tmp, ftmp);
1029                 felem_reduce(ftmp, tmp);
1030         }                       /* 2^32 - 2^16 */
1031         felem_mul(tmp, ftmp, e16);
1032         felem_reduce(ftmp, tmp);/* 2^32 - 2^0 */
1033         felem_assign(e32, ftmp);
1034         for (i = 0; i < 32; i++) {
1035                 felem_square(tmp, ftmp);
1036                 felem_reduce(ftmp, tmp);
1037         }                       /* 2^64 - 2^32 */
1038         felem_assign(e64, ftmp);
1039         felem_mul(tmp, ftmp, in);
1040         felem_reduce(ftmp, tmp);/* 2^64 - 2^32 + 2^0 */
1041         for (i = 0; i < 192; i++) {
1042                 felem_square(tmp, ftmp);
1043                 felem_reduce(ftmp, tmp);
1044         }                       /* 2^256 - 2^224 + 2^192 */
1045
1046         felem_mul(tmp, e64, e32);
1047         felem_reduce(ftmp2, tmp);       /* 2^64 - 2^0 */
1048         for (i = 0; i < 16; i++) {
1049                 felem_square(tmp, ftmp2);
1050                 felem_reduce(ftmp2, tmp);
1051         }                       /* 2^80 - 2^16 */
1052         felem_mul(tmp, ftmp2, e16);
1053         felem_reduce(ftmp2, tmp);       /* 2^80 - 2^0 */
1054         for (i = 0; i < 8; i++) {
1055                 felem_square(tmp, ftmp2);
1056                 felem_reduce(ftmp2, tmp);
1057         }                       /* 2^88 - 2^8 */
1058         felem_mul(tmp, ftmp2, e8);
1059         felem_reduce(ftmp2, tmp);       /* 2^88 - 2^0 */
1060         for (i = 0; i < 4; i++) {
1061                 felem_square(tmp, ftmp2);
1062                 felem_reduce(ftmp2, tmp);
1063         }                       /* 2^92 - 2^4 */
1064         felem_mul(tmp, ftmp2, e4);
1065         felem_reduce(ftmp2, tmp);       /* 2^92 - 2^0 */
1066         felem_square(tmp, ftmp2);
1067         felem_reduce(ftmp2, tmp);       /* 2^93 - 2^1 */
1068         felem_square(tmp, ftmp2);
1069         felem_reduce(ftmp2, tmp);       /* 2^94 - 2^2 */
1070         felem_mul(tmp, ftmp2, e2);
1071         felem_reduce(ftmp2, tmp);       /* 2^94 - 2^0 */
1072         felem_square(tmp, ftmp2);
1073         felem_reduce(ftmp2, tmp);       /* 2^95 - 2^1 */
1074         felem_square(tmp, ftmp2);
1075         felem_reduce(ftmp2, tmp);       /* 2^96 - 2^2 */
1076         felem_mul(tmp, ftmp2, in);
1077         felem_reduce(ftmp2, tmp);       /* 2^96 - 3 */
1078
1079         felem_mul(tmp, ftmp2, ftmp);
1080         felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
1081 }
1082
1083 static void
1084 smallfelem_inv_contract(smallfelem out, const smallfelem in)
1085 {
1086         felem tmp;
1087
1088         smallfelem_expand(tmp, in);
1089         felem_inv(tmp, tmp);
1090         felem_contract(out, tmp);
1091 }
1092
1093 /* Group operations
1094  * ----------------
1095  *
1096  * Building on top of the field operations we have the operations on the
1097  * elliptic curve group itself. Points on the curve are represented in Jacobian
1098  * coordinates */
1099
1100 /* point_double calculates 2*(x_in, y_in, z_in)
1101  *
1102  * The method is taken from:
1103  *   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
1104  *
1105  * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
1106  * while x_out == y_in is not (maybe this works, but it's not tested). */
1107 static void
1108 point_double(felem x_out, felem y_out, felem z_out,
1109     const felem x_in, const felem y_in, const felem z_in)
1110 {
1111         longfelem tmp, tmp2;
1112         felem delta, gamma, beta, alpha, ftmp, ftmp2;
1113         smallfelem small1, small2;
1114
1115         felem_assign(ftmp, x_in);
1116         /* ftmp[i] < 2^106 */
1117         felem_assign(ftmp2, x_in);
1118         /* ftmp2[i] < 2^106 */
1119
1120         /* delta = z^2 */
1121         felem_square(tmp, z_in);
1122         felem_reduce(delta, tmp);
1123         /* delta[i] < 2^101 */
1124
1125         /* gamma = y^2 */
1126         felem_square(tmp, y_in);
1127         felem_reduce(gamma, tmp);
1128         /* gamma[i] < 2^101 */
1129         felem_shrink(small1, gamma);
1130
1131         /* beta = x*gamma */
1132         felem_small_mul(tmp, small1, x_in);
1133         felem_reduce(beta, tmp);
1134         /* beta[i] < 2^101 */
1135
1136         /* alpha = 3*(x-delta)*(x+delta) */
1137         felem_diff(ftmp, delta);
1138         /* ftmp[i] < 2^105 + 2^106 < 2^107 */
1139         felem_sum(ftmp2, delta);
1140         /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
1141         felem_scalar(ftmp2, 3);
1142         /* ftmp2[i] < 3 * 2^107 < 2^109 */
1143         felem_mul(tmp, ftmp, ftmp2);
1144         felem_reduce(alpha, tmp);
1145         /* alpha[i] < 2^101 */
1146         felem_shrink(small2, alpha);
1147
1148         /* x' = alpha^2 - 8*beta */
1149         smallfelem_square(tmp, small2);
1150         felem_reduce(x_out, tmp);
1151         felem_assign(ftmp, beta);
1152         felem_scalar(ftmp, 8);
1153         /* ftmp[i] < 8 * 2^101 = 2^104 */
1154         felem_diff(x_out, ftmp);
1155         /* x_out[i] < 2^105 + 2^101 < 2^106 */
1156
1157         /* z' = (y + z)^2 - gamma - delta */
1158         felem_sum(delta, gamma);
1159         /* delta[i] < 2^101 + 2^101 = 2^102 */
1160         felem_assign(ftmp, y_in);
1161         felem_sum(ftmp, z_in);
1162         /* ftmp[i] < 2^106 + 2^106 = 2^107 */
1163         felem_square(tmp, ftmp);
1164         felem_reduce(z_out, tmp);
1165         felem_diff(z_out, delta);
1166         /* z_out[i] < 2^105 + 2^101 < 2^106 */
1167
1168         /* y' = alpha*(4*beta - x') - 8*gamma^2 */
1169         felem_scalar(beta, 4);
1170         /* beta[i] < 4 * 2^101 = 2^103 */
1171         felem_diff_zero107(beta, x_out);
1172         /* beta[i] < 2^107 + 2^103 < 2^108 */
1173         felem_small_mul(tmp, small2, beta);
1174         /* tmp[i] < 7 * 2^64 < 2^67 */
1175         smallfelem_square(tmp2, small1);
1176         /* tmp2[i] < 7 * 2^64 */
1177         longfelem_scalar(tmp2, 8);
1178         /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
1179         longfelem_diff(tmp, tmp2);
1180         /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1181         felem_reduce_zero105(y_out, tmp);
1182         /* y_out[i] < 2^106 */
1183 }
1184
1185 /* point_double_small is the same as point_double, except that it operates on
1186  * smallfelems */
1187 static void
1188 point_double_small(smallfelem x_out, smallfelem y_out, smallfelem z_out,
1189     const smallfelem x_in, const smallfelem y_in, const smallfelem z_in)
1190 {
1191         felem felem_x_out, felem_y_out, felem_z_out;
1192         felem felem_x_in, felem_y_in, felem_z_in;
1193
1194         smallfelem_expand(felem_x_in, x_in);
1195         smallfelem_expand(felem_y_in, y_in);
1196         smallfelem_expand(felem_z_in, z_in);
1197         point_double(felem_x_out, felem_y_out, felem_z_out,
1198             felem_x_in, felem_y_in, felem_z_in);
1199         felem_shrink(x_out, felem_x_out);
1200         felem_shrink(y_out, felem_y_out);
1201         felem_shrink(z_out, felem_z_out);
1202 }
1203
1204 /* copy_conditional copies in to out iff mask is all ones. */
1205 static void
1206 copy_conditional(felem out, const felem in, limb mask)
1207 {
1208         unsigned i;
1209         for (i = 0; i < NLIMBS; ++i) {
1210                 const limb tmp = mask & (in[i] ^ out[i]);
1211                 out[i] ^= tmp;
1212         }
1213 }
1214
1215 /* copy_small_conditional copies in to out iff mask is all ones. */
1216 static void
1217 copy_small_conditional(felem out, const smallfelem in, limb mask)
1218 {
1219         unsigned i;
1220         const u64 mask64 = mask;
1221         for (i = 0; i < NLIMBS; ++i) {
1222                 out[i] = ((limb) (in[i] & mask64)) | (out[i] & ~mask);
1223         }
1224 }
1225
1226 /* point_add calcuates (x1, y1, z1) + (x2, y2, z2)
1227  *
1228  * The method is taken from:
1229  *   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
1230  * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
1231  *
1232  * This function includes a branch for checking whether the two input points
1233  * are equal, (while not equal to the point at infinity). This case never
1234  * happens during single point multiplication, so there is no timing leak for
1235  * ECDH or ECDSA signing. */
1236 static void
1237 point_add(felem x3, felem y3, felem z3,
1238     const felem x1, const felem y1, const felem z1,
1239     const int mixed, const smallfelem x2, const smallfelem y2, const smallfelem z2)
1240 {
1241         felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
1242         longfelem tmp, tmp2;
1243         smallfelem small1, small2, small3, small4, small5;
1244         limb x_equal, y_equal, z1_is_zero, z2_is_zero;
1245
1246         felem_shrink(small3, z1);
1247
1248         z1_is_zero = smallfelem_is_zero(small3);
1249         z2_is_zero = smallfelem_is_zero(z2);
1250
1251         /* ftmp = z1z1 = z1**2 */
1252         smallfelem_square(tmp, small3);
1253         felem_reduce(ftmp, tmp);
1254         /* ftmp[i] < 2^101 */
1255         felem_shrink(small1, ftmp);
1256
1257         if (!mixed) {
1258                 /* ftmp2 = z2z2 = z2**2 */
1259                 smallfelem_square(tmp, z2);
1260                 felem_reduce(ftmp2, tmp);
1261                 /* ftmp2[i] < 2^101 */
1262                 felem_shrink(small2, ftmp2);
1263
1264                 felem_shrink(small5, x1);
1265
1266                 /* u1 = ftmp3 = x1*z2z2 */
1267                 smallfelem_mul(tmp, small5, small2);
1268                 felem_reduce(ftmp3, tmp);
1269                 /* ftmp3[i] < 2^101 */
1270
1271                 /* ftmp5 = z1 + z2 */
1272                 felem_assign(ftmp5, z1);
1273                 felem_small_sum(ftmp5, z2);
1274                 /* ftmp5[i] < 2^107 */
1275
1276                 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
1277                 felem_square(tmp, ftmp5);
1278                 felem_reduce(ftmp5, tmp);
1279                 /* ftmp2 = z2z2 + z1z1 */
1280                 felem_sum(ftmp2, ftmp);
1281                 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
1282                 felem_diff(ftmp5, ftmp2);
1283                 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */
1284
1285                 /* ftmp2 = z2 * z2z2 */
1286                 smallfelem_mul(tmp, small2, z2);
1287                 felem_reduce(ftmp2, tmp);
1288
1289                 /* s1 = ftmp2 = y1 * z2**3 */
1290                 felem_mul(tmp, y1, ftmp2);
1291                 felem_reduce(ftmp6, tmp);
1292                 /* ftmp6[i] < 2^101 */
1293         } else {
1294                 /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */
1295
1296                 /* u1 = ftmp3 = x1*z2z2 */
1297                 felem_assign(ftmp3, x1);
1298                 /* ftmp3[i] < 2^106 */
1299
1300                 /* ftmp5 = 2z1z2 */
1301                 felem_assign(ftmp5, z1);
1302                 felem_scalar(ftmp5, 2);
1303                 /* ftmp5[i] < 2*2^106 = 2^107 */
1304
1305                 /* s1 = ftmp2 = y1 * z2**3 */
1306                 felem_assign(ftmp6, y1);
1307                 /* ftmp6[i] < 2^106 */
1308         }
1309
1310         /* u2 = x2*z1z1 */
1311         smallfelem_mul(tmp, x2, small1);
1312         felem_reduce(ftmp4, tmp);
1313
1314         /* h = ftmp4 = u2 - u1 */
1315         felem_diff_zero107(ftmp4, ftmp3);
1316         /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
1317         felem_shrink(small4, ftmp4);
1318
1319         x_equal = smallfelem_is_zero(small4);
1320
1321         /* z_out = ftmp5 * h */
1322         felem_small_mul(tmp, small4, ftmp5);
1323         felem_reduce(z_out, tmp);
1324         /* z_out[i] < 2^101 */
1325
1326         /* ftmp = z1 * z1z1 */
1327         smallfelem_mul(tmp, small1, small3);
1328         felem_reduce(ftmp, tmp);
1329
1330         /* s2 = tmp = y2 * z1**3 */
1331         felem_small_mul(tmp, y2, ftmp);
1332         felem_reduce(ftmp5, tmp);
1333
1334         /* r = ftmp5 = (s2 - s1)*2 */
1335         felem_diff_zero107(ftmp5, ftmp6);
1336         /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
1337         felem_scalar(ftmp5, 2);
1338         /* ftmp5[i] < 2^109 */
1339         felem_shrink(small1, ftmp5);
1340         y_equal = smallfelem_is_zero(small1);
1341
1342         if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
1343                 point_double(x3, y3, z3, x1, y1, z1);
1344                 return;
1345         }
1346         /* I = ftmp = (2h)**2 */
1347         felem_assign(ftmp, ftmp4);
1348         felem_scalar(ftmp, 2);
1349         /* ftmp[i] < 2*2^108 = 2^109 */
1350         felem_square(tmp, ftmp);
1351         felem_reduce(ftmp, tmp);
1352
1353         /* J = ftmp2 = h * I */
1354         felem_mul(tmp, ftmp4, ftmp);
1355         felem_reduce(ftmp2, tmp);
1356
1357         /* V = ftmp4 = U1 * I */
1358         felem_mul(tmp, ftmp3, ftmp);
1359         felem_reduce(ftmp4, tmp);
1360
1361         /* x_out = r**2 - J - 2V */
1362         smallfelem_square(tmp, small1);
1363         felem_reduce(x_out, tmp);
1364         felem_assign(ftmp3, ftmp4);
1365         felem_scalar(ftmp4, 2);
1366         felem_sum(ftmp4, ftmp2);
1367         /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
1368         felem_diff(x_out, ftmp4);
1369         /* x_out[i] < 2^105 + 2^101 */
1370
1371         /* y_out = r(V-x_out) - 2 * s1 * J */
1372         felem_diff_zero107(ftmp3, x_out);
1373         /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
1374         felem_small_mul(tmp, small1, ftmp3);
1375         felem_mul(tmp2, ftmp6, ftmp2);
1376         longfelem_scalar(tmp2, 2);
1377         /* tmp2[i] < 2*2^67 = 2^68 */
1378         longfelem_diff(tmp, tmp2);
1379         /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
1380         felem_reduce_zero105(y_out, tmp);
1381         /* y_out[i] < 2^106 */
1382
1383         copy_small_conditional(x_out, x2, z1_is_zero);
1384         copy_conditional(x_out, x1, z2_is_zero);
1385         copy_small_conditional(y_out, y2, z1_is_zero);
1386         copy_conditional(y_out, y1, z2_is_zero);
1387         copy_small_conditional(z_out, z2, z1_is_zero);
1388         copy_conditional(z_out, z1, z2_is_zero);
1389         felem_assign(x3, x_out);
1390         felem_assign(y3, y_out);
1391         felem_assign(z3, z_out);
1392 }
1393
1394 /* point_add_small is the same as point_add, except that it operates on
1395  * smallfelems */
1396 static void
1397 point_add_small(smallfelem x3, smallfelem y3, smallfelem z3,
1398     smallfelem x1, smallfelem y1, smallfelem z1,
1399     smallfelem x2, smallfelem y2, smallfelem z2)
1400 {
1401         felem felem_x3, felem_y3, felem_z3;
1402         felem felem_x1, felem_y1, felem_z1;
1403         smallfelem_expand(felem_x1, x1);
1404         smallfelem_expand(felem_y1, y1);
1405         smallfelem_expand(felem_z1, z1);
1406         point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, x2, y2, z2);
1407         felem_shrink(x3, felem_x3);
1408         felem_shrink(y3, felem_y3);
1409         felem_shrink(z3, felem_z3);
1410 }
1411
1412 /* Base point pre computation
1413  * --------------------------
1414  *
1415  * Two different sorts of precomputed tables are used in the following code.
1416  * Each contain various points on the curve, where each point is three field
1417  * elements (x, y, z).
1418  *
1419  * For the base point table, z is usually 1 (0 for the point at infinity).
1420  * This table has 2 * 16 elements, starting with the following:
1421  * index | bits    | point
1422  * ------+---------+------------------------------
1423  *     0 | 0 0 0 0 | 0G
1424  *     1 | 0 0 0 1 | 1G
1425  *     2 | 0 0 1 0 | 2^64G
1426  *     3 | 0 0 1 1 | (2^64 + 1)G
1427  *     4 | 0 1 0 0 | 2^128G
1428  *     5 | 0 1 0 1 | (2^128 + 1)G
1429  *     6 | 0 1 1 0 | (2^128 + 2^64)G
1430  *     7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
1431  *     8 | 1 0 0 0 | 2^192G
1432  *     9 | 1 0 0 1 | (2^192 + 1)G
1433  *    10 | 1 0 1 0 | (2^192 + 2^64)G
1434  *    11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
1435  *    12 | 1 1 0 0 | (2^192 + 2^128)G
1436  *    13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
1437  *    14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
1438  *    15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
1439  * followed by a copy of this with each element multiplied by 2^32.
1440  *
1441  * The reason for this is so that we can clock bits into four different
1442  * locations when doing simple scalar multiplies against the base point,
1443  * and then another four locations using the second 16 elements.
1444  *
1445  * Tables for other points have table[i] = iG for i in 0 .. 16. */
1446
1447 /* gmul is the table of precomputed base points */
1448 static const smallfelem gmul[2][16][3] =
1449 {{{{0, 0, 0, 0},
1450 {0, 0, 0, 0},
1451 {0, 0, 0, 0}},
1452 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2, 0x6b17d1f2e12c4247},
1453 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16, 0x4fe342e2fe1a7f9b},
1454 {1, 0, 0, 0}},
1455 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de, 0x0fa822bc2811aaa5},
1456 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b, 0xbff44ae8f5dba80d},
1457 {1, 0, 0, 0}},
1458 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789, 0x300a4bbc89d6726f},
1459 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f, 0x72aac7e0d09b4644},
1460 {1, 0, 0, 0}},
1461 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e, 0x447d739beedb5e67},
1462 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7, 0x2d4825ab834131ee},
1463 {1, 0, 0, 0}},
1464 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60, 0xef9519328a9c72ff},
1465 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c, 0x611e9fc37dbb2c9b},
1466 {1, 0, 0, 0}},
1467 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf, 0x550663797b51f5d8},
1468 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5, 0x157164848aecb851},
1469 {1, 0, 0, 0}},
1470 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391, 0xeb5d7745b21141ea},
1471 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee, 0xeafd72ebdbecc17b},
1472 {1, 0, 0, 0}},
1473 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5, 0xa6d39677a7849276},
1474 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf, 0x674f84749b0b8816},
1475 {1, 0, 0, 0}},
1476 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb, 0x4e769e7672c9ddad},
1477 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281, 0x42b99082de830663},
1478 {1, 0, 0, 0}},
1479 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478, 0x78878ef61c6ce04d},
1480 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def, 0xb6cb3f5d7b72c321},
1481 {1, 0, 0, 0}},
1482 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae, 0x0c88bc4d716b1287},
1483 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa, 0xdd5ddea3f3901dc6},
1484 {1, 0, 0, 0}},
1485 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3, 0x68f344af6b317466},
1486 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3, 0x31b9c405f8540a20},
1487 {1, 0, 0, 0}},
1488 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0, 0x4052bf4b6f461db9},
1489 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8, 0xfecf4d5190b0fc61},
1490 {1, 0, 0, 0}},
1491 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a, 0x1eddbae2c802e41a},
1492 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0, 0x43104d86560ebcfc},
1493 {1, 0, 0, 0}},
1494 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a, 0xb48e26b484f7a21c},
1495 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668, 0xfac015404d4d3dab},
1496 {1, 0, 0, 0}}},
1497 {{{0, 0, 0, 0},
1498 {0, 0, 0, 0},
1499 {0, 0, 0, 0}},
1500 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da, 0x7fe36b40af22af89},
1501 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1, 0xe697d45825b63624},
1502 {1, 0, 0, 0}},
1503 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902, 0x4a5b506612a677a6},
1504 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40, 0xeb13461ceac089f1},
1505 {1, 0, 0, 0}},
1506 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857, 0x0781b8291c6a220a},
1507 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434, 0x690cde8df0151593},
1508 {1, 0, 0, 0}},
1509 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326, 0x8a535f566ec73617},
1510 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf, 0x0455c08468b08bd7},
1511 {1, 0, 0, 0}},
1512 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279, 0x06bada7ab77f8276},
1513 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70, 0x5b476dfd0e6cb18a},
1514 {1, 0, 0, 0}},
1515 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8, 0x3e29864e8a2ec908},
1516 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed, 0x239b90ea3dc31e7e},
1517 {1, 0, 0, 0}},
1518 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4, 0x820f4dd949f72ff7},
1519 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3, 0x140406ec783a05ec},
1520 {1, 0, 0, 0}},
1521 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe, 0x68f6b8542783dfee},
1522 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028, 0xcbe1feba92e40ce6},
1523 {1, 0, 0, 0}},
1524 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927, 0xd0b2f94d2f420109},
1525 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a, 0x971459828b0719e5},
1526 {1, 0, 0, 0}},
1527 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687, 0x961610004a866aba},
1528 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c, 0x7acb9fadcee75e44},
1529 {1, 0, 0, 0}},
1530 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea, 0x24eb9acca333bf5b},
1531 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d, 0x69f891c5acd079cc},
1532 {1, 0, 0, 0}},
1533 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514, 0xe51f547c5972a107},
1534 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06, 0x1c309a2b25bb1387},
1535 {1, 0, 0, 0}},
1536 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828, 0x20b87b8aa2c4e503},
1537 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044, 0xf5c6fa49919776be},
1538 {1, 0, 0, 0}},
1539 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56, 0x1ed7d1b9332010b9},
1540 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24, 0x3a2b03f03217257a},
1541 {1, 0, 0, 0}},
1542 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b, 0x15fee545c78dd9f6},
1543 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb, 0x4ab5b6b2b8753f81},
1544 {1, 0, 0, 0}}}};
1545
1546 /* select_point selects the |idx|th point from a precomputation table and
1547  * copies it to out. */
1548 static void
1549 select_point(const u64 idx, unsigned int size, const smallfelem pre_comp[16][3], smallfelem out[3])
1550 {
1551         unsigned i, j;
1552         u64 *outlimbs = &out[0][0];
1553         memset(outlimbs, 0, 3 * sizeof(smallfelem));
1554
1555         for (i = 0; i < size; i++) {
1556                 const u64 *inlimbs = (u64 *) & pre_comp[i][0][0];
1557                 u64 mask = i ^ idx;
1558                 mask |= mask >> 4;
1559                 mask |= mask >> 2;
1560                 mask |= mask >> 1;
1561                 mask &= 1;
1562                 mask--;
1563                 for (j = 0; j < NLIMBS * 3; j++)
1564                         outlimbs[j] |= inlimbs[j] & mask;
1565         }
1566 }
1567
1568 /* get_bit returns the |i|th bit in |in| */
1569 static char
1570 get_bit(const felem_bytearray in, int i)
1571 {
1572         if ((i < 0) || (i >= 256))
1573                 return 0;
1574         return (in[i >> 3] >> (i & 7)) & 1;
1575 }
1576
1577 /* Interleaved point multiplication using precomputed point multiples:
1578  * The small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[],
1579  * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
1580  * of the generator, using certain (large) precomputed multiples in g_pre_comp.
1581  * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
1582 static void
1583 batch_mul(felem x_out, felem y_out, felem z_out,
1584     const felem_bytearray scalars[], const unsigned num_points, const u8 * g_scalar,
1585     const int mixed, const smallfelem pre_comp[][17][3], const smallfelem g_pre_comp[2][16][3])
1586 {
1587         int i, skip;
1588         unsigned num, gen_mul = (g_scalar != NULL);
1589         felem nq[3], ftmp;
1590         smallfelem tmp[3];
1591         u64 bits;
1592         u8 sign, digit;
1593
1594         /* set nq to the point at infinity */
1595         memset(nq, 0, 3 * sizeof(felem));
1596
1597         /*
1598          * Loop over all scalars msb-to-lsb, interleaving additions of
1599          * multiples of the generator (two in each of the last 32 rounds) and
1600          * additions of other points multiples (every 5th round).
1601          */
1602         skip = 1;               /* save two point operations in the first
1603                                  * round */
1604         for (i = (num_points ? 255 : 31); i >= 0; --i) {
1605                 /* double */
1606                 if (!skip)
1607                         point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
1608
1609                 /* add multiples of the generator */
1610                 if (gen_mul && (i <= 31)) {
1611                         /* first, look 32 bits upwards */
1612                         bits = get_bit(g_scalar, i + 224) << 3;
1613                         bits |= get_bit(g_scalar, i + 160) << 2;
1614                         bits |= get_bit(g_scalar, i + 96) << 1;
1615                         bits |= get_bit(g_scalar, i + 32);
1616                         /* select the point to add, in constant time */
1617                         select_point(bits, 16, g_pre_comp[1], tmp);
1618
1619                         if (!skip) {
1620                                 point_add(nq[0], nq[1], nq[2],
1621                                     nq[0], nq[1], nq[2],
1622                                     1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1623                         } else {
1624                                 smallfelem_expand(nq[0], tmp[0]);
1625                                 smallfelem_expand(nq[1], tmp[1]);
1626                                 smallfelem_expand(nq[2], tmp[2]);
1627                                 skip = 0;
1628                         }
1629
1630                         /* second, look at the current position */
1631                         bits = get_bit(g_scalar, i + 192) << 3;
1632                         bits |= get_bit(g_scalar, i + 128) << 2;
1633                         bits |= get_bit(g_scalar, i + 64) << 1;
1634                         bits |= get_bit(g_scalar, i);
1635                         /* select the point to add, in constant time */
1636                         select_point(bits, 16, g_pre_comp[0], tmp);
1637                         point_add(nq[0], nq[1], nq[2],
1638                             nq[0], nq[1], nq[2],
1639                             1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
1640                 }
1641                 /* do other additions every 5 doublings */
1642                 if (num_points && (i % 5 == 0)) {
1643                         /* loop over all scalars */
1644                         for (num = 0; num < num_points; ++num) {
1645                                 bits = get_bit(scalars[num], i + 4) << 5;
1646                                 bits |= get_bit(scalars[num], i + 3) << 4;
1647                                 bits |= get_bit(scalars[num], i + 2) << 3;
1648                                 bits |= get_bit(scalars[num], i + 1) << 2;
1649                                 bits |= get_bit(scalars[num], i) << 1;
1650                                 bits |= get_bit(scalars[num], i - 1);
1651                                 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);
1652
1653                                 /*
1654                                  * select the point to add or subtract, in
1655                                  * constant time
1656                                  */
1657                                 select_point(digit, 17, pre_comp[num], tmp);
1658                                 smallfelem_neg(ftmp, tmp[1]);   /* (X, -Y, Z) is the
1659                                                                  * negative point */
1660                                 copy_small_conditional(ftmp, tmp[1], (((limb) sign) - 1));
1661                                 felem_contract(tmp[1], ftmp);
1662
1663                                 if (!skip) {
1664                                         point_add(nq[0], nq[1], nq[2],
1665                                             nq[0], nq[1], nq[2],
1666                                             mixed, tmp[0], tmp[1], tmp[2]);
1667                                 } else {
1668                                         smallfelem_expand(nq[0], tmp[0]);
1669                                         smallfelem_expand(nq[1], tmp[1]);
1670                                         smallfelem_expand(nq[2], tmp[2]);
1671                                         skip = 0;
1672                                 }
1673                         }
1674                 }
1675         }
1676         felem_assign(x_out, nq[0]);
1677         felem_assign(y_out, nq[1]);
1678         felem_assign(z_out, nq[2]);
1679 }
1680
1681 /* Precomputation for the group generator. */
1682 typedef struct {
1683         smallfelem g_pre_comp[2][16][3];
1684         int references;
1685 } NISTP256_PRE_COMP;
1686
1687 const EC_METHOD *
1688 EC_GFp_nistp256_method(void)
1689 {
1690         static const EC_METHOD ret = {
1691                 .flags = EC_FLAGS_DEFAULT_OCT,
1692                 .field_type = NID_X9_62_prime_field,
1693                 .group_init = ec_GFp_nistp256_group_init,
1694                 .group_finish = ec_GFp_simple_group_finish,
1695                 .group_clear_finish = ec_GFp_simple_group_clear_finish,
1696                 .group_copy = ec_GFp_nist_group_copy,
1697                 .group_set_curve = ec_GFp_nistp256_group_set_curve,
1698                 .group_get_curve = ec_GFp_simple_group_get_curve,
1699                 .group_get_degree = ec_GFp_simple_group_get_degree,
1700                 .group_check_discriminant =
1701                 ec_GFp_simple_group_check_discriminant,
1702                 .point_init = ec_GFp_simple_point_init,
1703                 .point_finish = ec_GFp_simple_point_finish,
1704                 .point_clear_finish = ec_GFp_simple_point_clear_finish,
1705                 .point_copy = ec_GFp_simple_point_copy,
1706                 .point_set_to_infinity = ec_GFp_simple_point_set_to_infinity,
1707                 .point_set_Jprojective_coordinates_GFp =
1708                 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1709                 .point_get_Jprojective_coordinates_GFp =
1710                 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1711                 .point_set_affine_coordinates =
1712                 ec_GFp_simple_point_set_affine_coordinates,
1713                 .point_get_affine_coordinates =
1714                 ec_GFp_nistp256_point_get_affine_coordinates,
1715                 .add = ec_GFp_simple_add,
1716                 .dbl = ec_GFp_simple_dbl,
1717                 .invert = ec_GFp_simple_invert,
1718                 .is_at_infinity = ec_GFp_simple_is_at_infinity,
1719                 .is_on_curve = ec_GFp_simple_is_on_curve,
1720                 .point_cmp = ec_GFp_simple_cmp,
1721                 .make_affine = ec_GFp_simple_make_affine,
1722                 .points_make_affine = ec_GFp_simple_points_make_affine,
1723                 .mul = ec_GFp_nistp256_points_mul,
1724                 .precompute_mult = ec_GFp_nistp256_precompute_mult,
1725                 .have_precompute_mult = ec_GFp_nistp256_have_precompute_mult,
1726                 .field_mul = ec_GFp_nist_field_mul,
1727                 .field_sqr = ec_GFp_nist_field_sqr
1728         };
1729
1730         return &ret;
1731 }
1732
1733 /******************************************************************************/
1734 /*                     FUNCTIONS TO MANAGE PRECOMPUTATION
1735  */
1736
1737 static NISTP256_PRE_COMP *
1738 nistp256_pre_comp_new()
1739 {
1740         NISTP256_PRE_COMP *ret = NULL;
1741         ret = malloc(sizeof *ret);
1742         if (!ret) {
1743                 ECerror(ERR_R_MALLOC_FAILURE);
1744                 return ret;
1745         }
1746         memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1747         ret->references = 1;
1748         return ret;
1749 }
1750
1751 static void *
1752 nistp256_pre_comp_dup(void *src_)
1753 {
1754         NISTP256_PRE_COMP *src = src_;
1755
1756         /* no need to actually copy, these objects never change! */
1757         CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1758
1759         return src_;
1760 }
1761
1762 static void
1763 nistp256_pre_comp_free(void *pre_)
1764 {
1765         int i;
1766         NISTP256_PRE_COMP *pre = pre_;
1767
1768         if (!pre)
1769                 return;
1770
1771         i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1772         if (i > 0)
1773                 return;
1774
1775         free(pre);
1776 }
1777
1778 static void
1779 nistp256_pre_comp_clear_free(void *pre_)
1780 {
1781         int i;
1782         NISTP256_PRE_COMP *pre = pre_;
1783
1784         if (!pre)
1785                 return;
1786
1787         i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1788         if (i > 0)
1789                 return;
1790
1791         freezero(pre, sizeof *pre);
1792 }
1793
1794 /******************************************************************************/
1795 /*                         OPENSSL EC_METHOD FUNCTIONS
1796  */
1797
1798 int
1799 ec_GFp_nistp256_group_init(EC_GROUP * group)
1800 {
1801         int ret;
1802         ret = ec_GFp_simple_group_init(group);
1803         group->a_is_minus3 = 1;
1804         return ret;
1805 }
1806
1807 int
1808 ec_GFp_nistp256_group_set_curve(EC_GROUP * group, const BIGNUM * p,
1809     const BIGNUM * a, const BIGNUM * b, BN_CTX * ctx)
1810 {
1811         int ret = 0;
1812         BN_CTX *new_ctx = NULL;
1813         BIGNUM *curve_p, *curve_a, *curve_b;
1814
1815         if (ctx == NULL)
1816                 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1817                         return 0;
1818         BN_CTX_start(ctx);
1819         if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1820             ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1821             ((curve_b = BN_CTX_get(ctx)) == NULL))
1822                 goto err;
1823         BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p);
1824         BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a);
1825         BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b);
1826         if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1827             (BN_cmp(curve_b, b))) {
1828                 ECerror(EC_R_WRONG_CURVE_PARAMETERS);
1829                 goto err;
1830         }
1831         group->field_mod_func = BN_nist_mod_256;
1832         ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1833 err:
1834         BN_CTX_end(ctx);
1835         BN_CTX_free(new_ctx);
1836         return ret;
1837 }
1838
1839 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1840  * (X', Y') = (X/Z^2, Y/Z^3) */
1841 int
1842 ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP * group,
1843     const EC_POINT * point, BIGNUM * x, BIGNUM * y, BN_CTX * ctx)
1844 {
1845         felem z1, z2, x_in, y_in;
1846         smallfelem x_out, y_out;
1847         longfelem tmp;
1848
1849         if (EC_POINT_is_at_infinity(group, point) > 0) {
1850                 ECerror(EC_R_POINT_AT_INFINITY);
1851                 return 0;
1852         }
1853         if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1854             (!BN_to_felem(z1, &point->Z)))
1855                 return 0;
1856         felem_inv(z2, z1);
1857         felem_square(tmp, z2);
1858         felem_reduce(z1, tmp);
1859         felem_mul(tmp, x_in, z1);
1860         felem_reduce(x_in, tmp);
1861         felem_contract(x_out, x_in);
1862         if (x != NULL) {
1863                 if (!smallfelem_to_BN(x, x_out)) {
1864                         ECerror(ERR_R_BN_LIB);
1865                         return 0;
1866                 }
1867         }
1868         felem_mul(tmp, z1, z2);
1869         felem_reduce(z1, tmp);
1870         felem_mul(tmp, y_in, z1);
1871         felem_reduce(y_in, tmp);
1872         felem_contract(y_out, y_in);
1873         if (y != NULL) {
1874                 if (!smallfelem_to_BN(y, y_out)) {
1875                         ECerror(ERR_R_BN_LIB);
1876                         return 0;
1877                 }
1878         }
1879         return 1;
1880 }
1881
1882 static void
1883 make_points_affine(size_t num, smallfelem points[ /* num */ ][3], smallfelem tmp_smallfelems[ /* num+1 */ ])
1884 {
1885         /*
1886          * Runs in constant time, unless an input is the point at infinity
1887          * (which normally shouldn't happen).
1888          */
1889         ec_GFp_nistp_points_make_affine_internal(
1890             num,
1891             points,
1892             sizeof(smallfelem),
1893             tmp_smallfelems,
1894             (void (*) (void *)) smallfelem_one,
1895             (int (*) (const void *)) smallfelem_is_zero_int,
1896             (void (*) (void *, const void *)) smallfelem_assign,
1897             (void (*) (void *, const void *)) smallfelem_square_contract,
1898             (void (*) (void *, const void *, const void *)) smallfelem_mul_contract,
1899             (void (*) (void *, const void *)) smallfelem_inv_contract,
1900             (void (*) (void *, const void *)) smallfelem_assign /* nothing to contract */ );
1901 }
1902
1903 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1904  * Result is stored in r (r can equal one of the inputs). */
1905 int
1906 ec_GFp_nistp256_points_mul(const EC_GROUP * group, EC_POINT * r,
1907     const BIGNUM * scalar, size_t num, const EC_POINT * points[],
1908     const BIGNUM * scalars[], BN_CTX * ctx)
1909 {
1910         int ret = 0;
1911         int j;
1912         int mixed = 0;
1913         BN_CTX *new_ctx = NULL;
1914         BIGNUM *x, *y, *z, *tmp_scalar;
1915         felem_bytearray g_secret;
1916         felem_bytearray *secrets = NULL;
1917         smallfelem(*pre_comp)[17][3] = NULL;
1918         smallfelem *tmp_smallfelems = NULL;
1919         felem_bytearray tmp;
1920         unsigned i, num_bytes;
1921         int have_pre_comp = 0;
1922         size_t num_points = num;
1923         smallfelem x_in, y_in, z_in;
1924         felem x_out, y_out, z_out;
1925         NISTP256_PRE_COMP *pre = NULL;
1926         const smallfelem(*g_pre_comp)[16][3] = NULL;
1927         EC_POINT *generator = NULL;
1928         const EC_POINT *p = NULL;
1929         const BIGNUM *p_scalar = NULL;
1930
1931         if (ctx == NULL)
1932                 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
1933                         return 0;
1934         BN_CTX_start(ctx);
1935         if (((x = BN_CTX_get(ctx)) == NULL) ||
1936             ((y = BN_CTX_get(ctx)) == NULL) ||
1937             ((z = BN_CTX_get(ctx)) == NULL) ||
1938             ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1939                 goto err;
1940
1941         if (scalar != NULL) {
1942                 pre = EC_EX_DATA_get_data(group->extra_data,
1943                     nistp256_pre_comp_dup, nistp256_pre_comp_free,
1944                     nistp256_pre_comp_clear_free);
1945                 if (pre)
1946                         /* we have precomputation, try to use it */
1947                         g_pre_comp = (const smallfelem(*)[16][3]) pre->g_pre_comp;
1948                 else
1949                         /* try to use the standard precomputation */
1950                         g_pre_comp = &gmul[0];
1951                 generator = EC_POINT_new(group);
1952                 if (generator == NULL)
1953                         goto err;
1954                 /* get the generator from precomputation */
1955                 if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) ||
1956                     !smallfelem_to_BN(y, g_pre_comp[0][1][1]) ||
1957                     !smallfelem_to_BN(z, g_pre_comp[0][1][2])) {
1958                         ECerror(ERR_R_BN_LIB);
1959                         goto err;
1960                 }
1961                 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1962                         generator, x, y, z, ctx))
1963                         goto err;
1964                 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1965                         /* precomputation matches generator */
1966                         have_pre_comp = 1;
1967                 else
1968                         /*
1969                          * we don't have valid precomputation: treat the
1970                          * generator as a random point
1971                          */
1972                         num_points++;
1973         }
1974         if (num_points > 0) {
1975                 if (num_points >= 3) {
1976                         /*
1977                          * unless we precompute multiples for just one or two
1978                          * points, converting those into affine form is time
1979                          * well spent
1980                          */
1981                         mixed = 1;
1982                 }
1983                 secrets = calloc(num_points, sizeof(felem_bytearray));
1984                 pre_comp = calloc(num_points, 17 * 3 * sizeof(smallfelem));
1985                 if (mixed) {
1986                         /* XXX should do more int overflow checking */
1987                         tmp_smallfelems = reallocarray(NULL,
1988                             (num_points * 17 + 1), sizeof(smallfelem));
1989                 }
1990                 if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_smallfelems == NULL))) {
1991                         ECerror(ERR_R_MALLOC_FAILURE);
1992                         goto err;
1993                 }
1994                 /*
1995                  * we treat NULL scalars as 0, and NULL points as points at
1996                  * infinity, i.e., they contribute nothing to the linear
1997                  * combination
1998                  */
1999                 for (i = 0; i < num_points; ++i) {
2000                         if (i == num)
2001                                 /*
2002                                  * we didn't have a valid precomputation, so
2003                                  * we pick the generator
2004                                  */
2005                         {
2006                                 p = EC_GROUP_get0_generator(group);
2007                                 p_scalar = scalar;
2008                         } else
2009                                 /* the i^th point */
2010                         {
2011                                 p = points[i];
2012                                 p_scalar = scalars[i];
2013                         }
2014                         if ((p_scalar != NULL) && (p != NULL)) {
2015                                 /* reduce scalar to 0 <= scalar < 2^256 */
2016                                 if ((BN_num_bits(p_scalar) > 256) || (BN_is_negative(p_scalar))) {
2017                                         /*
2018                                          * this is an unusual input, and we
2019                                          * don't guarantee constant-timeness
2020                                          */
2021                                         if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) {
2022                                                 ECerror(ERR_R_BN_LIB);
2023                                                 goto err;
2024                                         }
2025                                         num_bytes = BN_bn2bin(tmp_scalar, tmp);
2026                                 } else
2027                                         num_bytes = BN_bn2bin(p_scalar, tmp);
2028                                 flip_endian(secrets[i], tmp, num_bytes);
2029                                 /* precompute multiples */
2030                                 if ((!BN_to_felem(x_out, &p->X)) ||
2031                                     (!BN_to_felem(y_out, &p->Y)) ||
2032                                     (!BN_to_felem(z_out, &p->Z)))
2033                                         goto err;
2034                                 felem_shrink(pre_comp[i][1][0], x_out);
2035                                 felem_shrink(pre_comp[i][1][1], y_out);
2036                                 felem_shrink(pre_comp[i][1][2], z_out);
2037                                 for (j = 2; j <= 16; ++j) {
2038                                         if (j & 1) {
2039                                                 point_add_small(
2040                                                     pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
2041                                                     pre_comp[i][1][0], pre_comp[i][1][1], pre_comp[i][1][2],
2042                                                     pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], pre_comp[i][j - 1][2]);
2043                                         } else {
2044                                                 point_double_small(
2045                                                     pre_comp[i][j][0], pre_comp[i][j][1], pre_comp[i][j][2],
2046                                                     pre_comp[i][j / 2][0], pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]);
2047                                         }
2048                                 }
2049                         }
2050                 }
2051                 if (mixed)
2052                         make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems);
2053         }
2054         /* the scalar for the generator */
2055         if ((scalar != NULL) && (have_pre_comp)) {
2056                 memset(g_secret, 0, sizeof(g_secret));
2057                 /* reduce scalar to 0 <= scalar < 2^256 */
2058                 if ((BN_num_bits(scalar) > 256) || (BN_is_negative(scalar))) {
2059                         /*
2060                          * this is an unusual input, and we don't guarantee
2061                          * constant-timeness
2062                          */
2063                         if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) {
2064                                 ECerror(ERR_R_BN_LIB);
2065                                 goto err;
2066                         }
2067                         num_bytes = BN_bn2bin(tmp_scalar, tmp);
2068                 } else
2069                         num_bytes = BN_bn2bin(scalar, tmp);
2070                 flip_endian(g_secret, tmp, num_bytes);
2071                 /* do the multiplication with generator precomputation */
2072                 batch_mul(x_out, y_out, z_out,
2073                     (const felem_bytearray(*)) secrets, num_points,
2074                     g_secret,
2075                     mixed, (const smallfelem(*)[17][3]) pre_comp,
2076                     g_pre_comp);
2077         } else
2078                 /* do the multiplication without generator precomputation */
2079                 batch_mul(x_out, y_out, z_out,
2080                     (const felem_bytearray(*)) secrets, num_points,
2081                     NULL, mixed, (const smallfelem(*)[17][3]) pre_comp, NULL);
2082         /* reduce the output to its unique minimal representation */
2083         felem_contract(x_in, x_out);
2084         felem_contract(y_in, y_out);
2085         felem_contract(z_in, z_out);
2086         if ((!smallfelem_to_BN(x, x_in)) || (!smallfelem_to_BN(y, y_in)) ||
2087             (!smallfelem_to_BN(z, z_in))) {
2088                 ECerror(ERR_R_BN_LIB);
2089                 goto err;
2090         }
2091         ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
2092
2093 err:
2094         BN_CTX_end(ctx);
2095         EC_POINT_free(generator);
2096         BN_CTX_free(new_ctx);
2097         free(secrets);
2098         free(pre_comp);
2099         free(tmp_smallfelems);
2100         return ret;
2101 }
2102
2103 int
2104 ec_GFp_nistp256_precompute_mult(EC_GROUP * group, BN_CTX * ctx)
2105 {
2106         int ret = 0;
2107         NISTP256_PRE_COMP *pre = NULL;
2108         int i, j;
2109         BN_CTX *new_ctx = NULL;
2110         BIGNUM *x, *y;
2111         EC_POINT *generator = NULL;
2112         smallfelem tmp_smallfelems[32];
2113         felem x_tmp, y_tmp, z_tmp;
2114
2115         /* throw away old precomputation */
2116         EC_EX_DATA_free_data(&group->extra_data, nistp256_pre_comp_dup,
2117             nistp256_pre_comp_free, nistp256_pre_comp_clear_free);
2118         if (ctx == NULL)
2119                 if ((ctx = new_ctx = BN_CTX_new()) == NULL)
2120                         return 0;
2121         BN_CTX_start(ctx);
2122         if (((x = BN_CTX_get(ctx)) == NULL) ||
2123             ((y = BN_CTX_get(ctx)) == NULL))
2124                 goto err;
2125         /* get the generator */
2126         if (group->generator == NULL)
2127                 goto err;
2128         generator = EC_POINT_new(group);
2129         if (generator == NULL)
2130                 goto err;
2131         BN_bin2bn(nistp256_curve_params[3], sizeof(felem_bytearray), x);
2132         BN_bin2bn(nistp256_curve_params[4], sizeof(felem_bytearray), y);
2133         if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
2134                 goto err;
2135         if ((pre = nistp256_pre_comp_new()) == NULL)
2136                 goto err;
2137         /* if the generator is the standard one, use built-in precomputation */
2138         if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
2139                 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
2140                 ret = 1;
2141                 goto err;
2142         }
2143         if ((!BN_to_felem(x_tmp, &group->generator->X)) ||
2144             (!BN_to_felem(y_tmp, &group->generator->Y)) ||
2145             (!BN_to_felem(z_tmp, &group->generator->Z)))
2146                 goto err;
2147         felem_shrink(pre->g_pre_comp[0][1][0], x_tmp);
2148         felem_shrink(pre->g_pre_comp[0][1][1], y_tmp);
2149         felem_shrink(pre->g_pre_comp[0][1][2], z_tmp);
2150         /*
2151          * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G,
2152          * 2^96*G, 2^160*G, 2^224*G for the second one
2153          */
2154         for (i = 1; i <= 8; i <<= 1) {
2155                 point_double_small(
2156                     pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
2157                     pre->g_pre_comp[0][i][0], pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
2158                 for (j = 0; j < 31; ++j) {
2159                         point_double_small(
2160                             pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
2161                             pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
2162                 }
2163                 if (i == 8)
2164                         break;
2165                 point_double_small(
2166                     pre->g_pre_comp[0][2 * i][0], pre->g_pre_comp[0][2 * i][1], pre->g_pre_comp[0][2 * i][2],
2167                     pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
2168                 for (j = 0; j < 31; ++j) {
2169                         point_double_small(
2170                             pre->g_pre_comp[0][2 * i][0], pre->g_pre_comp[0][2 * i][1], pre->g_pre_comp[0][2 * i][2],
2171                             pre->g_pre_comp[0][2 * i][0], pre->g_pre_comp[0][2 * i][1], pre->g_pre_comp[0][2 * i][2]);
2172                 }
2173         }
2174         for (i = 0; i < 2; i++) {
2175                 /* g_pre_comp[i][0] is the point at infinity */
2176                 memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
2177                 /* the remaining multiples */
2178                 /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
2179                 point_add_small(
2180                     pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1], pre->g_pre_comp[i][6][2],
2181                     pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
2182                     pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]);
2183                 /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
2184                 point_add_small(
2185                     pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1], pre->g_pre_comp[i][10][2],
2186                     pre->g_pre_comp[i][8][0], pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
2187                     pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]);
2188                 /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
2189                 point_add_small(
2190                     pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
2191                     pre->g_pre_comp[i][8][0], pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
2192                     pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2]);
2193                 /*
2194                  * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G +
2195                  * 2^224*G
2196                  */
2197                 point_add_small(
2198                     pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1], pre->g_pre_comp[i][14][2],
2199                     pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
2200                     pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]);
2201                 for (j = 1; j < 8; ++j) {
2202                         /* odd multiples: add G resp. 2^32*G */
2203                         point_add_small(
2204                             pre->g_pre_comp[i][2 * j + 1][0], pre->g_pre_comp[i][2 * j + 1][1], pre->g_pre_comp[i][2 * j + 1][2],
2205                             pre->g_pre_comp[i][2 * j][0], pre->g_pre_comp[i][2 * j][1], pre->g_pre_comp[i][2 * j][2],
2206                             pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1], pre->g_pre_comp[i][1][2]);
2207                 }
2208         }
2209         make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_smallfelems);
2210
2211         if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp256_pre_comp_dup,
2212                 nistp256_pre_comp_free, nistp256_pre_comp_clear_free))
2213                 goto err;
2214         ret = 1;
2215         pre = NULL;
2216 err:
2217         BN_CTX_end(ctx);
2218         EC_POINT_free(generator);
2219         BN_CTX_free(new_ctx);
2220         nistp256_pre_comp_free(pre);
2221         return ret;
2222 }
2223
2224 int
2225 ec_GFp_nistp256_have_precompute_mult(const EC_GROUP * group)
2226 {
2227         if (EC_EX_DATA_get_data(group->extra_data, nistp256_pre_comp_dup,
2228                 nistp256_pre_comp_free, nistp256_pre_comp_clear_free)
2229             != NULL)
2230                 return 1;
2231         else
2232                 return 0;
2233 }
2234 #endif