1 /* Copyright (c) 2015, Google Inc. 2 * 3 * Permission to use, copy, modify, and/or distribute this software for any 4 * purpose with or without fee is hereby granted, provided that the above 5 * copyright notice and this permission notice appear in all copies. 6 * 7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY 10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION 12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN 13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ 14 15 /* A 64-bit implementation of the NIST P-256 elliptic curve point 16 * multiplication 17 * 18 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c. 19 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519 20 * work which got its smarts from Daniel J. Bernstein's work on the same. */ 21 22 #include <openssl/base.h> 23 24 #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) 25 26 #include <openssl/bn.h> 27 #include <openssl/ec.h> 28 #include <openssl/err.h> 29 #include <openssl/mem.h> 30 #include <openssl/obj.h> 31 32 #include <string.h> 33 34 #include "internal.h" 35 36 37 typedef uint8_t u8; 38 typedef uint64_t u64; 39 typedef int64_t s64; 40 typedef __uint128_t uint128_t; 41 typedef __int128_t int128_t; 42 43 /* The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We 44 * can serialise an element of this field into 32 bytes. We call this an 45 * felem_bytearray. */ 46 typedef u8 felem_bytearray[32]; 47 48 /* These are the parameters of P256, taken from FIPS 186-3, page 86. These 49 * values are big-endian. */ 50 static const felem_bytearray nistp256_curve_params[5] = { 51 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */ 52 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 53 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff}, 54 {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */ 55 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 56 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 57 0xfc}, /* b */ 58 {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, 0xb3, 0xeb, 0xbd, 0x55, 59 0x76, 0x98, 0x86, 0xbc, 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6, 60 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b}, 61 {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */ 62 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2, 0x77, 0x03, 0x7d, 0x81, 63 0x2d, 0xeb, 0x33, 0xa0, 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96}, 64 {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */ 65 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16, 0x2b, 0xce, 0x33, 0x57, 66 0x6b, 0x31, 0x5e, 0xce, 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}}; 67 68 /* The representation of field elements. 69 * ------------------------------------ 70 * 71 * We represent field elements with either four 128-bit values, eight 128-bit 72 * values, or four 64-bit values. The field element represented is: 73 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p) 74 * or: 75 * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p) 76 * 77 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits 78 * apart, but are 128-bits wide, the most significant bits of each limb overlap 79 * with the least significant bits of the next. 80 * 81 * A field element with four limbs is an 'felem'. One with eight limbs is a 82 * 'longfelem' 83 * 84 * A field element with four, 64-bit values is called a 'smallfelem'. Small 85 * values are used as intermediate values before multiplication. */ 86 87 #define NLIMBS 4 88 89 typedef uint128_t limb; 90 typedef limb felem[NLIMBS]; 91 typedef limb longfelem[NLIMBS * 2]; 92 typedef u64 smallfelem[NLIMBS]; 93 94 /* This is the value of the prime as four 64-bit words, little-endian. */ 95 static const u64 kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0, 96 0xffffffff00000001ul}; 97 static const u64 bottom63bits = 0x7ffffffffffffffful; 98 99 /* bin32_to_felem takes a little-endian byte array and converts it into felem 100 * form. This assumes that the CPU is little-endian. */ 101 static void bin32_to_felem(felem out, const u8 in[32]) { 102 out[0] = *((u64 *)&in[0]); 103 out[1] = *((u64 *)&in[8]); 104 out[2] = *((u64 *)&in[16]); 105 out[3] = *((u64 *)&in[24]); 106 } 107 108 /* smallfelem_to_bin32 takes a smallfelem and serialises into a little endian, 109 * 32 byte array. This assumes that the CPU is little-endian. */ 110 static void smallfelem_to_bin32(u8 out[32], const smallfelem in) { 111 *((u64 *)&out[0]) = in[0]; 112 *((u64 *)&out[8]) = in[1]; 113 *((u64 *)&out[16]) = in[2]; 114 *((u64 *)&out[24]) = in[3]; 115 } 116 117 /* To preserve endianness when using BN_bn2bin and BN_bin2bn. */ 118 static void flip_endian(u8 *out, const u8 *in, unsigned len) { 119 unsigned i; 120 for (i = 0; i < len; ++i) { 121 out[i] = in[len - 1 - i]; 122 } 123 } 124 125 /* BN_to_felem converts an OpenSSL BIGNUM into an felem. */ 126 static int BN_to_felem(felem out, const BIGNUM *bn) { 127 if (BN_is_negative(bn)) { 128 OPENSSL_PUT_ERROR(EC, BN_to_felem, EC_R_BIGNUM_OUT_OF_RANGE); 129 return 0; 130 } 131 132 felem_bytearray b_out; 133 /* BN_bn2bin eats leading zeroes */ 134 memset(b_out, 0, sizeof(b_out)); 135 unsigned num_bytes = BN_num_bytes(bn); 136 if (num_bytes > sizeof(b_out)) { 137 OPENSSL_PUT_ERROR(EC, BN_to_felem, EC_R_BIGNUM_OUT_OF_RANGE); 138 return 0; 139 } 140 141 felem_bytearray b_in; 142 num_bytes = BN_bn2bin(bn, b_in); 143 flip_endian(b_out, b_in, num_bytes); 144 bin32_to_felem(out, b_out); 145 return 1; 146 } 147 148 /* felem_to_BN converts an felem into an OpenSSL BIGNUM. */ 149 static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in) { 150 felem_bytearray b_in, b_out; 151 smallfelem_to_bin32(b_in, in); 152 flip_endian(b_out, b_in, sizeof(b_out)); 153 return BN_bin2bn(b_out, sizeof(b_out), out); 154 } 155 156 /* Field operations. */ 157 158 static void smallfelem_one(smallfelem out) { 159 out[0] = 1; 160 out[1] = 0; 161 out[2] = 0; 162 out[3] = 0; 163 } 164 165 static void smallfelem_assign(smallfelem out, const smallfelem in) { 166 out[0] = in[0]; 167 out[1] = in[1]; 168 out[2] = in[2]; 169 out[3] = in[3]; 170 } 171 172 static void felem_assign(felem out, const felem in) { 173 out[0] = in[0]; 174 out[1] = in[1]; 175 out[2] = in[2]; 176 out[3] = in[3]; 177 } 178 179 /* felem_sum sets out = out + in. */ 180 static void felem_sum(felem out, const felem in) { 181 out[0] += in[0]; 182 out[1] += in[1]; 183 out[2] += in[2]; 184 out[3] += in[3]; 185 } 186 187 /* felem_small_sum sets out = out + in. */ 188 static void felem_small_sum(felem out, const smallfelem in) { 189 out[0] += in[0]; 190 out[1] += in[1]; 191 out[2] += in[2]; 192 out[3] += in[3]; 193 } 194 195 /* felem_scalar sets out = out * scalar */ 196 static void felem_scalar(felem out, const u64 scalar) { 197 out[0] *= scalar; 198 out[1] *= scalar; 199 out[2] *= scalar; 200 out[3] *= scalar; 201 } 202 203 /* longfelem_scalar sets out = out * scalar */ 204 static void longfelem_scalar(longfelem out, const u64 scalar) { 205 out[0] *= scalar; 206 out[1] *= scalar; 207 out[2] *= scalar; 208 out[3] *= scalar; 209 out[4] *= scalar; 210 out[5] *= scalar; 211 out[6] *= scalar; 212 out[7] *= scalar; 213 } 214 215 #define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9) 216 #define two105 (((limb)1) << 105) 217 #define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9) 218 219 /* zero105 is 0 mod p */ 220 static const felem zero105 = {two105m41m9, two105, two105m41p9, two105m41p9}; 221 222 /* smallfelem_neg sets |out| to |-small| 223 * On exit: 224 * out[i] < out[i] + 2^105 */ 225 static void smallfelem_neg(felem out, const smallfelem small) { 226 /* In order to prevent underflow, we subtract from 0 mod p. */ 227 out[0] = zero105[0] - small[0]; 228 out[1] = zero105[1] - small[1]; 229 out[2] = zero105[2] - small[2]; 230 out[3] = zero105[3] - small[3]; 231 } 232 233 /* felem_diff subtracts |in| from |out| 234 * On entry: 235 * in[i] < 2^104 236 * On exit: 237 * out[i] < out[i] + 2^105. */ 238 static void felem_diff(felem out, const felem in) { 239 /* In order to prevent underflow, we add 0 mod p before subtracting. */ 240 out[0] += zero105[0]; 241 out[1] += zero105[1]; 242 out[2] += zero105[2]; 243 out[3] += zero105[3]; 244 245 out[0] -= in[0]; 246 out[1] -= in[1]; 247 out[2] -= in[2]; 248 out[3] -= in[3]; 249 } 250 251 #define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11) 252 #define two107 (((limb)1) << 107) 253 #define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11) 254 255 /* zero107 is 0 mod p */ 256 static const felem zero107 = {two107m43m11, two107, two107m43p11, two107m43p11}; 257 258 /* An alternative felem_diff for larger inputs |in| 259 * felem_diff_zero107 subtracts |in| from |out| 260 * On entry: 261 * in[i] < 2^106 262 * On exit: 263 * out[i] < out[i] + 2^107. */ 264 static void felem_diff_zero107(felem out, const felem in) { 265 /* In order to prevent underflow, we add 0 mod p before subtracting. */ 266 out[0] += zero107[0]; 267 out[1] += zero107[1]; 268 out[2] += zero107[2]; 269 out[3] += zero107[3]; 270 271 out[0] -= in[0]; 272 out[1] -= in[1]; 273 out[2] -= in[2]; 274 out[3] -= in[3]; 275 } 276 277 /* longfelem_diff subtracts |in| from |out| 278 * On entry: 279 * in[i] < 7*2^67 280 * On exit: 281 * out[i] < out[i] + 2^70 + 2^40. */ 282 static void longfelem_diff(longfelem out, const longfelem in) { 283 static const limb two70m8p6 = 284 (((limb)1) << 70) - (((limb)1) << 8) + (((limb)1) << 6); 285 static const limb two70p40 = (((limb)1) << 70) + (((limb)1) << 40); 286 static const limb two70 = (((limb)1) << 70); 287 static const limb two70m40m38p6 = (((limb)1) << 70) - (((limb)1) << 40) - 288 (((limb)1) << 38) + (((limb)1) << 6); 289 static const limb two70m6 = (((limb)1) << 70) - (((limb)1) << 6); 290 291 /* add 0 mod p to avoid underflow */ 292 out[0] += two70m8p6; 293 out[1] += two70p40; 294 out[2] += two70; 295 out[3] += two70m40m38p6; 296 out[4] += two70m6; 297 out[5] += two70m6; 298 out[6] += two70m6; 299 out[7] += two70m6; 300 301 /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */ 302 out[0] -= in[0]; 303 out[1] -= in[1]; 304 out[2] -= in[2]; 305 out[3] -= in[3]; 306 out[4] -= in[4]; 307 out[5] -= in[5]; 308 out[6] -= in[6]; 309 out[7] -= in[7]; 310 } 311 312 #define two64m0 (((limb)1) << 64) - 1 313 #define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1 314 #define two64m46 (((limb)1) << 64) - (((limb)1) << 46) 315 #define two64m32 (((limb)1) << 64) - (((limb)1) << 32) 316 317 /* zero110 is 0 mod p. */ 318 static const felem zero110 = {two64m0, two110p32m0, two64m46, two64m32}; 319 320 /* felem_shrink converts an felem into a smallfelem. The result isn't quite 321 * minimal as the value may be greater than p. 322 * 323 * On entry: 324 * in[i] < 2^109 325 * On exit: 326 * out[i] < 2^64. */ 327 static void felem_shrink(smallfelem out, const felem in) { 328 felem tmp; 329 u64 a, b, mask; 330 s64 high, low; 331 static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */ 332 333 /* Carry 2->3 */ 334 tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64)); 335 /* tmp[3] < 2^110 */ 336 337 tmp[2] = zero110[2] + (u64)in[2]; 338 tmp[0] = zero110[0] + in[0]; 339 tmp[1] = zero110[1] + in[1]; 340 /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */ 341 342 /* We perform two partial reductions where we eliminate the high-word of 343 * tmp[3]. We don't update the other words till the end. */ 344 a = tmp[3] >> 64; /* a < 2^46 */ 345 tmp[3] = (u64)tmp[3]; 346 tmp[3] -= a; 347 tmp[3] += ((limb)a) << 32; 348 /* tmp[3] < 2^79 */ 349 350 b = a; 351 a = tmp[3] >> 64; /* a < 2^15 */ 352 b += a; /* b < 2^46 + 2^15 < 2^47 */ 353 tmp[3] = (u64)tmp[3]; 354 tmp[3] -= a; 355 tmp[3] += ((limb)a) << 32; 356 /* tmp[3] < 2^64 + 2^47 */ 357 358 /* This adjusts the other two words to complete the two partial 359 * reductions. */ 360 tmp[0] += b; 361 tmp[1] -= (((limb)b) << 32); 362 363 /* In order to make space in tmp[3] for the carry from 2 -> 3, we 364 * conditionally subtract kPrime if tmp[3] is large enough. */ 365 high = tmp[3] >> 64; 366 /* As tmp[3] < 2^65, high is either 1 or 0 */ 367 high <<= 63; 368 high >>= 63; 369 /* high is: 370 * all ones if the high word of tmp[3] is 1 371 * all zeros if the high word of tmp[3] if 0 */ 372 low = tmp[3]; 373 mask = low >> 63; 374 /* mask is: 375 * all ones if the MSB of low is 1 376 * all zeros if the MSB of low if 0 */ 377 low &= bottom63bits; 378 low -= kPrime3Test; 379 /* if low was greater than kPrime3Test then the MSB is zero */ 380 low = ~low; 381 low >>= 63; 382 /* low is: 383 * all ones if low was > kPrime3Test 384 * all zeros if low was <= kPrime3Test */ 385 mask = (mask & low) | high; 386 tmp[0] -= mask & kPrime[0]; 387 tmp[1] -= mask & kPrime[1]; 388 /* kPrime[2] is zero, so omitted */ 389 tmp[3] -= mask & kPrime[3]; 390 /* tmp[3] < 2**64 - 2**32 + 1 */ 391 392 tmp[1] += ((u64)(tmp[0] >> 64)); 393 tmp[0] = (u64)tmp[0]; 394 tmp[2] += ((u64)(tmp[1] >> 64)); 395 tmp[1] = (u64)tmp[1]; 396 tmp[3] += ((u64)(tmp[2] >> 64)); 397 tmp[2] = (u64)tmp[2]; 398 /* tmp[i] < 2^64 */ 399 400 out[0] = tmp[0]; 401 out[1] = tmp[1]; 402 out[2] = tmp[2]; 403 out[3] = tmp[3]; 404 } 405 406 /* smallfelem_expand converts a smallfelem to an felem */ 407 static void smallfelem_expand(felem out, const smallfelem in) { 408 out[0] = in[0]; 409 out[1] = in[1]; 410 out[2] = in[2]; 411 out[3] = in[3]; 412 } 413 414 /* smallfelem_square sets |out| = |small|^2 415 * On entry: 416 * small[i] < 2^64 417 * On exit: 418 * out[i] < 7 * 2^64 < 2^67 */ 419 static void smallfelem_square(longfelem out, const smallfelem small) { 420 limb a; 421 u64 high, low; 422 423 a = ((uint128_t)small[0]) * small[0]; 424 low = a; 425 high = a >> 64; 426 out[0] = low; 427 out[1] = high; 428 429 a = ((uint128_t)small[0]) * small[1]; 430 low = a; 431 high = a >> 64; 432 out[1] += low; 433 out[1] += low; 434 out[2] = high; 435 436 a = ((uint128_t)small[0]) * small[2]; 437 low = a; 438 high = a >> 64; 439 out[2] += low; 440 out[2] *= 2; 441 out[3] = high; 442 443 a = ((uint128_t)small[0]) * small[3]; 444 low = a; 445 high = a >> 64; 446 out[3] += low; 447 out[4] = high; 448 449 a = ((uint128_t)small[1]) * small[2]; 450 low = a; 451 high = a >> 64; 452 out[3] += low; 453 out[3] *= 2; 454 out[4] += high; 455 456 a = ((uint128_t)small[1]) * small[1]; 457 low = a; 458 high = a >> 64; 459 out[2] += low; 460 out[3] += high; 461 462 a = ((uint128_t)small[1]) * small[3]; 463 low = a; 464 high = a >> 64; 465 out[4] += low; 466 out[4] *= 2; 467 out[5] = high; 468 469 a = ((uint128_t)small[2]) * small[3]; 470 low = a; 471 high = a >> 64; 472 out[5] += low; 473 out[5] *= 2; 474 out[6] = high; 475 out[6] += high; 476 477 a = ((uint128_t)small[2]) * small[2]; 478 low = a; 479 high = a >> 64; 480 out[4] += low; 481 out[5] += high; 482 483 a = ((uint128_t)small[3]) * small[3]; 484 low = a; 485 high = a >> 64; 486 out[6] += low; 487 out[7] = high; 488 } 489 490 /*felem_square sets |out| = |in|^2 491 * On entry: 492 * in[i] < 2^109 493 * On exit: 494 * out[i] < 7 * 2^64 < 2^67. */ 495 static void felem_square(longfelem out, const felem in) { 496 u64 small[4]; 497 felem_shrink(small, in); 498 smallfelem_square(out, small); 499 } 500 501 /* smallfelem_mul sets |out| = |small1| * |small2| 502 * On entry: 503 * small1[i] < 2^64 504 * small2[i] < 2^64 505 * On exit: 506 * out[i] < 7 * 2^64 < 2^67. */ 507 static void smallfelem_mul(longfelem out, const smallfelem small1, 508 const smallfelem small2) { 509 limb a; 510 u64 high, low; 511 512 a = ((uint128_t)small1[0]) * small2[0]; 513 low = a; 514 high = a >> 64; 515 out[0] = low; 516 out[1] = high; 517 518 a = ((uint128_t)small1[0]) * small2[1]; 519 low = a; 520 high = a >> 64; 521 out[1] += low; 522 out[2] = high; 523 524 a = ((uint128_t)small1[1]) * small2[0]; 525 low = a; 526 high = a >> 64; 527 out[1] += low; 528 out[2] += high; 529 530 a = ((uint128_t)small1[0]) * small2[2]; 531 low = a; 532 high = a >> 64; 533 out[2] += low; 534 out[3] = high; 535 536 a = ((uint128_t)small1[1]) * small2[1]; 537 low = a; 538 high = a >> 64; 539 out[2] += low; 540 out[3] += high; 541 542 a = ((uint128_t)small1[2]) * small2[0]; 543 low = a; 544 high = a >> 64; 545 out[2] += low; 546 out[3] += high; 547 548 a = ((uint128_t)small1[0]) * small2[3]; 549 low = a; 550 high = a >> 64; 551 out[3] += low; 552 out[4] = high; 553 554 a = ((uint128_t)small1[1]) * small2[2]; 555 low = a; 556 high = a >> 64; 557 out[3] += low; 558 out[4] += high; 559 560 a = ((uint128_t)small1[2]) * small2[1]; 561 low = a; 562 high = a >> 64; 563 out[3] += low; 564 out[4] += high; 565 566 a = ((uint128_t)small1[3]) * small2[0]; 567 low = a; 568 high = a >> 64; 569 out[3] += low; 570 out[4] += high; 571 572 a = ((uint128_t)small1[1]) * small2[3]; 573 low = a; 574 high = a >> 64; 575 out[4] += low; 576 out[5] = high; 577 578 a = ((uint128_t)small1[2]) * small2[2]; 579 low = a; 580 high = a >> 64; 581 out[4] += low; 582 out[5] += high; 583 584 a = ((uint128_t)small1[3]) * small2[1]; 585 low = a; 586 high = a >> 64; 587 out[4] += low; 588 out[5] += high; 589 590 a = ((uint128_t)small1[2]) * small2[3]; 591 low = a; 592 high = a >> 64; 593 out[5] += low; 594 out[6] = high; 595 596 a = ((uint128_t)small1[3]) * small2[2]; 597 low = a; 598 high = a >> 64; 599 out[5] += low; 600 out[6] += high; 601 602 a = ((uint128_t)small1[3]) * small2[3]; 603 low = a; 604 high = a >> 64; 605 out[6] += low; 606 out[7] = high; 607 } 608 609 /* felem_mul sets |out| = |in1| * |in2| 610 * On entry: 611 * in1[i] < 2^109 612 * in2[i] < 2^109 613 * On exit: 614 * out[i] < 7 * 2^64 < 2^67 */ 615 static void felem_mul(longfelem out, const felem in1, const felem in2) { 616 smallfelem small1, small2; 617 felem_shrink(small1, in1); 618 felem_shrink(small2, in2); 619 smallfelem_mul(out, small1, small2); 620 } 621 622 /* felem_small_mul sets |out| = |small1| * |in2| 623 * On entry: 624 * small1[i] < 2^64 625 * in2[i] < 2^109 626 * On exit: 627 * out[i] < 7 * 2^64 < 2^67 */ 628 static void felem_small_mul(longfelem out, const smallfelem small1, 629 const felem in2) { 630 smallfelem small2; 631 felem_shrink(small2, in2); 632 smallfelem_mul(out, small1, small2); 633 } 634 635 #define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4) 636 #define two100 (((limb)1) << 100) 637 #define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4) 638 639 /* zero100 is 0 mod p */ 640 static const felem zero100 = {two100m36m4, two100, two100m36p4, two100m36p4}; 641 642 /* Internal function for the different flavours of felem_reduce. 643 * felem_reduce_ reduces the higher coefficients in[4]-in[7]. 644 * On entry: 645 * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7] 646 * out[1] >= in[7] + 2^32*in[4] 647 * out[2] >= in[5] + 2^32*in[5] 648 * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6] 649 * On exit: 650 * out[0] <= out[0] + in[4] + 2^32*in[5] 651 * out[1] <= out[1] + in[5] + 2^33*in[6] 652 * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7] 653 * out[3] <= out[3] + 2^32*in[4] + 3*in[7] */ 654 static void felem_reduce_(felem out, const longfelem in) { 655 int128_t c; 656 /* combine common terms from below */ 657 c = in[4] + (in[5] << 32); 658 out[0] += c; 659 out[3] -= c; 660 661 c = in[5] - in[7]; 662 out[1] += c; 663 out[2] -= c; 664 665 /* the remaining terms */ 666 /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */ 667 out[1] -= (in[4] << 32); 668 out[3] += (in[4] << 32); 669 670 /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */ 671 out[2] -= (in[5] << 32); 672 673 /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */ 674 out[0] -= in[6]; 675 out[0] -= (in[6] << 32); 676 out[1] += (in[6] << 33); 677 out[2] += (in[6] * 2); 678 out[3] -= (in[6] << 32); 679 680 /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */ 681 out[0] -= in[7]; 682 out[0] -= (in[7] << 32); 683 out[2] += (in[7] << 33); 684 out[3] += (in[7] * 3); 685 } 686 687 /* felem_reduce converts a longfelem into an felem. 688 * To be called directly after felem_square or felem_mul. 689 * On entry: 690 * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64 691 * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64 692 * On exit: 693 * out[i] < 2^101 */ 694 static void felem_reduce(felem out, const longfelem in) { 695 out[0] = zero100[0] + in[0]; 696 out[1] = zero100[1] + in[1]; 697 out[2] = zero100[2] + in[2]; 698 out[3] = zero100[3] + in[3]; 699 700 felem_reduce_(out, in); 701 702 /* out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0 703 * out[1] > 2^100 - 2^64 - 7*2^96 > 0 704 * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0 705 * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0 706 * 707 * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101 708 * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101 709 * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101 710 * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101 */ 711 } 712 713 /* felem_reduce_zero105 converts a larger longfelem into an felem. 714 * On entry: 715 * in[0] < 2^71 716 * On exit: 717 * out[i] < 2^106 */ 718 static void felem_reduce_zero105(felem out, const longfelem in) { 719 out[0] = zero105[0] + in[0]; 720 out[1] = zero105[1] + in[1]; 721 out[2] = zero105[2] + in[2]; 722 out[3] = zero105[3] + in[3]; 723 724 felem_reduce_(out, in); 725 726 /* out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0 727 * out[1] > 2^105 - 2^71 - 2^103 > 0 728 * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0 729 * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0 730 * 731 * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 732 * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 733 * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106 734 * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106 */ 735 } 736 737 /* subtract_u64 sets *result = *result - v and *carry to one if the 738 * subtraction underflowed. */ 739 static void subtract_u64(u64 *result, u64 *carry, u64 v) { 740 uint128_t r = *result; 741 r -= v; 742 *carry = (r >> 64) & 1; 743 *result = (u64)r; 744 } 745 746 /* felem_contract converts |in| to its unique, minimal representation. On 747 * entry: in[i] < 2^109. */ 748 static void felem_contract(smallfelem out, const felem in) { 749 u64 all_equal_so_far = 0, result = 0; 750 751 felem_shrink(out, in); 752 /* small is minimal except that the value might be > p */ 753 754 all_equal_so_far--; 755 /* We are doing a constant time test if out >= kPrime. We need to compare 756 * each u64, from most-significant to least significant. For each one, if 757 * all words so far have been equal (m is all ones) then a non-equal 758 * result is the answer. Otherwise we continue. */ 759 unsigned i; 760 for (i = 3; i < 4; i--) { 761 u64 equal; 762 uint128_t a = ((uint128_t)kPrime[i]) - out[i]; 763 /* if out[i] > kPrime[i] then a will underflow and the high 64-bits 764 * will all be set. */ 765 result |= all_equal_so_far & ((u64)(a >> 64)); 766 767 /* if kPrime[i] == out[i] then |equal| will be all zeros and the 768 * decrement will make it all ones. */ 769 equal = kPrime[i] ^ out[i]; 770 equal--; 771 equal &= equal << 32; 772 equal &= equal << 16; 773 equal &= equal << 8; 774 equal &= equal << 4; 775 equal &= equal << 2; 776 equal &= equal << 1; 777 equal = ((s64)equal) >> 63; 778 779 all_equal_so_far &= equal; 780 } 781 782 /* if all_equal_so_far is still all ones then the two values are equal 783 * and so out >= kPrime is true. */ 784 result |= all_equal_so_far; 785 786 /* if out >= kPrime then we subtract kPrime. */ 787 u64 carry; 788 subtract_u64(&out[0], &carry, result & kPrime[0]); 789 subtract_u64(&out[1], &carry, carry); 790 subtract_u64(&out[2], &carry, carry); 791 subtract_u64(&out[3], &carry, carry); 792 793 subtract_u64(&out[1], &carry, result & kPrime[1]); 794 subtract_u64(&out[2], &carry, carry); 795 subtract_u64(&out[3], &carry, carry); 796 797 subtract_u64(&out[2], &carry, result & kPrime[2]); 798 subtract_u64(&out[3], &carry, carry); 799 800 subtract_u64(&out[3], &carry, result & kPrime[3]); 801 } 802 803 static void smallfelem_square_contract(smallfelem out, const smallfelem in) { 804 longfelem longtmp; 805 felem tmp; 806 807 smallfelem_square(longtmp, in); 808 felem_reduce(tmp, longtmp); 809 felem_contract(out, tmp); 810 } 811 812 static void smallfelem_mul_contract(smallfelem out, const smallfelem in1, 813 const smallfelem in2) { 814 longfelem longtmp; 815 felem tmp; 816 817 smallfelem_mul(longtmp, in1, in2); 818 felem_reduce(tmp, longtmp); 819 felem_contract(out, tmp); 820 } 821 822 /* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 823 * otherwise. 824 * On entry: 825 * small[i] < 2^64 */ 826 static limb smallfelem_is_zero(const smallfelem small) { 827 limb result; 828 u64 is_p; 829 830 u64 is_zero = small[0] | small[1] | small[2] | small[3]; 831 is_zero--; 832 is_zero &= is_zero << 32; 833 is_zero &= is_zero << 16; 834 is_zero &= is_zero << 8; 835 is_zero &= is_zero << 4; 836 is_zero &= is_zero << 2; 837 is_zero &= is_zero << 1; 838 is_zero = ((s64)is_zero) >> 63; 839 840 is_p = (small[0] ^ kPrime[0]) | (small[1] ^ kPrime[1]) | 841 (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]); 842 is_p--; 843 is_p &= is_p << 32; 844 is_p &= is_p << 16; 845 is_p &= is_p << 8; 846 is_p &= is_p << 4; 847 is_p &= is_p << 2; 848 is_p &= is_p << 1; 849 is_p = ((s64)is_p) >> 63; 850 851 is_zero |= is_p; 852 853 result = is_zero; 854 result |= ((limb)is_zero) << 64; 855 return result; 856 } 857 858 static int smallfelem_is_zero_int(const smallfelem small) { 859 return (int)(smallfelem_is_zero(small) & ((limb)1)); 860 } 861 862 /* felem_inv calculates |out| = |in|^{-1} 863 * 864 * Based on Fermat's Little Theorem: 865 * a^p = a (mod p) 866 * a^{p-1} = 1 (mod p) 867 * a^{p-2} = a^{-1} (mod p) */ 868 static void felem_inv(felem out, const felem in) { 869 felem ftmp, ftmp2; 870 /* each e_I will hold |in|^{2^I - 1} */ 871 felem e2, e4, e8, e16, e32, e64; 872 longfelem tmp; 873 unsigned i; 874 875 felem_square(tmp, in); 876 felem_reduce(ftmp, tmp); /* 2^1 */ 877 felem_mul(tmp, in, ftmp); 878 felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */ 879 felem_assign(e2, ftmp); 880 felem_square(tmp, ftmp); 881 felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */ 882 felem_square(tmp, ftmp); 883 felem_reduce(ftmp, tmp); /* 2^4 - 2^2 */ 884 felem_mul(tmp, ftmp, e2); 885 felem_reduce(ftmp, tmp); /* 2^4 - 2^0 */ 886 felem_assign(e4, ftmp); 887 felem_square(tmp, ftmp); 888 felem_reduce(ftmp, tmp); /* 2^5 - 2^1 */ 889 felem_square(tmp, ftmp); 890 felem_reduce(ftmp, tmp); /* 2^6 - 2^2 */ 891 felem_square(tmp, ftmp); 892 felem_reduce(ftmp, tmp); /* 2^7 - 2^3 */ 893 felem_square(tmp, ftmp); 894 felem_reduce(ftmp, tmp); /* 2^8 - 2^4 */ 895 felem_mul(tmp, ftmp, e4); 896 felem_reduce(ftmp, tmp); /* 2^8 - 2^0 */ 897 felem_assign(e8, ftmp); 898 for (i = 0; i < 8; i++) { 899 felem_square(tmp, ftmp); 900 felem_reduce(ftmp, tmp); 901 } /* 2^16 - 2^8 */ 902 felem_mul(tmp, ftmp, e8); 903 felem_reduce(ftmp, tmp); /* 2^16 - 2^0 */ 904 felem_assign(e16, ftmp); 905 for (i = 0; i < 16; i++) { 906 felem_square(tmp, ftmp); 907 felem_reduce(ftmp, tmp); 908 } /* 2^32 - 2^16 */ 909 felem_mul(tmp, ftmp, e16); 910 felem_reduce(ftmp, tmp); /* 2^32 - 2^0 */ 911 felem_assign(e32, ftmp); 912 for (i = 0; i < 32; i++) { 913 felem_square(tmp, ftmp); 914 felem_reduce(ftmp, tmp); 915 } /* 2^64 - 2^32 */ 916 felem_assign(e64, ftmp); 917 felem_mul(tmp, ftmp, in); 918 felem_reduce(ftmp, tmp); /* 2^64 - 2^32 + 2^0 */ 919 for (i = 0; i < 192; i++) { 920 felem_square(tmp, ftmp); 921 felem_reduce(ftmp, tmp); 922 } /* 2^256 - 2^224 + 2^192 */ 923 924 felem_mul(tmp, e64, e32); 925 felem_reduce(ftmp2, tmp); /* 2^64 - 2^0 */ 926 for (i = 0; i < 16; i++) { 927 felem_square(tmp, ftmp2); 928 felem_reduce(ftmp2, tmp); 929 } /* 2^80 - 2^16 */ 930 felem_mul(tmp, ftmp2, e16); 931 felem_reduce(ftmp2, tmp); /* 2^80 - 2^0 */ 932 for (i = 0; i < 8; i++) { 933 felem_square(tmp, ftmp2); 934 felem_reduce(ftmp2, tmp); 935 } /* 2^88 - 2^8 */ 936 felem_mul(tmp, ftmp2, e8); 937 felem_reduce(ftmp2, tmp); /* 2^88 - 2^0 */ 938 for (i = 0; i < 4; i++) { 939 felem_square(tmp, ftmp2); 940 felem_reduce(ftmp2, tmp); 941 } /* 2^92 - 2^4 */ 942 felem_mul(tmp, ftmp2, e4); 943 felem_reduce(ftmp2, tmp); /* 2^92 - 2^0 */ 944 felem_square(tmp, ftmp2); 945 felem_reduce(ftmp2, tmp); /* 2^93 - 2^1 */ 946 felem_square(tmp, ftmp2); 947 felem_reduce(ftmp2, tmp); /* 2^94 - 2^2 */ 948 felem_mul(tmp, ftmp2, e2); 949 felem_reduce(ftmp2, tmp); /* 2^94 - 2^0 */ 950 felem_square(tmp, ftmp2); 951 felem_reduce(ftmp2, tmp); /* 2^95 - 2^1 */ 952 felem_square(tmp, ftmp2); 953 felem_reduce(ftmp2, tmp); /* 2^96 - 2^2 */ 954 felem_mul(tmp, ftmp2, in); 955 felem_reduce(ftmp2, tmp); /* 2^96 - 3 */ 956 957 felem_mul(tmp, ftmp2, ftmp); 958 felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */ 959 } 960 961 static void smallfelem_inv_contract(smallfelem out, const smallfelem in) { 962 felem tmp; 963 964 smallfelem_expand(tmp, in); 965 felem_inv(tmp, tmp); 966 felem_contract(out, tmp); 967 } 968 969 /* Group operations 970 * ---------------- 971 * 972 * Building on top of the field operations we have the operations on the 973 * elliptic curve group itself. Points on the curve are represented in Jacobian 974 * coordinates. */ 975 976 /* point_double calculates 2*(x_in, y_in, z_in) 977 * 978 * The method is taken from: 979 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b 980 * 981 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. 982 * while x_out == y_in is not (maybe this works, but it's not tested). */ 983 static void point_double(felem x_out, felem y_out, felem z_out, 984 const felem x_in, const felem y_in, const felem z_in) { 985 longfelem tmp, tmp2; 986 felem delta, gamma, beta, alpha, ftmp, ftmp2; 987 smallfelem small1, small2; 988 989 felem_assign(ftmp, x_in); 990 /* ftmp[i] < 2^106 */ 991 felem_assign(ftmp2, x_in); 992 /* ftmp2[i] < 2^106 */ 993 994 /* delta = z^2 */ 995 felem_square(tmp, z_in); 996 felem_reduce(delta, tmp); 997 /* delta[i] < 2^101 */ 998 999 /* gamma = y^2 */ 1000 felem_square(tmp, y_in); 1001 felem_reduce(gamma, tmp); 1002 /* gamma[i] < 2^101 */ 1003 felem_shrink(small1, gamma); 1004 1005 /* beta = x*gamma */ 1006 felem_small_mul(tmp, small1, x_in); 1007 felem_reduce(beta, tmp); 1008 /* beta[i] < 2^101 */ 1009 1010 /* alpha = 3*(x-delta)*(x+delta) */ 1011 felem_diff(ftmp, delta); 1012 /* ftmp[i] < 2^105 + 2^106 < 2^107 */ 1013 felem_sum(ftmp2, delta); 1014 /* ftmp2[i] < 2^105 + 2^106 < 2^107 */ 1015 felem_scalar(ftmp2, 3); 1016 /* ftmp2[i] < 3 * 2^107 < 2^109 */ 1017 felem_mul(tmp, ftmp, ftmp2); 1018 felem_reduce(alpha, tmp); 1019 /* alpha[i] < 2^101 */ 1020 felem_shrink(small2, alpha); 1021 1022 /* x' = alpha^2 - 8*beta */ 1023 smallfelem_square(tmp, small2); 1024 felem_reduce(x_out, tmp); 1025 felem_assign(ftmp, beta); 1026 felem_scalar(ftmp, 8); 1027 /* ftmp[i] < 8 * 2^101 = 2^104 */ 1028 felem_diff(x_out, ftmp); 1029 /* x_out[i] < 2^105 + 2^101 < 2^106 */ 1030 1031 /* z' = (y + z)^2 - gamma - delta */ 1032 felem_sum(delta, gamma); 1033 /* delta[i] < 2^101 + 2^101 = 2^102 */ 1034 felem_assign(ftmp, y_in); 1035 felem_sum(ftmp, z_in); 1036 /* ftmp[i] < 2^106 + 2^106 = 2^107 */ 1037 felem_square(tmp, ftmp); 1038 felem_reduce(z_out, tmp); 1039 felem_diff(z_out, delta); 1040 /* z_out[i] < 2^105 + 2^101 < 2^106 */ 1041 1042 /* y' = alpha*(4*beta - x') - 8*gamma^2 */ 1043 felem_scalar(beta, 4); 1044 /* beta[i] < 4 * 2^101 = 2^103 */ 1045 felem_diff_zero107(beta, x_out); 1046 /* beta[i] < 2^107 + 2^103 < 2^108 */ 1047 felem_small_mul(tmp, small2, beta); 1048 /* tmp[i] < 7 * 2^64 < 2^67 */ 1049 smallfelem_square(tmp2, small1); 1050 /* tmp2[i] < 7 * 2^64 */ 1051 longfelem_scalar(tmp2, 8); 1052 /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */ 1053 longfelem_diff(tmp, tmp2); 1054 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */ 1055 felem_reduce_zero105(y_out, tmp); 1056 /* y_out[i] < 2^106 */ 1057 } 1058 1059 /* point_double_small is the same as point_double, except that it operates on 1060 * smallfelems. */ 1061 static void point_double_small(smallfelem x_out, smallfelem y_out, 1062 smallfelem z_out, const smallfelem x_in, 1063 const smallfelem y_in, const smallfelem z_in) { 1064 felem felem_x_out, felem_y_out, felem_z_out; 1065 felem felem_x_in, felem_y_in, felem_z_in; 1066 1067 smallfelem_expand(felem_x_in, x_in); 1068 smallfelem_expand(felem_y_in, y_in); 1069 smallfelem_expand(felem_z_in, z_in); 1070 point_double(felem_x_out, felem_y_out, felem_z_out, felem_x_in, felem_y_in, 1071 felem_z_in); 1072 felem_shrink(x_out, felem_x_out); 1073 felem_shrink(y_out, felem_y_out); 1074 felem_shrink(z_out, felem_z_out); 1075 } 1076 1077 /* copy_conditional copies in to out iff mask is all ones. */ 1078 static void copy_conditional(felem out, const felem in, limb mask) { 1079 unsigned i; 1080 for (i = 0; i < NLIMBS; ++i) { 1081 const limb tmp = mask & (in[i] ^ out[i]); 1082 out[i] ^= tmp; 1083 } 1084 } 1085 1086 /* copy_small_conditional copies in to out iff mask is all ones. */ 1087 static void copy_small_conditional(felem out, const smallfelem in, limb mask) { 1088 unsigned i; 1089 const u64 mask64 = mask; 1090 for (i = 0; i < NLIMBS; ++i) { 1091 out[i] = ((limb)(in[i] & mask64)) | (out[i] & ~mask); 1092 } 1093 } 1094 1095 /* point_add calcuates (x1, y1, z1) + (x2, y2, z2) 1096 * 1097 * The method is taken from: 1098 * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, 1099 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). 1100 * 1101 * This function includes a branch for checking whether the two input points 1102 * are equal, (while not equal to the point at infinity). This case never 1103 * happens during single point multiplication, so there is no timing leak for 1104 * ECDH or ECDSA signing. */ 1105 static void point_add(felem x3, felem y3, felem z3, const felem x1, 1106 const felem y1, const felem z1, const int mixed, 1107 const smallfelem x2, const smallfelem y2, 1108 const smallfelem z2) { 1109 felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out; 1110 longfelem tmp, tmp2; 1111 smallfelem small1, small2, small3, small4, small5; 1112 limb x_equal, y_equal, z1_is_zero, z2_is_zero; 1113 1114 felem_shrink(small3, z1); 1115 1116 z1_is_zero = smallfelem_is_zero(small3); 1117 z2_is_zero = smallfelem_is_zero(z2); 1118 1119 /* ftmp = z1z1 = z1**2 */ 1120 smallfelem_square(tmp, small3); 1121 felem_reduce(ftmp, tmp); 1122 /* ftmp[i] < 2^101 */ 1123 felem_shrink(small1, ftmp); 1124 1125 if (!mixed) { 1126 /* ftmp2 = z2z2 = z2**2 */ 1127 smallfelem_square(tmp, z2); 1128 felem_reduce(ftmp2, tmp); 1129 /* ftmp2[i] < 2^101 */ 1130 felem_shrink(small2, ftmp2); 1131 1132 felem_shrink(small5, x1); 1133 1134 /* u1 = ftmp3 = x1*z2z2 */ 1135 smallfelem_mul(tmp, small5, small2); 1136 felem_reduce(ftmp3, tmp); 1137 /* ftmp3[i] < 2^101 */ 1138 1139 /* ftmp5 = z1 + z2 */ 1140 felem_assign(ftmp5, z1); 1141 felem_small_sum(ftmp5, z2); 1142 /* ftmp5[i] < 2^107 */ 1143 1144 /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */ 1145 felem_square(tmp, ftmp5); 1146 felem_reduce(ftmp5, tmp); 1147 /* ftmp2 = z2z2 + z1z1 */ 1148 felem_sum(ftmp2, ftmp); 1149 /* ftmp2[i] < 2^101 + 2^101 = 2^102 */ 1150 felem_diff(ftmp5, ftmp2); 1151 /* ftmp5[i] < 2^105 + 2^101 < 2^106 */ 1152 1153 /* ftmp2 = z2 * z2z2 */ 1154 smallfelem_mul(tmp, small2, z2); 1155 felem_reduce(ftmp2, tmp); 1156 1157 /* s1 = ftmp2 = y1 * z2**3 */ 1158 felem_mul(tmp, y1, ftmp2); 1159 felem_reduce(ftmp6, tmp); 1160 /* ftmp6[i] < 2^101 */ 1161 } else { 1162 /* We'll assume z2 = 1 (special case z2 = 0 is handled later). */ 1163 1164 /* u1 = ftmp3 = x1*z2z2 */ 1165 felem_assign(ftmp3, x1); 1166 /* ftmp3[i] < 2^106 */ 1167 1168 /* ftmp5 = 2z1z2 */ 1169 felem_assign(ftmp5, z1); 1170 felem_scalar(ftmp5, 2); 1171 /* ftmp5[i] < 2*2^106 = 2^107 */ 1172 1173 /* s1 = ftmp2 = y1 * z2**3 */ 1174 felem_assign(ftmp6, y1); 1175 /* ftmp6[i] < 2^106 */ 1176 } 1177 1178 /* u2 = x2*z1z1 */ 1179 smallfelem_mul(tmp, x2, small1); 1180 felem_reduce(ftmp4, tmp); 1181 1182 /* h = ftmp4 = u2 - u1 */ 1183 felem_diff_zero107(ftmp4, ftmp3); 1184 /* ftmp4[i] < 2^107 + 2^101 < 2^108 */ 1185 felem_shrink(small4, ftmp4); 1186 1187 x_equal = smallfelem_is_zero(small4); 1188 1189 /* z_out = ftmp5 * h */ 1190 felem_small_mul(tmp, small4, ftmp5); 1191 felem_reduce(z_out, tmp); 1192 /* z_out[i] < 2^101 */ 1193 1194 /* ftmp = z1 * z1z1 */ 1195 smallfelem_mul(tmp, small1, small3); 1196 felem_reduce(ftmp, tmp); 1197 1198 /* s2 = tmp = y2 * z1**3 */ 1199 felem_small_mul(tmp, y2, ftmp); 1200 felem_reduce(ftmp5, tmp); 1201 1202 /* r = ftmp5 = (s2 - s1)*2 */ 1203 felem_diff_zero107(ftmp5, ftmp6); 1204 /* ftmp5[i] < 2^107 + 2^107 = 2^108 */ 1205 felem_scalar(ftmp5, 2); 1206 /* ftmp5[i] < 2^109 */ 1207 felem_shrink(small1, ftmp5); 1208 y_equal = smallfelem_is_zero(small1); 1209 1210 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { 1211 point_double(x3, y3, z3, x1, y1, z1); 1212 return; 1213 } 1214 1215 /* I = ftmp = (2h)**2 */ 1216 felem_assign(ftmp, ftmp4); 1217 felem_scalar(ftmp, 2); 1218 /* ftmp[i] < 2*2^108 = 2^109 */ 1219 felem_square(tmp, ftmp); 1220 felem_reduce(ftmp, tmp); 1221 1222 /* J = ftmp2 = h * I */ 1223 felem_mul(tmp, ftmp4, ftmp); 1224 felem_reduce(ftmp2, tmp); 1225 1226 /* V = ftmp4 = U1 * I */ 1227 felem_mul(tmp, ftmp3, ftmp); 1228 felem_reduce(ftmp4, tmp); 1229 1230 /* x_out = r**2 - J - 2V */ 1231 smallfelem_square(tmp, small1); 1232 felem_reduce(x_out, tmp); 1233 felem_assign(ftmp3, ftmp4); 1234 felem_scalar(ftmp4, 2); 1235 felem_sum(ftmp4, ftmp2); 1236 /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */ 1237 felem_diff(x_out, ftmp4); 1238 /* x_out[i] < 2^105 + 2^101 */ 1239 1240 /* y_out = r(V-x_out) - 2 * s1 * J */ 1241 felem_diff_zero107(ftmp3, x_out); 1242 /* ftmp3[i] < 2^107 + 2^101 < 2^108 */ 1243 felem_small_mul(tmp, small1, ftmp3); 1244 felem_mul(tmp2, ftmp6, ftmp2); 1245 longfelem_scalar(tmp2, 2); 1246 /* tmp2[i] < 2*2^67 = 2^68 */ 1247 longfelem_diff(tmp, tmp2); 1248 /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */ 1249 felem_reduce_zero105(y_out, tmp); 1250 /* y_out[i] < 2^106 */ 1251 1252 copy_small_conditional(x_out, x2, z1_is_zero); 1253 copy_conditional(x_out, x1, z2_is_zero); 1254 copy_small_conditional(y_out, y2, z1_is_zero); 1255 copy_conditional(y_out, y1, z2_is_zero); 1256 copy_small_conditional(z_out, z2, z1_is_zero); 1257 copy_conditional(z_out, z1, z2_is_zero); 1258 felem_assign(x3, x_out); 1259 felem_assign(y3, y_out); 1260 felem_assign(z3, z_out); 1261 } 1262 1263 /* point_add_small is the same as point_add, except that it operates on 1264 * smallfelems. */ 1265 static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3, 1266 smallfelem x1, smallfelem y1, smallfelem z1, 1267 smallfelem x2, smallfelem y2, smallfelem z2) { 1268 felem felem_x3, felem_y3, felem_z3; 1269 felem felem_x1, felem_y1, felem_z1; 1270 smallfelem_expand(felem_x1, x1); 1271 smallfelem_expand(felem_y1, y1); 1272 smallfelem_expand(felem_z1, z1); 1273 point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, x2, 1274 y2, z2); 1275 felem_shrink(x3, felem_x3); 1276 felem_shrink(y3, felem_y3); 1277 felem_shrink(z3, felem_z3); 1278 } 1279 1280 /* Base point pre computation 1281 * -------------------------- 1282 * 1283 * Two different sorts of precomputed tables are used in the following code. 1284 * Each contain various points on the curve, where each point is three field 1285 * elements (x, y, z). 1286 * 1287 * For the base point table, z is usually 1 (0 for the point at infinity). 1288 * This table has 2 * 16 elements, starting with the following: 1289 * index | bits | point 1290 * ------+---------+------------------------------ 1291 * 0 | 0 0 0 0 | 0G 1292 * 1 | 0 0 0 1 | 1G 1293 * 2 | 0 0 1 0 | 2^64G 1294 * 3 | 0 0 1 1 | (2^64 + 1)G 1295 * 4 | 0 1 0 0 | 2^128G 1296 * 5 | 0 1 0 1 | (2^128 + 1)G 1297 * 6 | 0 1 1 0 | (2^128 + 2^64)G 1298 * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G 1299 * 8 | 1 0 0 0 | 2^192G 1300 * 9 | 1 0 0 1 | (2^192 + 1)G 1301 * 10 | 1 0 1 0 | (2^192 + 2^64)G 1302 * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G 1303 * 12 | 1 1 0 0 | (2^192 + 2^128)G 1304 * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G 1305 * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G 1306 * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G 1307 * followed by a copy of this with each element multiplied by 2^32. 1308 * 1309 * The reason for this is so that we can clock bits into four different 1310 * locations when doing simple scalar multiplies against the base point, 1311 * and then another four locations using the second 16 elements. 1312 * 1313 * Tables for other points have table[i] = iG for i in 0 .. 16. */ 1314 1315 /* gmul is the table of precomputed base points */ 1316 static const smallfelem gmul[2][16][3] = { 1317 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 1318 {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2, 1319 0x6b17d1f2e12c4247}, 1320 {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16, 1321 0x4fe342e2fe1a7f9b}, 1322 {1, 0, 0, 0}}, 1323 {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de, 1324 0x0fa822bc2811aaa5}, 1325 {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b, 1326 0xbff44ae8f5dba80d}, 1327 {1, 0, 0, 0}}, 1328 {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789, 1329 0x300a4bbc89d6726f}, 1330 {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f, 1331 0x72aac7e0d09b4644}, 1332 {1, 0, 0, 0}}, 1333 {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e, 1334 0x447d739beedb5e67}, 1335 {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7, 1336 0x2d4825ab834131ee}, 1337 {1, 0, 0, 0}}, 1338 {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60, 1339 0xef9519328a9c72ff}, 1340 {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c, 1341 0x611e9fc37dbb2c9b}, 1342 {1, 0, 0, 0}}, 1343 {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf, 1344 0x550663797b51f5d8}, 1345 {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5, 1346 0x157164848aecb851}, 1347 {1, 0, 0, 0}}, 1348 {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391, 1349 0xeb5d7745b21141ea}, 1350 {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee, 1351 0xeafd72ebdbecc17b}, 1352 {1, 0, 0, 0}}, 1353 {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5, 1354 0xa6d39677a7849276}, 1355 {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf, 1356 0x674f84749b0b8816}, 1357 {1, 0, 0, 0}}, 1358 {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb, 1359 0x4e769e7672c9ddad}, 1360 {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281, 1361 0x42b99082de830663}, 1362 {1, 0, 0, 0}}, 1363 {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478, 1364 0x78878ef61c6ce04d}, 1365 {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def, 1366 0xb6cb3f5d7b72c321}, 1367 {1, 0, 0, 0}}, 1368 {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae, 1369 0x0c88bc4d716b1287}, 1370 {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa, 1371 0xdd5ddea3f3901dc6}, 1372 {1, 0, 0, 0}}, 1373 {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3, 1374 0x68f344af6b317466}, 1375 {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3, 1376 0x31b9c405f8540a20}, 1377 {1, 0, 0, 0}}, 1378 {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0, 1379 0x4052bf4b6f461db9}, 1380 {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8, 1381 0xfecf4d5190b0fc61}, 1382 {1, 0, 0, 0}}, 1383 {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a, 1384 0x1eddbae2c802e41a}, 1385 {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0, 1386 0x43104d86560ebcfc}, 1387 {1, 0, 0, 0}}, 1388 {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a, 1389 0xb48e26b484f7a21c}, 1390 {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668, 1391 0xfac015404d4d3dab}, 1392 {1, 0, 0, 0}}}, 1393 {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, 1394 {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da, 1395 0x7fe36b40af22af89}, 1396 {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1, 1397 0xe697d45825b63624}, 1398 {1, 0, 0, 0}}, 1399 {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902, 1400 0x4a5b506612a677a6}, 1401 {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40, 1402 0xeb13461ceac089f1}, 1403 {1, 0, 0, 0}}, 1404 {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857, 1405 0x0781b8291c6a220a}, 1406 {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434, 1407 0x690cde8df0151593}, 1408 {1, 0, 0, 0}}, 1409 {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326, 1410 0x8a535f566ec73617}, 1411 {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf, 1412 0x0455c08468b08bd7}, 1413 {1, 0, 0, 0}}, 1414 {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279, 1415 0x06bada7ab77f8276}, 1416 {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70, 1417 0x5b476dfd0e6cb18a}, 1418 {1, 0, 0, 0}}, 1419 {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8, 1420 0x3e29864e8a2ec908}, 1421 {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed, 1422 0x239b90ea3dc31e7e}, 1423 {1, 0, 0, 0}}, 1424 {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4, 1425 0x820f4dd949f72ff7}, 1426 {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3, 1427 0x140406ec783a05ec}, 1428 {1, 0, 0, 0}}, 1429 {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe, 1430 0x68f6b8542783dfee}, 1431 {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028, 1432 0xcbe1feba92e40ce6}, 1433 {1, 0, 0, 0}}, 1434 {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927, 1435 0xd0b2f94d2f420109}, 1436 {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a, 1437 0x971459828b0719e5}, 1438 {1, 0, 0, 0}}, 1439 {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687, 1440 0x961610004a866aba}, 1441 {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c, 1442 0x7acb9fadcee75e44}, 1443 {1, 0, 0, 0}}, 1444 {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea, 1445 0x24eb9acca333bf5b}, 1446 {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d, 1447 0x69f891c5acd079cc}, 1448 {1, 0, 0, 0}}, 1449 {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514, 1450 0xe51f547c5972a107}, 1451 {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06, 1452 0x1c309a2b25bb1387}, 1453 {1, 0, 0, 0}}, 1454 {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828, 1455 0x20b87b8aa2c4e503}, 1456 {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044, 1457 0xf5c6fa49919776be}, 1458 {1, 0, 0, 0}}, 1459 {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56, 1460 0x1ed7d1b9332010b9}, 1461 {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24, 1462 0x3a2b03f03217257a}, 1463 {1, 0, 0, 0}}, 1464 {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b, 1465 0x15fee545c78dd9f6}, 1466 {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb, 1467 0x4ab5b6b2b8753f81}, 1468 {1, 0, 0, 0}}}}; 1469 1470 /* select_point selects the |idx|th point from a precomputation table and 1471 * copies it to out. */ 1472 static void select_point(const u64 idx, unsigned int size, 1473 const smallfelem pre_comp[16][3], smallfelem out[3]) { 1474 unsigned i, j; 1475 u64 *outlimbs = &out[0][0]; 1476 memset(outlimbs, 0, 3 * sizeof(smallfelem)); 1477 1478 for (i = 0; i < size; i++) { 1479 const u64 *inlimbs = (u64 *)&pre_comp[i][0][0]; 1480 u64 mask = i ^ idx; 1481 mask |= mask >> 4; 1482 mask |= mask >> 2; 1483 mask |= mask >> 1; 1484 mask &= 1; 1485 mask--; 1486 for (j = 0; j < NLIMBS * 3; j++) { 1487 outlimbs[j] |= inlimbs[j] & mask; 1488 } 1489 } 1490 } 1491 1492 /* get_bit returns the |i|th bit in |in| */ 1493 static char get_bit(const felem_bytearray in, int i) { 1494 if (i < 0 || i >= 256) { 1495 return 0; 1496 } 1497 return (in[i >> 3] >> (i & 7)) & 1; 1498 } 1499 1500 /* Interleaved point multiplication using precomputed point multiples: The 1501 * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars 1502 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the 1503 * generator, using certain (large) precomputed multiples in g_pre_comp. 1504 * Output point (X, Y, Z) is stored in x_out, y_out, z_out. */ 1505 static void batch_mul(felem x_out, felem y_out, felem z_out, 1506 const felem_bytearray scalars[], 1507 const unsigned num_points, const u8 *g_scalar, 1508 const int mixed, const smallfelem pre_comp[][17][3], 1509 const smallfelem g_pre_comp[2][16][3]) { 1510 int i, skip; 1511 unsigned num, gen_mul = (g_scalar != NULL); 1512 felem nq[3], ftmp; 1513 smallfelem tmp[3]; 1514 u64 bits; 1515 u8 sign, digit; 1516 1517 /* set nq to the point at infinity */ 1518 memset(nq, 0, 3 * sizeof(felem)); 1519 1520 /* Loop over all scalars msb-to-lsb, interleaving additions of multiples 1521 * of the generator (two in each of the last 32 rounds) and additions of 1522 * other points multiples (every 5th round). */ 1523 1524 skip = 1; /* save two point operations in the first 1525 * round */ 1526 for (i = (num_points ? 255 : 31); i >= 0; --i) { 1527 /* double */ 1528 if (!skip) { 1529 point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); 1530 } 1531 1532 /* add multiples of the generator */ 1533 if (gen_mul && i <= 31) { 1534 /* first, look 32 bits upwards */ 1535 bits = get_bit(g_scalar, i + 224) << 3; 1536 bits |= get_bit(g_scalar, i + 160) << 2; 1537 bits |= get_bit(g_scalar, i + 96) << 1; 1538 bits |= get_bit(g_scalar, i + 32); 1539 /* select the point to add, in constant time */ 1540 select_point(bits, 16, g_pre_comp[1], tmp); 1541 1542 if (!skip) { 1543 /* Arg 1 below is for "mixed" */ 1544 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], 1545 tmp[2]); 1546 } else { 1547 smallfelem_expand(nq[0], tmp[0]); 1548 smallfelem_expand(nq[1], tmp[1]); 1549 smallfelem_expand(nq[2], tmp[2]); 1550 skip = 0; 1551 } 1552 1553 /* second, look at the current position */ 1554 bits = get_bit(g_scalar, i + 192) << 3; 1555 bits |= get_bit(g_scalar, i + 128) << 2; 1556 bits |= get_bit(g_scalar, i + 64) << 1; 1557 bits |= get_bit(g_scalar, i); 1558 /* select the point to add, in constant time */ 1559 select_point(bits, 16, g_pre_comp[0], tmp); 1560 /* Arg 1 below is for "mixed" */ 1561 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], 1562 tmp[2]); 1563 } 1564 1565 /* do other additions every 5 doublings */ 1566 if (num_points && (i % 5 == 0)) { 1567 /* loop over all scalars */ 1568 for (num = 0; num < num_points; ++num) { 1569 bits = get_bit(scalars[num], i + 4) << 5; 1570 bits |= get_bit(scalars[num], i + 3) << 4; 1571 bits |= get_bit(scalars[num], i + 2) << 3; 1572 bits |= get_bit(scalars[num], i + 1) << 2; 1573 bits |= get_bit(scalars[num], i) << 1; 1574 bits |= get_bit(scalars[num], i - 1); 1575 ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); 1576 1577 /* select the point to add or subtract, in constant time. */ 1578 select_point(digit, 17, pre_comp[num], tmp); 1579 smallfelem_neg(ftmp, tmp[1]); /* (X, -Y, Z) is the negative 1580 * point */ 1581 copy_small_conditional(ftmp, tmp[1], (((limb)sign) - 1)); 1582 felem_contract(tmp[1], ftmp); 1583 1584 if (!skip) { 1585 point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], mixed, tmp[0], 1586 tmp[1], tmp[2]); 1587 } else { 1588 smallfelem_expand(nq[0], tmp[0]); 1589 smallfelem_expand(nq[1], tmp[1]); 1590 smallfelem_expand(nq[2], tmp[2]); 1591 skip = 0; 1592 } 1593 } 1594 } 1595 } 1596 felem_assign(x_out, nq[0]); 1597 felem_assign(y_out, nq[1]); 1598 felem_assign(z_out, nq[2]); 1599 } 1600 1601 /* Precomputation for the group generator. */ 1602 typedef struct { 1603 smallfelem g_pre_comp[2][16][3]; 1604 } NISTP256_PRE_COMP; 1605 1606 /******************************************************************************/ 1607 /* 1608 * OPENSSL EC_METHOD FUNCTIONS 1609 */ 1610 1611 int ec_GFp_nistp256_group_init(EC_GROUP *group) { 1612 int ret = ec_GFp_simple_group_init(group); 1613 group->a_is_minus3 = 1; 1614 return ret; 1615 } 1616 1617 int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p, 1618 const BIGNUM *a, const BIGNUM *b, 1619 BN_CTX *ctx) { 1620 int ret = 0; 1621 BN_CTX *new_ctx = NULL; 1622 BIGNUM *curve_p, *curve_a, *curve_b; 1623 1624 if (ctx == NULL) { 1625 if ((ctx = new_ctx = BN_CTX_new()) == NULL) { 1626 return 0; 1627 } 1628 } 1629 BN_CTX_start(ctx); 1630 if (((curve_p = BN_CTX_get(ctx)) == NULL) || 1631 ((curve_a = BN_CTX_get(ctx)) == NULL) || 1632 ((curve_b = BN_CTX_get(ctx)) == NULL)) { 1633 goto err; 1634 } 1635 BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p); 1636 BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a); 1637 BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b); 1638 if (BN_cmp(curve_p, p) || 1639 BN_cmp(curve_a, a) || 1640 BN_cmp(curve_b, b)) { 1641 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_group_set_curve, 1642 EC_R_WRONG_CURVE_PARAMETERS); 1643 goto err; 1644 } 1645 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); 1646 1647 err: 1648 BN_CTX_end(ctx); 1649 BN_CTX_free(new_ctx); 1650 return ret; 1651 } 1652 1653 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = 1654 * (X/Z^2, Y/Z^3). */ 1655 int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group, 1656 const EC_POINT *point, 1657 BIGNUM *x, BIGNUM *y, 1658 BN_CTX *ctx) { 1659 felem z1, z2, x_in, y_in; 1660 smallfelem x_out, y_out; 1661 longfelem tmp; 1662 1663 if (EC_POINT_is_at_infinity(group, point)) { 1664 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates, 1665 EC_R_POINT_AT_INFINITY); 1666 return 0; 1667 } 1668 if (!BN_to_felem(x_in, &point->X) || 1669 !BN_to_felem(y_in, &point->Y) || 1670 !BN_to_felem(z1, &point->Z)) { 1671 return 0; 1672 } 1673 felem_inv(z2, z1); 1674 felem_square(tmp, z2); 1675 felem_reduce(z1, tmp); 1676 felem_mul(tmp, x_in, z1); 1677 felem_reduce(x_in, tmp); 1678 felem_contract(x_out, x_in); 1679 if (x != NULL && !smallfelem_to_BN(x, x_out)) { 1680 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates, 1681 ERR_R_BN_LIB); 1682 return 0; 1683 } 1684 felem_mul(tmp, z1, z2); 1685 felem_reduce(z1, tmp); 1686 felem_mul(tmp, y_in, z1); 1687 felem_reduce(y_in, tmp); 1688 felem_contract(y_out, y_in); 1689 if (y != NULL && !smallfelem_to_BN(y, y_out)) { 1690 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates, 1691 ERR_R_BN_LIB); 1692 return 0; 1693 } 1694 return 1; 1695 } 1696 1697 /* points below is of size |num|, and tmp_smallfelems is of size |num+1| */ 1698 static void make_points_affine(size_t num, smallfelem points[][3], 1699 smallfelem tmp_smallfelems[]) { 1700 /* Runs in constant time, unless an input is the point at infinity (which 1701 * normally shouldn't happen). */ 1702 ec_GFp_nistp_points_make_affine_internal( 1703 num, points, sizeof(smallfelem), tmp_smallfelems, 1704 (void (*)(void *))smallfelem_one, 1705 (int (*)(const void *))smallfelem_is_zero_int, 1706 (void (*)(void *, const void *))smallfelem_assign, 1707 (void (*)(void *, const void *))smallfelem_square_contract, 1708 (void (*)(void *, const void *, const void *))smallfelem_mul_contract, 1709 (void (*)(void *, const void *))smallfelem_inv_contract, 1710 /* nothing to contract */ 1711 (void (*)(void *, const void *))smallfelem_assign); 1712 } 1713 1714 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL 1715 * values Result is stored in r (r can equal one of the inputs). */ 1716 int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r, 1717 const BIGNUM *scalar, size_t num, 1718 const EC_POINT *points[], 1719 const BIGNUM *scalars[], BN_CTX *ctx) { 1720 int ret = 0; 1721 int j; 1722 int mixed = 0; 1723 BN_CTX *new_ctx = NULL; 1724 BIGNUM *x, *y, *z, *tmp_scalar; 1725 felem_bytearray g_secret; 1726 felem_bytearray *secrets = NULL; 1727 smallfelem(*pre_comp)[17][3] = NULL; 1728 smallfelem *tmp_smallfelems = NULL; 1729 felem_bytearray tmp; 1730 unsigned i, num_bytes; 1731 int have_pre_comp = 0; 1732 size_t num_points = num; 1733 smallfelem x_in, y_in, z_in; 1734 felem x_out, y_out, z_out; 1735 const smallfelem(*g_pre_comp)[16][3] = NULL; 1736 EC_POINT *generator = NULL; 1737 const EC_POINT *p = NULL; 1738 const BIGNUM *p_scalar = NULL; 1739 1740 if (ctx == NULL) { 1741 ctx = new_ctx = BN_CTX_new(); 1742 if (ctx == NULL) { 1743 return 0; 1744 } 1745 } 1746 1747 BN_CTX_start(ctx); 1748 if ((x = BN_CTX_get(ctx)) == NULL || 1749 (y = BN_CTX_get(ctx)) == NULL || 1750 (z = BN_CTX_get(ctx)) == NULL || 1751 (tmp_scalar = BN_CTX_get(ctx)) == NULL) { 1752 goto err; 1753 } 1754 1755 if (scalar != NULL) { 1756 /* try to use the standard precomputation */ 1757 g_pre_comp = &gmul[0]; 1758 generator = EC_POINT_new(group); 1759 if (generator == NULL) { 1760 goto err; 1761 } 1762 /* get the generator from precomputation */ 1763 if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) || 1764 !smallfelem_to_BN(y, g_pre_comp[0][1][1]) || 1765 !smallfelem_to_BN(z, g_pre_comp[0][1][2])) { 1766 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); 1767 goto err; 1768 } 1769 if (!ec_point_set_Jprojective_coordinates_GFp(group, generator, x, y, z, 1770 ctx)) { 1771 goto err; 1772 } 1773 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { 1774 /* precomputation matches generator */ 1775 have_pre_comp = 1; 1776 } else { 1777 /* we don't have valid precomputation: treat the generator as a 1778 * random point. */ 1779 num_points++; 1780 } 1781 } 1782 1783 if (num_points > 0) { 1784 if (num_points >= 3) { 1785 /* unless we precompute multiples for just one or two points, 1786 * converting those into affine form is time well spent */ 1787 mixed = 1; 1788 } 1789 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); 1790 pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(smallfelem)); 1791 if (mixed) { 1792 tmp_smallfelems = 1793 OPENSSL_malloc((num_points * 17 + 1) * sizeof(smallfelem)); 1794 } 1795 if (secrets == NULL || pre_comp == NULL || 1796 (mixed && tmp_smallfelems == NULL)) { 1797 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_MALLOC_FAILURE); 1798 goto err; 1799 } 1800 1801 /* we treat NULL scalars as 0, and NULL points as points at infinity, 1802 * i.e., they contribute nothing to the linear combination. */ 1803 memset(secrets, 0, num_points * sizeof(felem_bytearray)); 1804 memset(pre_comp, 0, num_points * 17 * 3 * sizeof(smallfelem)); 1805 for (i = 0; i < num_points; ++i) { 1806 if (i == num) { 1807 /* we didn't have a valid precomputation, so we pick the generator. */ 1808 p = EC_GROUP_get0_generator(group); 1809 p_scalar = scalar; 1810 } else { 1811 /* the i^th point */ 1812 p = points[i]; 1813 p_scalar = scalars[i]; 1814 } 1815 if (p_scalar != NULL && p != NULL) { 1816 /* reduce scalar to 0 <= scalar < 2^256 */ 1817 if (BN_num_bits(p_scalar) > 256 || BN_is_negative(p_scalar)) { 1818 /* this is an unusual input, and we don't guarantee 1819 * constant-timeness. */ 1820 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { 1821 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); 1822 goto err; 1823 } 1824 num_bytes = BN_bn2bin(tmp_scalar, tmp); 1825 } else { 1826 num_bytes = BN_bn2bin(p_scalar, tmp); 1827 } 1828 flip_endian(secrets[i], tmp, num_bytes); 1829 /* precompute multiples */ 1830 if (!BN_to_felem(x_out, &p->X) || 1831 !BN_to_felem(y_out, &p->Y) || 1832 !BN_to_felem(z_out, &p->Z)) { 1833 goto err; 1834 } 1835 felem_shrink(pre_comp[i][1][0], x_out); 1836 felem_shrink(pre_comp[i][1][1], y_out); 1837 felem_shrink(pre_comp[i][1][2], z_out); 1838 for (j = 2; j <= 16; ++j) { 1839 if (j & 1) { 1840 point_add_small(pre_comp[i][j][0], pre_comp[i][j][1], 1841 pre_comp[i][j][2], pre_comp[i][1][0], 1842 pre_comp[i][1][1], pre_comp[i][1][2], 1843 pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], 1844 pre_comp[i][j - 1][2]); 1845 } else { 1846 point_double_small(pre_comp[i][j][0], pre_comp[i][j][1], 1847 pre_comp[i][j][2], pre_comp[i][j / 2][0], 1848 pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]); 1849 } 1850 } 1851 } 1852 } 1853 if (mixed) { 1854 make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems); 1855 } 1856 } 1857 1858 /* the scalar for the generator */ 1859 if (scalar != NULL && have_pre_comp) { 1860 memset(g_secret, 0, sizeof(g_secret)); 1861 /* reduce scalar to 0 <= scalar < 2^256 */ 1862 if (BN_num_bits(scalar) > 256 || BN_is_negative(scalar)) { 1863 /* this is an unusual input, and we don't guarantee 1864 * constant-timeness. */ 1865 if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) { 1866 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); 1867 goto err; 1868 } 1869 num_bytes = BN_bn2bin(tmp_scalar, tmp); 1870 } else { 1871 num_bytes = BN_bn2bin(scalar, tmp); 1872 } 1873 flip_endian(g_secret, tmp, num_bytes); 1874 /* do the multiplication with generator precomputation */ 1875 batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, 1876 num_points, g_secret, mixed, (const smallfelem(*)[17][3])pre_comp, 1877 g_pre_comp); 1878 } else { 1879 /* do the multiplication without generator precomputation */ 1880 batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, 1881 num_points, NULL, mixed, (const smallfelem(*)[17][3])pre_comp, 1882 NULL); 1883 } 1884 1885 /* reduce the output to its unique minimal representation */ 1886 felem_contract(x_in, x_out); 1887 felem_contract(y_in, y_out); 1888 felem_contract(z_in, z_out); 1889 if (!smallfelem_to_BN(x, x_in) || 1890 !smallfelem_to_BN(y, y_in) || 1891 !smallfelem_to_BN(z, z_in)) { 1892 OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); 1893 goto err; 1894 } 1895 ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); 1896 1897 err: 1898 BN_CTX_end(ctx); 1899 EC_POINT_free(generator); 1900 BN_CTX_free(new_ctx); 1901 OPENSSL_free(secrets); 1902 OPENSSL_free(pre_comp); 1903 OPENSSL_free(tmp_smallfelems); 1904 return ret; 1905 } 1906 1907 const EC_METHOD *EC_GFp_nistp256_method(void) { 1908 static const EC_METHOD ret = { 1909 EC_FLAGS_DEFAULT_OCT, 1910 ec_GFp_nistp256_group_init, 1911 ec_GFp_simple_group_finish, 1912 ec_GFp_simple_group_clear_finish, 1913 ec_GFp_simple_group_copy, ec_GFp_nistp256_group_set_curve, 1914 ec_GFp_simple_group_get_curve, ec_GFp_simple_group_get_degree, 1915 ec_GFp_simple_group_check_discriminant, ec_GFp_simple_point_init, 1916 ec_GFp_simple_point_finish, ec_GFp_simple_point_clear_finish, 1917 ec_GFp_simple_point_copy, ec_GFp_simple_point_set_to_infinity, 1918 ec_GFp_simple_set_Jprojective_coordinates_GFp, 1919 ec_GFp_simple_get_Jprojective_coordinates_GFp, 1920 ec_GFp_simple_point_set_affine_coordinates, 1921 ec_GFp_nistp256_point_get_affine_coordinates, 1922 0 /* point_set_compressed_coordinates */, 0 /* point2oct */, 1923 0 /* oct2point */, ec_GFp_simple_add, ec_GFp_simple_dbl, 1924 ec_GFp_simple_invert, ec_GFp_simple_is_at_infinity, 1925 ec_GFp_simple_is_on_curve, ec_GFp_simple_cmp, ec_GFp_simple_make_affine, 1926 ec_GFp_simple_points_make_affine, ec_GFp_nistp256_points_mul, 1927 0 /* precompute_mult */, 0 /* have_precompute_mult */, 1928 ec_GFp_simple_field_mul, ec_GFp_simple_field_sqr, 0 /* field_div */, 1929 0 /* field_encode */, 0 /* field_decode */, 0 /* field_set_to_one */ 1930 }; 1931 1932 return &ret; 1933 } 1934 1935 #endif /* 64_BIT && !WINDOWS */ 1936
