1 /* 2 * Copyright 2016-2019 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the OpenSSL license (the "License"). You may not use 5 * this file except in compliance with the License. You can obtain a copy 6 * in the file LICENSE in the source distribution or at 7 * https://www.openssl.org/source/license.html 8 */ 9 10 #include <string.h> 11 #include "ec_local.h" 12 #include <openssl/sha.h> 13 14 #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \ 15 defined(_M_AMD64) || defined(_M_X64)) 16 17 # define BASE_2_64_IMPLEMENTED 18 19 typedef uint64_t fe64[4]; 20 21 int x25519_fe64_eligible(void); 22 23 /* 24 * Following subroutines perform corresponding operations modulo 25 * 2^256-38, i.e. double the curve modulus. However, inputs and 26 * outputs are permitted to be partially reduced, i.e. to remain 27 * in [0..2^256) range. It's all tied up in final fe64_tobytes 28 * that performs full reduction modulo 2^255-19. 29 * 30 * There are no reference C implementations for these. 31 */ 32 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g); 33 void x25519_fe64_sqr(fe64 h, const fe64 f); 34 void x25519_fe64_mul121666(fe64 h, fe64 f); 35 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g); 36 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g); 37 void x25519_fe64_tobytes(uint8_t *s, const fe64 f); 38 # define fe64_mul x25519_fe64_mul 39 # define fe64_sqr x25519_fe64_sqr 40 # define fe64_mul121666 x25519_fe64_mul121666 41 # define fe64_add x25519_fe64_add 42 # define fe64_sub x25519_fe64_sub 43 # define fe64_tobytes x25519_fe64_tobytes 44 45 static uint64_t load_8(const uint8_t *in) 46 { 47 uint64_t result; 48 49 result = in[0]; 50 result |= ((uint64_t)in[1]) << 8; 51 result |= ((uint64_t)in[2]) << 16; 52 result |= ((uint64_t)in[3]) << 24; 53 result |= ((uint64_t)in[4]) << 32; 54 result |= ((uint64_t)in[5]) << 40; 55 result |= ((uint64_t)in[6]) << 48; 56 result |= ((uint64_t)in[7]) << 56; 57 58 return result; 59 } 60 61 static void fe64_frombytes(fe64 h, const uint8_t *s) 62 { 63 h[0] = load_8(s); 64 h[1] = load_8(s + 8); 65 h[2] = load_8(s + 16); 66 h[3] = load_8(s + 24) & 0x7fffffffffffffff; 67 } 68 69 static void fe64_0(fe64 h) 70 { 71 h[0] = 0; 72 h[1] = 0; 73 h[2] = 0; 74 h[3] = 0; 75 } 76 77 static void fe64_1(fe64 h) 78 { 79 h[0] = 1; 80 h[1] = 0; 81 h[2] = 0; 82 h[3] = 0; 83 } 84 85 static void fe64_copy(fe64 h, const fe64 f) 86 { 87 h[0] = f[0]; 88 h[1] = f[1]; 89 h[2] = f[2]; 90 h[3] = f[3]; 91 } 92 93 static void fe64_cswap(fe64 f, fe64 g, unsigned int b) 94 { 95 int i; 96 uint64_t mask = 0 - (uint64_t)b; 97 98 for (i = 0; i < 4; i++) { 99 uint64_t x = f[i] ^ g[i]; 100 x &= mask; 101 f[i] ^= x; 102 g[i] ^= x; 103 } 104 } 105 106 static void fe64_invert(fe64 out, const fe64 z) 107 { 108 fe64 t0; 109 fe64 t1; 110 fe64 t2; 111 fe64 t3; 112 int i; 113 114 /* 115 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 116 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 117 */ 118 119 /* t0 = z ** 2 */ 120 fe64_sqr(t0, z); 121 122 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 123 fe64_sqr(t1, t0); 124 fe64_sqr(t1, t1); 125 126 /* t1 = z * t1 = z ** 9 */ 127 fe64_mul(t1, z, t1); 128 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 129 fe64_mul(t0, t0, t1); 130 131 /* t2 = t0 ** 2 = z ** 22 */ 132 fe64_sqr(t2, t0); 133 134 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 135 fe64_mul(t1, t1, t2); 136 137 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 138 fe64_sqr(t2, t1); 139 for (i = 1; i < 5; ++i) 140 fe64_sqr(t2, t2); 141 142 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 143 fe64_mul(t1, t2, t1); 144 145 /* Continuing similarly... */ 146 147 /* t2 = z ** (2 ** 20 - 1) */ 148 fe64_sqr(t2, t1); 149 for (i = 1; i < 10; ++i) 150 fe64_sqr(t2, t2); 151 152 fe64_mul(t2, t2, t1); 153 154 /* t2 = z ** (2 ** 40 - 1) */ 155 fe64_sqr(t3, t2); 156 for (i = 1; i < 20; ++i) 157 fe64_sqr(t3, t3); 158 159 fe64_mul(t2, t3, t2); 160 161 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 162 for (i = 0; i < 10; ++i) 163 fe64_sqr(t2, t2); 164 165 /* t1 = z ** (2 ** 50 - 1) */ 166 fe64_mul(t1, t2, t1); 167 168 /* t2 = z ** (2 ** 100 - 1) */ 169 fe64_sqr(t2, t1); 170 for (i = 1; i < 50; ++i) 171 fe64_sqr(t2, t2); 172 173 fe64_mul(t2, t2, t1); 174 175 /* t2 = z ** (2 ** 200 - 1) */ 176 fe64_sqr(t3, t2); 177 for (i = 1; i < 100; ++i) 178 fe64_sqr(t3, t3); 179 180 fe64_mul(t2, t3, t2); 181 182 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 183 for (i = 0; i < 50; ++i) 184 fe64_sqr(t2, t2); 185 186 /* t1 = z ** (2 ** 250 - 1) */ 187 fe64_mul(t1, t2, t1); 188 189 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 190 for (i = 0; i < 5; ++i) 191 fe64_sqr(t1, t1); 192 193 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 194 fe64_mul(out, t1, t0); 195 } 196 197 /* 198 * Duplicate of original x25519_scalar_mult_generic, but using 199 * fe64_* subroutines. 200 */ 201 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32], 202 const uint8_t point[32]) 203 { 204 fe64 x1, x2, z2, x3, z3, tmp0, tmp1; 205 uint8_t e[32]; 206 unsigned swap = 0; 207 int pos; 208 209 memcpy(e, scalar, 32); 210 e[0] &= 0xf8; 211 e[31] &= 0x7f; 212 e[31] |= 0x40; 213 fe64_frombytes(x1, point); 214 fe64_1(x2); 215 fe64_0(z2); 216 fe64_copy(x3, x1); 217 fe64_1(z3); 218 219 for (pos = 254; pos >= 0; --pos) { 220 unsigned int b = 1 & (e[pos / 8] >> (pos & 7)); 221 222 swap ^= b; 223 fe64_cswap(x2, x3, swap); 224 fe64_cswap(z2, z3, swap); 225 swap = b; 226 fe64_sub(tmp0, x3, z3); 227 fe64_sub(tmp1, x2, z2); 228 fe64_add(x2, x2, z2); 229 fe64_add(z2, x3, z3); 230 fe64_mul(z3, x2, tmp0); 231 fe64_mul(z2, z2, tmp1); 232 fe64_sqr(tmp0, tmp1); 233 fe64_sqr(tmp1, x2); 234 fe64_add(x3, z3, z2); 235 fe64_sub(z2, z3, z2); 236 fe64_mul(x2, tmp1, tmp0); 237 fe64_sub(tmp1, tmp1, tmp0); 238 fe64_sqr(z2, z2); 239 fe64_mul121666(z3, tmp1); 240 fe64_sqr(x3, x3); 241 fe64_add(tmp0, tmp0, z3); 242 fe64_mul(z3, x1, z2); 243 fe64_mul(z2, tmp1, tmp0); 244 } 245 246 fe64_invert(z2, z2); 247 fe64_mul(x2, x2, z2); 248 fe64_tobytes(out, x2); 249 250 OPENSSL_cleanse(e, sizeof(e)); 251 } 252 #endif 253 254 #if defined(X25519_ASM) \ 255 || ( (defined(__SIZEOF_INT128__) && __SIZEOF_INT128__ == 16) \ 256 && !defined(__sparc__) \ 257 && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \ 258 && !(defined(__ANDROID__) && !defined(__clang__)) ) 259 /* 260 * Base 2^51 implementation. It's virtually no different from reference 261 * base 2^25.5 implementation in respect to lax boundary conditions for 262 * intermediate values and even individual limbs. So that whatever you 263 * know about the reference, applies even here... 264 */ 265 # define BASE_2_51_IMPLEMENTED 266 267 typedef uint64_t fe51[5]; 268 269 static const uint64_t MASK51 = 0x7ffffffffffff; 270 271 static uint64_t load_7(const uint8_t *in) 272 { 273 uint64_t result; 274 275 result = in[0]; 276 result |= ((uint64_t)in[1]) << 8; 277 result |= ((uint64_t)in[2]) << 16; 278 result |= ((uint64_t)in[3]) << 24; 279 result |= ((uint64_t)in[4]) << 32; 280 result |= ((uint64_t)in[5]) << 40; 281 result |= ((uint64_t)in[6]) << 48; 282 283 return result; 284 } 285 286 static uint64_t load_6(const uint8_t *in) 287 { 288 uint64_t result; 289 290 result = in[0]; 291 result |= ((uint64_t)in[1]) << 8; 292 result |= ((uint64_t)in[2]) << 16; 293 result |= ((uint64_t)in[3]) << 24; 294 result |= ((uint64_t)in[4]) << 32; 295 result |= ((uint64_t)in[5]) << 40; 296 297 return result; 298 } 299 300 static void fe51_frombytes(fe51 h, const uint8_t *s) 301 { 302 uint64_t h0 = load_7(s); /* 56 bits */ 303 uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */ 304 uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */ 305 uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */ 306 uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */ 307 308 h1 |= h0 >> 51; h0 &= MASK51; 309 h2 |= h1 >> 51; h1 &= MASK51; 310 h3 |= h2 >> 51; h2 &= MASK51; 311 h4 |= h3 >> 51; h3 &= MASK51; 312 313 h[0] = h0; 314 h[1] = h1; 315 h[2] = h2; 316 h[3] = h3; 317 h[4] = h4; 318 } 319 320 static void fe51_tobytes(uint8_t *s, const fe51 h) 321 { 322 uint64_t h0 = h[0]; 323 uint64_t h1 = h[1]; 324 uint64_t h2 = h[2]; 325 uint64_t h3 = h[3]; 326 uint64_t h4 = h[4]; 327 uint64_t q; 328 329 /* compare to modulus */ 330 q = (h0 + 19) >> 51; 331 q = (h1 + q) >> 51; 332 q = (h2 + q) >> 51; 333 q = (h3 + q) >> 51; 334 q = (h4 + q) >> 51; 335 336 /* full reduce */ 337 h0 += 19 * q; 338 h1 += h0 >> 51; h0 &= MASK51; 339 h2 += h1 >> 51; h1 &= MASK51; 340 h3 += h2 >> 51; h2 &= MASK51; 341 h4 += h3 >> 51; h3 &= MASK51; 342 h4 &= MASK51; 343 344 /* smash */ 345 s[0] = (uint8_t)(h0 >> 0); 346 s[1] = (uint8_t)(h0 >> 8); 347 s[2] = (uint8_t)(h0 >> 16); 348 s[3] = (uint8_t)(h0 >> 24); 349 s[4] = (uint8_t)(h0 >> 32); 350 s[5] = (uint8_t)(h0 >> 40); 351 s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3)); 352 s[7] = (uint8_t)(h1 >> 5); 353 s[8] = (uint8_t)(h1 >> 13); 354 s[9] = (uint8_t)(h1 >> 21); 355 s[10] = (uint8_t)(h1 >> 29); 356 s[11] = (uint8_t)(h1 >> 37); 357 s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6)); 358 s[13] = (uint8_t)(h2 >> 2); 359 s[14] = (uint8_t)(h2 >> 10); 360 s[15] = (uint8_t)(h2 >> 18); 361 s[16] = (uint8_t)(h2 >> 26); 362 s[17] = (uint8_t)(h2 >> 34); 363 s[18] = (uint8_t)(h2 >> 42); 364 s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1)); 365 s[20] = (uint8_t)(h3 >> 7); 366 s[21] = (uint8_t)(h3 >> 15); 367 s[22] = (uint8_t)(h3 >> 23); 368 s[23] = (uint8_t)(h3 >> 31); 369 s[24] = (uint8_t)(h3 >> 39); 370 s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4)); 371 s[26] = (uint8_t)(h4 >> 4); 372 s[27] = (uint8_t)(h4 >> 12); 373 s[28] = (uint8_t)(h4 >> 20); 374 s[29] = (uint8_t)(h4 >> 28); 375 s[30] = (uint8_t)(h4 >> 36); 376 s[31] = (uint8_t)(h4 >> 44); 377 } 378 379 # if defined(X25519_ASM) 380 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g); 381 void x25519_fe51_sqr(fe51 h, const fe51 f); 382 void x25519_fe51_mul121666(fe51 h, fe51 f); 383 # define fe51_mul x25519_fe51_mul 384 # define fe51_sq x25519_fe51_sqr 385 # define fe51_mul121666 x25519_fe51_mul121666 386 # else 387 388 typedef __uint128_t u128; 389 390 static void fe51_mul(fe51 h, const fe51 f, const fe51 g) 391 { 392 u128 h0, h1, h2, h3, h4; 393 uint64_t f_i, g0, g1, g2, g3, g4; 394 395 f_i = f[0]; 396 h0 = (u128)f_i * (g0 = g[0]); 397 h1 = (u128)f_i * (g1 = g[1]); 398 h2 = (u128)f_i * (g2 = g[2]); 399 h3 = (u128)f_i * (g3 = g[3]); 400 h4 = (u128)f_i * (g4 = g[4]); 401 402 f_i = f[1]; 403 h0 += (u128)f_i * (g4 *= 19); 404 h1 += (u128)f_i * g0; 405 h2 += (u128)f_i * g1; 406 h3 += (u128)f_i * g2; 407 h4 += (u128)f_i * g3; 408 409 f_i = f[2]; 410 h0 += (u128)f_i * (g3 *= 19); 411 h1 += (u128)f_i * g4; 412 h2 += (u128)f_i * g0; 413 h3 += (u128)f_i * g1; 414 h4 += (u128)f_i * g2; 415 416 f_i = f[3]; 417 h0 += (u128)f_i * (g2 *= 19); 418 h1 += (u128)f_i * g3; 419 h2 += (u128)f_i * g4; 420 h3 += (u128)f_i * g0; 421 h4 += (u128)f_i * g1; 422 423 f_i = f[4]; 424 h0 += (u128)f_i * (g1 *= 19); 425 h1 += (u128)f_i * g2; 426 h2 += (u128)f_i * g3; 427 h3 += (u128)f_i * g4; 428 h4 += (u128)f_i * g0; 429 430 /* partial [lazy] reduction */ 431 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51; 432 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51; 433 434 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51; 435 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51; 436 437 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51; 438 g3 += g2 >> 51; g2 &= MASK51; 439 g1 += g0 >> 51; g0 &= MASK51; 440 441 h[0] = g0; 442 h[1] = g1; 443 h[2] = g2; 444 h[3] = g3; 445 h[4] = g4; 446 } 447 448 static void fe51_sq(fe51 h, const fe51 f) 449 { 450 # if defined(OPENSSL_SMALL_FOOTPRINT) 451 fe51_mul(h, f, f); 452 # else 453 /* dedicated squaring gives 16-25% overall improvement */ 454 uint64_t g0 = f[0]; 455 uint64_t g1 = f[1]; 456 uint64_t g2 = f[2]; 457 uint64_t g3 = f[3]; 458 uint64_t g4 = f[4]; 459 u128 h0, h1, h2, h3, h4; 460 461 h0 = (u128)g0 * g0; g0 *= 2; 462 h1 = (u128)g0 * g1; 463 h2 = (u128)g0 * g2; 464 h3 = (u128)g0 * g3; 465 h4 = (u128)g0 * g4; 466 467 g0 = g4; /* borrow g0 */ 468 h3 += (u128)g0 * (g4 *= 19); 469 470 h2 += (u128)g1 * g1; g1 *= 2; 471 h3 += (u128)g1 * g2; 472 h4 += (u128)g1 * g3; 473 h0 += (u128)g1 * g4; 474 475 g0 = g3; /* borrow g0 */ 476 h1 += (u128)g0 * (g3 *= 19); 477 h2 += (u128)(g0 * 2) * g4; 478 479 h4 += (u128)g2 * g2; g2 *= 2; 480 h0 += (u128)g2 * g3; 481 h1 += (u128)g2 * g4; 482 483 /* partial [lazy] reduction */ 484 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51; 485 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51; 486 487 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51; 488 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51; 489 490 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51; 491 g3 += g2 >> 51; g2 &= MASK51; 492 g1 += g0 >> 51; g0 &= MASK51; 493 494 h[0] = g0; 495 h[1] = g1; 496 h[2] = g2; 497 h[3] = g3; 498 h[4] = g4; 499 # endif 500 } 501 502 static void fe51_mul121666(fe51 h, fe51 f) 503 { 504 u128 h0 = f[0] * (u128)121666; 505 u128 h1 = f[1] * (u128)121666; 506 u128 h2 = f[2] * (u128)121666; 507 u128 h3 = f[3] * (u128)121666; 508 u128 h4 = f[4] * (u128)121666; 509 uint64_t g0, g1, g2, g3, g4; 510 511 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51; 512 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51; 513 514 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51; 515 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51; 516 517 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51; 518 g3 += g2 >> 51; g2 &= MASK51; 519 g1 += g0 >> 51; g0 &= MASK51; 520 521 h[0] = g0; 522 h[1] = g1; 523 h[2] = g2; 524 h[3] = g3; 525 h[4] = g4; 526 } 527 # endif 528 529 static void fe51_add(fe51 h, const fe51 f, const fe51 g) 530 { 531 h[0] = f[0] + g[0]; 532 h[1] = f[1] + g[1]; 533 h[2] = f[2] + g[2]; 534 h[3] = f[3] + g[3]; 535 h[4] = f[4] + g[4]; 536 } 537 538 static void fe51_sub(fe51 h, const fe51 f, const fe51 g) 539 { 540 /* 541 * Add 2*modulus to ensure that result remains positive 542 * even if subtrahend is partially reduced. 543 */ 544 h[0] = (f[0] + 0xfffffffffffda) - g[0]; 545 h[1] = (f[1] + 0xffffffffffffe) - g[1]; 546 h[2] = (f[2] + 0xffffffffffffe) - g[2]; 547 h[3] = (f[3] + 0xffffffffffffe) - g[3]; 548 h[4] = (f[4] + 0xffffffffffffe) - g[4]; 549 } 550 551 static void fe51_0(fe51 h) 552 { 553 h[0] = 0; 554 h[1] = 0; 555 h[2] = 0; 556 h[3] = 0; 557 h[4] = 0; 558 } 559 560 static void fe51_1(fe51 h) 561 { 562 h[0] = 1; 563 h[1] = 0; 564 h[2] = 0; 565 h[3] = 0; 566 h[4] = 0; 567 } 568 569 static void fe51_copy(fe51 h, const fe51 f) 570 { 571 h[0] = f[0]; 572 h[1] = f[1]; 573 h[2] = f[2]; 574 h[3] = f[3]; 575 h[4] = f[4]; 576 } 577 578 static void fe51_cswap(fe51 f, fe51 g, unsigned int b) 579 { 580 int i; 581 uint64_t mask = 0 - (uint64_t)b; 582 583 for (i = 0; i < 5; i++) { 584 int64_t x = f[i] ^ g[i]; 585 x &= mask; 586 f[i] ^= x; 587 g[i] ^= x; 588 } 589 } 590 591 static void fe51_invert(fe51 out, const fe51 z) 592 { 593 fe51 t0; 594 fe51 t1; 595 fe51 t2; 596 fe51 t3; 597 int i; 598 599 /* 600 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 601 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 602 */ 603 604 /* t0 = z ** 2 */ 605 fe51_sq(t0, z); 606 607 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 608 fe51_sq(t1, t0); 609 fe51_sq(t1, t1); 610 611 /* t1 = z * t1 = z ** 9 */ 612 fe51_mul(t1, z, t1); 613 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 614 fe51_mul(t0, t0, t1); 615 616 /* t2 = t0 ** 2 = z ** 22 */ 617 fe51_sq(t2, t0); 618 619 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 620 fe51_mul(t1, t1, t2); 621 622 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 623 fe51_sq(t2, t1); 624 for (i = 1; i < 5; ++i) 625 fe51_sq(t2, t2); 626 627 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 628 fe51_mul(t1, t2, t1); 629 630 /* Continuing similarly... */ 631 632 /* t2 = z ** (2 ** 20 - 1) */ 633 fe51_sq(t2, t1); 634 for (i = 1; i < 10; ++i) 635 fe51_sq(t2, t2); 636 637 fe51_mul(t2, t2, t1); 638 639 /* t2 = z ** (2 ** 40 - 1) */ 640 fe51_sq(t3, t2); 641 for (i = 1; i < 20; ++i) 642 fe51_sq(t3, t3); 643 644 fe51_mul(t2, t3, t2); 645 646 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 647 for (i = 0; i < 10; ++i) 648 fe51_sq(t2, t2); 649 650 /* t1 = z ** (2 ** 50 - 1) */ 651 fe51_mul(t1, t2, t1); 652 653 /* t2 = z ** (2 ** 100 - 1) */ 654 fe51_sq(t2, t1); 655 for (i = 1; i < 50; ++i) 656 fe51_sq(t2, t2); 657 658 fe51_mul(t2, t2, t1); 659 660 /* t2 = z ** (2 ** 200 - 1) */ 661 fe51_sq(t3, t2); 662 for (i = 1; i < 100; ++i) 663 fe51_sq(t3, t3); 664 665 fe51_mul(t2, t3, t2); 666 667 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 668 for (i = 0; i < 50; ++i) 669 fe51_sq(t2, t2); 670 671 /* t1 = z ** (2 ** 250 - 1) */ 672 fe51_mul(t1, t2, t1); 673 674 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 675 for (i = 0; i < 5; ++i) 676 fe51_sq(t1, t1); 677 678 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 679 fe51_mul(out, t1, t0); 680 } 681 682 /* 683 * Duplicate of original x25519_scalar_mult_generic, but using 684 * fe51_* subroutines. 685 */ 686 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32], 687 const uint8_t point[32]) 688 { 689 fe51 x1, x2, z2, x3, z3, tmp0, tmp1; 690 uint8_t e[32]; 691 unsigned swap = 0; 692 int pos; 693 694 # ifdef BASE_2_64_IMPLEMENTED 695 if (x25519_fe64_eligible()) { 696 x25519_scalar_mulx(out, scalar, point); 697 return; 698 } 699 # endif 700 701 memcpy(e, scalar, 32); 702 e[0] &= 0xf8; 703 e[31] &= 0x7f; 704 e[31] |= 0x40; 705 fe51_frombytes(x1, point); 706 fe51_1(x2); 707 fe51_0(z2); 708 fe51_copy(x3, x1); 709 fe51_1(z3); 710 711 for (pos = 254; pos >= 0; --pos) { 712 unsigned int b = 1 & (e[pos / 8] >> (pos & 7)); 713 714 swap ^= b; 715 fe51_cswap(x2, x3, swap); 716 fe51_cswap(z2, z3, swap); 717 swap = b; 718 fe51_sub(tmp0, x3, z3); 719 fe51_sub(tmp1, x2, z2); 720 fe51_add(x2, x2, z2); 721 fe51_add(z2, x3, z3); 722 fe51_mul(z3, tmp0, x2); 723 fe51_mul(z2, z2, tmp1); 724 fe51_sq(tmp0, tmp1); 725 fe51_sq(tmp1, x2); 726 fe51_add(x3, z3, z2); 727 fe51_sub(z2, z3, z2); 728 fe51_mul(x2, tmp1, tmp0); 729 fe51_sub(tmp1, tmp1, tmp0); 730 fe51_sq(z2, z2); 731 fe51_mul121666(z3, tmp1); 732 fe51_sq(x3, x3); 733 fe51_add(tmp0, tmp0, z3); 734 fe51_mul(z3, x1, z2); 735 fe51_mul(z2, tmp1, tmp0); 736 } 737 738 fe51_invert(z2, z2); 739 fe51_mul(x2, x2, z2); 740 fe51_tobytes(out, x2); 741 742 OPENSSL_cleanse(e, sizeof(e)); 743 } 744 #endif 745 746 /* 747 * Reference base 2^25.5 implementation. 748 * 749 * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP 750 * 20141124 (http://bench.cr.yp.to/supercop.html). 751 * 752 * The field functions are shared by Ed25519 and X25519 where possible. 753 */ 754 755 /* 756 * fe means field element. Here the field is \Z/(2^255-19). An element t, 757 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 758 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on 759 * context. 760 */ 761 typedef int32_t fe[10]; 762 763 static const int64_t kBottom21Bits = 0x1fffffLL; 764 static const int64_t kBottom25Bits = 0x1ffffffLL; 765 static const int64_t kBottom26Bits = 0x3ffffffLL; 766 static const int64_t kTop39Bits = 0xfffffffffe000000LL; 767 static const int64_t kTop38Bits = 0xfffffffffc000000LL; 768 769 static uint64_t load_3(const uint8_t *in) 770 { 771 uint64_t result; 772 773 result = ((uint64_t)in[0]); 774 result |= ((uint64_t)in[1]) << 8; 775 result |= ((uint64_t)in[2]) << 16; 776 return result; 777 } 778 779 static uint64_t load_4(const uint8_t *in) 780 { 781 uint64_t result; 782 783 result = ((uint64_t)in[0]); 784 result |= ((uint64_t)in[1]) << 8; 785 result |= ((uint64_t)in[2]) << 16; 786 result |= ((uint64_t)in[3]) << 24; 787 return result; 788 } 789 790 static void fe_frombytes(fe h, const uint8_t *s) 791 { 792 /* Ignores top bit of h. */ 793 int64_t h0 = load_4(s); 794 int64_t h1 = load_3(s + 4) << 6; 795 int64_t h2 = load_3(s + 7) << 5; 796 int64_t h3 = load_3(s + 10) << 3; 797 int64_t h4 = load_3(s + 13) << 2; 798 int64_t h5 = load_4(s + 16); 799 int64_t h6 = load_3(s + 20) << 7; 800 int64_t h7 = load_3(s + 23) << 5; 801 int64_t h8 = load_3(s + 26) << 4; 802 int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2; 803 int64_t carry0; 804 int64_t carry1; 805 int64_t carry2; 806 int64_t carry3; 807 int64_t carry4; 808 int64_t carry5; 809 int64_t carry6; 810 int64_t carry7; 811 int64_t carry8; 812 int64_t carry9; 813 814 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 815 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 816 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 817 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 818 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 819 820 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 821 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 822 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 823 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 824 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 825 826 h[0] = (int32_t)h0; 827 h[1] = (int32_t)h1; 828 h[2] = (int32_t)h2; 829 h[3] = (int32_t)h3; 830 h[4] = (int32_t)h4; 831 h[5] = (int32_t)h5; 832 h[6] = (int32_t)h6; 833 h[7] = (int32_t)h7; 834 h[8] = (int32_t)h8; 835 h[9] = (int32_t)h9; 836 } 837 838 /* 839 * Preconditions: 840 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 841 * 842 * Write p=2^255-19; q=floor(h/p). 843 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). 844 * 845 * Proof: 846 * Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. 847 * Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4. 848 * 849 * Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). 850 * Then 0<y<1. 851 * 852 * Write r=h-pq. 853 * Have 0<=r<=p-1=2^255-20. 854 * Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1. 855 * 856 * Write x=r+19(2^-255)r+y. 857 * Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q. 858 * 859 * Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1)) 860 * so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q. 861 */ 862 static void fe_tobytes(uint8_t *s, const fe h) 863 { 864 int32_t h0 = h[0]; 865 int32_t h1 = h[1]; 866 int32_t h2 = h[2]; 867 int32_t h3 = h[3]; 868 int32_t h4 = h[4]; 869 int32_t h5 = h[5]; 870 int32_t h6 = h[6]; 871 int32_t h7 = h[7]; 872 int32_t h8 = h[8]; 873 int32_t h9 = h[9]; 874 int32_t q; 875 876 q = (19 * h9 + (((int32_t) 1) << 24)) >> 25; 877 q = (h0 + q) >> 26; 878 q = (h1 + q) >> 25; 879 q = (h2 + q) >> 26; 880 q = (h3 + q) >> 25; 881 q = (h4 + q) >> 26; 882 q = (h5 + q) >> 25; 883 q = (h6 + q) >> 26; 884 q = (h7 + q) >> 25; 885 q = (h8 + q) >> 26; 886 q = (h9 + q) >> 25; 887 888 /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */ 889 h0 += 19 * q; 890 /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */ 891 892 h1 += h0 >> 26; h0 &= kBottom26Bits; 893 h2 += h1 >> 25; h1 &= kBottom25Bits; 894 h3 += h2 >> 26; h2 &= kBottom26Bits; 895 h4 += h3 >> 25; h3 &= kBottom25Bits; 896 h5 += h4 >> 26; h4 &= kBottom26Bits; 897 h6 += h5 >> 25; h5 &= kBottom25Bits; 898 h7 += h6 >> 26; h6 &= kBottom26Bits; 899 h8 += h7 >> 25; h7 &= kBottom25Bits; 900 h9 += h8 >> 26; h8 &= kBottom26Bits; 901 h9 &= kBottom25Bits; 902 /* h10 = carry9 */ 903 904 /* 905 * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. 906 * Have h0+...+2^230 h9 between 0 and 2^255-1; 907 * evidently 2^255 h10-2^255 q = 0. 908 * Goal: Output h0+...+2^230 h9. 909 */ 910 s[ 0] = (uint8_t) (h0 >> 0); 911 s[ 1] = (uint8_t) (h0 >> 8); 912 s[ 2] = (uint8_t) (h0 >> 16); 913 s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2)); 914 s[ 4] = (uint8_t) (h1 >> 6); 915 s[ 5] = (uint8_t) (h1 >> 14); 916 s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3)); 917 s[ 7] = (uint8_t) (h2 >> 5); 918 s[ 8] = (uint8_t) (h2 >> 13); 919 s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5)); 920 s[10] = (uint8_t) (h3 >> 3); 921 s[11] = (uint8_t) (h3 >> 11); 922 s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6)); 923 s[13] = (uint8_t) (h4 >> 2); 924 s[14] = (uint8_t) (h4 >> 10); 925 s[15] = (uint8_t) (h4 >> 18); 926 s[16] = (uint8_t) (h5 >> 0); 927 s[17] = (uint8_t) (h5 >> 8); 928 s[18] = (uint8_t) (h5 >> 16); 929 s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1)); 930 s[20] = (uint8_t) (h6 >> 7); 931 s[21] = (uint8_t) (h6 >> 15); 932 s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3)); 933 s[23] = (uint8_t) (h7 >> 5); 934 s[24] = (uint8_t) (h7 >> 13); 935 s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4)); 936 s[26] = (uint8_t) (h8 >> 4); 937 s[27] = (uint8_t) (h8 >> 12); 938 s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6)); 939 s[29] = (uint8_t) (h9 >> 2); 940 s[30] = (uint8_t) (h9 >> 10); 941 s[31] = (uint8_t) (h9 >> 18); 942 } 943 944 /* h = f */ 945 static void fe_copy(fe h, const fe f) 946 { 947 memmove(h, f, sizeof(int32_t) * 10); 948 } 949 950 /* h = 0 */ 951 static void fe_0(fe h) 952 { 953 memset(h, 0, sizeof(int32_t) * 10); 954 } 955 956 /* h = 1 */ 957 static void fe_1(fe h) 958 { 959 memset(h, 0, sizeof(int32_t) * 10); 960 h[0] = 1; 961 } 962 963 /* 964 * h = f + g 965 * 966 * Can overlap h with f or g. 967 * 968 * Preconditions: 969 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 970 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 971 * 972 * Postconditions: 973 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 974 */ 975 static void fe_add(fe h, const fe f, const fe g) 976 { 977 unsigned i; 978 979 for (i = 0; i < 10; i++) { 980 h[i] = f[i] + g[i]; 981 } 982 } 983 984 /* 985 * h = f - g 986 * 987 * Can overlap h with f or g. 988 * 989 * Preconditions: 990 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 991 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 992 * 993 * Postconditions: 994 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 995 */ 996 static void fe_sub(fe h, const fe f, const fe g) 997 { 998 unsigned i; 999 1000 for (i = 0; i < 10; i++) { 1001 h[i] = f[i] - g[i]; 1002 } 1003 } 1004 1005 /* 1006 * h = f * g 1007 * 1008 * Can overlap h with f or g. 1009 * 1010 * Preconditions: 1011 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1012 * |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1013 * 1014 * Postconditions: 1015 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1016 * 1017 * Notes on implementation strategy: 1018 * 1019 * Using schoolbook multiplication. 1020 * Karatsuba would save a little in some cost models. 1021 * 1022 * Most multiplications by 2 and 19 are 32-bit precomputations; 1023 * cheaper than 64-bit postcomputations. 1024 * 1025 * There is one remaining multiplication by 19 in the carry chain; 1026 * one *19 precomputation can be merged into this, 1027 * but the resulting data flow is considerably less clean. 1028 * 1029 * There are 12 carries below. 1030 * 10 of them are 2-way parallelizable and vectorizable. 1031 * Can get away with 11 carries, but then data flow is much deeper. 1032 * 1033 * With tighter constraints on inputs can squeeze carries into int32. 1034 */ 1035 static void fe_mul(fe h, const fe f, const fe g) 1036 { 1037 int32_t f0 = f[0]; 1038 int32_t f1 = f[1]; 1039 int32_t f2 = f[2]; 1040 int32_t f3 = f[3]; 1041 int32_t f4 = f[4]; 1042 int32_t f5 = f[5]; 1043 int32_t f6 = f[6]; 1044 int32_t f7 = f[7]; 1045 int32_t f8 = f[8]; 1046 int32_t f9 = f[9]; 1047 int32_t g0 = g[0]; 1048 int32_t g1 = g[1]; 1049 int32_t g2 = g[2]; 1050 int32_t g3 = g[3]; 1051 int32_t g4 = g[4]; 1052 int32_t g5 = g[5]; 1053 int32_t g6 = g[6]; 1054 int32_t g7 = g[7]; 1055 int32_t g8 = g[8]; 1056 int32_t g9 = g[9]; 1057 int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */ 1058 int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */ 1059 int32_t g3_19 = 19 * g3; 1060 int32_t g4_19 = 19 * g4; 1061 int32_t g5_19 = 19 * g5; 1062 int32_t g6_19 = 19 * g6; 1063 int32_t g7_19 = 19 * g7; 1064 int32_t g8_19 = 19 * g8; 1065 int32_t g9_19 = 19 * g9; 1066 int32_t f1_2 = 2 * f1; 1067 int32_t f3_2 = 2 * f3; 1068 int32_t f5_2 = 2 * f5; 1069 int32_t f7_2 = 2 * f7; 1070 int32_t f9_2 = 2 * f9; 1071 int64_t f0g0 = f0 * (int64_t) g0; 1072 int64_t f0g1 = f0 * (int64_t) g1; 1073 int64_t f0g2 = f0 * (int64_t) g2; 1074 int64_t f0g3 = f0 * (int64_t) g3; 1075 int64_t f0g4 = f0 * (int64_t) g4; 1076 int64_t f0g5 = f0 * (int64_t) g5; 1077 int64_t f0g6 = f0 * (int64_t) g6; 1078 int64_t f0g7 = f0 * (int64_t) g7; 1079 int64_t f0g8 = f0 * (int64_t) g8; 1080 int64_t f0g9 = f0 * (int64_t) g9; 1081 int64_t f1g0 = f1 * (int64_t) g0; 1082 int64_t f1g1_2 = f1_2 * (int64_t) g1; 1083 int64_t f1g2 = f1 * (int64_t) g2; 1084 int64_t f1g3_2 = f1_2 * (int64_t) g3; 1085 int64_t f1g4 = f1 * (int64_t) g4; 1086 int64_t f1g5_2 = f1_2 * (int64_t) g5; 1087 int64_t f1g6 = f1 * (int64_t) g6; 1088 int64_t f1g7_2 = f1_2 * (int64_t) g7; 1089 int64_t f1g8 = f1 * (int64_t) g8; 1090 int64_t f1g9_38 = f1_2 * (int64_t) g9_19; 1091 int64_t f2g0 = f2 * (int64_t) g0; 1092 int64_t f2g1 = f2 * (int64_t) g1; 1093 int64_t f2g2 = f2 * (int64_t) g2; 1094 int64_t f2g3 = f2 * (int64_t) g3; 1095 int64_t f2g4 = f2 * (int64_t) g4; 1096 int64_t f2g5 = f2 * (int64_t) g5; 1097 int64_t f2g6 = f2 * (int64_t) g6; 1098 int64_t f2g7 = f2 * (int64_t) g7; 1099 int64_t f2g8_19 = f2 * (int64_t) g8_19; 1100 int64_t f2g9_19 = f2 * (int64_t) g9_19; 1101 int64_t f3g0 = f3 * (int64_t) g0; 1102 int64_t f3g1_2 = f3_2 * (int64_t) g1; 1103 int64_t f3g2 = f3 * (int64_t) g2; 1104 int64_t f3g3_2 = f3_2 * (int64_t) g3; 1105 int64_t f3g4 = f3 * (int64_t) g4; 1106 int64_t f3g5_2 = f3_2 * (int64_t) g5; 1107 int64_t f3g6 = f3 * (int64_t) g6; 1108 int64_t f3g7_38 = f3_2 * (int64_t) g7_19; 1109 int64_t f3g8_19 = f3 * (int64_t) g8_19; 1110 int64_t f3g9_38 = f3_2 * (int64_t) g9_19; 1111 int64_t f4g0 = f4 * (int64_t) g0; 1112 int64_t f4g1 = f4 * (int64_t) g1; 1113 int64_t f4g2 = f4 * (int64_t) g2; 1114 int64_t f4g3 = f4 * (int64_t) g3; 1115 int64_t f4g4 = f4 * (int64_t) g4; 1116 int64_t f4g5 = f4 * (int64_t) g5; 1117 int64_t f4g6_19 = f4 * (int64_t) g6_19; 1118 int64_t f4g7_19 = f4 * (int64_t) g7_19; 1119 int64_t f4g8_19 = f4 * (int64_t) g8_19; 1120 int64_t f4g9_19 = f4 * (int64_t) g9_19; 1121 int64_t f5g0 = f5 * (int64_t) g0; 1122 int64_t f5g1_2 = f5_2 * (int64_t) g1; 1123 int64_t f5g2 = f5 * (int64_t) g2; 1124 int64_t f5g3_2 = f5_2 * (int64_t) g3; 1125 int64_t f5g4 = f5 * (int64_t) g4; 1126 int64_t f5g5_38 = f5_2 * (int64_t) g5_19; 1127 int64_t f5g6_19 = f5 * (int64_t) g6_19; 1128 int64_t f5g7_38 = f5_2 * (int64_t) g7_19; 1129 int64_t f5g8_19 = f5 * (int64_t) g8_19; 1130 int64_t f5g9_38 = f5_2 * (int64_t) g9_19; 1131 int64_t f6g0 = f6 * (int64_t) g0; 1132 int64_t f6g1 = f6 * (int64_t) g1; 1133 int64_t f6g2 = f6 * (int64_t) g2; 1134 int64_t f6g3 = f6 * (int64_t) g3; 1135 int64_t f6g4_19 = f6 * (int64_t) g4_19; 1136 int64_t f6g5_19 = f6 * (int64_t) g5_19; 1137 int64_t f6g6_19 = f6 * (int64_t) g6_19; 1138 int64_t f6g7_19 = f6 * (int64_t) g7_19; 1139 int64_t f6g8_19 = f6 * (int64_t) g8_19; 1140 int64_t f6g9_19 = f6 * (int64_t) g9_19; 1141 int64_t f7g0 = f7 * (int64_t) g0; 1142 int64_t f7g1_2 = f7_2 * (int64_t) g1; 1143 int64_t f7g2 = f7 * (int64_t) g2; 1144 int64_t f7g3_38 = f7_2 * (int64_t) g3_19; 1145 int64_t f7g4_19 = f7 * (int64_t) g4_19; 1146 int64_t f7g5_38 = f7_2 * (int64_t) g5_19; 1147 int64_t f7g6_19 = f7 * (int64_t) g6_19; 1148 int64_t f7g7_38 = f7_2 * (int64_t) g7_19; 1149 int64_t f7g8_19 = f7 * (int64_t) g8_19; 1150 int64_t f7g9_38 = f7_2 * (int64_t) g9_19; 1151 int64_t f8g0 = f8 * (int64_t) g0; 1152 int64_t f8g1 = f8 * (int64_t) g1; 1153 int64_t f8g2_19 = f8 * (int64_t) g2_19; 1154 int64_t f8g3_19 = f8 * (int64_t) g3_19; 1155 int64_t f8g4_19 = f8 * (int64_t) g4_19; 1156 int64_t f8g5_19 = f8 * (int64_t) g5_19; 1157 int64_t f8g6_19 = f8 * (int64_t) g6_19; 1158 int64_t f8g7_19 = f8 * (int64_t) g7_19; 1159 int64_t f8g8_19 = f8 * (int64_t) g8_19; 1160 int64_t f8g9_19 = f8 * (int64_t) g9_19; 1161 int64_t f9g0 = f9 * (int64_t) g0; 1162 int64_t f9g1_38 = f9_2 * (int64_t) g1_19; 1163 int64_t f9g2_19 = f9 * (int64_t) g2_19; 1164 int64_t f9g3_38 = f9_2 * (int64_t) g3_19; 1165 int64_t f9g4_19 = f9 * (int64_t) g4_19; 1166 int64_t f9g5_38 = f9_2 * (int64_t) g5_19; 1167 int64_t f9g6_19 = f9 * (int64_t) g6_19; 1168 int64_t f9g7_38 = f9_2 * (int64_t) g7_19; 1169 int64_t f9g8_19 = f9 * (int64_t) g8_19; 1170 int64_t f9g9_38 = f9_2 * (int64_t) g9_19; 1171 int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38; 1172 int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19; 1173 int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38; 1174 int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19; 1175 int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38; 1176 int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19; 1177 int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38; 1178 int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19; 1179 int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38; 1180 int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 ; 1181 int64_t carry0; 1182 int64_t carry1; 1183 int64_t carry2; 1184 int64_t carry3; 1185 int64_t carry4; 1186 int64_t carry5; 1187 int64_t carry6; 1188 int64_t carry7; 1189 int64_t carry8; 1190 int64_t carry9; 1191 1192 /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38)) 1193 * i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8 1194 * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19)) 1195 * i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */ 1196 1197 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1198 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1199 /* |h0| <= 2^25 */ 1200 /* |h4| <= 2^25 */ 1201 /* |h1| <= 1.71*2^59 */ 1202 /* |h5| <= 1.71*2^59 */ 1203 1204 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 1205 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 1206 /* |h1| <= 2^24; from now on fits into int32 */ 1207 /* |h5| <= 2^24; from now on fits into int32 */ 1208 /* |h2| <= 1.41*2^60 */ 1209 /* |h6| <= 1.41*2^60 */ 1210 1211 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 1212 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 1213 /* |h2| <= 2^25; from now on fits into int32 unchanged */ 1214 /* |h6| <= 2^25; from now on fits into int32 unchanged */ 1215 /* |h3| <= 1.71*2^59 */ 1216 /* |h7| <= 1.71*2^59 */ 1217 1218 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 1219 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 1220 /* |h3| <= 2^24; from now on fits into int32 unchanged */ 1221 /* |h7| <= 2^24; from now on fits into int32 unchanged */ 1222 /* |h4| <= 1.72*2^34 */ 1223 /* |h8| <= 1.41*2^60 */ 1224 1225 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1226 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 1227 /* |h4| <= 2^25; from now on fits into int32 unchanged */ 1228 /* |h8| <= 2^25; from now on fits into int32 unchanged */ 1229 /* |h5| <= 1.01*2^24 */ 1230 /* |h9| <= 1.71*2^59 */ 1231 1232 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 1233 /* |h9| <= 2^24; from now on fits into int32 unchanged */ 1234 /* |h0| <= 1.1*2^39 */ 1235 1236 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1237 /* |h0| <= 2^25; from now on fits into int32 unchanged */ 1238 /* |h1| <= 1.01*2^24 */ 1239 1240 h[0] = (int32_t)h0; 1241 h[1] = (int32_t)h1; 1242 h[2] = (int32_t)h2; 1243 h[3] = (int32_t)h3; 1244 h[4] = (int32_t)h4; 1245 h[5] = (int32_t)h5; 1246 h[6] = (int32_t)h6; 1247 h[7] = (int32_t)h7; 1248 h[8] = (int32_t)h8; 1249 h[9] = (int32_t)h9; 1250 } 1251 1252 /* 1253 * h = f * f 1254 * 1255 * Can overlap h with f. 1256 * 1257 * Preconditions: 1258 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1259 * 1260 * Postconditions: 1261 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1262 * 1263 * See fe_mul.c for discussion of implementation strategy. 1264 */ 1265 static void fe_sq(fe h, const fe f) 1266 { 1267 int32_t f0 = f[0]; 1268 int32_t f1 = f[1]; 1269 int32_t f2 = f[2]; 1270 int32_t f3 = f[3]; 1271 int32_t f4 = f[4]; 1272 int32_t f5 = f[5]; 1273 int32_t f6 = f[6]; 1274 int32_t f7 = f[7]; 1275 int32_t f8 = f[8]; 1276 int32_t f9 = f[9]; 1277 int32_t f0_2 = 2 * f0; 1278 int32_t f1_2 = 2 * f1; 1279 int32_t f2_2 = 2 * f2; 1280 int32_t f3_2 = 2 * f3; 1281 int32_t f4_2 = 2 * f4; 1282 int32_t f5_2 = 2 * f5; 1283 int32_t f6_2 = 2 * f6; 1284 int32_t f7_2 = 2 * f7; 1285 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */ 1286 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */ 1287 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */ 1288 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */ 1289 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */ 1290 int64_t f0f0 = f0 * (int64_t) f0; 1291 int64_t f0f1_2 = f0_2 * (int64_t) f1; 1292 int64_t f0f2_2 = f0_2 * (int64_t) f2; 1293 int64_t f0f3_2 = f0_2 * (int64_t) f3; 1294 int64_t f0f4_2 = f0_2 * (int64_t) f4; 1295 int64_t f0f5_2 = f0_2 * (int64_t) f5; 1296 int64_t f0f6_2 = f0_2 * (int64_t) f6; 1297 int64_t f0f7_2 = f0_2 * (int64_t) f7; 1298 int64_t f0f8_2 = f0_2 * (int64_t) f8; 1299 int64_t f0f9_2 = f0_2 * (int64_t) f9; 1300 int64_t f1f1_2 = f1_2 * (int64_t) f1; 1301 int64_t f1f2_2 = f1_2 * (int64_t) f2; 1302 int64_t f1f3_4 = f1_2 * (int64_t) f3_2; 1303 int64_t f1f4_2 = f1_2 * (int64_t) f4; 1304 int64_t f1f5_4 = f1_2 * (int64_t) f5_2; 1305 int64_t f1f6_2 = f1_2 * (int64_t) f6; 1306 int64_t f1f7_4 = f1_2 * (int64_t) f7_2; 1307 int64_t f1f8_2 = f1_2 * (int64_t) f8; 1308 int64_t f1f9_76 = f1_2 * (int64_t) f9_38; 1309 int64_t f2f2 = f2 * (int64_t) f2; 1310 int64_t f2f3_2 = f2_2 * (int64_t) f3; 1311 int64_t f2f4_2 = f2_2 * (int64_t) f4; 1312 int64_t f2f5_2 = f2_2 * (int64_t) f5; 1313 int64_t f2f6_2 = f2_2 * (int64_t) f6; 1314 int64_t f2f7_2 = f2_2 * (int64_t) f7; 1315 int64_t f2f8_38 = f2_2 * (int64_t) f8_19; 1316 int64_t f2f9_38 = f2 * (int64_t) f9_38; 1317 int64_t f3f3_2 = f3_2 * (int64_t) f3; 1318 int64_t f3f4_2 = f3_2 * (int64_t) f4; 1319 int64_t f3f5_4 = f3_2 * (int64_t) f5_2; 1320 int64_t f3f6_2 = f3_2 * (int64_t) f6; 1321 int64_t f3f7_76 = f3_2 * (int64_t) f7_38; 1322 int64_t f3f8_38 = f3_2 * (int64_t) f8_19; 1323 int64_t f3f9_76 = f3_2 * (int64_t) f9_38; 1324 int64_t f4f4 = f4 * (int64_t) f4; 1325 int64_t f4f5_2 = f4_2 * (int64_t) f5; 1326 int64_t f4f6_38 = f4_2 * (int64_t) f6_19; 1327 int64_t f4f7_38 = f4 * (int64_t) f7_38; 1328 int64_t f4f8_38 = f4_2 * (int64_t) f8_19; 1329 int64_t f4f9_38 = f4 * (int64_t) f9_38; 1330 int64_t f5f5_38 = f5 * (int64_t) f5_38; 1331 int64_t f5f6_38 = f5_2 * (int64_t) f6_19; 1332 int64_t f5f7_76 = f5_2 * (int64_t) f7_38; 1333 int64_t f5f8_38 = f5_2 * (int64_t) f8_19; 1334 int64_t f5f9_76 = f5_2 * (int64_t) f9_38; 1335 int64_t f6f6_19 = f6 * (int64_t) f6_19; 1336 int64_t f6f7_38 = f6 * (int64_t) f7_38; 1337 int64_t f6f8_38 = f6_2 * (int64_t) f8_19; 1338 int64_t f6f9_38 = f6 * (int64_t) f9_38; 1339 int64_t f7f7_38 = f7 * (int64_t) f7_38; 1340 int64_t f7f8_38 = f7_2 * (int64_t) f8_19; 1341 int64_t f7f9_76 = f7_2 * (int64_t) f9_38; 1342 int64_t f8f8_19 = f8 * (int64_t) f8_19; 1343 int64_t f8f9_38 = f8 * (int64_t) f9_38; 1344 int64_t f9f9_38 = f9 * (int64_t) f9_38; 1345 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38; 1346 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38; 1347 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19; 1348 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38; 1349 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38; 1350 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38; 1351 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19; 1352 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38; 1353 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38; 1354 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2; 1355 int64_t carry0; 1356 int64_t carry1; 1357 int64_t carry2; 1358 int64_t carry3; 1359 int64_t carry4; 1360 int64_t carry5; 1361 int64_t carry6; 1362 int64_t carry7; 1363 int64_t carry8; 1364 int64_t carry9; 1365 1366 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1367 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1368 1369 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 1370 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 1371 1372 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 1373 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 1374 1375 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 1376 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 1377 1378 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1379 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 1380 1381 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 1382 1383 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1384 1385 h[0] = (int32_t)h0; 1386 h[1] = (int32_t)h1; 1387 h[2] = (int32_t)h2; 1388 h[3] = (int32_t)h3; 1389 h[4] = (int32_t)h4; 1390 h[5] = (int32_t)h5; 1391 h[6] = (int32_t)h6; 1392 h[7] = (int32_t)h7; 1393 h[8] = (int32_t)h8; 1394 h[9] = (int32_t)h9; 1395 } 1396 1397 static void fe_invert(fe out, const fe z) 1398 { 1399 fe t0; 1400 fe t1; 1401 fe t2; 1402 fe t3; 1403 int i; 1404 1405 /* 1406 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 1407 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 1408 */ 1409 1410 /* t0 = z ** 2 */ 1411 fe_sq(t0, z); 1412 1413 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 1414 fe_sq(t1, t0); 1415 fe_sq(t1, t1); 1416 1417 /* t1 = z * t1 = z ** 9 */ 1418 fe_mul(t1, z, t1); 1419 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 1420 fe_mul(t0, t0, t1); 1421 1422 /* t2 = t0 ** 2 = z ** 22 */ 1423 fe_sq(t2, t0); 1424 1425 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 1426 fe_mul(t1, t1, t2); 1427 1428 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 1429 fe_sq(t2, t1); 1430 for (i = 1; i < 5; ++i) { 1431 fe_sq(t2, t2); 1432 } 1433 1434 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 1435 fe_mul(t1, t2, t1); 1436 1437 /* Continuing similarly... */ 1438 1439 /* t2 = z ** (2 ** 20 - 1) */ 1440 fe_sq(t2, t1); 1441 for (i = 1; i < 10; ++i) { 1442 fe_sq(t2, t2); 1443 } 1444 fe_mul(t2, t2, t1); 1445 1446 /* t2 = z ** (2 ** 40 - 1) */ 1447 fe_sq(t3, t2); 1448 for (i = 1; i < 20; ++i) { 1449 fe_sq(t3, t3); 1450 } 1451 fe_mul(t2, t3, t2); 1452 1453 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 1454 for (i = 0; i < 10; ++i) { 1455 fe_sq(t2, t2); 1456 } 1457 /* t1 = z ** (2 ** 50 - 1) */ 1458 fe_mul(t1, t2, t1); 1459 1460 /* t2 = z ** (2 ** 100 - 1) */ 1461 fe_sq(t2, t1); 1462 for (i = 1; i < 50; ++i) { 1463 fe_sq(t2, t2); 1464 } 1465 fe_mul(t2, t2, t1); 1466 1467 /* t2 = z ** (2 ** 200 - 1) */ 1468 fe_sq(t3, t2); 1469 for (i = 1; i < 100; ++i) { 1470 fe_sq(t3, t3); 1471 } 1472 fe_mul(t2, t3, t2); 1473 1474 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 1475 fe_sq(t2, t2); 1476 for (i = 1; i < 50; ++i) { 1477 fe_sq(t2, t2); 1478 } 1479 1480 /* t1 = z ** (2 ** 250 - 1) */ 1481 fe_mul(t1, t2, t1); 1482 1483 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 1484 fe_sq(t1, t1); 1485 for (i = 1; i < 5; ++i) { 1486 fe_sq(t1, t1); 1487 } 1488 1489 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 1490 fe_mul(out, t1, t0); 1491 } 1492 1493 /* 1494 * h = -f 1495 * 1496 * Preconditions: 1497 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1498 * 1499 * Postconditions: 1500 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1501 */ 1502 static void fe_neg(fe h, const fe f) 1503 { 1504 unsigned i; 1505 1506 for (i = 0; i < 10; i++) { 1507 h[i] = -f[i]; 1508 } 1509 } 1510 1511 /* 1512 * Replace (f,g) with (g,g) if b == 1; 1513 * replace (f,g) with (f,g) if b == 0. 1514 * 1515 * Preconditions: b in {0,1}. 1516 */ 1517 static void fe_cmov(fe f, const fe g, unsigned b) 1518 { 1519 size_t i; 1520 1521 b = 0-b; 1522 for (i = 0; i < 10; i++) { 1523 int32_t x = f[i] ^ g[i]; 1524 x &= b; 1525 f[i] ^= x; 1526 } 1527 } 1528 1529 /* 1530 * return 0 if f == 0 1531 * return 1 if f != 0 1532 * 1533 * Preconditions: 1534 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1535 */ 1536 static int fe_isnonzero(const fe f) 1537 { 1538 uint8_t s[32]; 1539 static const uint8_t zero[32] = {0}; 1540 1541 fe_tobytes(s, f); 1542 1543 return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0; 1544 } 1545 1546 /* 1547 * return 1 if f is in {1,3,5,...,q-2} 1548 * return 0 if f is in {0,2,4,...,q-1} 1549 * 1550 * Preconditions: 1551 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1552 */ 1553 static int fe_isnegative(const fe f) 1554 { 1555 uint8_t s[32]; 1556 1557 fe_tobytes(s, f); 1558 return s[0] & 1; 1559 } 1560 1561 /* 1562 * h = 2 * f * f 1563 * 1564 * Can overlap h with f. 1565 * 1566 * Preconditions: 1567 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1568 * 1569 * Postconditions: 1570 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1571 * 1572 * See fe_mul.c for discussion of implementation strategy. 1573 */ 1574 static void fe_sq2(fe h, const fe f) 1575 { 1576 int32_t f0 = f[0]; 1577 int32_t f1 = f[1]; 1578 int32_t f2 = f[2]; 1579 int32_t f3 = f[3]; 1580 int32_t f4 = f[4]; 1581 int32_t f5 = f[5]; 1582 int32_t f6 = f[6]; 1583 int32_t f7 = f[7]; 1584 int32_t f8 = f[8]; 1585 int32_t f9 = f[9]; 1586 int32_t f0_2 = 2 * f0; 1587 int32_t f1_2 = 2 * f1; 1588 int32_t f2_2 = 2 * f2; 1589 int32_t f3_2 = 2 * f3; 1590 int32_t f4_2 = 2 * f4; 1591 int32_t f5_2 = 2 * f5; 1592 int32_t f6_2 = 2 * f6; 1593 int32_t f7_2 = 2 * f7; 1594 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */ 1595 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */ 1596 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */ 1597 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */ 1598 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */ 1599 int64_t f0f0 = f0 * (int64_t) f0; 1600 int64_t f0f1_2 = f0_2 * (int64_t) f1; 1601 int64_t f0f2_2 = f0_2 * (int64_t) f2; 1602 int64_t f0f3_2 = f0_2 * (int64_t) f3; 1603 int64_t f0f4_2 = f0_2 * (int64_t) f4; 1604 int64_t f0f5_2 = f0_2 * (int64_t) f5; 1605 int64_t f0f6_2 = f0_2 * (int64_t) f6; 1606 int64_t f0f7_2 = f0_2 * (int64_t) f7; 1607 int64_t f0f8_2 = f0_2 * (int64_t) f8; 1608 int64_t f0f9_2 = f0_2 * (int64_t) f9; 1609 int64_t f1f1_2 = f1_2 * (int64_t) f1; 1610 int64_t f1f2_2 = f1_2 * (int64_t) f2; 1611 int64_t f1f3_4 = f1_2 * (int64_t) f3_2; 1612 int64_t f1f4_2 = f1_2 * (int64_t) f4; 1613 int64_t f1f5_4 = f1_2 * (int64_t) f5_2; 1614 int64_t f1f6_2 = f1_2 * (int64_t) f6; 1615 int64_t f1f7_4 = f1_2 * (int64_t) f7_2; 1616 int64_t f1f8_2 = f1_2 * (int64_t) f8; 1617 int64_t f1f9_76 = f1_2 * (int64_t) f9_38; 1618 int64_t f2f2 = f2 * (int64_t) f2; 1619 int64_t f2f3_2 = f2_2 * (int64_t) f3; 1620 int64_t f2f4_2 = f2_2 * (int64_t) f4; 1621 int64_t f2f5_2 = f2_2 * (int64_t) f5; 1622 int64_t f2f6_2 = f2_2 * (int64_t) f6; 1623 int64_t f2f7_2 = f2_2 * (int64_t) f7; 1624 int64_t f2f8_38 = f2_2 * (int64_t) f8_19; 1625 int64_t f2f9_38 = f2 * (int64_t) f9_38; 1626 int64_t f3f3_2 = f3_2 * (int64_t) f3; 1627 int64_t f3f4_2 = f3_2 * (int64_t) f4; 1628 int64_t f3f5_4 = f3_2 * (int64_t) f5_2; 1629 int64_t f3f6_2 = f3_2 * (int64_t) f6; 1630 int64_t f3f7_76 = f3_2 * (int64_t) f7_38; 1631 int64_t f3f8_38 = f3_2 * (int64_t) f8_19; 1632 int64_t f3f9_76 = f3_2 * (int64_t) f9_38; 1633 int64_t f4f4 = f4 * (int64_t) f4; 1634 int64_t f4f5_2 = f4_2 * (int64_t) f5; 1635 int64_t f4f6_38 = f4_2 * (int64_t) f6_19; 1636 int64_t f4f7_38 = f4 * (int64_t) f7_38; 1637 int64_t f4f8_38 = f4_2 * (int64_t) f8_19; 1638 int64_t f4f9_38 = f4 * (int64_t) f9_38; 1639 int64_t f5f5_38 = f5 * (int64_t) f5_38; 1640 int64_t f5f6_38 = f5_2 * (int64_t) f6_19; 1641 int64_t f5f7_76 = f5_2 * (int64_t) f7_38; 1642 int64_t f5f8_38 = f5_2 * (int64_t) f8_19; 1643 int64_t f5f9_76 = f5_2 * (int64_t) f9_38; 1644 int64_t f6f6_19 = f6 * (int64_t) f6_19; 1645 int64_t f6f7_38 = f6 * (int64_t) f7_38; 1646 int64_t f6f8_38 = f6_2 * (int64_t) f8_19; 1647 int64_t f6f9_38 = f6 * (int64_t) f9_38; 1648 int64_t f7f7_38 = f7 * (int64_t) f7_38; 1649 int64_t f7f8_38 = f7_2 * (int64_t) f8_19; 1650 int64_t f7f9_76 = f7_2 * (int64_t) f9_38; 1651 int64_t f8f8_19 = f8 * (int64_t) f8_19; 1652 int64_t f8f9_38 = f8 * (int64_t) f9_38; 1653 int64_t f9f9_38 = f9 * (int64_t) f9_38; 1654 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38; 1655 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38; 1656 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19; 1657 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38; 1658 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38; 1659 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38; 1660 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19; 1661 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38; 1662 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38; 1663 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2; 1664 int64_t carry0; 1665 int64_t carry1; 1666 int64_t carry2; 1667 int64_t carry3; 1668 int64_t carry4; 1669 int64_t carry5; 1670 int64_t carry6; 1671 int64_t carry7; 1672 int64_t carry8; 1673 int64_t carry9; 1674 1675 h0 += h0; 1676 h1 += h1; 1677 h2 += h2; 1678 h3 += h3; 1679 h4 += h4; 1680 h5 += h5; 1681 h6 += h6; 1682 h7 += h7; 1683 h8 += h8; 1684 h9 += h9; 1685 1686 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1687 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1688 1689 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 1690 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 1691 1692 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 1693 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 1694 1695 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 1696 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 1697 1698 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1699 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 1700 1701 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 1702 1703 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1704 1705 h[0] = (int32_t)h0; 1706 h[1] = (int32_t)h1; 1707 h[2] = (int32_t)h2; 1708 h[3] = (int32_t)h3; 1709 h[4] = (int32_t)h4; 1710 h[5] = (int32_t)h5; 1711 h[6] = (int32_t)h6; 1712 h[7] = (int32_t)h7; 1713 h[8] = (int32_t)h8; 1714 h[9] = (int32_t)h9; 1715 } 1716 1717 static void fe_pow22523(fe out, const fe z) 1718 { 1719 fe t0; 1720 fe t1; 1721 fe t2; 1722 int i; 1723 1724 fe_sq(t0, z); 1725 fe_sq(t1, t0); 1726 for (i = 1; i < 2; ++i) { 1727 fe_sq(t1, t1); 1728 } 1729 fe_mul(t1, z, t1); 1730 fe_mul(t0, t0, t1); 1731 fe_sq(t0, t0); 1732 fe_mul(t0, t1, t0); 1733 fe_sq(t1, t0); 1734 for (i = 1; i < 5; ++i) { 1735 fe_sq(t1, t1); 1736 } 1737 fe_mul(t0, t1, t0); 1738 fe_sq(t1, t0); 1739 for (i = 1; i < 10; ++i) { 1740 fe_sq(t1, t1); 1741 } 1742 fe_mul(t1, t1, t0); 1743 fe_sq(t2, t1); 1744 for (i = 1; i < 20; ++i) { 1745 fe_sq(t2, t2); 1746 } 1747 fe_mul(t1, t2, t1); 1748 fe_sq(t1, t1); 1749 for (i = 1; i < 10; ++i) { 1750 fe_sq(t1, t1); 1751 } 1752 fe_mul(t0, t1, t0); 1753 fe_sq(t1, t0); 1754 for (i = 1; i < 50; ++i) { 1755 fe_sq(t1, t1); 1756 } 1757 fe_mul(t1, t1, t0); 1758 fe_sq(t2, t1); 1759 for (i = 1; i < 100; ++i) { 1760 fe_sq(t2, t2); 1761 } 1762 fe_mul(t1, t2, t1); 1763 fe_sq(t1, t1); 1764 for (i = 1; i < 50; ++i) { 1765 fe_sq(t1, t1); 1766 } 1767 fe_mul(t0, t1, t0); 1768 fe_sq(t0, t0); 1769 for (i = 1; i < 2; ++i) { 1770 fe_sq(t0, t0); 1771 } 1772 fe_mul(out, t0, z); 1773 } 1774 1775 /* 1776 * ge means group element. 1777 * 1778 * Here the group is the set of pairs (x,y) of field elements (see fe.h) 1779 * satisfying -x^2 + y^2 = 1 + d x^2y^2 1780 * where d = -121665/121666. 1781 * 1782 * Representations: 1783 * ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z 1784 * ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT 1785 * ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T 1786 * ge_precomp (Duif): (y+x,y-x,2dxy) 1787 */ 1788 typedef struct { 1789 fe X; 1790 fe Y; 1791 fe Z; 1792 } ge_p2; 1793 1794 typedef struct { 1795 fe X; 1796 fe Y; 1797 fe Z; 1798 fe T; 1799 } ge_p3; 1800 1801 typedef struct { 1802 fe X; 1803 fe Y; 1804 fe Z; 1805 fe T; 1806 } ge_p1p1; 1807 1808 typedef struct { 1809 fe yplusx; 1810 fe yminusx; 1811 fe xy2d; 1812 } ge_precomp; 1813 1814 typedef struct { 1815 fe YplusX; 1816 fe YminusX; 1817 fe Z; 1818 fe T2d; 1819 } ge_cached; 1820 1821 static void ge_tobytes(uint8_t *s, const ge_p2 *h) 1822 { 1823 fe recip; 1824 fe x; 1825 fe y; 1826 1827 fe_invert(recip, h->Z); 1828 fe_mul(x, h->X, recip); 1829 fe_mul(y, h->Y, recip); 1830 fe_tobytes(s, y); 1831 s[31] ^= fe_isnegative(x) << 7; 1832 } 1833 1834 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h) 1835 { 1836 fe recip; 1837 fe x; 1838 fe y; 1839 1840 fe_invert(recip, h->Z); 1841 fe_mul(x, h->X, recip); 1842 fe_mul(y, h->Y, recip); 1843 fe_tobytes(s, y); 1844 s[31] ^= fe_isnegative(x) << 7; 1845 } 1846 1847 static const fe d = { 1848 -10913610, 13857413, -15372611, 6949391, 114729, 1849 -8787816, -6275908, -3247719, -18696448, -12055116 1850 }; 1851 1852 static const fe sqrtm1 = { 1853 -32595792, -7943725, 9377950, 3500415, 12389472, 1854 -272473, -25146209, -2005654, 326686, 11406482 1855 }; 1856 1857 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s) 1858 { 1859 fe u; 1860 fe v; 1861 fe v3; 1862 fe vxx; 1863 fe check; 1864 1865 fe_frombytes(h->Y, s); 1866 fe_1(h->Z); 1867 fe_sq(u, h->Y); 1868 fe_mul(v, u, d); 1869 fe_sub(u, u, h->Z); /* u = y^2-1 */ 1870 fe_add(v, v, h->Z); /* v = dy^2+1 */ 1871 1872 fe_sq(v3, v); 1873 fe_mul(v3, v3, v); /* v3 = v^3 */ 1874 fe_sq(h->X, v3); 1875 fe_mul(h->X, h->X, v); 1876 fe_mul(h->X, h->X, u); /* x = uv^7 */ 1877 1878 fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */ 1879 fe_mul(h->X, h->X, v3); 1880 fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */ 1881 1882 fe_sq(vxx, h->X); 1883 fe_mul(vxx, vxx, v); 1884 fe_sub(check, vxx, u); /* vx^2-u */ 1885 if (fe_isnonzero(check)) { 1886 fe_add(check, vxx, u); /* vx^2+u */ 1887 if (fe_isnonzero(check)) { 1888 return -1; 1889 } 1890 fe_mul(h->X, h->X, sqrtm1); 1891 } 1892 1893 if (fe_isnegative(h->X) != (s[31] >> 7)) { 1894 fe_neg(h->X, h->X); 1895 } 1896 1897 fe_mul(h->T, h->X, h->Y); 1898 return 0; 1899 } 1900 1901 static void ge_p2_0(ge_p2 *h) 1902 { 1903 fe_0(h->X); 1904 fe_1(h->Y); 1905 fe_1(h->Z); 1906 } 1907 1908 static void ge_p3_0(ge_p3 *h) 1909 { 1910 fe_0(h->X); 1911 fe_1(h->Y); 1912 fe_1(h->Z); 1913 fe_0(h->T); 1914 } 1915 1916 static void ge_precomp_0(ge_precomp *h) 1917 { 1918 fe_1(h->yplusx); 1919 fe_1(h->yminusx); 1920 fe_0(h->xy2d); 1921 } 1922 1923 /* r = p */ 1924 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) 1925 { 1926 fe_copy(r->X, p->X); 1927 fe_copy(r->Y, p->Y); 1928 fe_copy(r->Z, p->Z); 1929 } 1930 1931 static const fe d2 = { 1932 -21827239, -5839606, -30745221, 13898782, 229458, 1933 15978800, -12551817, -6495438, 29715968, 9444199 1934 }; 1935 1936 /* r = p */ 1937 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p) 1938 { 1939 fe_add(r->YplusX, p->Y, p->X); 1940 fe_sub(r->YminusX, p->Y, p->X); 1941 fe_copy(r->Z, p->Z); 1942 fe_mul(r->T2d, p->T, d2); 1943 } 1944 1945 /* r = p */ 1946 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) 1947 { 1948 fe_mul(r->X, p->X, p->T); 1949 fe_mul(r->Y, p->Y, p->Z); 1950 fe_mul(r->Z, p->Z, p->T); 1951 } 1952 1953 /* r = p */ 1954 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p) 1955 { 1956 fe_mul(r->X, p->X, p->T); 1957 fe_mul(r->Y, p->Y, p->Z); 1958 fe_mul(r->Z, p->Z, p->T); 1959 fe_mul(r->T, p->X, p->Y); 1960 } 1961 1962 /* r = 2 * p */ 1963 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p) 1964 { 1965 fe t0; 1966 1967 fe_sq(r->X, p->X); 1968 fe_sq(r->Z, p->Y); 1969 fe_sq2(r->T, p->Z); 1970 fe_add(r->Y, p->X, p->Y); 1971 fe_sq(t0, r->Y); 1972 fe_add(r->Y, r->Z, r->X); 1973 fe_sub(r->Z, r->Z, r->X); 1974 fe_sub(r->X, t0, r->Y); 1975 fe_sub(r->T, r->T, r->Z); 1976 } 1977 1978 /* r = 2 * p */ 1979 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p) 1980 { 1981 ge_p2 q; 1982 ge_p3_to_p2(&q, p); 1983 ge_p2_dbl(r, &q); 1984 } 1985 1986 /* r = p + q */ 1987 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) 1988 { 1989 fe t0; 1990 1991 fe_add(r->X, p->Y, p->X); 1992 fe_sub(r->Y, p->Y, p->X); 1993 fe_mul(r->Z, r->X, q->yplusx); 1994 fe_mul(r->Y, r->Y, q->yminusx); 1995 fe_mul(r->T, q->xy2d, p->T); 1996 fe_add(t0, p->Z, p->Z); 1997 fe_sub(r->X, r->Z, r->Y); 1998 fe_add(r->Y, r->Z, r->Y); 1999 fe_add(r->Z, t0, r->T); 2000 fe_sub(r->T, t0, r->T); 2001 } 2002 2003 /* r = p - q */ 2004 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) 2005 { 2006 fe t0; 2007 2008 fe_add(r->X, p->Y, p->X); 2009 fe_sub(r->Y, p->Y, p->X); 2010 fe_mul(r->Z, r->X, q->yminusx); 2011 fe_mul(r->Y, r->Y, q->yplusx); 2012 fe_mul(r->T, q->xy2d, p->T); 2013 fe_add(t0, p->Z, p->Z); 2014 fe_sub(r->X, r->Z, r->Y); 2015 fe_add(r->Y, r->Z, r->Y); 2016 fe_sub(r->Z, t0, r->T); 2017 fe_add(r->T, t0, r->T); 2018 } 2019 2020 /* r = p + q */ 2021 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) 2022 { 2023 fe t0; 2024 2025 fe_add(r->X, p->Y, p->X); 2026 fe_sub(r->Y, p->Y, p->X); 2027 fe_mul(r->Z, r->X, q->YplusX); 2028 fe_mul(r->Y, r->Y, q->YminusX); 2029 fe_mul(r->T, q->T2d, p->T); 2030 fe_mul(r->X, p->Z, q->Z); 2031 fe_add(t0, r->X, r->X); 2032 fe_sub(r->X, r->Z, r->Y); 2033 fe_add(r->Y, r->Z, r->Y); 2034 fe_add(r->Z, t0, r->T); 2035 fe_sub(r->T, t0, r->T); 2036 } 2037 2038 /* r = p - q */ 2039 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) 2040 { 2041 fe t0; 2042 2043 fe_add(r->X, p->Y, p->X); 2044 fe_sub(r->Y, p->Y, p->X); 2045 fe_mul(r->Z, r->X, q->YminusX); 2046 fe_mul(r->Y, r->Y, q->YplusX); 2047 fe_mul(r->T, q->T2d, p->T); 2048 fe_mul(r->X, p->Z, q->Z); 2049 fe_add(t0, r->X, r->X); 2050 fe_sub(r->X, r->Z, r->Y); 2051 fe_add(r->Y, r->Z, r->Y); 2052 fe_sub(r->Z, t0, r->T); 2053 fe_add(r->T, t0, r->T); 2054 } 2055 2056 static uint8_t equal(signed char b, signed char c) 2057 { 2058 uint8_t ub = b; 2059 uint8_t uc = c; 2060 uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */ 2061 uint32_t y = x; /* 0: yes; 1..255: no */ 2062 y -= 1; /* 4294967295: yes; 0..254: no */ 2063 y >>= 31; /* 1: yes; 0: no */ 2064 return y; 2065 } 2066 2067 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b) 2068 { 2069 fe_cmov(t->yplusx, u->yplusx, b); 2070 fe_cmov(t->yminusx, u->yminusx, b); 2071 fe_cmov(t->xy2d, u->xy2d, b); 2072 } 2073 2074 /* k25519Precomp[i][j] = (j+1)*256^i*B */ 2075 static const ge_precomp k25519Precomp[32][8] = { 2076 { 2077 { 2078 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 2079 27544626, -11754271, -6079156, 2047605}, 2080 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 2081 5043384, 19500929, -15469378}, 2082 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 2083 29287919, 11864899, -24514362, -4438546}, 2084 }, 2085 { 2086 {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, 2087 -11717903, -3814571, -358445, -10211303}, 2088 {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, 2089 -15616551, 11189268, -26829678, -5319081}, 2090 {26966642, 11152617, 32442495, 15396054, 14353839, -12752335, 2091 -3128826, -9541118, -15472047, -4166697}, 2092 }, 2093 { 2094 {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 2095 27787600, -14772189, 28944400, -1550024}, 2096 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 2097 16354577, -11775962, 7689662, 11199574}, 2098 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 2099 7512774, 10017326, -17749093, -9920357}, 2100 }, 2101 { 2102 {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, 2103 -28926210, 15006023, 3284568, -6276540}, 2104 {23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 2105 7464579, 9656445, 13059162, 10374397}, 2106 {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, 2107 -3839045, -641708, -101325}, 2108 }, 2109 { 2110 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 2111 32867885, 14515107, -15438304, 10819380}, 2112 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 2113 12483688, -12668491, 5581306}, 2114 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 2115 28542350, 13850243, -23678021, -15815942}, 2116 }, 2117 { 2118 {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, 2119 -19188627, -15224819, -9818940, -12085777}, 2120 {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, 2121 -15689887, 1762328, 14866737}, 2122 {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, 2123 -28236412, 3959421, 27914454, 4383652}, 2124 }, 2125 { 2126 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 2127 5230134, -23952439, -15175766}, 2128 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 2129 20654025, 16520125, 30598449, 7715701}, 2130 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, 2131 -31400660, 1370708, 29794553, -1409300}, 2132 }, 2133 { 2134 {14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 2135 30290735, 10876454, -33154098, 2381726}, 2136 {-7195431, -2655363, -14730155, 462251, -27724326, 3941372, 2137 -6236617, 3696005, -32300832, 15351955}, 2138 {27431194, 8222322, 16448760, -3907995, -18707002, 11938355, 2139 -32961401, -2970515, 29551813, 10109425}, 2140 }, 2141 }, 2142 { 2143 { 2144 {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, 2145 -2378284, -1627556, 10092783, -4764171}, 2146 {27939166, 14210322, 4677035, 16277044, -22964462, -12398139, 2147 -32508754, 12005538, -17810127, 12803510}, 2148 {17228999, -15661624, -1233527, 300140, -1224870, -11714777, 2149 30364213, -9038194, 18016357, 4397660}, 2150 }, 2151 { 2152 {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, 2153 -26619106, 14544525, -17477504, 982639}, 2154 {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, 2155 -4120128, -21047696, 9934963}, 2156 {5793303, 16271923, -24131614, -10116404, 29188560, 1206517, 2157 -14747930, 4559895, -30123922, -10897950}, 2158 }, 2159 { 2160 {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 2161 24191034, 4541697, -13338309, 5500568}, 2162 {12650548, -1497113, 9052871, 11355358, -17680037, -8400164, 2163 -17430592, 12264343, 10874051, 13524335}, 2164 {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 2165 5080568, -22528059, 5376628}, 2166 }, 2167 { 2168 {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, 2169 -22321305, -9447443, 4535768, 1569007}, 2170 {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, 2171 -30494562, 3044290, 31848280, 12543772}, 2172 {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, 2173 -27377195, -2062731, 7718482, 14474653}, 2174 }, 2175 { 2176 {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, 2177 -7236665, 24316168, -5253567}, 2178 {13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 2179 33040651, -13424532, -20729456, 8321686}, 2180 {21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 2181 23845965, -11874838, -9984458, 608372}, 2182 }, 2183 { 2184 {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 2185 1123968, -6780577, 27229399, 23887}, 2186 {-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 2187 12797024, -6440308, -1633405, 16678954}, 2188 {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, 2189 -1508144, -4795045, -17169265, 4904953}, 2190 }, 2191 { 2192 {24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 2193 5169211, 16191880, 2128236, -4326833}, 2194 {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247, 2195 -29806336, 916033, -6882542, -2986532}, 2196 {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 2197 285431, 2763829, 15736322, 4143876}, 2198 }, 2199 { 2200 {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, 2201 -14594663, 23527084, -16458268}, 2202 {33431127, -11130478, -17838966, -15626900, 8909499, 8376530, 2203 -32625340, 4087881, -15188911, -14416214}, 2204 {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055, 2205 4357868, -4774191, -16323038}, 2206 }, 2207 }, 2208 { 2209 { 2210 {6721966, 13833823, -23523388, -1551314, 26354293, -11863321, 2211 23365147, -3949732, 7390890, 2759800}, 2212 {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353, 2213 -4264057, 1244380, -12919645}, 2214 {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 2215 9208236, 15886429, 16489664}, 2216 }, 2217 { 2218 {1996075, 10375649, 14346367, 13311202, -6874135, -16438411, 2219 -13693198, 398369, -30606455, -712933}, 2220 {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363, 2221 13348553, 12076947, -30836462, 5113182}, 2222 {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344, 2223 -30341101, -7336386, 13847711, 5387222}, 2224 }, 2225 { 2226 {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611, 2227 8763061, 3617786, -19600662, 10370991}, 2228 {20246567, -14369378, 22358229, -543712, 18507283, -10413996, 2229 14554437, -8746092, 32232924, 16763880}, 2230 {9648505, 10094563, 26416693, 14745928, -30374318, -6472621, 2231 11094161, 15689506, 3140038, -16510092}, 2232 }, 2233 { 2234 {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685, 2235 -27224800, 9448613, -28774454, 366295}, 2236 {19153450, 11523972, -11096490, -6503142, -24647631, 5420647, 2237 28344573, 8041113, 719605, 11671788}, 2238 {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916, 2239 -15266516, 27000813, -10195553}, 2240 }, 2241 { 2242 {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065, 2243 5336097, 6750977, -14521026}, 2244 {11836410, -3979488, 26297894, 16080799, 23455045, 15735944, 2245 1695823, -8819122, 8169720, 16220347}, 2246 {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763, 2247 -11144307, -2627664, -5990708, -14166033}, 2248 }, 2249 { 2250 {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004, 2251 27884329, 2847284, 2655861, 1738395}, 2252 {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821, 2253 21651608, -3239336, -19087449, -11005278}, 2254 {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092, 2255 5821408, 10478196, 8544890}, 2256 }, 2257 { 2258 {32173121, -16129311, 24896207, 3921497, 22579056, -3410854, 2259 19270449, 12217473, 17789017, -3395995}, 2260 {-30552961, -2228401, -15578829, -10147201, 13243889, 517024, 2261 15479401, -3853233, 30460520, 1052596}, 2262 {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687, 2263 27491595, -4612359, 3179268, -9478891}, 2264 }, 2265 { 2266 {31947069, -14366651, -4640583, -15339921, -15125977, -6039709, 2267 -14756777, -16411740, 19072640, -9511060}, 2268 {11685058, 11822410, 3158003, -13952594, 33402194, -4165066, 2269 5977896, -5215017, 473099, 5040608}, 2270 {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760, 2271 28326862, 1721092, -19558642, -3131606}, 2272 }, 2273 }, 2274 { 2275 { 2276 {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786, 2277 8076149, -27868496, 11538389}, 2278 {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211, 2279 8754525, 7446702, -5676054, 5797016}, 2280 {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199, 2281 2014099, -9050574, -2369172, -5877341}, 2282 }, 2283 { 2284 {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559, 2285 1192730, -3714199, 15123619, 10811505}, 2286 {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363, 2287 15776356, -28886779, -11974553}, 2288 {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569, 2289 -20654173, -16484855, 4714547, -9600655}, 2290 }, 2291 { 2292 {15200332, 8368572, 19679101, 15970074, -31872674, 1959451, 2293 24611599, -4543832, -11745876, 12340220}, 2294 {12876937, -10480056, 33134381, 6590940, -6307776, 14872440, 2295 9613953, 8241152, 15370987, 9608631}, 2296 {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868, 2297 15866074, -28210621, -8814099}, 2298 }, 2299 { 2300 {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233, 2301 858697, 20571223, 8420556}, 2302 {14620715, 13067227, -15447274, 8264467, 14106269, 15080814, 2303 33531827, 12516406, -21574435, -12476749}, 2304 {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519, 2305 7256740, 8791136, 15069930}, 2306 }, 2307 { 2308 {1276410, -9371918, 22949635, -16322807, -23493039, -5702186, 2309 14711875, 4874229, -30663140, -2331391}, 2310 {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175, 2311 -7912378, -33069337, 9234253}, 2312 {20590503, -9018988, 31529744, -7352666, -2706834, 10650548, 2313 31559055, -11609587, 18979186, 13396066}, 2314 }, 2315 { 2316 {24474287, 4968103, 22267082, 4407354, 24063882, -8325180, 2317 -18816887, 13594782, 33514650, 7021958}, 2318 {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421, 2319 -25948728, -3916677, -21480480, 12868082}, 2320 {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208, 2321 -21446107, 2244500, -12455797, -8089383}, 2322 }, 2323 { 2324 {-30595528, 13793479, -5852820, 319136, -25723172, -6263899, 2325 33086546, 8957937, -15233648, 5540521}, 2326 {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908, 2327 -23710744, -1568984, -16128528, -14962807}, 2328 {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819, 2329 892185, -11513277, -15205948}, 2330 }, 2331 { 2332 {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819, 2333 4763127, -19179614, 5867134}, 2334 {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500, 2335 27846559, 5931263, -29749703, -16108455}, 2336 {27461885, -2977536, 22380810, 1815854, -23033753, -3031938, 2337 7283490, -15148073, -19526700, 7734629}, 2338 }, 2339 }, 2340 { 2341 { 2342 {-8010264, -9590817, -11120403, 6196038, 29344158, -13430885, 2343 7585295, -3176626, 18549497, 15302069}, 2344 {-32658337, -6171222, -7672793, -11051681, 6258878, 13504381, 2345 10458790, -6418461, -8872242, 8424746}, 2346 {24687205, 8613276, -30667046, -3233545, 1863892, -1830544, 2347 19206234, 7134917, -11284482, -828919}, 2348 }, 2349 { 2350 {11334899, -9218022, 8025293, 12707519, 17523892, -10476071, 2351 10243738, -14685461, -5066034, 16498837}, 2352 {8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925, 2353 -14124238, 6536641, 10543906}, 2354 {-28946384, 15479763, -17466835, 568876, -1497683, 11223454, 2355 -2669190, -16625574, -27235709, 8876771}, 2356 }, 2357 { 2358 {-25742899, -12566864, -15649966, -846607, -33026686, -796288, 2359 -33481822, 15824474, -604426, -9039817}, 2360 {10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697, 2361 -4890037, 1657394, 3084098}, 2362 {10477963, -7470260, 12119566, -13250805, 29016247, -5365589, 2363 31280319, 14396151, -30233575, 15272409}, 2364 }, 2365 { 2366 {-12288309, 3169463, 28813183, 16658753, 25116432, -5630466, 2367 -25173957, -12636138, -25014757, 1950504}, 2368 {-26180358, 9489187, 11053416, -14746161, -31053720, 5825630, 2369 -8384306, -8767532, 15341279, 8373727}, 2370 {28685821, 7759505, -14378516, -12002860, -31971820, 4079242, 2371 298136, -10232602, -2878207, 15190420}, 2372 }, 2373 { 2374 {-32932876, 13806336, -14337485, -15794431, -24004620, 10940928, 2375 8669718, 2742393, -26033313, -6875003}, 2376 {-1580388, -11729417, -25979658, -11445023, -17411874, -10912854, 2377 9291594, -16247779, -12154742, 6048605}, 2378 {-30305315, 14843444, 1539301, 11864366, 20201677, 1900163, 2379 13934231, 5128323, 11213262, 9168384}, 2380 }, 2381 { 2382 {-26280513, 11007847, 19408960, -940758, -18592965, -4328580, 2383 -5088060, -11105150, 20470157, -16398701}, 2384 {-23136053, 9282192, 14855179, -15390078, -7362815, -14408560, 2385 -22783952, 14461608, 14042978, 5230683}, 2386 {29969567, -2741594, 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-11572403, 22828471, 13290673, 4087 -7125085, 12469656, 29111212, -5451014}, 4088 }, 4089 { 4090 {24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 4091 24367466, 6388839, -10295587, 452383}, 4092 {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, 4093 -24236251, -5915248, 15766062, 8407814}, 4094 {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, 4095 -8917023, -4388953, -8067909, 2276718}, 4096 }, 4097 { 4098 {30157918, 12924066, -17712050, 9245753, 19895028, 3368142, 4099 -23827587, 5096219, 22740376, -7303417}, 4100 {2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 4101 24051124, 13742383, -15637599, 13295222}, 4102 {33338237, -8505733, 12532113, 7977527, 9106186, -1715251, 4103 -17720195, -4612972, -4451357, -14669444}, 4104 }, 4105 { 4106 {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, 4107 -2469266, -4141880, 7770569, 9620597}, 4108 {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, 4109 -1694323, -33502340, -14767970}, 4110 {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 4111 1220118, 30494170, -11440799}, 4112 }, 4113 { 4114 {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, 4115 -26739026, 926050, -1684339, -13333647}, 4116 {13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 4117 9021034, 9078865, 3353509, 4033511}, 4118 {-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 4119 23161163, 8839127, 27485041, 7356032}, 4120 }, 4121 }, 4122 { 4123 { 4124 {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 4125 2625015, 28431036, -16771834}, 4126 {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, 4127 -22545972, 14150565, 15970762, 4099461}, 4128 {29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 4129 13617293, -9937143, 11465739, 8317062}, 4130 }, 4131 { 4132 {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 4133 14898637, 3848455, 20969334, -5157516}, 4134 {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, 4135 -21610826, -3649888, 11177095, 14989547}, 4136 {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 4137 13515641, 2581286, -28487508, 9930240}, 4138 }, 4139 { 4140 {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 4141 18345767, -13403753, 16291481, -5314038}, 4142 {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 4143 6957617, 4368891, 9788741}, 4144 {16660756, 7281060, -10830758, 12911820, 20108584, -8101676, 4145 -21722536, -8613148, 16250552, -11111103}, 4146 }, 4147 { 4148 {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 4149 10604807, -30190403, 4782747}, 4150 {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, 4151 -9981571, 4383045, 22546403, 437323}, 4152 {31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 4153 27343084, 2786261, -30633590, -14097016}, 4154 }, 4155 { 4156 {-14467279, -683715, -33374107, 7448552, 19294360, 14334329, 4157 -19690631, 2355319, -19284671, -6114373}, 4158 {15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 4159 18952177, 15496498, -29380133, 11754228}, 4160 {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 4161 7141596, 11724556, 22761615, -10134141}, 4162 }, 4163 { 4164 {16918416, 11729663, -18083579, 3022987, -31015732, -13339659, 4165 -28741185, -12227393, 32851222, 11717399}, 4166 {11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 4167 31474879, 3483633, -1193175, -4030831}, 4168 {-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 4169 33142652, 6546660, -19985279, -3948376}, 4170 }, 4171 { 4172 {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, 4173 -8537131, -12833048, -30772034, -15486313}, 4174 {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, 4175 -31135347, -16049879, 10928917, 3011958}, 4176 {-6957757, -15594337, 31696059, 334240, 29576716, 14796075, 4177 -30831056, -12805180, 18008031, 10258577}, 4178 }, 4179 { 4180 {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, 4181 -1853465, 1367120, 25127874, 6671743}, 4182 {29701166, -14373934, -10878120, 9279288, -17568, 13127210, 4183 21382910, 11042292, 25838796, 4642684}, 4184 {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 4185 30468147, -13900640, 18423289, 4177476}, 4186 }, 4187 }, 4188 }; 4189 4190 static uint8_t negative(signed char b) 4191 { 4192 uint32_t x = b; 4193 4194 x >>= 31; /* 1: yes; 0: no */ 4195 return x; 4196 } 4197 4198 static void table_select(ge_precomp *t, int pos, signed char b) 4199 { 4200 ge_precomp minust; 4201 uint8_t bnegative = negative(b); 4202 uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1); 4203 4204 ge_precomp_0(t); 4205 cmov(t, &k25519Precomp[pos][0], equal(babs, 1)); 4206 cmov(t, &k25519Precomp[pos][1], equal(babs, 2)); 4207 cmov(t, &k25519Precomp[pos][2], equal(babs, 3)); 4208 cmov(t, &k25519Precomp[pos][3], equal(babs, 4)); 4209 cmov(t, &k25519Precomp[pos][4], equal(babs, 5)); 4210 cmov(t, &k25519Precomp[pos][5], equal(babs, 6)); 4211 cmov(t, &k25519Precomp[pos][6], equal(babs, 7)); 4212 cmov(t, &k25519Precomp[pos][7], equal(babs, 8)); 4213 fe_copy(minust.yplusx, t->yminusx); 4214 fe_copy(minust.yminusx, t->yplusx); 4215 fe_neg(minust.xy2d, t->xy2d); 4216 cmov(t, &minust, bnegative); 4217 } 4218 4219 /* 4220 * h = a * B 4221 * 4222 * where a = a[0]+256*a[1]+...+256^31 a[31] 4223 * B is the Ed25519 base point (x,4/5) with x positive. 4224 * 4225 * Preconditions: 4226 * a[31] <= 127 4227 */ 4228 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a) 4229 { 4230 signed char e[64]; 4231 signed char carry; 4232 ge_p1p1 r; 4233 ge_p2 s; 4234 ge_precomp t; 4235 int i; 4236 4237 for (i = 0; i < 32; ++i) { 4238 e[2 * i + 0] = (a[i] >> 0) & 15; 4239 e[2 * i + 1] = (a[i] >> 4) & 15; 4240 } 4241 /* each e[i] is between 0 and 15 */ 4242 /* e[63] is between 0 and 7 */ 4243 4244 carry = 0; 4245 for (i = 0; i < 63; ++i) { 4246 e[i] += carry; 4247 carry = e[i] + 8; 4248 carry >>= 4; 4249 e[i] -= carry << 4; 4250 } 4251 e[63] += carry; 4252 /* each e[i] is between -8 and 8 */ 4253 4254 ge_p3_0(h); 4255 for (i = 1; i < 64; i += 2) { 4256 table_select(&t, i / 2, e[i]); 4257 ge_madd(&r, h, &t); 4258 ge_p1p1_to_p3(h, &r); 4259 } 4260 4261 ge_p3_dbl(&r, h); 4262 ge_p1p1_to_p2(&s, &r); 4263 ge_p2_dbl(&r, &s); 4264 ge_p1p1_to_p2(&s, &r); 4265 ge_p2_dbl(&r, &s); 4266 ge_p1p1_to_p2(&s, &r); 4267 ge_p2_dbl(&r, &s); 4268 ge_p1p1_to_p3(h, &r); 4269 4270 for (i = 0; i < 64; i += 2) { 4271 table_select(&t, i / 2, e[i]); 4272 ge_madd(&r, h, &t); 4273 ge_p1p1_to_p3(h, &r); 4274 } 4275 4276 OPENSSL_cleanse(e, sizeof(e)); 4277 } 4278 4279 #if !defined(BASE_2_51_IMPLEMENTED) 4280 /* 4281 * Replace (f,g) with (g,f) if b == 1; 4282 * replace (f,g) with (f,g) if b == 0. 4283 * 4284 * Preconditions: b in {0,1}. 4285 */ 4286 static void fe_cswap(fe f, fe g, unsigned int b) 4287 { 4288 size_t i; 4289 4290 b = 0-b; 4291 for (i = 0; i < 10; i++) { 4292 int32_t x = f[i] ^ g[i]; 4293 x &= b; 4294 f[i] ^= x; 4295 g[i] ^= x; 4296 } 4297 } 4298 4299 /* 4300 * h = f * 121666 4301 * 4302 * Can overlap h with f. 4303 * 4304 * Preconditions: 4305 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 4306 * 4307 * Postconditions: 4308 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 4309 */ 4310 static void fe_mul121666(fe h, fe f) 4311 { 4312 int32_t f0 = f[0]; 4313 int32_t f1 = f[1]; 4314 int32_t f2 = f[2]; 4315 int32_t f3 = f[3]; 4316 int32_t f4 = f[4]; 4317 int32_t f5 = f[5]; 4318 int32_t f6 = f[6]; 4319 int32_t f7 = f[7]; 4320 int32_t f8 = f[8]; 4321 int32_t f9 = f[9]; 4322 int64_t h0 = f0 * (int64_t) 121666; 4323 int64_t h1 = f1 * (int64_t) 121666; 4324 int64_t h2 = f2 * (int64_t) 121666; 4325 int64_t h3 = f3 * (int64_t) 121666; 4326 int64_t h4 = f4 * (int64_t) 121666; 4327 int64_t h5 = f5 * (int64_t) 121666; 4328 int64_t h6 = f6 * (int64_t) 121666; 4329 int64_t h7 = f7 * (int64_t) 121666; 4330 int64_t h8 = f8 * (int64_t) 121666; 4331 int64_t h9 = f9 * (int64_t) 121666; 4332 int64_t carry0; 4333 int64_t carry1; 4334 int64_t carry2; 4335 int64_t carry3; 4336 int64_t carry4; 4337 int64_t carry5; 4338 int64_t carry6; 4339 int64_t carry7; 4340 int64_t carry8; 4341 int64_t carry9; 4342 4343 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 4344 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 4345 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 4346 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 4347 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 4348 4349 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 4350 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 4351 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 4352 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 4353 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 4354 4355 h[0] = (int32_t)h0; 4356 h[1] = (int32_t)h1; 4357 h[2] = (int32_t)h2; 4358 h[3] = (int32_t)h3; 4359 h[4] = (int32_t)h4; 4360 h[5] = (int32_t)h5; 4361 h[6] = (int32_t)h6; 4362 h[7] = (int32_t)h7; 4363 h[8] = (int32_t)h8; 4364 h[9] = (int32_t)h9; 4365 } 4366 4367 static void x25519_scalar_mult_generic(uint8_t out[32], 4368 const uint8_t scalar[32], 4369 const uint8_t point[32]) { 4370 fe x1, x2, z2, x3, z3, tmp0, tmp1; 4371 uint8_t e[32]; 4372 unsigned swap = 0; 4373 int pos; 4374 4375 memcpy(e, scalar, 32); 4376 e[0] &= 248; 4377 e[31] &= 127; 4378 e[31] |= 64; 4379 fe_frombytes(x1, point); 4380 fe_1(x2); 4381 fe_0(z2); 4382 fe_copy(x3, x1); 4383 fe_1(z3); 4384 4385 for (pos = 254; pos >= 0; --pos) { 4386 unsigned b = 1 & (e[pos / 8] >> (pos & 7)); 4387 swap ^= b; 4388 fe_cswap(x2, x3, swap); 4389 fe_cswap(z2, z3, swap); 4390 swap = b; 4391 fe_sub(tmp0, x3, z3); 4392 fe_sub(tmp1, x2, z2); 4393 fe_add(x2, x2, z2); 4394 fe_add(z2, x3, z3); 4395 fe_mul(z3, tmp0, x2); 4396 fe_mul(z2, z2, tmp1); 4397 fe_sq(tmp0, tmp1); 4398 fe_sq(tmp1, x2); 4399 fe_add(x3, z3, z2); 4400 fe_sub(z2, z3, z2); 4401 fe_mul(x2, tmp1, tmp0); 4402 fe_sub(tmp1, tmp1, tmp0); 4403 fe_sq(z2, z2); 4404 fe_mul121666(z3, tmp1); 4405 fe_sq(x3, x3); 4406 fe_add(tmp0, tmp0, z3); 4407 fe_mul(z3, x1, z2); 4408 fe_mul(z2, tmp1, tmp0); 4409 } 4410 4411 fe_invert(z2, z2); 4412 fe_mul(x2, x2, z2); 4413 fe_tobytes(out, x2); 4414 4415 OPENSSL_cleanse(e, sizeof(e)); 4416 } 4417 4418 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32], 4419 const uint8_t point[32]) { 4420 x25519_scalar_mult_generic(out, scalar, point); 4421 } 4422 #endif 4423 4424 static void slide(signed char *r, const uint8_t *a) 4425 { 4426 int i; 4427 int b; 4428 int k; 4429 4430 for (i = 0; i < 256; ++i) { 4431 r[i] = 1 & (a[i >> 3] >> (i & 7)); 4432 } 4433 4434 for (i = 0; i < 256; ++i) { 4435 if (r[i]) { 4436 for (b = 1; b <= 6 && i + b < 256; ++b) { 4437 if (r[i + b]) { 4438 if (r[i] + (r[i + b] << b) <= 15) { 4439 r[i] += r[i + b] << b; 4440 r[i + b] = 0; 4441 } else if (r[i] - (r[i + b] << b) >= -15) { 4442 r[i] -= r[i + b] << b; 4443 for (k = i + b; k < 256; ++k) { 4444 if (!r[k]) { 4445 r[k] = 1; 4446 break; 4447 } 4448 r[k] = 0; 4449 } 4450 } else { 4451 break; 4452 } 4453 } 4454 } 4455 } 4456 } 4457 } 4458 4459 static const ge_precomp Bi[8] = { 4460 { 4461 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, 4462 -11754271, -6079156, 2047605}, 4463 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 4464 5043384, 19500929, -15469378}, 4465 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 4466 11864899, -24514362, -4438546}, 4467 }, 4468 { 4469 {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, 4470 -14772189, 28944400, -1550024}, 4471 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, 4472 -11775962, 7689662, 11199574}, 4473 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 4474 10017326, -17749093, -9920357}, 4475 }, 4476 { 4477 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 4478 14515107, -15438304, 10819380}, 4479 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 4480 12483688, -12668491, 5581306}, 4481 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 4482 13850243, -23678021, -15815942}, 4483 }, 4484 { 4485 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 4486 5230134, -23952439, -15175766}, 4487 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 4488 16520125, 30598449, 7715701}, 4489 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 4490 1370708, 29794553, -1409300}, 4491 }, 4492 { 4493 {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, 4494 -1361450, -13062696, 13821877}, 4495 {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, 4496 -7212327, 18853322, -14220951}, 4497 {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, 4498 -10431137, 2207753, -3209784}, 4499 }, 4500 { 4501 {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, 4502 -663000, -31111463, -16132436}, 4503 {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 4504 15725684, 171356, 6466918}, 4505 {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, 4506 -14088058, -30714912, 16193877}, 4507 }, 4508 { 4509 {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4510 4729455, -18074513, 9256800}, 4511 {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 4512 9761698, -19827198, 630305}, 4513 {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, 4514 -15960994, -2449256, -14291300}, 4515 }, 4516 { 4517 {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 4518 15033784, 25105118, -7894876}, 4519 {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 4520 1573892, -2625887, 2198790, -15804619}, 4521 {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, 4522 -16236442, -32461234, -12290683}, 4523 }, 4524 }; 4525 4526 /* 4527 * r = a * A + b * B 4528 * 4529 * where a = a[0]+256*a[1]+...+256^31 a[31]. 4530 * and b = b[0]+256*b[1]+...+256^31 b[31]. 4531 * B is the Ed25519 base point (x,4/5) with x positive. 4532 */ 4533 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a, 4534 const ge_p3 *A, const uint8_t *b) 4535 { 4536 signed char aslide[256]; 4537 signed char bslide[256]; 4538 ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */ 4539 ge_p1p1 t; 4540 ge_p3 u; 4541 ge_p3 A2; 4542 int i; 4543 4544 slide(aslide, a); 4545 slide(bslide, b); 4546 4547 ge_p3_to_cached(&Ai[0], A); 4548 ge_p3_dbl(&t, A); 4549 ge_p1p1_to_p3(&A2, &t); 4550 ge_add(&t, &A2, &Ai[0]); 4551 ge_p1p1_to_p3(&u, &t); 4552 ge_p3_to_cached(&Ai[1], &u); 4553 ge_add(&t, &A2, &Ai[1]); 4554 ge_p1p1_to_p3(&u, &t); 4555 ge_p3_to_cached(&Ai[2], &u); 4556 ge_add(&t, &A2, &Ai[2]); 4557 ge_p1p1_to_p3(&u, &t); 4558 ge_p3_to_cached(&Ai[3], &u); 4559 ge_add(&t, &A2, &Ai[3]); 4560 ge_p1p1_to_p3(&u, &t); 4561 ge_p3_to_cached(&Ai[4], &u); 4562 ge_add(&t, &A2, &Ai[4]); 4563 ge_p1p1_to_p3(&u, &t); 4564 ge_p3_to_cached(&Ai[5], &u); 4565 ge_add(&t, &A2, &Ai[5]); 4566 ge_p1p1_to_p3(&u, &t); 4567 ge_p3_to_cached(&Ai[6], &u); 4568 ge_add(&t, &A2, &Ai[6]); 4569 ge_p1p1_to_p3(&u, &t); 4570 ge_p3_to_cached(&Ai[7], &u); 4571 4572 ge_p2_0(r); 4573 4574 for (i = 255; i >= 0; --i) { 4575 if (aslide[i] || bslide[i]) { 4576 break; 4577 } 4578 } 4579 4580 for (; i >= 0; --i) { 4581 ge_p2_dbl(&t, r); 4582 4583 if (aslide[i] > 0) { 4584 ge_p1p1_to_p3(&u, &t); 4585 ge_add(&t, &u, &Ai[aslide[i] / 2]); 4586 } else if (aslide[i] < 0) { 4587 ge_p1p1_to_p3(&u, &t); 4588 ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]); 4589 } 4590 4591 if (bslide[i] > 0) { 4592 ge_p1p1_to_p3(&u, &t); 4593 ge_madd(&t, &u, &Bi[bslide[i] / 2]); 4594 } else if (bslide[i] < 0) { 4595 ge_p1p1_to_p3(&u, &t); 4596 ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]); 4597 } 4598 4599 ge_p1p1_to_p2(r, &t); 4600 } 4601 } 4602 4603 /* 4604 * The set of scalars is \Z/l 4605 * where l = 2^252 + 27742317777372353535851937790883648493. 4606 * 4607 * Input: 4608 * s[0]+256*s[1]+...+256^63*s[63] = s 4609 * 4610 * Output: 4611 * s[0]+256*s[1]+...+256^31*s[31] = s mod l 4612 * where l = 2^252 + 27742317777372353535851937790883648493. 4613 * Overwrites s in place. 4614 */ 4615 static void x25519_sc_reduce(uint8_t *s) 4616 { 4617 int64_t s0 = kBottom21Bits & load_3(s); 4618 int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5); 4619 int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2); 4620 int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7); 4621 int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4); 4622 int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1); 4623 int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6); 4624 int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3); 4625 int64_t s8 = kBottom21Bits & load_3(s + 21); 4626 int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5); 4627 int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2); 4628 int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7); 4629 int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4); 4630 int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1); 4631 int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6); 4632 int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3); 4633 int64_t s16 = kBottom21Bits & load_3(s + 42); 4634 int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5); 4635 int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2); 4636 int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7); 4637 int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4); 4638 int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1); 4639 int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6); 4640 int64_t s23 = (load_4(s + 60) >> 3); 4641 int64_t carry0; 4642 int64_t carry1; 4643 int64_t carry2; 4644 int64_t carry3; 4645 int64_t carry4; 4646 int64_t carry5; 4647 int64_t carry6; 4648 int64_t carry7; 4649 int64_t carry8; 4650 int64_t carry9; 4651 int64_t carry10; 4652 int64_t carry11; 4653 int64_t carry12; 4654 int64_t carry13; 4655 int64_t carry14; 4656 int64_t carry15; 4657 int64_t carry16; 4658 4659 s11 += s23 * 666643; 4660 s12 += s23 * 470296; 4661 s13 += s23 * 654183; 4662 s14 -= s23 * 997805; 4663 s15 += s23 * 136657; 4664 s16 -= s23 * 683901; 4665 s23 = 0; 4666 4667 s10 += s22 * 666643; 4668 s11 += s22 * 470296; 4669 s12 += s22 * 654183; 4670 s13 -= s22 * 997805; 4671 s14 += s22 * 136657; 4672 s15 -= s22 * 683901; 4673 s22 = 0; 4674 4675 s9 += s21 * 666643; 4676 s10 += s21 * 470296; 4677 s11 += s21 * 654183; 4678 s12 -= s21 * 997805; 4679 s13 += s21 * 136657; 4680 s14 -= s21 * 683901; 4681 s21 = 0; 4682 4683 s8 += s20 * 666643; 4684 s9 += s20 * 470296; 4685 s10 += s20 * 654183; 4686 s11 -= s20 * 997805; 4687 s12 += s20 * 136657; 4688 s13 -= s20 * 683901; 4689 s20 = 0; 4690 4691 s7 += s19 * 666643; 4692 s8 += s19 * 470296; 4693 s9 += s19 * 654183; 4694 s10 -= s19 * 997805; 4695 s11 += s19 * 136657; 4696 s12 -= s19 * 683901; 4697 s19 = 0; 4698 4699 s6 += s18 * 666643; 4700 s7 += s18 * 470296; 4701 s8 += s18 * 654183; 4702 s9 -= s18 * 997805; 4703 s10 += s18 * 136657; 4704 s11 -= s18 * 683901; 4705 s18 = 0; 4706 4707 carry6 = (s6 + (1 << 20)) >> 21; 4708 s7 += carry6; 4709 s6 -= carry6 * (1 << 21); 4710 carry8 = (s8 + (1 << 20)) >> 21; 4711 s9 += carry8; 4712 s8 -= carry8 * (1 << 21); 4713 carry10 = (s10 + (1 << 20)) >> 21; 4714 s11 += carry10; 4715 s10 -= carry10 * (1 << 21); 4716 carry12 = (s12 + (1 << 20)) >> 21; 4717 s13 += carry12; 4718 s12 -= carry12 * (1 << 21); 4719 carry14 = (s14 + (1 << 20)) >> 21; 4720 s15 += carry14; 4721 s14 -= carry14 * (1 << 21); 4722 carry16 = (s16 + (1 << 20)) >> 21; 4723 s17 += carry16; 4724 s16 -= carry16 * (1 << 21); 4725 4726 carry7 = (s7 + (1 << 20)) >> 21; 4727 s8 += carry7; 4728 s7 -= carry7 * (1 << 21); 4729 carry9 = (s9 + (1 << 20)) >> 21; 4730 s10 += carry9; 4731 s9 -= carry9 * (1 << 21); 4732 carry11 = (s11 + (1 << 20)) >> 21; 4733 s12 += carry11; 4734 s11 -= carry11 * (1 << 21); 4735 carry13 = (s13 + (1 << 20)) >> 21; 4736 s14 += carry13; 4737 s13 -= carry13 * (1 << 21); 4738 carry15 = (s15 + (1 << 20)) >> 21; 4739 s16 += carry15; 4740 s15 -= carry15 * (1 << 21); 4741 4742 s5 += s17 * 666643; 4743 s6 += s17 * 470296; 4744 s7 += s17 * 654183; 4745 s8 -= s17 * 997805; 4746 s9 += s17 * 136657; 4747 s10 -= s17 * 683901; 4748 s17 = 0; 4749 4750 s4 += s16 * 666643; 4751 s5 += s16 * 470296; 4752 s6 += s16 * 654183; 4753 s7 -= s16 * 997805; 4754 s8 += s16 * 136657; 4755 s9 -= s16 * 683901; 4756 s16 = 0; 4757 4758 s3 += s15 * 666643; 4759 s4 += s15 * 470296; 4760 s5 += s15 * 654183; 4761 s6 -= s15 * 997805; 4762 s7 += s15 * 136657; 4763 s8 -= s15 * 683901; 4764 s15 = 0; 4765 4766 s2 += s14 * 666643; 4767 s3 += s14 * 470296; 4768 s4 += s14 * 654183; 4769 s5 -= s14 * 997805; 4770 s6 += s14 * 136657; 4771 s7 -= s14 * 683901; 4772 s14 = 0; 4773 4774 s1 += s13 * 666643; 4775 s2 += s13 * 470296; 4776 s3 += s13 * 654183; 4777 s4 -= s13 * 997805; 4778 s5 += s13 * 136657; 4779 s6 -= s13 * 683901; 4780 s13 = 0; 4781 4782 s0 += s12 * 666643; 4783 s1 += s12 * 470296; 4784 s2 += s12 * 654183; 4785 s3 -= s12 * 997805; 4786 s4 += s12 * 136657; 4787 s5 -= s12 * 683901; 4788 s12 = 0; 4789 4790 carry0 = (s0 + (1 << 20)) >> 21; 4791 s1 += carry0; 4792 s0 -= carry0 * (1 << 21); 4793 carry2 = (s2 + (1 << 20)) >> 21; 4794 s3 += carry2; 4795 s2 -= carry2 * (1 << 21); 4796 carry4 = (s4 + (1 << 20)) >> 21; 4797 s5 += carry4; 4798 s4 -= carry4 * (1 << 21); 4799 carry6 = (s6 + (1 << 20)) >> 21; 4800 s7 += carry6; 4801 s6 -= carry6 * (1 << 21); 4802 carry8 = (s8 + (1 << 20)) >> 21; 4803 s9 += carry8; 4804 s8 -= carry8 * (1 << 21); 4805 carry10 = (s10 + (1 << 20)) >> 21; 4806 s11 += carry10; 4807 s10 -= carry10 * (1 << 21); 4808 4809 carry1 = (s1 + (1 << 20)) >> 21; 4810 s2 += carry1; 4811 s1 -= carry1 * (1 << 21); 4812 carry3 = (s3 + (1 << 20)) >> 21; 4813 s4 += carry3; 4814 s3 -= carry3 * (1 << 21); 4815 carry5 = (s5 + (1 << 20)) >> 21; 4816 s6 += carry5; 4817 s5 -= carry5 * (1 << 21); 4818 carry7 = (s7 + (1 << 20)) >> 21; 4819 s8 += carry7; 4820 s7 -= carry7 * (1 << 21); 4821 carry9 = (s9 + (1 << 20)) >> 21; 4822 s10 += carry9; 4823 s9 -= carry9 * (1 << 21); 4824 carry11 = (s11 + (1 << 20)) >> 21; 4825 s12 += carry11; 4826 s11 -= carry11 * (1 << 21); 4827 4828 s0 += s12 * 666643; 4829 s1 += s12 * 470296; 4830 s2 += s12 * 654183; 4831 s3 -= s12 * 997805; 4832 s4 += s12 * 136657; 4833 s5 -= s12 * 683901; 4834 s12 = 0; 4835 4836 carry0 = s0 >> 21; 4837 s1 += carry0; 4838 s0 -= carry0 * (1 << 21); 4839 carry1 = s1 >> 21; 4840 s2 += carry1; 4841 s1 -= carry1 * (1 << 21); 4842 carry2 = s2 >> 21; 4843 s3 += carry2; 4844 s2 -= carry2 * (1 << 21); 4845 carry3 = s3 >> 21; 4846 s4 += carry3; 4847 s3 -= carry3 * (1 << 21); 4848 carry4 = s4 >> 21; 4849 s5 += carry4; 4850 s4 -= carry4 * (1 << 21); 4851 carry5 = s5 >> 21; 4852 s6 += carry5; 4853 s5 -= carry5 * (1 << 21); 4854 carry6 = s6 >> 21; 4855 s7 += carry6; 4856 s6 -= carry6 * (1 << 21); 4857 carry7 = s7 >> 21; 4858 s8 += carry7; 4859 s7 -= carry7 * (1 << 21); 4860 carry8 = s8 >> 21; 4861 s9 += carry8; 4862 s8 -= carry8 * (1 << 21); 4863 carry9 = s9 >> 21; 4864 s10 += carry9; 4865 s9 -= carry9 * (1 << 21); 4866 carry10 = s10 >> 21; 4867 s11 += carry10; 4868 s10 -= carry10 * (1 << 21); 4869 carry11 = s11 >> 21; 4870 s12 += carry11; 4871 s11 -= carry11 * (1 << 21); 4872 4873 s0 += s12 * 666643; 4874 s1 += s12 * 470296; 4875 s2 += s12 * 654183; 4876 s3 -= s12 * 997805; 4877 s4 += s12 * 136657; 4878 s5 -= s12 * 683901; 4879 s12 = 0; 4880 4881 carry0 = s0 >> 21; 4882 s1 += carry0; 4883 s0 -= carry0 * (1 << 21); 4884 carry1 = s1 >> 21; 4885 s2 += carry1; 4886 s1 -= carry1 * (1 << 21); 4887 carry2 = s2 >> 21; 4888 s3 += carry2; 4889 s2 -= carry2 * (1 << 21); 4890 carry3 = s3 >> 21; 4891 s4 += carry3; 4892 s3 -= carry3 * (1 << 21); 4893 carry4 = s4 >> 21; 4894 s5 += carry4; 4895 s4 -= carry4 * (1 << 21); 4896 carry5 = s5 >> 21; 4897 s6 += carry5; 4898 s5 -= carry5 * (1 << 21); 4899 carry6 = s6 >> 21; 4900 s7 += carry6; 4901 s6 -= carry6 * (1 << 21); 4902 carry7 = s7 >> 21; 4903 s8 += carry7; 4904 s7 -= carry7 * (1 << 21); 4905 carry8 = s8 >> 21; 4906 s9 += carry8; 4907 s8 -= carry8 * (1 << 21); 4908 carry9 = s9 >> 21; 4909 s10 += carry9; 4910 s9 -= carry9 * (1 << 21); 4911 carry10 = s10 >> 21; 4912 s11 += carry10; 4913 s10 -= carry10 * (1 << 21); 4914 4915 s[ 0] = (uint8_t) (s0 >> 0); 4916 s[ 1] = (uint8_t) (s0 >> 8); 4917 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5)); 4918 s[ 3] = (uint8_t) (s1 >> 3); 4919 s[ 4] = (uint8_t) (s1 >> 11); 4920 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2)); 4921 s[ 6] = (uint8_t) (s2 >> 6); 4922 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7)); 4923 s[ 8] = (uint8_t) (s3 >> 1); 4924 s[ 9] = (uint8_t) (s3 >> 9); 4925 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4)); 4926 s[11] = (uint8_t) (s4 >> 4); 4927 s[12] = (uint8_t) (s4 >> 12); 4928 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1)); 4929 s[14] = (uint8_t) (s5 >> 7); 4930 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6)); 4931 s[16] = (uint8_t) (s6 >> 2); 4932 s[17] = (uint8_t) (s6 >> 10); 4933 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3)); 4934 s[19] = (uint8_t) (s7 >> 5); 4935 s[20] = (uint8_t) (s7 >> 13); 4936 s[21] = (uint8_t) (s8 >> 0); 4937 s[22] = (uint8_t) (s8 >> 8); 4938 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5)); 4939 s[24] = (uint8_t) (s9 >> 3); 4940 s[25] = (uint8_t) (s9 >> 11); 4941 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2)); 4942 s[27] = (uint8_t) (s10 >> 6); 4943 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7)); 4944 s[29] = (uint8_t) (s11 >> 1); 4945 s[30] = (uint8_t) (s11 >> 9); 4946 s[31] = (uint8_t) (s11 >> 17); 4947 } 4948 4949 /* 4950 * Input: 4951 * a[0]+256*a[1]+...+256^31*a[31] = a 4952 * b[0]+256*b[1]+...+256^31*b[31] = b 4953 * c[0]+256*c[1]+...+256^31*c[31] = c 4954 * 4955 * Output: 4956 * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l 4957 * where l = 2^252 + 27742317777372353535851937790883648493. 4958 */ 4959 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b, 4960 const uint8_t *c) 4961 { 4962 int64_t a0 = kBottom21Bits & load_3(a); 4963 int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5); 4964 int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2); 4965 int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7); 4966 int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4); 4967 int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1); 4968 int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6); 4969 int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3); 4970 int64_t a8 = kBottom21Bits & load_3(a + 21); 4971 int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5); 4972 int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2); 4973 int64_t a11 = (load_4(a + 28) >> 7); 4974 int64_t b0 = kBottom21Bits & load_3(b); 4975 int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5); 4976 int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2); 4977 int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7); 4978 int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4); 4979 int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1); 4980 int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6); 4981 int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3); 4982 int64_t b8 = kBottom21Bits & load_3(b + 21); 4983 int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5); 4984 int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2); 4985 int64_t b11 = (load_4(b + 28) >> 7); 4986 int64_t c0 = kBottom21Bits & load_3(c); 4987 int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5); 4988 int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2); 4989 int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7); 4990 int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4); 4991 int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1); 4992 int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6); 4993 int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3); 4994 int64_t c8 = kBottom21Bits & load_3(c + 21); 4995 int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5); 4996 int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2); 4997 int64_t c11 = (load_4(c + 28) >> 7); 4998 int64_t s0; 4999 int64_t s1; 5000 int64_t s2; 5001 int64_t s3; 5002 int64_t s4; 5003 int64_t s5; 5004 int64_t s6; 5005 int64_t s7; 5006 int64_t s8; 5007 int64_t s9; 5008 int64_t s10; 5009 int64_t s11; 5010 int64_t s12; 5011 int64_t s13; 5012 int64_t s14; 5013 int64_t s15; 5014 int64_t s16; 5015 int64_t s17; 5016 int64_t s18; 5017 int64_t s19; 5018 int64_t s20; 5019 int64_t s21; 5020 int64_t s22; 5021 int64_t s23; 5022 int64_t carry0; 5023 int64_t carry1; 5024 int64_t carry2; 5025 int64_t carry3; 5026 int64_t carry4; 5027 int64_t carry5; 5028 int64_t carry6; 5029 int64_t carry7; 5030 int64_t carry8; 5031 int64_t carry9; 5032 int64_t carry10; 5033 int64_t carry11; 5034 int64_t carry12; 5035 int64_t carry13; 5036 int64_t carry14; 5037 int64_t carry15; 5038 int64_t carry16; 5039 int64_t carry17; 5040 int64_t carry18; 5041 int64_t carry19; 5042 int64_t carry20; 5043 int64_t carry21; 5044 int64_t carry22; 5045 5046 s0 = c0 + a0 * b0; 5047 s1 = c1 + a0 * b1 + a1 * b0; 5048 s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0; 5049 s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0; 5050 s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0; 5051 s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0; 5052 s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0; 5053 s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0; 5054 s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0; 5055 s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0; 5056 s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0; 5057 s11 = c11 + a0 * b11 + a1 * b10 + a2 * b9 + a3 * b8 + a4 * b7 + a5 * b6 + a6 * b5 + a7 * b4 + a8 * b3 + a9 * b2 + a10 * b1 + a11 * b0; 5058 s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1; 5059 s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2; 5060 s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3; 5061 s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4; 5062 s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5; 5063 s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6; 5064 s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7; 5065 s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8; 5066 s20 = a9 * b11 + a10 * b10 + a11 * b9; 5067 s21 = a10 * b11 + a11 * b10; 5068 s22 = a11 * b11; 5069 s23 = 0; 5070 5071 carry0 = (s0 + (1 << 20)) >> 21; 5072 s1 += carry0; 5073 s0 -= carry0 * (1 << 21); 5074 carry2 = (s2 + (1 << 20)) >> 21; 5075 s3 += carry2; 5076 s2 -= carry2 * (1 << 21); 5077 carry4 = (s4 + (1 << 20)) >> 21; 5078 s5 += carry4; 5079 s4 -= carry4 * (1 << 21); 5080 carry6 = (s6 + (1 << 20)) >> 21; 5081 s7 += carry6; 5082 s6 -= carry6 * (1 << 21); 5083 carry8 = (s8 + (1 << 20)) >> 21; 5084 s9 += carry8; 5085 s8 -= carry8 * (1 << 21); 5086 carry10 = (s10 + (1 << 20)) >> 21; 5087 s11 += carry10; 5088 s10 -= carry10 * (1 << 21); 5089 carry12 = (s12 + (1 << 20)) >> 21; 5090 s13 += carry12; 5091 s12 -= carry12 * (1 << 21); 5092 carry14 = (s14 + (1 << 20)) >> 21; 5093 s15 += carry14; 5094 s14 -= carry14 * (1 << 21); 5095 carry16 = (s16 + (1 << 20)) >> 21; 5096 s17 += carry16; 5097 s16 -= carry16 * (1 << 21); 5098 carry18 = (s18 + (1 << 20)) >> 21; 5099 s19 += carry18; 5100 s18 -= carry18 * (1 << 21); 5101 carry20 = (s20 + (1 << 20)) >> 21; 5102 s21 += carry20; 5103 s20 -= carry20 * (1 << 21); 5104 carry22 = (s22 + (1 << 20)) >> 21; 5105 s23 += carry22; 5106 s22 -= carry22 * (1 << 21); 5107 5108 carry1 = (s1 + (1 << 20)) >> 21; 5109 s2 += carry1; 5110 s1 -= carry1 * (1 << 21); 5111 carry3 = (s3 + (1 << 20)) >> 21; 5112 s4 += carry3; 5113 s3 -= carry3 * (1 << 21); 5114 carry5 = (s5 + (1 << 20)) >> 21; 5115 s6 += carry5; 5116 s5 -= carry5 * (1 << 21); 5117 carry7 = (s7 + (1 << 20)) >> 21; 5118 s8 += carry7; 5119 s7 -= carry7 * (1 << 21); 5120 carry9 = (s9 + (1 << 20)) >> 21; 5121 s10 += carry9; 5122 s9 -= carry9 * (1 << 21); 5123 carry11 = (s11 + (1 << 20)) >> 21; 5124 s12 += carry11; 5125 s11 -= carry11 * (1 << 21); 5126 carry13 = (s13 + (1 << 20)) >> 21; 5127 s14 += carry13; 5128 s13 -= carry13 * (1 << 21); 5129 carry15 = (s15 + (1 << 20)) >> 21; 5130 s16 += carry15; 5131 s15 -= carry15 * (1 << 21); 5132 carry17 = (s17 + (1 << 20)) >> 21; 5133 s18 += carry17; 5134 s17 -= carry17 * (1 << 21); 5135 carry19 = (s19 + (1 << 20)) >> 21; 5136 s20 += carry19; 5137 s19 -= carry19 * (1 << 21); 5138 carry21 = (s21 + (1 << 20)) >> 21; 5139 s22 += carry21; 5140 s21 -= carry21 * (1 << 21); 5141 5142 s11 += s23 * 666643; 5143 s12 += s23 * 470296; 5144 s13 += s23 * 654183; 5145 s14 -= s23 * 997805; 5146 s15 += s23 * 136657; 5147 s16 -= s23 * 683901; 5148 s23 = 0; 5149 5150 s10 += s22 * 666643; 5151 s11 += s22 * 470296; 5152 s12 += s22 * 654183; 5153 s13 -= s22 * 997805; 5154 s14 += s22 * 136657; 5155 s15 -= s22 * 683901; 5156 s22 = 0; 5157 5158 s9 += s21 * 666643; 5159 s10 += s21 * 470296; 5160 s11 += s21 * 654183; 5161 s12 -= s21 * 997805; 5162 s13 += s21 * 136657; 5163 s14 -= s21 * 683901; 5164 s21 = 0; 5165 5166 s8 += s20 * 666643; 5167 s9 += s20 * 470296; 5168 s10 += s20 * 654183; 5169 s11 -= s20 * 997805; 5170 s12 += s20 * 136657; 5171 s13 -= s20 * 683901; 5172 s20 = 0; 5173 5174 s7 += s19 * 666643; 5175 s8 += s19 * 470296; 5176 s9 += s19 * 654183; 5177 s10 -= s19 * 997805; 5178 s11 += s19 * 136657; 5179 s12 -= s19 * 683901; 5180 s19 = 0; 5181 5182 s6 += s18 * 666643; 5183 s7 += s18 * 470296; 5184 s8 += s18 * 654183; 5185 s9 -= s18 * 997805; 5186 s10 += s18 * 136657; 5187 s11 -= s18 * 683901; 5188 s18 = 0; 5189 5190 carry6 = (s6 + (1 << 20)) >> 21; 5191 s7 += carry6; 5192 s6 -= carry6 * (1 << 21); 5193 carry8 = (s8 + (1 << 20)) >> 21; 5194 s9 += carry8; 5195 s8 -= carry8 * (1 << 21); 5196 carry10 = (s10 + (1 << 20)) >> 21; 5197 s11 += carry10; 5198 s10 -= carry10 * (1 << 21); 5199 carry12 = (s12 + (1 << 20)) >> 21; 5200 s13 += carry12; 5201 s12 -= carry12 * (1 << 21); 5202 carry14 = (s14 + (1 << 20)) >> 21; 5203 s15 += carry14; 5204 s14 -= carry14 * (1 << 21); 5205 carry16 = (s16 + (1 << 20)) >> 21; 5206 s17 += carry16; 5207 s16 -= carry16 * (1 << 21); 5208 5209 carry7 = (s7 + (1 << 20)) >> 21; 5210 s8 += carry7; 5211 s7 -= carry7 * (1 << 21); 5212 carry9 = (s9 + (1 << 20)) >> 21; 5213 s10 += carry9; 5214 s9 -= carry9 * (1 << 21); 5215 carry11 = (s11 + (1 << 20)) >> 21; 5216 s12 += carry11; 5217 s11 -= carry11 * (1 << 21); 5218 carry13 = (s13 + (1 << 20)) >> 21; 5219 s14 += carry13; 5220 s13 -= carry13 * (1 << 21); 5221 carry15 = (s15 + (1 << 20)) >> 21; 5222 s16 += carry15; 5223 s15 -= carry15 * (1 << 21); 5224 5225 s5 += s17 * 666643; 5226 s6 += s17 * 470296; 5227 s7 += s17 * 654183; 5228 s8 -= s17 * 997805; 5229 s9 += s17 * 136657; 5230 s10 -= s17 * 683901; 5231 s17 = 0; 5232 5233 s4 += s16 * 666643; 5234 s5 += s16 * 470296; 5235 s6 += s16 * 654183; 5236 s7 -= s16 * 997805; 5237 s8 += s16 * 136657; 5238 s9 -= s16 * 683901; 5239 s16 = 0; 5240 5241 s3 += s15 * 666643; 5242 s4 += s15 * 470296; 5243 s5 += s15 * 654183; 5244 s6 -= s15 * 997805; 5245 s7 += s15 * 136657; 5246 s8 -= s15 * 683901; 5247 s15 = 0; 5248 5249 s2 += s14 * 666643; 5250 s3 += s14 * 470296; 5251 s4 += s14 * 654183; 5252 s5 -= s14 * 997805; 5253 s6 += s14 * 136657; 5254 s7 -= s14 * 683901; 5255 s14 = 0; 5256 5257 s1 += s13 * 666643; 5258 s2 += s13 * 470296; 5259 s3 += s13 * 654183; 5260 s4 -= s13 * 997805; 5261 s5 += s13 * 136657; 5262 s6 -= s13 * 683901; 5263 s13 = 0; 5264 5265 s0 += s12 * 666643; 5266 s1 += s12 * 470296; 5267 s2 += s12 * 654183; 5268 s3 -= s12 * 997805; 5269 s4 += s12 * 136657; 5270 s5 -= s12 * 683901; 5271 s12 = 0; 5272 5273 carry0 = (s0 + (1 << 20)) >> 21; 5274 s1 += carry0; 5275 s0 -= carry0 * (1 << 21); 5276 carry2 = (s2 + (1 << 20)) >> 21; 5277 s3 += carry2; 5278 s2 -= carry2 * (1 << 21); 5279 carry4 = (s4 + (1 << 20)) >> 21; 5280 s5 += carry4; 5281 s4 -= carry4 * (1 << 21); 5282 carry6 = (s6 + (1 << 20)) >> 21; 5283 s7 += carry6; 5284 s6 -= carry6 * (1 << 21); 5285 carry8 = (s8 + (1 << 20)) >> 21; 5286 s9 += carry8; 5287 s8 -= carry8 * (1 << 21); 5288 carry10 = (s10 + (1 << 20)) >> 21; 5289 s11 += carry10; 5290 s10 -= carry10 * (1 << 21); 5291 5292 carry1 = (s1 + (1 << 20)) >> 21; 5293 s2 += carry1; 5294 s1 -= carry1 * (1 << 21); 5295 carry3 = (s3 + (1 << 20)) >> 21; 5296 s4 += carry3; 5297 s3 -= carry3 * (1 << 21); 5298 carry5 = (s5 + (1 << 20)) >> 21; 5299 s6 += carry5; 5300 s5 -= carry5 * (1 << 21); 5301 carry7 = (s7 + (1 << 20)) >> 21; 5302 s8 += carry7; 5303 s7 -= carry7 * (1 << 21); 5304 carry9 = (s9 + (1 << 20)) >> 21; 5305 s10 += carry9; 5306 s9 -= carry9 * (1 << 21); 5307 carry11 = (s11 + (1 << 20)) >> 21; 5308 s12 += carry11; 5309 s11 -= carry11 * (1 << 21); 5310 5311 s0 += s12 * 666643; 5312 s1 += s12 * 470296; 5313 s2 += s12 * 654183; 5314 s3 -= s12 * 997805; 5315 s4 += s12 * 136657; 5316 s5 -= s12 * 683901; 5317 s12 = 0; 5318 5319 carry0 = s0 >> 21; 5320 s1 += carry0; 5321 s0 -= carry0 * (1 << 21); 5322 carry1 = s1 >> 21; 5323 s2 += carry1; 5324 s1 -= carry1 * (1 << 21); 5325 carry2 = s2 >> 21; 5326 s3 += carry2; 5327 s2 -= carry2 * (1 << 21); 5328 carry3 = s3 >> 21; 5329 s4 += carry3; 5330 s3 -= carry3 * (1 << 21); 5331 carry4 = s4 >> 21; 5332 s5 += carry4; 5333 s4 -= carry4 * (1 << 21); 5334 carry5 = s5 >> 21; 5335 s6 += carry5; 5336 s5 -= carry5 * (1 << 21); 5337 carry6 = s6 >> 21; 5338 s7 += carry6; 5339 s6 -= carry6 * (1 << 21); 5340 carry7 = s7 >> 21; 5341 s8 += carry7; 5342 s7 -= carry7 * (1 << 21); 5343 carry8 = s8 >> 21; 5344 s9 += carry8; 5345 s8 -= carry8 * (1 << 21); 5346 carry9 = s9 >> 21; 5347 s10 += carry9; 5348 s9 -= carry9 * (1 << 21); 5349 carry10 = s10 >> 21; 5350 s11 += carry10; 5351 s10 -= carry10 * (1 << 21); 5352 carry11 = s11 >> 21; 5353 s12 += carry11; 5354 s11 -= carry11 * (1 << 21); 5355 5356 s0 += s12 * 666643; 5357 s1 += s12 * 470296; 5358 s2 += s12 * 654183; 5359 s3 -= s12 * 997805; 5360 s4 += s12 * 136657; 5361 s5 -= s12 * 683901; 5362 s12 = 0; 5363 5364 carry0 = s0 >> 21; 5365 s1 += carry0; 5366 s0 -= carry0 * (1 << 21); 5367 carry1 = s1 >> 21; 5368 s2 += carry1; 5369 s1 -= carry1 * (1 << 21); 5370 carry2 = s2 >> 21; 5371 s3 += carry2; 5372 s2 -= carry2 * (1 << 21); 5373 carry3 = s3 >> 21; 5374 s4 += carry3; 5375 s3 -= carry3 * (1 << 21); 5376 carry4 = s4 >> 21; 5377 s5 += carry4; 5378 s4 -= carry4 * (1 << 21); 5379 carry5 = s5 >> 21; 5380 s6 += carry5; 5381 s5 -= carry5 * (1 << 21); 5382 carry6 = s6 >> 21; 5383 s7 += carry6; 5384 s6 -= carry6 * (1 << 21); 5385 carry7 = s7 >> 21; 5386 s8 += carry7; 5387 s7 -= carry7 * (1 << 21); 5388 carry8 = s8 >> 21; 5389 s9 += carry8; 5390 s8 -= carry8 * (1 << 21); 5391 carry9 = s9 >> 21; 5392 s10 += carry9; 5393 s9 -= carry9 * (1 << 21); 5394 carry10 = s10 >> 21; 5395 s11 += carry10; 5396 s10 -= carry10 * (1 << 21); 5397 5398 s[ 0] = (uint8_t) (s0 >> 0); 5399 s[ 1] = (uint8_t) (s0 >> 8); 5400 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5)); 5401 s[ 3] = (uint8_t) (s1 >> 3); 5402 s[ 4] = (uint8_t) (s1 >> 11); 5403 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2)); 5404 s[ 6] = (uint8_t) (s2 >> 6); 5405 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7)); 5406 s[ 8] = (uint8_t) (s3 >> 1); 5407 s[ 9] = (uint8_t) (s3 >> 9); 5408 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4)); 5409 s[11] = (uint8_t) (s4 >> 4); 5410 s[12] = (uint8_t) (s4 >> 12); 5411 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1)); 5412 s[14] = (uint8_t) (s5 >> 7); 5413 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6)); 5414 s[16] = (uint8_t) (s6 >> 2); 5415 s[17] = (uint8_t) (s6 >> 10); 5416 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3)); 5417 s[19] = (uint8_t) (s7 >> 5); 5418 s[20] = (uint8_t) (s7 >> 13); 5419 s[21] = (uint8_t) (s8 >> 0); 5420 s[22] = (uint8_t) (s8 >> 8); 5421 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5)); 5422 s[24] = (uint8_t) (s9 >> 3); 5423 s[25] = (uint8_t) (s9 >> 11); 5424 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2)); 5425 s[27] = (uint8_t) (s10 >> 6); 5426 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7)); 5427 s[29] = (uint8_t) (s11 >> 1); 5428 s[30] = (uint8_t) (s11 >> 9); 5429 s[31] = (uint8_t) (s11 >> 17); 5430 } 5431 5432 int ED25519_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len, 5433 const uint8_t public_key[32], const uint8_t private_key[32]) 5434 { 5435 uint8_t az[SHA512_DIGEST_LENGTH]; 5436 uint8_t nonce[SHA512_DIGEST_LENGTH]; 5437 ge_p3 R; 5438 uint8_t hram[SHA512_DIGEST_LENGTH]; 5439 SHA512_CTX hash_ctx; 5440 5441 SHA512_Init(&hash_ctx); 5442 SHA512_Update(&hash_ctx, private_key, 32); 5443 SHA512_Final(az, &hash_ctx); 5444 5445 az[0] &= 248; 5446 az[31] &= 63; 5447 az[31] |= 64; 5448 5449 SHA512_Init(&hash_ctx); 5450 SHA512_Update(&hash_ctx, az + 32, 32); 5451 SHA512_Update(&hash_ctx, message, message_len); 5452 SHA512_Final(nonce, &hash_ctx); 5453 5454 x25519_sc_reduce(nonce); 5455 ge_scalarmult_base(&R, nonce); 5456 ge_p3_tobytes(out_sig, &R); 5457 5458 SHA512_Init(&hash_ctx); 5459 SHA512_Update(&hash_ctx, out_sig, 32); 5460 SHA512_Update(&hash_ctx, public_key, 32); 5461 SHA512_Update(&hash_ctx, message, message_len); 5462 SHA512_Final(hram, &hash_ctx); 5463 5464 x25519_sc_reduce(hram); 5465 sc_muladd(out_sig + 32, hram, az, nonce); 5466 5467 OPENSSL_cleanse(&hash_ctx, sizeof(hash_ctx)); 5468 OPENSSL_cleanse(nonce, sizeof(nonce)); 5469 OPENSSL_cleanse(az, sizeof(az)); 5470 5471 return 1; 5472 } 5473 5474 static const char allzeroes[15]; 5475 5476 int ED25519_verify(const uint8_t *message, size_t message_len, 5477 const uint8_t signature[64], const uint8_t public_key[32]) 5478 { 5479 int i; 5480 ge_p3 A; 5481 const uint8_t *r, *s; 5482 SHA512_CTX hash_ctx; 5483 ge_p2 R; 5484 uint8_t rcheck[32]; 5485 uint8_t h[SHA512_DIGEST_LENGTH]; 5486 /* 27742317777372353535851937790883648493 in little endian format */ 5487 const uint8_t l_low[16] = { 5488 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2, 5489 0xDE, 0xF9, 0xDE, 0x14 5490 }; 5491 5492 r = signature; 5493 s = signature + 32; 5494 5495 /* 5496 * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493 5497 * 5498 * If not the signature is publicly invalid. Since it's public we can do the 5499 * check in variable time. 5500 * 5501 * First check the most significant byte 5502 */ 5503 if (s[31] > 0x10) 5504 return 0; 5505 if (s[31] == 0x10) { 5506 /* 5507 * Most significant byte indicates a value close to 2^252 so check the 5508 * rest 5509 */ 5510 if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0) 5511 return 0; 5512 for (i = 15; i >= 0; i--) { 5513 if (s[i] < l_low[i]) 5514 break; 5515 if (s[i] > l_low[i]) 5516 return 0; 5517 } 5518 if (i < 0) 5519 return 0; 5520 } 5521 5522 if (ge_frombytes_vartime(&A, public_key) != 0) { 5523 return 0; 5524 } 5525 5526 fe_neg(A.X, A.X); 5527 fe_neg(A.T, A.T); 5528 5529 SHA512_Init(&hash_ctx); 5530 SHA512_Update(&hash_ctx, r, 32); 5531 SHA512_Update(&hash_ctx, public_key, 32); 5532 SHA512_Update(&hash_ctx, message, message_len); 5533 SHA512_Final(h, &hash_ctx); 5534 5535 x25519_sc_reduce(h); 5536 5537 ge_double_scalarmult_vartime(&R, h, &A, s); 5538 5539 ge_tobytes(rcheck, &R); 5540 5541 return CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0; 5542 } 5543 5544 void ED25519_public_from_private(uint8_t out_public_key[32], 5545 const uint8_t private_key[32]) 5546 { 5547 uint8_t az[SHA512_DIGEST_LENGTH]; 5548 ge_p3 A; 5549 5550 SHA512(private_key, 32, az); 5551 5552 az[0] &= 248; 5553 az[31] &= 63; 5554 az[31] |= 64; 5555 5556 ge_scalarmult_base(&A, az); 5557 ge_p3_tobytes(out_public_key, &A); 5558 5559 OPENSSL_cleanse(az, sizeof(az)); 5560 } 5561 5562 int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32], 5563 const uint8_t peer_public_value[32]) 5564 { 5565 static const uint8_t kZeros[32] = {0}; 5566 x25519_scalar_mult(out_shared_key, private_key, peer_public_value); 5567 /* The all-zero output results when the input is a point of small order. */ 5568 return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0; 5569 } 5570 5571 void X25519_public_from_private(uint8_t out_public_value[32], 5572 const uint8_t private_key[32]) 5573 { 5574 uint8_t e[32]; 5575 ge_p3 A; 5576 fe zplusy, zminusy, zminusy_inv; 5577 5578 memcpy(e, private_key, 32); 5579 e[0] &= 248; 5580 e[31] &= 127; 5581 e[31] |= 64; 5582 5583 ge_scalarmult_base(&A, e); 5584 5585 /* 5586 * We only need the u-coordinate of the curve25519 point. 5587 * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives 5588 * u=(Z+Y)/(Z-Y). 5589 */ 5590 fe_add(zplusy, A.Z, A.Y); 5591 fe_sub(zminusy, A.Z, A.Y); 5592 fe_invert(zminusy_inv, zminusy); 5593 fe_mul(zplusy, zplusy, zminusy_inv); 5594 fe_tobytes(out_public_value, zplusy); 5595 5596 OPENSSL_cleanse(e, sizeof(e)); 5597 } 5598