1 /* 2 * Copyright 2016-2021 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the Apache License 2.0 (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 /* 11 * ECDSA low level APIs are deprecated for public use, but still ok for 12 * internal use. 13 */ 14 #include "internal/deprecated.h" 15 16 #include <string.h> 17 #include "crypto/ecx.h" 18 #include "ec_local.h" 19 #include <openssl/evp.h> 20 #include <openssl/sha.h> 21 22 #include "internal/numbers.h" 23 24 #if defined(X25519_ASM) && (defined(__x86_64) || defined(__x86_64__) || \ 25 defined(_M_AMD64) || defined(_M_X64)) 26 27 # define BASE_2_64_IMPLEMENTED 28 29 typedef uint64_t fe64[4]; 30 31 int x25519_fe64_eligible(void); 32 33 /* 34 * Following subroutines perform corresponding operations modulo 35 * 2^256-38, i.e. double the curve modulus. However, inputs and 36 * outputs are permitted to be partially reduced, i.e. to remain 37 * in [0..2^256) range. It's all tied up in final fe64_tobytes 38 * that performs full reduction modulo 2^255-19. 39 * 40 * There are no reference C implementations for these. 41 */ 42 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g); 43 void x25519_fe64_sqr(fe64 h, const fe64 f); 44 void x25519_fe64_mul121666(fe64 h, fe64 f); 45 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g); 46 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g); 47 void x25519_fe64_tobytes(uint8_t *s, const fe64 f); 48 # define fe64_mul x25519_fe64_mul 49 # define fe64_sqr x25519_fe64_sqr 50 # define fe64_mul121666 x25519_fe64_mul121666 51 # define fe64_add x25519_fe64_add 52 # define fe64_sub x25519_fe64_sub 53 # define fe64_tobytes x25519_fe64_tobytes 54 55 static uint64_t load_8(const uint8_t *in) 56 { 57 uint64_t result; 58 59 result = in[0]; 60 result |= ((uint64_t)in[1]) << 8; 61 result |= ((uint64_t)in[2]) << 16; 62 result |= ((uint64_t)in[3]) << 24; 63 result |= ((uint64_t)in[4]) << 32; 64 result |= ((uint64_t)in[5]) << 40; 65 result |= ((uint64_t)in[6]) << 48; 66 result |= ((uint64_t)in[7]) << 56; 67 68 return result; 69 } 70 71 static void fe64_frombytes(fe64 h, const uint8_t *s) 72 { 73 h[0] = load_8(s); 74 h[1] = load_8(s + 8); 75 h[2] = load_8(s + 16); 76 h[3] = load_8(s + 24) & 0x7fffffffffffffff; 77 } 78 79 static void fe64_0(fe64 h) 80 { 81 h[0] = 0; 82 h[1] = 0; 83 h[2] = 0; 84 h[3] = 0; 85 } 86 87 static void fe64_1(fe64 h) 88 { 89 h[0] = 1; 90 h[1] = 0; 91 h[2] = 0; 92 h[3] = 0; 93 } 94 95 static void fe64_copy(fe64 h, const fe64 f) 96 { 97 h[0] = f[0]; 98 h[1] = f[1]; 99 h[2] = f[2]; 100 h[3] = f[3]; 101 } 102 103 static void fe64_cswap(fe64 f, fe64 g, unsigned int b) 104 { 105 int i; 106 uint64_t mask = 0 - (uint64_t)b; 107 108 for (i = 0; i < 4; i++) { 109 uint64_t x = f[i] ^ g[i]; 110 x &= mask; 111 f[i] ^= x; 112 g[i] ^= x; 113 } 114 } 115 116 static void fe64_invert(fe64 out, const fe64 z) 117 { 118 fe64 t0; 119 fe64 t1; 120 fe64 t2; 121 fe64 t3; 122 int i; 123 124 /* 125 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 126 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 127 */ 128 129 /* t0 = z ** 2 */ 130 fe64_sqr(t0, z); 131 132 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 133 fe64_sqr(t1, t0); 134 fe64_sqr(t1, t1); 135 136 /* t1 = z * t1 = z ** 9 */ 137 fe64_mul(t1, z, t1); 138 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 139 fe64_mul(t0, t0, t1); 140 141 /* t2 = t0 ** 2 = z ** 22 */ 142 fe64_sqr(t2, t0); 143 144 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 145 fe64_mul(t1, t1, t2); 146 147 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 148 fe64_sqr(t2, t1); 149 for (i = 1; i < 5; ++i) 150 fe64_sqr(t2, t2); 151 152 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 153 fe64_mul(t1, t2, t1); 154 155 /* Continuing similarly... */ 156 157 /* t2 = z ** (2 ** 20 - 1) */ 158 fe64_sqr(t2, t1); 159 for (i = 1; i < 10; ++i) 160 fe64_sqr(t2, t2); 161 162 fe64_mul(t2, t2, t1); 163 164 /* t2 = z ** (2 ** 40 - 1) */ 165 fe64_sqr(t3, t2); 166 for (i = 1; i < 20; ++i) 167 fe64_sqr(t3, t3); 168 169 fe64_mul(t2, t3, t2); 170 171 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 172 for (i = 0; i < 10; ++i) 173 fe64_sqr(t2, t2); 174 175 /* t1 = z ** (2 ** 50 - 1) */ 176 fe64_mul(t1, t2, t1); 177 178 /* t2 = z ** (2 ** 100 - 1) */ 179 fe64_sqr(t2, t1); 180 for (i = 1; i < 50; ++i) 181 fe64_sqr(t2, t2); 182 183 fe64_mul(t2, t2, t1); 184 185 /* t2 = z ** (2 ** 200 - 1) */ 186 fe64_sqr(t3, t2); 187 for (i = 1; i < 100; ++i) 188 fe64_sqr(t3, t3); 189 190 fe64_mul(t2, t3, t2); 191 192 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 193 for (i = 0; i < 50; ++i) 194 fe64_sqr(t2, t2); 195 196 /* t1 = z ** (2 ** 250 - 1) */ 197 fe64_mul(t1, t2, t1); 198 199 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 200 for (i = 0; i < 5; ++i) 201 fe64_sqr(t1, t1); 202 203 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 204 fe64_mul(out, t1, t0); 205 } 206 207 /* 208 * Duplicate of original x25519_scalar_mult_generic, but using 209 * fe64_* subroutines. 210 */ 211 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32], 212 const uint8_t point[32]) 213 { 214 fe64 x1, x2, z2, x3, z3, tmp0, tmp1; 215 uint8_t e[32]; 216 unsigned swap = 0; 217 int pos; 218 219 memcpy(e, scalar, 32); 220 e[0] &= 0xf8; 221 e[31] &= 0x7f; 222 e[31] |= 0x40; 223 fe64_frombytes(x1, point); 224 fe64_1(x2); 225 fe64_0(z2); 226 fe64_copy(x3, x1); 227 fe64_1(z3); 228 229 for (pos = 254; pos >= 0; --pos) { 230 unsigned int b = 1 & (e[pos / 8] >> (pos & 7)); 231 232 swap ^= b; 233 fe64_cswap(x2, x3, swap); 234 fe64_cswap(z2, z3, swap); 235 swap = b; 236 fe64_sub(tmp0, x3, z3); 237 fe64_sub(tmp1, x2, z2); 238 fe64_add(x2, x2, z2); 239 fe64_add(z2, x3, z3); 240 fe64_mul(z3, x2, tmp0); 241 fe64_mul(z2, z2, tmp1); 242 fe64_sqr(tmp0, tmp1); 243 fe64_sqr(tmp1, x2); 244 fe64_add(x3, z3, z2); 245 fe64_sub(z2, z3, z2); 246 fe64_mul(x2, tmp1, tmp0); 247 fe64_sub(tmp1, tmp1, tmp0); 248 fe64_sqr(z2, z2); 249 fe64_mul121666(z3, tmp1); 250 fe64_sqr(x3, x3); 251 fe64_add(tmp0, tmp0, z3); 252 fe64_mul(z3, x1, z2); 253 fe64_mul(z2, tmp1, tmp0); 254 } 255 256 fe64_invert(z2, z2); 257 fe64_mul(x2, x2, z2); 258 fe64_tobytes(out, x2); 259 260 OPENSSL_cleanse(e, sizeof(e)); 261 } 262 #endif 263 264 #if defined(X25519_ASM) \ 265 || ( defined(INT128_MAX) \ 266 && !defined(__sparc__) \ 267 && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \ 268 && !(defined(__ANDROID__) && !defined(__clang__)) ) 269 /* 270 * Base 2^51 implementation. It's virtually no different from reference 271 * base 2^25.5 implementation in respect to lax boundary conditions for 272 * intermediate values and even individual limbs. So that whatever you 273 * know about the reference, applies even here... 274 */ 275 # define BASE_2_51_IMPLEMENTED 276 277 typedef uint64_t fe51[5]; 278 279 static const uint64_t MASK51 = 0x7ffffffffffff; 280 281 static uint64_t load_7(const uint8_t *in) 282 { 283 uint64_t result; 284 285 result = in[0]; 286 result |= ((uint64_t)in[1]) << 8; 287 result |= ((uint64_t)in[2]) << 16; 288 result |= ((uint64_t)in[3]) << 24; 289 result |= ((uint64_t)in[4]) << 32; 290 result |= ((uint64_t)in[5]) << 40; 291 result |= ((uint64_t)in[6]) << 48; 292 293 return result; 294 } 295 296 static uint64_t load_6(const uint8_t *in) 297 { 298 uint64_t result; 299 300 result = in[0]; 301 result |= ((uint64_t)in[1]) << 8; 302 result |= ((uint64_t)in[2]) << 16; 303 result |= ((uint64_t)in[3]) << 24; 304 result |= ((uint64_t)in[4]) << 32; 305 result |= ((uint64_t)in[5]) << 40; 306 307 return result; 308 } 309 310 static void fe51_frombytes(fe51 h, const uint8_t *s) 311 { 312 uint64_t h0 = load_7(s); /* 56 bits */ 313 uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */ 314 uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */ 315 uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */ 316 uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */ 317 318 h1 |= h0 >> 51; h0 &= MASK51; 319 h2 |= h1 >> 51; h1 &= MASK51; 320 h3 |= h2 >> 51; h2 &= MASK51; 321 h4 |= h3 >> 51; h3 &= MASK51; 322 323 h[0] = h0; 324 h[1] = h1; 325 h[2] = h2; 326 h[3] = h3; 327 h[4] = h4; 328 } 329 330 static void fe51_tobytes(uint8_t *s, const fe51 h) 331 { 332 uint64_t h0 = h[0]; 333 uint64_t h1 = h[1]; 334 uint64_t h2 = h[2]; 335 uint64_t h3 = h[3]; 336 uint64_t h4 = h[4]; 337 uint64_t q; 338 339 /* compare to modulus */ 340 q = (h0 + 19) >> 51; 341 q = (h1 + q) >> 51; 342 q = (h2 + q) >> 51; 343 q = (h3 + q) >> 51; 344 q = (h4 + q) >> 51; 345 346 /* full reduce */ 347 h0 += 19 * q; 348 h1 += h0 >> 51; h0 &= MASK51; 349 h2 += h1 >> 51; h1 &= MASK51; 350 h3 += h2 >> 51; h2 &= MASK51; 351 h4 += h3 >> 51; h3 &= MASK51; 352 h4 &= MASK51; 353 354 /* smash */ 355 s[0] = (uint8_t)(h0 >> 0); 356 s[1] = (uint8_t)(h0 >> 8); 357 s[2] = (uint8_t)(h0 >> 16); 358 s[3] = (uint8_t)(h0 >> 24); 359 s[4] = (uint8_t)(h0 >> 32); 360 s[5] = (uint8_t)(h0 >> 40); 361 s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3)); 362 s[7] = (uint8_t)(h1 >> 5); 363 s[8] = (uint8_t)(h1 >> 13); 364 s[9] = (uint8_t)(h1 >> 21); 365 s[10] = (uint8_t)(h1 >> 29); 366 s[11] = (uint8_t)(h1 >> 37); 367 s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6)); 368 s[13] = (uint8_t)(h2 >> 2); 369 s[14] = (uint8_t)(h2 >> 10); 370 s[15] = (uint8_t)(h2 >> 18); 371 s[16] = (uint8_t)(h2 >> 26); 372 s[17] = (uint8_t)(h2 >> 34); 373 s[18] = (uint8_t)(h2 >> 42); 374 s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1)); 375 s[20] = (uint8_t)(h3 >> 7); 376 s[21] = (uint8_t)(h3 >> 15); 377 s[22] = (uint8_t)(h3 >> 23); 378 s[23] = (uint8_t)(h3 >> 31); 379 s[24] = (uint8_t)(h3 >> 39); 380 s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4)); 381 s[26] = (uint8_t)(h4 >> 4); 382 s[27] = (uint8_t)(h4 >> 12); 383 s[28] = (uint8_t)(h4 >> 20); 384 s[29] = (uint8_t)(h4 >> 28); 385 s[30] = (uint8_t)(h4 >> 36); 386 s[31] = (uint8_t)(h4 >> 44); 387 } 388 389 # if defined(X25519_ASM) 390 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g); 391 void x25519_fe51_sqr(fe51 h, const fe51 f); 392 void x25519_fe51_mul121666(fe51 h, fe51 f); 393 # define fe51_mul x25519_fe51_mul 394 # define fe51_sq x25519_fe51_sqr 395 # define fe51_mul121666 x25519_fe51_mul121666 396 # else 397 398 typedef uint128_t u128; 399 400 static void fe51_mul(fe51 h, const fe51 f, const fe51 g) 401 { 402 u128 h0, h1, h2, h3, h4; 403 uint64_t f_i, g0, g1, g2, g3, g4; 404 405 f_i = f[0]; 406 h0 = (u128)f_i * (g0 = g[0]); 407 h1 = (u128)f_i * (g1 = g[1]); 408 h2 = (u128)f_i * (g2 = g[2]); 409 h3 = (u128)f_i * (g3 = g[3]); 410 h4 = (u128)f_i * (g4 = g[4]); 411 412 f_i = f[1]; 413 h0 += (u128)f_i * (g4 *= 19); 414 h1 += (u128)f_i * g0; 415 h2 += (u128)f_i * g1; 416 h3 += (u128)f_i * g2; 417 h4 += (u128)f_i * g3; 418 419 f_i = f[2]; 420 h0 += (u128)f_i * (g3 *= 19); 421 h1 += (u128)f_i * g4; 422 h2 += (u128)f_i * g0; 423 h3 += (u128)f_i * g1; 424 h4 += (u128)f_i * g2; 425 426 f_i = f[3]; 427 h0 += (u128)f_i * (g2 *= 19); 428 h1 += (u128)f_i * g3; 429 h2 += (u128)f_i * g4; 430 h3 += (u128)f_i * g0; 431 h4 += (u128)f_i * g1; 432 433 f_i = f[4]; 434 h0 += (u128)f_i * (g1 *= 19); 435 h1 += (u128)f_i * g2; 436 h2 += (u128)f_i * g3; 437 h3 += (u128)f_i * g4; 438 h4 += (u128)f_i * g0; 439 440 /* partial [lazy] reduction */ 441 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51; 442 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51; 443 444 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51; 445 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51; 446 447 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51; 448 g3 += g2 >> 51; g2 &= MASK51; 449 g1 += g0 >> 51; g0 &= MASK51; 450 451 h[0] = g0; 452 h[1] = g1; 453 h[2] = g2; 454 h[3] = g3; 455 h[4] = g4; 456 } 457 458 static void fe51_sq(fe51 h, const fe51 f) 459 { 460 # if defined(OPENSSL_SMALL_FOOTPRINT) 461 fe51_mul(h, f, f); 462 # else 463 /* dedicated squaring gives 16-25% overall improvement */ 464 uint64_t g0 = f[0]; 465 uint64_t g1 = f[1]; 466 uint64_t g2 = f[2]; 467 uint64_t g3 = f[3]; 468 uint64_t g4 = f[4]; 469 u128 h0, h1, h2, h3, h4; 470 471 h0 = (u128)g0 * g0; g0 *= 2; 472 h1 = (u128)g0 * g1; 473 h2 = (u128)g0 * g2; 474 h3 = (u128)g0 * g3; 475 h4 = (u128)g0 * g4; 476 477 g0 = g4; /* borrow g0 */ 478 h3 += (u128)g0 * (g4 *= 19); 479 480 h2 += (u128)g1 * g1; g1 *= 2; 481 h3 += (u128)g1 * g2; 482 h4 += (u128)g1 * g3; 483 h0 += (u128)g1 * g4; 484 485 g0 = g3; /* borrow g0 */ 486 h1 += (u128)g0 * (g3 *= 19); 487 h2 += (u128)(g0 * 2) * g4; 488 489 h4 += (u128)g2 * g2; g2 *= 2; 490 h0 += (u128)g2 * g3; 491 h1 += (u128)g2 * g4; 492 493 /* partial [lazy] reduction */ 494 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51; 495 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51; 496 497 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51; 498 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51; 499 500 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51; 501 g3 += g2 >> 51; g2 &= MASK51; 502 g1 += g0 >> 51; g0 &= MASK51; 503 504 h[0] = g0; 505 h[1] = g1; 506 h[2] = g2; 507 h[3] = g3; 508 h[4] = g4; 509 # endif 510 } 511 512 static void fe51_mul121666(fe51 h, fe51 f) 513 { 514 u128 h0 = f[0] * (u128)121666; 515 u128 h1 = f[1] * (u128)121666; 516 u128 h2 = f[2] * (u128)121666; 517 u128 h3 = f[3] * (u128)121666; 518 u128 h4 = f[4] * (u128)121666; 519 uint64_t g0, g1, g2, g3, g4; 520 521 h3 += (uint64_t)(h2 >> 51); g2 = (uint64_t)h2 & MASK51; 522 h1 += (uint64_t)(h0 >> 51); g0 = (uint64_t)h0 & MASK51; 523 524 h4 += (uint64_t)(h3 >> 51); g3 = (uint64_t)h3 & MASK51; 525 g2 += (uint64_t)(h1 >> 51); g1 = (uint64_t)h1 & MASK51; 526 527 g0 += (uint64_t)(h4 >> 51) * 19; g4 = (uint64_t)h4 & MASK51; 528 g3 += g2 >> 51; g2 &= MASK51; 529 g1 += g0 >> 51; g0 &= MASK51; 530 531 h[0] = g0; 532 h[1] = g1; 533 h[2] = g2; 534 h[3] = g3; 535 h[4] = g4; 536 } 537 # endif 538 539 static void fe51_add(fe51 h, const fe51 f, const fe51 g) 540 { 541 h[0] = f[0] + g[0]; 542 h[1] = f[1] + g[1]; 543 h[2] = f[2] + g[2]; 544 h[3] = f[3] + g[3]; 545 h[4] = f[4] + g[4]; 546 } 547 548 static void fe51_sub(fe51 h, const fe51 f, const fe51 g) 549 { 550 /* 551 * Add 2*modulus to ensure that result remains positive 552 * even if subtrahend is partially reduced. 553 */ 554 h[0] = (f[0] + 0xfffffffffffda) - g[0]; 555 h[1] = (f[1] + 0xffffffffffffe) - g[1]; 556 h[2] = (f[2] + 0xffffffffffffe) - g[2]; 557 h[3] = (f[3] + 0xffffffffffffe) - g[3]; 558 h[4] = (f[4] + 0xffffffffffffe) - g[4]; 559 } 560 561 static void fe51_0(fe51 h) 562 { 563 h[0] = 0; 564 h[1] = 0; 565 h[2] = 0; 566 h[3] = 0; 567 h[4] = 0; 568 } 569 570 static void fe51_1(fe51 h) 571 { 572 h[0] = 1; 573 h[1] = 0; 574 h[2] = 0; 575 h[3] = 0; 576 h[4] = 0; 577 } 578 579 static void fe51_copy(fe51 h, const fe51 f) 580 { 581 h[0] = f[0]; 582 h[1] = f[1]; 583 h[2] = f[2]; 584 h[3] = f[3]; 585 h[4] = f[4]; 586 } 587 588 static void fe51_cswap(fe51 f, fe51 g, unsigned int b) 589 { 590 int i; 591 uint64_t mask = 0 - (uint64_t)b; 592 593 for (i = 0; i < 5; i++) { 594 int64_t x = f[i] ^ g[i]; 595 x &= mask; 596 f[i] ^= x; 597 g[i] ^= x; 598 } 599 } 600 601 static void fe51_invert(fe51 out, const fe51 z) 602 { 603 fe51 t0; 604 fe51 t1; 605 fe51 t2; 606 fe51 t3; 607 int i; 608 609 /* 610 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 611 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 612 */ 613 614 /* t0 = z ** 2 */ 615 fe51_sq(t0, z); 616 617 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 618 fe51_sq(t1, t0); 619 fe51_sq(t1, t1); 620 621 /* t1 = z * t1 = z ** 9 */ 622 fe51_mul(t1, z, t1); 623 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 624 fe51_mul(t0, t0, t1); 625 626 /* t2 = t0 ** 2 = z ** 22 */ 627 fe51_sq(t2, t0); 628 629 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 630 fe51_mul(t1, t1, t2); 631 632 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 633 fe51_sq(t2, t1); 634 for (i = 1; i < 5; ++i) 635 fe51_sq(t2, t2); 636 637 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 638 fe51_mul(t1, t2, t1); 639 640 /* Continuing similarly... */ 641 642 /* t2 = z ** (2 ** 20 - 1) */ 643 fe51_sq(t2, t1); 644 for (i = 1; i < 10; ++i) 645 fe51_sq(t2, t2); 646 647 fe51_mul(t2, t2, t1); 648 649 /* t2 = z ** (2 ** 40 - 1) */ 650 fe51_sq(t3, t2); 651 for (i = 1; i < 20; ++i) 652 fe51_sq(t3, t3); 653 654 fe51_mul(t2, t3, t2); 655 656 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 657 for (i = 0; i < 10; ++i) 658 fe51_sq(t2, t2); 659 660 /* t1 = z ** (2 ** 50 - 1) */ 661 fe51_mul(t1, t2, t1); 662 663 /* t2 = z ** (2 ** 100 - 1) */ 664 fe51_sq(t2, t1); 665 for (i = 1; i < 50; ++i) 666 fe51_sq(t2, t2); 667 668 fe51_mul(t2, t2, t1); 669 670 /* t2 = z ** (2 ** 200 - 1) */ 671 fe51_sq(t3, t2); 672 for (i = 1; i < 100; ++i) 673 fe51_sq(t3, t3); 674 675 fe51_mul(t2, t3, t2); 676 677 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 678 for (i = 0; i < 50; ++i) 679 fe51_sq(t2, t2); 680 681 /* t1 = z ** (2 ** 250 - 1) */ 682 fe51_mul(t1, t2, t1); 683 684 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 685 for (i = 0; i < 5; ++i) 686 fe51_sq(t1, t1); 687 688 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 689 fe51_mul(out, t1, t0); 690 } 691 692 /* 693 * Duplicate of original x25519_scalar_mult_generic, but using 694 * fe51_* subroutines. 695 */ 696 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32], 697 const uint8_t point[32]) 698 { 699 fe51 x1, x2, z2, x3, z3, tmp0, tmp1; 700 uint8_t e[32]; 701 unsigned swap = 0; 702 int pos; 703 704 # ifdef BASE_2_64_IMPLEMENTED 705 if (x25519_fe64_eligible()) { 706 x25519_scalar_mulx(out, scalar, point); 707 return; 708 } 709 # endif 710 711 memcpy(e, scalar, 32); 712 e[0] &= 0xf8; 713 e[31] &= 0x7f; 714 e[31] |= 0x40; 715 fe51_frombytes(x1, point); 716 fe51_1(x2); 717 fe51_0(z2); 718 fe51_copy(x3, x1); 719 fe51_1(z3); 720 721 for (pos = 254; pos >= 0; --pos) { 722 unsigned int b = 1 & (e[pos / 8] >> (pos & 7)); 723 724 swap ^= b; 725 fe51_cswap(x2, x3, swap); 726 fe51_cswap(z2, z3, swap); 727 swap = b; 728 fe51_sub(tmp0, x3, z3); 729 fe51_sub(tmp1, x2, z2); 730 fe51_add(x2, x2, z2); 731 fe51_add(z2, x3, z3); 732 fe51_mul(z3, tmp0, x2); 733 fe51_mul(z2, z2, tmp1); 734 fe51_sq(tmp0, tmp1); 735 fe51_sq(tmp1, x2); 736 fe51_add(x3, z3, z2); 737 fe51_sub(z2, z3, z2); 738 fe51_mul(x2, tmp1, tmp0); 739 fe51_sub(tmp1, tmp1, tmp0); 740 fe51_sq(z2, z2); 741 fe51_mul121666(z3, tmp1); 742 fe51_sq(x3, x3); 743 fe51_add(tmp0, tmp0, z3); 744 fe51_mul(z3, x1, z2); 745 fe51_mul(z2, tmp1, tmp0); 746 } 747 748 fe51_invert(z2, z2); 749 fe51_mul(x2, x2, z2); 750 fe51_tobytes(out, x2); 751 752 OPENSSL_cleanse(e, sizeof(e)); 753 } 754 #endif 755 756 /* 757 * Reference base 2^25.5 implementation. 758 * 759 * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP 760 * 20141124 (http://bench.cr.yp.to/supercop.html). 761 * 762 * The field functions are shared by Ed25519 and X25519 where possible. 763 */ 764 765 /* 766 * fe means field element. Here the field is \Z/(2^255-19). An element t, 767 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 768 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on 769 * context. 770 */ 771 typedef int32_t fe[10]; 772 773 static const int64_t kBottom21Bits = 0x1fffffLL; 774 static const int64_t kBottom25Bits = 0x1ffffffLL; 775 static const int64_t kBottom26Bits = 0x3ffffffLL; 776 static const int64_t kTop39Bits = 0xfffffffffe000000LL; 777 static const int64_t kTop38Bits = 0xfffffffffc000000LL; 778 779 static uint64_t load_3(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 return result; 787 } 788 789 static uint64_t load_4(const uint8_t *in) 790 { 791 uint64_t result; 792 793 result = ((uint64_t)in[0]); 794 result |= ((uint64_t)in[1]) << 8; 795 result |= ((uint64_t)in[2]) << 16; 796 result |= ((uint64_t)in[3]) << 24; 797 return result; 798 } 799 800 static void fe_frombytes(fe h, const uint8_t *s) 801 { 802 /* Ignores top bit of h. */ 803 int64_t h0 = load_4(s); 804 int64_t h1 = load_3(s + 4) << 6; 805 int64_t h2 = load_3(s + 7) << 5; 806 int64_t h3 = load_3(s + 10) << 3; 807 int64_t h4 = load_3(s + 13) << 2; 808 int64_t h5 = load_4(s + 16); 809 int64_t h6 = load_3(s + 20) << 7; 810 int64_t h7 = load_3(s + 23) << 5; 811 int64_t h8 = load_3(s + 26) << 4; 812 int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2; 813 int64_t carry0; 814 int64_t carry1; 815 int64_t carry2; 816 int64_t carry3; 817 int64_t carry4; 818 int64_t carry5; 819 int64_t carry6; 820 int64_t carry7; 821 int64_t carry8; 822 int64_t carry9; 823 824 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 825 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 826 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 827 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 828 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 829 830 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 831 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 832 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 833 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 834 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 835 836 h[0] = (int32_t)h0; 837 h[1] = (int32_t)h1; 838 h[2] = (int32_t)h2; 839 h[3] = (int32_t)h3; 840 h[4] = (int32_t)h4; 841 h[5] = (int32_t)h5; 842 h[6] = (int32_t)h6; 843 h[7] = (int32_t)h7; 844 h[8] = (int32_t)h8; 845 h[9] = (int32_t)h9; 846 } 847 848 /* 849 * Preconditions: 850 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 851 * 852 * Write p=2^255-19; q=floor(h/p). 853 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). 854 * 855 * Proof: 856 * Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. 857 * Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4. 858 * 859 * Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). 860 * Then 0<y<1. 861 * 862 * Write r=h-pq. 863 * Have 0<=r<=p-1=2^255-20. 864 * Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1. 865 * 866 * Write x=r+19(2^-255)r+y. 867 * Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q. 868 * 869 * Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1)) 870 * so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q. 871 */ 872 static void fe_tobytes(uint8_t *s, const fe h) 873 { 874 int32_t h0 = h[0]; 875 int32_t h1 = h[1]; 876 int32_t h2 = h[2]; 877 int32_t h3 = h[3]; 878 int32_t h4 = h[4]; 879 int32_t h5 = h[5]; 880 int32_t h6 = h[6]; 881 int32_t h7 = h[7]; 882 int32_t h8 = h[8]; 883 int32_t h9 = h[9]; 884 int32_t q; 885 886 q = (19 * h9 + (((int32_t) 1) << 24)) >> 25; 887 q = (h0 + q) >> 26; 888 q = (h1 + q) >> 25; 889 q = (h2 + q) >> 26; 890 q = (h3 + q) >> 25; 891 q = (h4 + q) >> 26; 892 q = (h5 + q) >> 25; 893 q = (h6 + q) >> 26; 894 q = (h7 + q) >> 25; 895 q = (h8 + q) >> 26; 896 q = (h9 + q) >> 25; 897 898 /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */ 899 h0 += 19 * q; 900 /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */ 901 902 h1 += h0 >> 26; h0 &= kBottom26Bits; 903 h2 += h1 >> 25; h1 &= kBottom25Bits; 904 h3 += h2 >> 26; h2 &= kBottom26Bits; 905 h4 += h3 >> 25; h3 &= kBottom25Bits; 906 h5 += h4 >> 26; h4 &= kBottom26Bits; 907 h6 += h5 >> 25; h5 &= kBottom25Bits; 908 h7 += h6 >> 26; h6 &= kBottom26Bits; 909 h8 += h7 >> 25; h7 &= kBottom25Bits; 910 h9 += h8 >> 26; h8 &= kBottom26Bits; 911 h9 &= kBottom25Bits; 912 /* h10 = carry9 */ 913 914 /* 915 * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. 916 * Have h0+...+2^230 h9 between 0 and 2^255-1; 917 * evidently 2^255 h10-2^255 q = 0. 918 * Goal: Output h0+...+2^230 h9. 919 */ 920 s[ 0] = (uint8_t) (h0 >> 0); 921 s[ 1] = (uint8_t) (h0 >> 8); 922 s[ 2] = (uint8_t) (h0 >> 16); 923 s[ 3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2)); 924 s[ 4] = (uint8_t) (h1 >> 6); 925 s[ 5] = (uint8_t) (h1 >> 14); 926 s[ 6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3)); 927 s[ 7] = (uint8_t) (h2 >> 5); 928 s[ 8] = (uint8_t) (h2 >> 13); 929 s[ 9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5)); 930 s[10] = (uint8_t) (h3 >> 3); 931 s[11] = (uint8_t) (h3 >> 11); 932 s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6)); 933 s[13] = (uint8_t) (h4 >> 2); 934 s[14] = (uint8_t) (h4 >> 10); 935 s[15] = (uint8_t) (h4 >> 18); 936 s[16] = (uint8_t) (h5 >> 0); 937 s[17] = (uint8_t) (h5 >> 8); 938 s[18] = (uint8_t) (h5 >> 16); 939 s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1)); 940 s[20] = (uint8_t) (h6 >> 7); 941 s[21] = (uint8_t) (h6 >> 15); 942 s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3)); 943 s[23] = (uint8_t) (h7 >> 5); 944 s[24] = (uint8_t) (h7 >> 13); 945 s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4)); 946 s[26] = (uint8_t) (h8 >> 4); 947 s[27] = (uint8_t) (h8 >> 12); 948 s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6)); 949 s[29] = (uint8_t) (h9 >> 2); 950 s[30] = (uint8_t) (h9 >> 10); 951 s[31] = (uint8_t) (h9 >> 18); 952 } 953 954 /* h = f */ 955 static void fe_copy(fe h, const fe f) 956 { 957 memmove(h, f, sizeof(int32_t) * 10); 958 } 959 960 /* h = 0 */ 961 static void fe_0(fe h) 962 { 963 memset(h, 0, sizeof(int32_t) * 10); 964 } 965 966 /* h = 1 */ 967 static void fe_1(fe h) 968 { 969 memset(h, 0, sizeof(int32_t) * 10); 970 h[0] = 1; 971 } 972 973 /* 974 * h = f + g 975 * 976 * Can overlap h with f or g. 977 * 978 * Preconditions: 979 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 980 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 981 * 982 * Postconditions: 983 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 984 */ 985 static void fe_add(fe h, const fe f, const fe g) 986 { 987 unsigned i; 988 989 for (i = 0; i < 10; i++) { 990 h[i] = f[i] + g[i]; 991 } 992 } 993 994 /* 995 * h = f - g 996 * 997 * Can overlap h with f or g. 998 * 999 * Preconditions: 1000 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1001 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1002 * 1003 * Postconditions: 1004 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1005 */ 1006 static void fe_sub(fe h, const fe f, const fe g) 1007 { 1008 unsigned i; 1009 1010 for (i = 0; i < 10; i++) { 1011 h[i] = f[i] - g[i]; 1012 } 1013 } 1014 1015 /* 1016 * h = f * g 1017 * 1018 * Can overlap h with f or g. 1019 * 1020 * Preconditions: 1021 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1022 * |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1023 * 1024 * Postconditions: 1025 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1026 * 1027 * Notes on implementation strategy: 1028 * 1029 * Using schoolbook multiplication. 1030 * Karatsuba would save a little in some cost models. 1031 * 1032 * Most multiplications by 2 and 19 are 32-bit precomputations; 1033 * cheaper than 64-bit postcomputations. 1034 * 1035 * There is one remaining multiplication by 19 in the carry chain; 1036 * one *19 precomputation can be merged into this, 1037 * but the resulting data flow is considerably less clean. 1038 * 1039 * There are 12 carries below. 1040 * 10 of them are 2-way parallelizable and vectorizable. 1041 * Can get away with 11 carries, but then data flow is much deeper. 1042 * 1043 * With tighter constraints on inputs can squeeze carries into int32. 1044 */ 1045 static void fe_mul(fe h, const fe f, const fe g) 1046 { 1047 int32_t f0 = f[0]; 1048 int32_t f1 = f[1]; 1049 int32_t f2 = f[2]; 1050 int32_t f3 = f[3]; 1051 int32_t f4 = f[4]; 1052 int32_t f5 = f[5]; 1053 int32_t f6 = f[6]; 1054 int32_t f7 = f[7]; 1055 int32_t f8 = f[8]; 1056 int32_t f9 = f[9]; 1057 int32_t g0 = g[0]; 1058 int32_t g1 = g[1]; 1059 int32_t g2 = g[2]; 1060 int32_t g3 = g[3]; 1061 int32_t g4 = g[4]; 1062 int32_t g5 = g[5]; 1063 int32_t g6 = g[6]; 1064 int32_t g7 = g[7]; 1065 int32_t g8 = g[8]; 1066 int32_t g9 = g[9]; 1067 int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */ 1068 int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */ 1069 int32_t g3_19 = 19 * g3; 1070 int32_t g4_19 = 19 * g4; 1071 int32_t g5_19 = 19 * g5; 1072 int32_t g6_19 = 19 * g6; 1073 int32_t g7_19 = 19 * g7; 1074 int32_t g8_19 = 19 * g8; 1075 int32_t g9_19 = 19 * g9; 1076 int32_t f1_2 = 2 * f1; 1077 int32_t f3_2 = 2 * f3; 1078 int32_t f5_2 = 2 * f5; 1079 int32_t f7_2 = 2 * f7; 1080 int32_t f9_2 = 2 * f9; 1081 int64_t f0g0 = f0 * (int64_t) g0; 1082 int64_t f0g1 = f0 * (int64_t) g1; 1083 int64_t f0g2 = f0 * (int64_t) g2; 1084 int64_t f0g3 = f0 * (int64_t) g3; 1085 int64_t f0g4 = f0 * (int64_t) g4; 1086 int64_t f0g5 = f0 * (int64_t) g5; 1087 int64_t f0g6 = f0 * (int64_t) g6; 1088 int64_t f0g7 = f0 * (int64_t) g7; 1089 int64_t f0g8 = f0 * (int64_t) g8; 1090 int64_t f0g9 = f0 * (int64_t) g9; 1091 int64_t f1g0 = f1 * (int64_t) g0; 1092 int64_t f1g1_2 = f1_2 * (int64_t) g1; 1093 int64_t f1g2 = f1 * (int64_t) g2; 1094 int64_t f1g3_2 = f1_2 * (int64_t) g3; 1095 int64_t f1g4 = f1 * (int64_t) g4; 1096 int64_t f1g5_2 = f1_2 * (int64_t) g5; 1097 int64_t f1g6 = f1 * (int64_t) g6; 1098 int64_t f1g7_2 = f1_2 * (int64_t) g7; 1099 int64_t f1g8 = f1 * (int64_t) g8; 1100 int64_t f1g9_38 = f1_2 * (int64_t) g9_19; 1101 int64_t f2g0 = f2 * (int64_t) g0; 1102 int64_t f2g1 = f2 * (int64_t) g1; 1103 int64_t f2g2 = f2 * (int64_t) g2; 1104 int64_t f2g3 = f2 * (int64_t) g3; 1105 int64_t f2g4 = f2 * (int64_t) g4; 1106 int64_t f2g5 = f2 * (int64_t) g5; 1107 int64_t f2g6 = f2 * (int64_t) g6; 1108 int64_t f2g7 = f2 * (int64_t) g7; 1109 int64_t f2g8_19 = f2 * (int64_t) g8_19; 1110 int64_t f2g9_19 = f2 * (int64_t) g9_19; 1111 int64_t f3g0 = f3 * (int64_t) g0; 1112 int64_t f3g1_2 = f3_2 * (int64_t) g1; 1113 int64_t f3g2 = f3 * (int64_t) g2; 1114 int64_t f3g3_2 = f3_2 * (int64_t) g3; 1115 int64_t f3g4 = f3 * (int64_t) g4; 1116 int64_t f3g5_2 = f3_2 * (int64_t) g5; 1117 int64_t f3g6 = f3 * (int64_t) g6; 1118 int64_t f3g7_38 = f3_2 * (int64_t) g7_19; 1119 int64_t f3g8_19 = f3 * (int64_t) g8_19; 1120 int64_t f3g9_38 = f3_2 * (int64_t) g9_19; 1121 int64_t f4g0 = f4 * (int64_t) g0; 1122 int64_t f4g1 = f4 * (int64_t) g1; 1123 int64_t f4g2 = f4 * (int64_t) g2; 1124 int64_t f4g3 = f4 * (int64_t) g3; 1125 int64_t f4g4 = f4 * (int64_t) g4; 1126 int64_t f4g5 = f4 * (int64_t) g5; 1127 int64_t f4g6_19 = f4 * (int64_t) g6_19; 1128 int64_t f4g7_19 = f4 * (int64_t) g7_19; 1129 int64_t f4g8_19 = f4 * (int64_t) g8_19; 1130 int64_t f4g9_19 = f4 * (int64_t) g9_19; 1131 int64_t f5g0 = f5 * (int64_t) g0; 1132 int64_t f5g1_2 = f5_2 * (int64_t) g1; 1133 int64_t f5g2 = f5 * (int64_t) g2; 1134 int64_t f5g3_2 = f5_2 * (int64_t) g3; 1135 int64_t f5g4 = f5 * (int64_t) g4; 1136 int64_t f5g5_38 = f5_2 * (int64_t) g5_19; 1137 int64_t f5g6_19 = f5 * (int64_t) g6_19; 1138 int64_t f5g7_38 = f5_2 * (int64_t) g7_19; 1139 int64_t f5g8_19 = f5 * (int64_t) g8_19; 1140 int64_t f5g9_38 = f5_2 * (int64_t) g9_19; 1141 int64_t f6g0 = f6 * (int64_t) g0; 1142 int64_t f6g1 = f6 * (int64_t) g1; 1143 int64_t f6g2 = f6 * (int64_t) g2; 1144 int64_t f6g3 = f6 * (int64_t) g3; 1145 int64_t f6g4_19 = f6 * (int64_t) g4_19; 1146 int64_t f6g5_19 = f6 * (int64_t) g5_19; 1147 int64_t f6g6_19 = f6 * (int64_t) g6_19; 1148 int64_t f6g7_19 = f6 * (int64_t) g7_19; 1149 int64_t f6g8_19 = f6 * (int64_t) g8_19; 1150 int64_t f6g9_19 = f6 * (int64_t) g9_19; 1151 int64_t f7g0 = f7 * (int64_t) g0; 1152 int64_t f7g1_2 = f7_2 * (int64_t) g1; 1153 int64_t f7g2 = f7 * (int64_t) g2; 1154 int64_t f7g3_38 = f7_2 * (int64_t) g3_19; 1155 int64_t f7g4_19 = f7 * (int64_t) g4_19; 1156 int64_t f7g5_38 = f7_2 * (int64_t) g5_19; 1157 int64_t f7g6_19 = f7 * (int64_t) g6_19; 1158 int64_t f7g7_38 = f7_2 * (int64_t) g7_19; 1159 int64_t f7g8_19 = f7 * (int64_t) g8_19; 1160 int64_t f7g9_38 = f7_2 * (int64_t) g9_19; 1161 int64_t f8g0 = f8 * (int64_t) g0; 1162 int64_t f8g1 = f8 * (int64_t) g1; 1163 int64_t f8g2_19 = f8 * (int64_t) g2_19; 1164 int64_t f8g3_19 = f8 * (int64_t) g3_19; 1165 int64_t f8g4_19 = f8 * (int64_t) g4_19; 1166 int64_t f8g5_19 = f8 * (int64_t) g5_19; 1167 int64_t f8g6_19 = f8 * (int64_t) g6_19; 1168 int64_t f8g7_19 = f8 * (int64_t) g7_19; 1169 int64_t f8g8_19 = f8 * (int64_t) g8_19; 1170 int64_t f8g9_19 = f8 * (int64_t) g9_19; 1171 int64_t f9g0 = f9 * (int64_t) g0; 1172 int64_t f9g1_38 = f9_2 * (int64_t) g1_19; 1173 int64_t f9g2_19 = f9 * (int64_t) g2_19; 1174 int64_t f9g3_38 = f9_2 * (int64_t) g3_19; 1175 int64_t f9g4_19 = f9 * (int64_t) g4_19; 1176 int64_t f9g5_38 = f9_2 * (int64_t) g5_19; 1177 int64_t f9g6_19 = f9 * (int64_t) g6_19; 1178 int64_t f9g7_38 = f9_2 * (int64_t) g7_19; 1179 int64_t f9g8_19 = f9 * (int64_t) g8_19; 1180 int64_t f9g9_38 = f9_2 * (int64_t) g9_19; 1181 int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38; 1182 int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19; 1183 int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38; 1184 int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19; 1185 int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38; 1186 int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19; 1187 int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38; 1188 int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19; 1189 int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38; 1190 int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0 ; 1191 int64_t carry0; 1192 int64_t carry1; 1193 int64_t carry2; 1194 int64_t carry3; 1195 int64_t carry4; 1196 int64_t carry5; 1197 int64_t carry6; 1198 int64_t carry7; 1199 int64_t carry8; 1200 int64_t carry9; 1201 1202 /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38)) 1203 * i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8 1204 * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19)) 1205 * i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */ 1206 1207 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1208 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1209 /* |h0| <= 2^25 */ 1210 /* |h4| <= 2^25 */ 1211 /* |h1| <= 1.71*2^59 */ 1212 /* |h5| <= 1.71*2^59 */ 1213 1214 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 1215 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 1216 /* |h1| <= 2^24; from now on fits into int32 */ 1217 /* |h5| <= 2^24; from now on fits into int32 */ 1218 /* |h2| <= 1.41*2^60 */ 1219 /* |h6| <= 1.41*2^60 */ 1220 1221 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 1222 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 1223 /* |h2| <= 2^25; from now on fits into int32 unchanged */ 1224 /* |h6| <= 2^25; from now on fits into int32 unchanged */ 1225 /* |h3| <= 1.71*2^59 */ 1226 /* |h7| <= 1.71*2^59 */ 1227 1228 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 1229 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 1230 /* |h3| <= 2^24; from now on fits into int32 unchanged */ 1231 /* |h7| <= 2^24; from now on fits into int32 unchanged */ 1232 /* |h4| <= 1.72*2^34 */ 1233 /* |h8| <= 1.41*2^60 */ 1234 1235 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1236 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 1237 /* |h4| <= 2^25; from now on fits into int32 unchanged */ 1238 /* |h8| <= 2^25; from now on fits into int32 unchanged */ 1239 /* |h5| <= 1.01*2^24 */ 1240 /* |h9| <= 1.71*2^59 */ 1241 1242 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 1243 /* |h9| <= 2^24; from now on fits into int32 unchanged */ 1244 /* |h0| <= 1.1*2^39 */ 1245 1246 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1247 /* |h0| <= 2^25; from now on fits into int32 unchanged */ 1248 /* |h1| <= 1.01*2^24 */ 1249 1250 h[0] = (int32_t)h0; 1251 h[1] = (int32_t)h1; 1252 h[2] = (int32_t)h2; 1253 h[3] = (int32_t)h3; 1254 h[4] = (int32_t)h4; 1255 h[5] = (int32_t)h5; 1256 h[6] = (int32_t)h6; 1257 h[7] = (int32_t)h7; 1258 h[8] = (int32_t)h8; 1259 h[9] = (int32_t)h9; 1260 } 1261 1262 /* 1263 * h = f * f 1264 * 1265 * Can overlap h with f. 1266 * 1267 * Preconditions: 1268 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1269 * 1270 * Postconditions: 1271 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1272 * 1273 * See fe_mul.c for discussion of implementation strategy. 1274 */ 1275 static void fe_sq(fe h, const fe f) 1276 { 1277 int32_t f0 = f[0]; 1278 int32_t f1 = f[1]; 1279 int32_t f2 = f[2]; 1280 int32_t f3 = f[3]; 1281 int32_t f4 = f[4]; 1282 int32_t f5 = f[5]; 1283 int32_t f6 = f[6]; 1284 int32_t f7 = f[7]; 1285 int32_t f8 = f[8]; 1286 int32_t f9 = f[9]; 1287 int32_t f0_2 = 2 * f0; 1288 int32_t f1_2 = 2 * f1; 1289 int32_t f2_2 = 2 * f2; 1290 int32_t f3_2 = 2 * f3; 1291 int32_t f4_2 = 2 * f4; 1292 int32_t f5_2 = 2 * f5; 1293 int32_t f6_2 = 2 * f6; 1294 int32_t f7_2 = 2 * f7; 1295 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */ 1296 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */ 1297 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */ 1298 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */ 1299 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */ 1300 int64_t f0f0 = f0 * (int64_t) f0; 1301 int64_t f0f1_2 = f0_2 * (int64_t) f1; 1302 int64_t f0f2_2 = f0_2 * (int64_t) f2; 1303 int64_t f0f3_2 = f0_2 * (int64_t) f3; 1304 int64_t f0f4_2 = f0_2 * (int64_t) f4; 1305 int64_t f0f5_2 = f0_2 * (int64_t) f5; 1306 int64_t f0f6_2 = f0_2 * (int64_t) f6; 1307 int64_t f0f7_2 = f0_2 * (int64_t) f7; 1308 int64_t f0f8_2 = f0_2 * (int64_t) f8; 1309 int64_t f0f9_2 = f0_2 * (int64_t) f9; 1310 int64_t f1f1_2 = f1_2 * (int64_t) f1; 1311 int64_t f1f2_2 = f1_2 * (int64_t) f2; 1312 int64_t f1f3_4 = f1_2 * (int64_t) f3_2; 1313 int64_t f1f4_2 = f1_2 * (int64_t) f4; 1314 int64_t f1f5_4 = f1_2 * (int64_t) f5_2; 1315 int64_t f1f6_2 = f1_2 * (int64_t) f6; 1316 int64_t f1f7_4 = f1_2 * (int64_t) f7_2; 1317 int64_t f1f8_2 = f1_2 * (int64_t) f8; 1318 int64_t f1f9_76 = f1_2 * (int64_t) f9_38; 1319 int64_t f2f2 = f2 * (int64_t) f2; 1320 int64_t f2f3_2 = f2_2 * (int64_t) f3; 1321 int64_t f2f4_2 = f2_2 * (int64_t) f4; 1322 int64_t f2f5_2 = f2_2 * (int64_t) f5; 1323 int64_t f2f6_2 = f2_2 * (int64_t) f6; 1324 int64_t f2f7_2 = f2_2 * (int64_t) f7; 1325 int64_t f2f8_38 = f2_2 * (int64_t) f8_19; 1326 int64_t f2f9_38 = f2 * (int64_t) f9_38; 1327 int64_t f3f3_2 = f3_2 * (int64_t) f3; 1328 int64_t f3f4_2 = f3_2 * (int64_t) f4; 1329 int64_t f3f5_4 = f3_2 * (int64_t) f5_2; 1330 int64_t f3f6_2 = f3_2 * (int64_t) f6; 1331 int64_t f3f7_76 = f3_2 * (int64_t) f7_38; 1332 int64_t f3f8_38 = f3_2 * (int64_t) f8_19; 1333 int64_t f3f9_76 = f3_2 * (int64_t) f9_38; 1334 int64_t f4f4 = f4 * (int64_t) f4; 1335 int64_t f4f5_2 = f4_2 * (int64_t) f5; 1336 int64_t f4f6_38 = f4_2 * (int64_t) f6_19; 1337 int64_t f4f7_38 = f4 * (int64_t) f7_38; 1338 int64_t f4f8_38 = f4_2 * (int64_t) f8_19; 1339 int64_t f4f9_38 = f4 * (int64_t) f9_38; 1340 int64_t f5f5_38 = f5 * (int64_t) f5_38; 1341 int64_t f5f6_38 = f5_2 * (int64_t) f6_19; 1342 int64_t f5f7_76 = f5_2 * (int64_t) f7_38; 1343 int64_t f5f8_38 = f5_2 * (int64_t) f8_19; 1344 int64_t f5f9_76 = f5_2 * (int64_t) f9_38; 1345 int64_t f6f6_19 = f6 * (int64_t) f6_19; 1346 int64_t f6f7_38 = f6 * (int64_t) f7_38; 1347 int64_t f6f8_38 = f6_2 * (int64_t) f8_19; 1348 int64_t f6f9_38 = f6 * (int64_t) f9_38; 1349 int64_t f7f7_38 = f7 * (int64_t) f7_38; 1350 int64_t f7f8_38 = f7_2 * (int64_t) f8_19; 1351 int64_t f7f9_76 = f7_2 * (int64_t) f9_38; 1352 int64_t f8f8_19 = f8 * (int64_t) f8_19; 1353 int64_t f8f9_38 = f8 * (int64_t) f9_38; 1354 int64_t f9f9_38 = f9 * (int64_t) f9_38; 1355 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38; 1356 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38; 1357 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19; 1358 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38; 1359 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38; 1360 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38; 1361 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19; 1362 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38; 1363 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38; 1364 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2; 1365 int64_t carry0; 1366 int64_t carry1; 1367 int64_t carry2; 1368 int64_t carry3; 1369 int64_t carry4; 1370 int64_t carry5; 1371 int64_t carry6; 1372 int64_t carry7; 1373 int64_t carry8; 1374 int64_t carry9; 1375 1376 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1377 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1378 1379 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 1380 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 1381 1382 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 1383 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 1384 1385 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 1386 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 1387 1388 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1389 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 1390 1391 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 1392 1393 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1394 1395 h[0] = (int32_t)h0; 1396 h[1] = (int32_t)h1; 1397 h[2] = (int32_t)h2; 1398 h[3] = (int32_t)h3; 1399 h[4] = (int32_t)h4; 1400 h[5] = (int32_t)h5; 1401 h[6] = (int32_t)h6; 1402 h[7] = (int32_t)h7; 1403 h[8] = (int32_t)h8; 1404 h[9] = (int32_t)h9; 1405 } 1406 1407 static void fe_invert(fe out, const fe z) 1408 { 1409 fe t0; 1410 fe t1; 1411 fe t2; 1412 fe t3; 1413 int i; 1414 1415 /* 1416 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 1417 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 1418 */ 1419 1420 /* t0 = z ** 2 */ 1421 fe_sq(t0, z); 1422 1423 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 1424 fe_sq(t1, t0); 1425 fe_sq(t1, t1); 1426 1427 /* t1 = z * t1 = z ** 9 */ 1428 fe_mul(t1, z, t1); 1429 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 1430 fe_mul(t0, t0, t1); 1431 1432 /* t2 = t0 ** 2 = z ** 22 */ 1433 fe_sq(t2, t0); 1434 1435 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 1436 fe_mul(t1, t1, t2); 1437 1438 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 1439 fe_sq(t2, t1); 1440 for (i = 1; i < 5; ++i) { 1441 fe_sq(t2, t2); 1442 } 1443 1444 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 1445 fe_mul(t1, t2, t1); 1446 1447 /* Continuing similarly... */ 1448 1449 /* t2 = z ** (2 ** 20 - 1) */ 1450 fe_sq(t2, t1); 1451 for (i = 1; i < 10; ++i) { 1452 fe_sq(t2, t2); 1453 } 1454 fe_mul(t2, t2, t1); 1455 1456 /* t2 = z ** (2 ** 40 - 1) */ 1457 fe_sq(t3, t2); 1458 for (i = 1; i < 20; ++i) { 1459 fe_sq(t3, t3); 1460 } 1461 fe_mul(t2, t3, t2); 1462 1463 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 1464 for (i = 0; i < 10; ++i) { 1465 fe_sq(t2, t2); 1466 } 1467 /* t1 = z ** (2 ** 50 - 1) */ 1468 fe_mul(t1, t2, t1); 1469 1470 /* t2 = z ** (2 ** 100 - 1) */ 1471 fe_sq(t2, t1); 1472 for (i = 1; i < 50; ++i) { 1473 fe_sq(t2, t2); 1474 } 1475 fe_mul(t2, t2, t1); 1476 1477 /* t2 = z ** (2 ** 200 - 1) */ 1478 fe_sq(t3, t2); 1479 for (i = 1; i < 100; ++i) { 1480 fe_sq(t3, t3); 1481 } 1482 fe_mul(t2, t3, t2); 1483 1484 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 1485 fe_sq(t2, t2); 1486 for (i = 1; i < 50; ++i) { 1487 fe_sq(t2, t2); 1488 } 1489 1490 /* t1 = z ** (2 ** 250 - 1) */ 1491 fe_mul(t1, t2, t1); 1492 1493 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 1494 fe_sq(t1, t1); 1495 for (i = 1; i < 5; ++i) { 1496 fe_sq(t1, t1); 1497 } 1498 1499 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 1500 fe_mul(out, t1, t0); 1501 } 1502 1503 /* 1504 * h = -f 1505 * 1506 * Preconditions: 1507 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1508 * 1509 * Postconditions: 1510 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1511 */ 1512 static void fe_neg(fe h, const fe f) 1513 { 1514 unsigned i; 1515 1516 for (i = 0; i < 10; i++) { 1517 h[i] = -f[i]; 1518 } 1519 } 1520 1521 /* 1522 * Replace (f,g) with (g,g) if b == 1; 1523 * replace (f,g) with (f,g) if b == 0. 1524 * 1525 * Preconditions: b in {0,1}. 1526 */ 1527 static void fe_cmov(fe f, const fe g, unsigned b) 1528 { 1529 size_t i; 1530 1531 b = 0-b; 1532 for (i = 0; i < 10; i++) { 1533 int32_t x = f[i] ^ g[i]; 1534 x &= b; 1535 f[i] ^= x; 1536 } 1537 } 1538 1539 /* 1540 * return 0 if f == 0 1541 * return 1 if f != 0 1542 * 1543 * Preconditions: 1544 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1545 */ 1546 static int fe_isnonzero(const fe f) 1547 { 1548 uint8_t s[32]; 1549 static const uint8_t zero[32] = {0}; 1550 1551 fe_tobytes(s, f); 1552 1553 return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0; 1554 } 1555 1556 /* 1557 * return 1 if f is in {1,3,5,...,q-2} 1558 * return 0 if f is in {0,2,4,...,q-1} 1559 * 1560 * Preconditions: 1561 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1562 */ 1563 static int fe_isnegative(const fe f) 1564 { 1565 uint8_t s[32]; 1566 1567 fe_tobytes(s, f); 1568 return s[0] & 1; 1569 } 1570 1571 /* 1572 * h = 2 * f * f 1573 * 1574 * Can overlap h with f. 1575 * 1576 * Preconditions: 1577 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1578 * 1579 * Postconditions: 1580 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1581 * 1582 * See fe_mul.c for discussion of implementation strategy. 1583 */ 1584 static void fe_sq2(fe h, const fe f) 1585 { 1586 int32_t f0 = f[0]; 1587 int32_t f1 = f[1]; 1588 int32_t f2 = f[2]; 1589 int32_t f3 = f[3]; 1590 int32_t f4 = f[4]; 1591 int32_t f5 = f[5]; 1592 int32_t f6 = f[6]; 1593 int32_t f7 = f[7]; 1594 int32_t f8 = f[8]; 1595 int32_t f9 = f[9]; 1596 int32_t f0_2 = 2 * f0; 1597 int32_t f1_2 = 2 * f1; 1598 int32_t f2_2 = 2 * f2; 1599 int32_t f3_2 = 2 * f3; 1600 int32_t f4_2 = 2 * f4; 1601 int32_t f5_2 = 2 * f5; 1602 int32_t f6_2 = 2 * f6; 1603 int32_t f7_2 = 2 * f7; 1604 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */ 1605 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */ 1606 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */ 1607 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */ 1608 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */ 1609 int64_t f0f0 = f0 * (int64_t) f0; 1610 int64_t f0f1_2 = f0_2 * (int64_t) f1; 1611 int64_t f0f2_2 = f0_2 * (int64_t) f2; 1612 int64_t f0f3_2 = f0_2 * (int64_t) f3; 1613 int64_t f0f4_2 = f0_2 * (int64_t) f4; 1614 int64_t f0f5_2 = f0_2 * (int64_t) f5; 1615 int64_t f0f6_2 = f0_2 * (int64_t) f6; 1616 int64_t f0f7_2 = f0_2 * (int64_t) f7; 1617 int64_t f0f8_2 = f0_2 * (int64_t) f8; 1618 int64_t f0f9_2 = f0_2 * (int64_t) f9; 1619 int64_t f1f1_2 = f1_2 * (int64_t) f1; 1620 int64_t f1f2_2 = f1_2 * (int64_t) f2; 1621 int64_t f1f3_4 = f1_2 * (int64_t) f3_2; 1622 int64_t f1f4_2 = f1_2 * (int64_t) f4; 1623 int64_t f1f5_4 = f1_2 * (int64_t) f5_2; 1624 int64_t f1f6_2 = f1_2 * (int64_t) f6; 1625 int64_t f1f7_4 = f1_2 * (int64_t) f7_2; 1626 int64_t f1f8_2 = f1_2 * (int64_t) f8; 1627 int64_t f1f9_76 = f1_2 * (int64_t) f9_38; 1628 int64_t f2f2 = f2 * (int64_t) f2; 1629 int64_t f2f3_2 = f2_2 * (int64_t) f3; 1630 int64_t f2f4_2 = f2_2 * (int64_t) f4; 1631 int64_t f2f5_2 = f2_2 * (int64_t) f5; 1632 int64_t f2f6_2 = f2_2 * (int64_t) f6; 1633 int64_t f2f7_2 = f2_2 * (int64_t) f7; 1634 int64_t f2f8_38 = f2_2 * (int64_t) f8_19; 1635 int64_t f2f9_38 = f2 * (int64_t) f9_38; 1636 int64_t f3f3_2 = f3_2 * (int64_t) f3; 1637 int64_t f3f4_2 = f3_2 * (int64_t) f4; 1638 int64_t f3f5_4 = f3_2 * (int64_t) f5_2; 1639 int64_t f3f6_2 = f3_2 * (int64_t) f6; 1640 int64_t f3f7_76 = f3_2 * (int64_t) f7_38; 1641 int64_t f3f8_38 = f3_2 * (int64_t) f8_19; 1642 int64_t f3f9_76 = f3_2 * (int64_t) f9_38; 1643 int64_t f4f4 = f4 * (int64_t) f4; 1644 int64_t f4f5_2 = f4_2 * (int64_t) f5; 1645 int64_t f4f6_38 = f4_2 * (int64_t) f6_19; 1646 int64_t f4f7_38 = f4 * (int64_t) f7_38; 1647 int64_t f4f8_38 = f4_2 * (int64_t) f8_19; 1648 int64_t f4f9_38 = f4 * (int64_t) f9_38; 1649 int64_t f5f5_38 = f5 * (int64_t) f5_38; 1650 int64_t f5f6_38 = f5_2 * (int64_t) f6_19; 1651 int64_t f5f7_76 = f5_2 * (int64_t) f7_38; 1652 int64_t f5f8_38 = f5_2 * (int64_t) f8_19; 1653 int64_t f5f9_76 = f5_2 * (int64_t) f9_38; 1654 int64_t f6f6_19 = f6 * (int64_t) f6_19; 1655 int64_t f6f7_38 = f6 * (int64_t) f7_38; 1656 int64_t f6f8_38 = f6_2 * (int64_t) f8_19; 1657 int64_t f6f9_38 = f6 * (int64_t) f9_38; 1658 int64_t f7f7_38 = f7 * (int64_t) f7_38; 1659 int64_t f7f8_38 = f7_2 * (int64_t) f8_19; 1660 int64_t f7f9_76 = f7_2 * (int64_t) f9_38; 1661 int64_t f8f8_19 = f8 * (int64_t) f8_19; 1662 int64_t f8f9_38 = f8 * (int64_t) f9_38; 1663 int64_t f9f9_38 = f9 * (int64_t) f9_38; 1664 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38; 1665 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38; 1666 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19; 1667 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38; 1668 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38; 1669 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38; 1670 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19; 1671 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38; 1672 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38; 1673 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2; 1674 int64_t carry0; 1675 int64_t carry1; 1676 int64_t carry2; 1677 int64_t carry3; 1678 int64_t carry4; 1679 int64_t carry5; 1680 int64_t carry6; 1681 int64_t carry7; 1682 int64_t carry8; 1683 int64_t carry9; 1684 1685 h0 += h0; 1686 h1 += h1; 1687 h2 += h2; 1688 h3 += h3; 1689 h4 += h4; 1690 h5 += h5; 1691 h6 += h6; 1692 h7 += h7; 1693 h8 += h8; 1694 h9 += h9; 1695 1696 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1697 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1698 1699 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 1700 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 1701 1702 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 1703 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 1704 1705 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 1706 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 1707 1708 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 1709 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 1710 1711 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 1712 1713 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 1714 1715 h[0] = (int32_t)h0; 1716 h[1] = (int32_t)h1; 1717 h[2] = (int32_t)h2; 1718 h[3] = (int32_t)h3; 1719 h[4] = (int32_t)h4; 1720 h[5] = (int32_t)h5; 1721 h[6] = (int32_t)h6; 1722 h[7] = (int32_t)h7; 1723 h[8] = (int32_t)h8; 1724 h[9] = (int32_t)h9; 1725 } 1726 1727 static void fe_pow22523(fe out, const fe z) 1728 { 1729 fe t0; 1730 fe t1; 1731 fe t2; 1732 int i; 1733 1734 fe_sq(t0, z); 1735 fe_sq(t1, t0); 1736 for (i = 1; i < 2; ++i) { 1737 fe_sq(t1, t1); 1738 } 1739 fe_mul(t1, z, t1); 1740 fe_mul(t0, t0, t1); 1741 fe_sq(t0, t0); 1742 fe_mul(t0, t1, t0); 1743 fe_sq(t1, t0); 1744 for (i = 1; i < 5; ++i) { 1745 fe_sq(t1, t1); 1746 } 1747 fe_mul(t0, t1, t0); 1748 fe_sq(t1, t0); 1749 for (i = 1; i < 10; ++i) { 1750 fe_sq(t1, t1); 1751 } 1752 fe_mul(t1, t1, t0); 1753 fe_sq(t2, t1); 1754 for (i = 1; i < 20; ++i) { 1755 fe_sq(t2, t2); 1756 } 1757 fe_mul(t1, t2, t1); 1758 fe_sq(t1, t1); 1759 for (i = 1; i < 10; ++i) { 1760 fe_sq(t1, t1); 1761 } 1762 fe_mul(t0, t1, t0); 1763 fe_sq(t1, t0); 1764 for (i = 1; i < 50; ++i) { 1765 fe_sq(t1, t1); 1766 } 1767 fe_mul(t1, t1, t0); 1768 fe_sq(t2, t1); 1769 for (i = 1; i < 100; ++i) { 1770 fe_sq(t2, t2); 1771 } 1772 fe_mul(t1, t2, t1); 1773 fe_sq(t1, t1); 1774 for (i = 1; i < 50; ++i) { 1775 fe_sq(t1, t1); 1776 } 1777 fe_mul(t0, t1, t0); 1778 fe_sq(t0, t0); 1779 for (i = 1; i < 2; ++i) { 1780 fe_sq(t0, t0); 1781 } 1782 fe_mul(out, t0, z); 1783 } 1784 1785 /* 1786 * ge means group element. 1787 * 1788 * Here the group is the set of pairs (x,y) of field elements (see fe.h) 1789 * satisfying -x^2 + y^2 = 1 + d x^2y^2 1790 * where d = -121665/121666. 1791 * 1792 * Representations: 1793 * ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z 1794 * ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT 1795 * ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T 1796 * ge_precomp (Duif): (y+x,y-x,2dxy) 1797 */ 1798 typedef struct { 1799 fe X; 1800 fe Y; 1801 fe Z; 1802 } ge_p2; 1803 1804 typedef struct { 1805 fe X; 1806 fe Y; 1807 fe Z; 1808 fe T; 1809 } ge_p3; 1810 1811 typedef struct { 1812 fe X; 1813 fe Y; 1814 fe Z; 1815 fe T; 1816 } ge_p1p1; 1817 1818 typedef struct { 1819 fe yplusx; 1820 fe yminusx; 1821 fe xy2d; 1822 } ge_precomp; 1823 1824 typedef struct { 1825 fe YplusX; 1826 fe YminusX; 1827 fe Z; 1828 fe T2d; 1829 } ge_cached; 1830 1831 static void ge_tobytes(uint8_t *s, const ge_p2 *h) 1832 { 1833 fe recip; 1834 fe x; 1835 fe y; 1836 1837 fe_invert(recip, h->Z); 1838 fe_mul(x, h->X, recip); 1839 fe_mul(y, h->Y, recip); 1840 fe_tobytes(s, y); 1841 s[31] ^= fe_isnegative(x) << 7; 1842 } 1843 1844 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h) 1845 { 1846 fe recip; 1847 fe x; 1848 fe y; 1849 1850 fe_invert(recip, h->Z); 1851 fe_mul(x, h->X, recip); 1852 fe_mul(y, h->Y, recip); 1853 fe_tobytes(s, y); 1854 s[31] ^= fe_isnegative(x) << 7; 1855 } 1856 1857 static const fe d = { 1858 -10913610, 13857413, -15372611, 6949391, 114729, 1859 -8787816, -6275908, -3247719, -18696448, -12055116 1860 }; 1861 1862 static const fe sqrtm1 = { 1863 -32595792, -7943725, 9377950, 3500415, 12389472, 1864 -272473, -25146209, -2005654, 326686, 11406482 1865 }; 1866 1867 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s) 1868 { 1869 fe u; 1870 fe v; 1871 fe v3; 1872 fe vxx; 1873 fe check; 1874 1875 fe_frombytes(h->Y, s); 1876 fe_1(h->Z); 1877 fe_sq(u, h->Y); 1878 fe_mul(v, u, d); 1879 fe_sub(u, u, h->Z); /* u = y^2-1 */ 1880 fe_add(v, v, h->Z); /* v = dy^2+1 */ 1881 1882 fe_sq(v3, v); 1883 fe_mul(v3, v3, v); /* v3 = v^3 */ 1884 fe_sq(h->X, v3); 1885 fe_mul(h->X, h->X, v); 1886 fe_mul(h->X, h->X, u); /* x = uv^7 */ 1887 1888 fe_pow22523(h->X, h->X); /* x = (uv^7)^((q-5)/8) */ 1889 fe_mul(h->X, h->X, v3); 1890 fe_mul(h->X, h->X, u); /* x = uv^3(uv^7)^((q-5)/8) */ 1891 1892 fe_sq(vxx, h->X); 1893 fe_mul(vxx, vxx, v); 1894 fe_sub(check, vxx, u); /* vx^2-u */ 1895 if (fe_isnonzero(check)) { 1896 fe_add(check, vxx, u); /* vx^2+u */ 1897 if (fe_isnonzero(check)) { 1898 return -1; 1899 } 1900 fe_mul(h->X, h->X, sqrtm1); 1901 } 1902 1903 if (fe_isnegative(h->X) != (s[31] >> 7)) { 1904 fe_neg(h->X, h->X); 1905 } 1906 1907 fe_mul(h->T, h->X, h->Y); 1908 return 0; 1909 } 1910 1911 static void ge_p2_0(ge_p2 *h) 1912 { 1913 fe_0(h->X); 1914 fe_1(h->Y); 1915 fe_1(h->Z); 1916 } 1917 1918 static void ge_p3_0(ge_p3 *h) 1919 { 1920 fe_0(h->X); 1921 fe_1(h->Y); 1922 fe_1(h->Z); 1923 fe_0(h->T); 1924 } 1925 1926 static void ge_precomp_0(ge_precomp *h) 1927 { 1928 fe_1(h->yplusx); 1929 fe_1(h->yminusx); 1930 fe_0(h->xy2d); 1931 } 1932 1933 /* r = p */ 1934 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) 1935 { 1936 fe_copy(r->X, p->X); 1937 fe_copy(r->Y, p->Y); 1938 fe_copy(r->Z, p->Z); 1939 } 1940 1941 static const fe d2 = { 1942 -21827239, -5839606, -30745221, 13898782, 229458, 1943 15978800, -12551817, -6495438, 29715968, 9444199 1944 }; 1945 1946 /* r = p */ 1947 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p) 1948 { 1949 fe_add(r->YplusX, p->Y, p->X); 1950 fe_sub(r->YminusX, p->Y, p->X); 1951 fe_copy(r->Z, p->Z); 1952 fe_mul(r->T2d, p->T, d2); 1953 } 1954 1955 /* r = p */ 1956 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) 1957 { 1958 fe_mul(r->X, p->X, p->T); 1959 fe_mul(r->Y, p->Y, p->Z); 1960 fe_mul(r->Z, p->Z, p->T); 1961 } 1962 1963 /* r = p */ 1964 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p) 1965 { 1966 fe_mul(r->X, p->X, p->T); 1967 fe_mul(r->Y, p->Y, p->Z); 1968 fe_mul(r->Z, p->Z, p->T); 1969 fe_mul(r->T, p->X, p->Y); 1970 } 1971 1972 /* r = 2 * p */ 1973 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p) 1974 { 1975 fe t0; 1976 1977 fe_sq(r->X, p->X); 1978 fe_sq(r->Z, p->Y); 1979 fe_sq2(r->T, p->Z); 1980 fe_add(r->Y, p->X, p->Y); 1981 fe_sq(t0, r->Y); 1982 fe_add(r->Y, r->Z, r->X); 1983 fe_sub(r->Z, r->Z, r->X); 1984 fe_sub(r->X, t0, r->Y); 1985 fe_sub(r->T, r->T, r->Z); 1986 } 1987 1988 /* r = 2 * p */ 1989 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p) 1990 { 1991 ge_p2 q; 1992 ge_p3_to_p2(&q, p); 1993 ge_p2_dbl(r, &q); 1994 } 1995 1996 /* r = p + q */ 1997 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) 1998 { 1999 fe t0; 2000 2001 fe_add(r->X, p->Y, p->X); 2002 fe_sub(r->Y, p->Y, p->X); 2003 fe_mul(r->Z, r->X, q->yplusx); 2004 fe_mul(r->Y, r->Y, q->yminusx); 2005 fe_mul(r->T, q->xy2d, p->T); 2006 fe_add(t0, p->Z, p->Z); 2007 fe_sub(r->X, r->Z, r->Y); 2008 fe_add(r->Y, r->Z, r->Y); 2009 fe_add(r->Z, t0, r->T); 2010 fe_sub(r->T, t0, r->T); 2011 } 2012 2013 /* r = p - q */ 2014 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) 2015 { 2016 fe t0; 2017 2018 fe_add(r->X, p->Y, p->X); 2019 fe_sub(r->Y, p->Y, p->X); 2020 fe_mul(r->Z, r->X, q->yminusx); 2021 fe_mul(r->Y, r->Y, q->yplusx); 2022 fe_mul(r->T, q->xy2d, p->T); 2023 fe_add(t0, p->Z, p->Z); 2024 fe_sub(r->X, r->Z, r->Y); 2025 fe_add(r->Y, r->Z, r->Y); 2026 fe_sub(r->Z, t0, r->T); 2027 fe_add(r->T, t0, r->T); 2028 } 2029 2030 /* r = p + q */ 2031 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) 2032 { 2033 fe t0; 2034 2035 fe_add(r->X, p->Y, p->X); 2036 fe_sub(r->Y, p->Y, p->X); 2037 fe_mul(r->Z, r->X, q->YplusX); 2038 fe_mul(r->Y, r->Y, q->YminusX); 2039 fe_mul(r->T, q->T2d, p->T); 2040 fe_mul(r->X, p->Z, q->Z); 2041 fe_add(t0, r->X, r->X); 2042 fe_sub(r->X, r->Z, r->Y); 2043 fe_add(r->Y, r->Z, r->Y); 2044 fe_add(r->Z, t0, r->T); 2045 fe_sub(r->T, t0, r->T); 2046 } 2047 2048 /* r = p - q */ 2049 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) 2050 { 2051 fe t0; 2052 2053 fe_add(r->X, p->Y, p->X); 2054 fe_sub(r->Y, p->Y, p->X); 2055 fe_mul(r->Z, r->X, q->YminusX); 2056 fe_mul(r->Y, r->Y, q->YplusX); 2057 fe_mul(r->T, q->T2d, p->T); 2058 fe_mul(r->X, p->Z, q->Z); 2059 fe_add(t0, r->X, r->X); 2060 fe_sub(r->X, r->Z, r->Y); 2061 fe_add(r->Y, r->Z, r->Y); 2062 fe_sub(r->Z, t0, r->T); 2063 fe_add(r->T, t0, r->T); 2064 } 2065 2066 static uint8_t equal(signed char b, signed char c) 2067 { 2068 uint8_t ub = b; 2069 uint8_t uc = c; 2070 uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */ 2071 uint32_t y = x; /* 0: yes; 1..255: no */ 2072 y -= 1; /* 4294967295: yes; 0..254: no */ 2073 y >>= 31; /* 1: yes; 0: no */ 2074 return y; 2075 } 2076 2077 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b) 2078 { 2079 fe_cmov(t->yplusx, u->yplusx, b); 2080 fe_cmov(t->yminusx, u->yminusx, b); 2081 fe_cmov(t->xy2d, u->xy2d, b); 2082 } 2083 2084 /* k25519Precomp[i][j] = (j+1)*256^i*B */ 2085 static const ge_precomp k25519Precomp[32][8] = { 2086 { 2087 { 2088 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 2089 27544626, -11754271, -6079156, 2047605}, 2090 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 2091 5043384, 19500929, -15469378}, 2092 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 2093 29287919, 11864899, -24514362, -4438546}, 2094 }, 2095 { 2096 {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, 2097 -11717903, -3814571, -358445, -10211303}, 2098 {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, 2099 -15616551, 11189268, -26829678, -5319081}, 2100 {26966642, 11152617, 32442495, 15396054, 14353839, -12752335, 2101 -3128826, -9541118, -15472047, -4166697}, 2102 }, 2103 { 2104 {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 2105 27787600, -14772189, 28944400, -1550024}, 2106 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 2107 16354577, -11775962, 7689662, 11199574}, 2108 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 2109 7512774, 10017326, -17749093, -9920357}, 2110 }, 2111 { 2112 {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, 2113 -28926210, 15006023, 3284568, -6276540}, 2114 {23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 2115 7464579, 9656445, 13059162, 10374397}, 2116 {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, 2117 -3839045, -641708, -101325}, 2118 }, 2119 { 2120 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 2121 32867885, 14515107, -15438304, 10819380}, 2122 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 2123 12483688, -12668491, 5581306}, 2124 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 2125 28542350, 13850243, -23678021, -15815942}, 2126 }, 2127 { 2128 {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, 2129 -19188627, -15224819, -9818940, -12085777}, 2130 {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, 2131 -15689887, 1762328, 14866737}, 2132 {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, 2133 -28236412, 3959421, 27914454, 4383652}, 2134 }, 2135 { 2136 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 2137 5230134, -23952439, -15175766}, 2138 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 2139 20654025, 16520125, 30598449, 7715701}, 2140 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, 2141 -31400660, 1370708, 29794553, -1409300}, 2142 }, 2143 { 2144 {14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 2145 30290735, 10876454, -33154098, 2381726}, 2146 {-7195431, -2655363, -14730155, 462251, -27724326, 3941372, 2147 -6236617, 3696005, -32300832, 15351955}, 2148 {27431194, 8222322, 16448760, -3907995, -18707002, 11938355, 2149 -32961401, -2970515, 29551813, 10109425}, 2150 }, 2151 }, 2152 { 2153 { 2154 {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, 2155 -2378284, -1627556, 10092783, -4764171}, 2156 {27939166, 14210322, 4677035, 16277044, -22964462, -12398139, 2157 -32508754, 12005538, -17810127, 12803510}, 2158 {17228999, -15661624, -1233527, 300140, -1224870, -11714777, 2159 30364213, -9038194, 18016357, 4397660}, 2160 }, 2161 { 2162 {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, 2163 -26619106, 14544525, -17477504, 982639}, 2164 {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, 2165 -4120128, -21047696, 9934963}, 2166 {5793303, 16271923, -24131614, -10116404, 29188560, 1206517, 2167 -14747930, 4559895, -30123922, -10897950}, 2168 }, 2169 { 2170 {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 2171 24191034, 4541697, -13338309, 5500568}, 2172 {12650548, -1497113, 9052871, 11355358, -17680037, -8400164, 2173 -17430592, 12264343, 10874051, 13524335}, 2174 {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 2175 5080568, -22528059, 5376628}, 2176 }, 2177 { 2178 {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, 2179 -22321305, -9447443, 4535768, 1569007}, 2180 {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, 2181 -30494562, 3044290, 31848280, 12543772}, 2182 {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, 2183 -27377195, -2062731, 7718482, 14474653}, 2184 }, 2185 { 2186 {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, 2187 -7236665, 24316168, -5253567}, 2188 {13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 2189 33040651, -13424532, -20729456, 8321686}, 2190 {21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 2191 23845965, -11874838, -9984458, 608372}, 2192 }, 2193 { 2194 {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 2195 1123968, -6780577, 27229399, 23887}, 2196 {-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 2197 12797024, -6440308, -1633405, 16678954}, 2198 {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, 2199 -1508144, -4795045, -17169265, 4904953}, 2200 }, 2201 { 2202 {24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 2203 5169211, 16191880, 2128236, -4326833}, 2204 {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247, 2205 -29806336, 916033, -6882542, -2986532}, 2206 {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 2207 285431, 2763829, 15736322, 4143876}, 2208 }, 2209 { 2210 {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, 2211 -14594663, 23527084, -16458268}, 2212 {33431127, -11130478, -17838966, -15626900, 8909499, 8376530, 2213 -32625340, 4087881, -15188911, -14416214}, 2214 {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055, 2215 4357868, -4774191, -16323038}, 2216 }, 2217 }, 2218 { 2219 { 2220 {6721966, 13833823, -23523388, -1551314, 26354293, -11863321, 2221 23365147, -3949732, 7390890, 2759800}, 2222 {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353, 2223 -4264057, 1244380, -12919645}, 2224 {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 2225 9208236, 15886429, 16489664}, 2226 }, 2227 { 2228 {1996075, 10375649, 14346367, 13311202, -6874135, -16438411, 2229 -13693198, 398369, -30606455, -712933}, 2230 {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363, 2231 13348553, 12076947, -30836462, 5113182}, 2232 {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344, 2233 -30341101, -7336386, 13847711, 5387222}, 2234 }, 2235 { 2236 {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611, 2237 8763061, 3617786, -19600662, 10370991}, 2238 {20246567, -14369378, 22358229, -543712, 18507283, -10413996, 2239 14554437, -8746092, 32232924, 16763880}, 2240 {9648505, 10094563, 26416693, 14745928, -30374318, -6472621, 2241 11094161, 15689506, 3140038, -16510092}, 2242 }, 2243 { 2244 {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685, 2245 -27224800, 9448613, -28774454, 366295}, 2246 {19153450, 11523972, -11096490, -6503142, -24647631, 5420647, 2247 28344573, 8041113, 719605, 11671788}, 2248 {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916, 2249 -15266516, 27000813, -10195553}, 2250 }, 2251 { 2252 {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065, 2253 5336097, 6750977, -14521026}, 2254 {11836410, -3979488, 26297894, 16080799, 23455045, 15735944, 2255 1695823, -8819122, 8169720, 16220347}, 2256 {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763, 2257 -11144307, -2627664, -5990708, -14166033}, 2258 }, 2259 { 2260 {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004, 2261 27884329, 2847284, 2655861, 1738395}, 2262 {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821, 2263 21651608, -3239336, -19087449, -11005278}, 2264 {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092, 2265 5821408, 10478196, 8544890}, 2266 }, 2267 { 2268 {32173121, -16129311, 24896207, 3921497, 22579056, -3410854, 2269 19270449, 12217473, 17789017, -3395995}, 2270 {-30552961, -2228401, -15578829, -10147201, 13243889, 517024, 2271 15479401, -3853233, 30460520, 1052596}, 2272 {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687, 2273 27491595, -4612359, 3179268, -9478891}, 2274 }, 2275 { 2276 {31947069, -14366651, -4640583, -15339921, -15125977, -6039709, 2277 -14756777, -16411740, 19072640, -9511060}, 2278 {11685058, 11822410, 3158003, -13952594, 33402194, -4165066, 2279 5977896, -5215017, 473099, 5040608}, 2280 {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760, 2281 28326862, 1721092, -19558642, -3131606}, 2282 }, 2283 }, 2284 { 2285 { 2286 {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786, 2287 8076149, -27868496, 11538389}, 2288 {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211, 2289 8754525, 7446702, -5676054, 5797016}, 2290 {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199, 2291 2014099, -9050574, -2369172, -5877341}, 2292 }, 2293 { 2294 {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559, 2295 1192730, -3714199, 15123619, 10811505}, 2296 {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363, 2297 15776356, -28886779, -11974553}, 2298 {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569, 2299 -20654173, -16484855, 4714547, -9600655}, 2300 }, 2301 { 2302 {15200332, 8368572, 19679101, 15970074, -31872674, 1959451, 2303 24611599, -4543832, -11745876, 12340220}, 2304 {12876937, -10480056, 33134381, 6590940, -6307776, 14872440, 2305 9613953, 8241152, 15370987, 9608631}, 2306 {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868, 2307 15866074, -28210621, -8814099}, 2308 }, 2309 { 2310 {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233, 2311 858697, 20571223, 8420556}, 2312 {14620715, 13067227, -15447274, 8264467, 14106269, 15080814, 2313 33531827, 12516406, -21574435, -12476749}, 2314 {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519, 2315 7256740, 8791136, 15069930}, 2316 }, 2317 { 2318 {1276410, -9371918, 22949635, -16322807, -23493039, -5702186, 2319 14711875, 4874229, -30663140, -2331391}, 2320 {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175, 2321 -7912378, -33069337, 9234253}, 2322 {20590503, -9018988, 31529744, -7352666, -2706834, 10650548, 2323 31559055, -11609587, 18979186, 13396066}, 2324 }, 2325 { 2326 {24474287, 4968103, 22267082, 4407354, 24063882, -8325180, 2327 -18816887, 13594782, 33514650, 7021958}, 2328 {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421, 2329 -25948728, -3916677, -21480480, 12868082}, 2330 {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208, 2331 -21446107, 2244500, -12455797, -8089383}, 2332 }, 2333 { 2334 {-30595528, 13793479, -5852820, 319136, -25723172, -6263899, 2335 33086546, 8957937, -15233648, 5540521}, 2336 {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908, 2337 -23710744, -1568984, -16128528, -14962807}, 2338 {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819, 2339 892185, -11513277, -15205948}, 2340 }, 2341 { 2342 {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819, 2343 4763127, -19179614, 5867134}, 2344 {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500, 2345 27846559, 5931263, -29749703, -16108455}, 2346 {27461885, -2977536, 22380810, 1815854, -23033753, -3031938, 2347 7283490, -15148073, -19526700, 7734629}, 2348 }, 2349 }, 2350 { 2351 { 2352 {-8010264, -9590817, -11120403, 6196038, 29344158, -13430885, 2353 7585295, -3176626, 18549497, 15302069}, 2354 {-32658337, -6171222, -7672793, -11051681, 6258878, 13504381, 2355 10458790, -6418461, -8872242, 8424746}, 2356 {24687205, 8613276, -30667046, -3233545, 1863892, -1830544, 2357 19206234, 7134917, -11284482, -828919}, 2358 }, 2359 { 2360 {11334899, -9218022, 8025293, 12707519, 17523892, -10476071, 2361 10243738, -14685461, -5066034, 16498837}, 2362 {8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925, 2363 -14124238, 6536641, 10543906}, 2364 {-28946384, 15479763, -17466835, 568876, -1497683, 11223454, 2365 -2669190, -16625574, -27235709, 8876771}, 2366 }, 2367 { 2368 {-25742899, -12566864, -15649966, -846607, -33026686, -796288, 2369 -33481822, 15824474, -604426, -9039817}, 2370 {10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697, 2371 -4890037, 1657394, 3084098}, 2372 {10477963, -7470260, 12119566, -13250805, 29016247, -5365589, 2373 31280319, 14396151, 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2333242}, 4074 }, 4075 { 4076 {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134, 4077 -6679750, -12670638, 24350578, -13450001}, 4078 {-4116307, -11271533, -23886186, 4843615, -30088339, 690623, 4079 -31536088, -10406836, 8317860, 12352766}, 4080 {18200138, -14475911, -33087759, -2696619, -23702521, -9102511, 4081 -23552096, -2287550, 20712163, 6719373}, 4082 }, 4083 { 4084 {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, 4085 -3763210, 26224235, -3297458}, 4086 {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 4087 21728352, 9493610, 18620611, -16428628}, 4088 {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965, 4089 -5269471, -9725556, -30701573, -16479657}, 4090 }, 4091 { 4092 {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 4093 12248509, -5240639, 13735342, 1934062}, 4094 {25089769, 6742589, 17081145, -13406266, 21909293, -16067981, 4095 -15136294, -3765346, -21277997, 5473616}, 4096 {31883677, -7961101, 1083432, -11572403, 22828471, 13290673, 4097 -7125085, 12469656, 29111212, -5451014}, 4098 }, 4099 { 4100 {24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 4101 24367466, 6388839, -10295587, 452383}, 4102 {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, 4103 -24236251, -5915248, 15766062, 8407814}, 4104 {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, 4105 -8917023, -4388953, -8067909, 2276718}, 4106 }, 4107 { 4108 {30157918, 12924066, -17712050, 9245753, 19895028, 3368142, 4109 -23827587, 5096219, 22740376, -7303417}, 4110 {2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 4111 24051124, 13742383, -15637599, 13295222}, 4112 {33338237, -8505733, 12532113, 7977527, 9106186, -1715251, 4113 -17720195, -4612972, -4451357, -14669444}, 4114 }, 4115 { 4116 {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, 4117 -2469266, -4141880, 7770569, 9620597}, 4118 {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, 4119 -1694323, -33502340, -14767970}, 4120 {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 4121 1220118, 30494170, -11440799}, 4122 }, 4123 { 4124 {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, 4125 -26739026, 926050, -1684339, -13333647}, 4126 {13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 4127 9021034, 9078865, 3353509, 4033511}, 4128 {-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 4129 23161163, 8839127, 27485041, 7356032}, 4130 }, 4131 }, 4132 { 4133 { 4134 {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 4135 2625015, 28431036, -16771834}, 4136 {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, 4137 -22545972, 14150565, 15970762, 4099461}, 4138 {29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 4139 13617293, -9937143, 11465739, 8317062}, 4140 }, 4141 { 4142 {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 4143 14898637, 3848455, 20969334, -5157516}, 4144 {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, 4145 -21610826, -3649888, 11177095, 14989547}, 4146 {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 4147 13515641, 2581286, -28487508, 9930240}, 4148 }, 4149 { 4150 {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 4151 18345767, -13403753, 16291481, -5314038}, 4152 {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 4153 6957617, 4368891, 9788741}, 4154 {16660756, 7281060, -10830758, 12911820, 20108584, -8101676, 4155 -21722536, -8613148, 16250552, -11111103}, 4156 }, 4157 { 4158 {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 4159 10604807, -30190403, 4782747}, 4160 {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, 4161 -9981571, 4383045, 22546403, 437323}, 4162 {31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 4163 27343084, 2786261, -30633590, -14097016}, 4164 }, 4165 { 4166 {-14467279, -683715, -33374107, 7448552, 19294360, 14334329, 4167 -19690631, 2355319, -19284671, -6114373}, 4168 {15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 4169 18952177, 15496498, -29380133, 11754228}, 4170 {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 4171 7141596, 11724556, 22761615, -10134141}, 4172 }, 4173 { 4174 {16918416, 11729663, -18083579, 3022987, -31015732, -13339659, 4175 -28741185, -12227393, 32851222, 11717399}, 4176 {11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 4177 31474879, 3483633, -1193175, -4030831}, 4178 {-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 4179 33142652, 6546660, -19985279, -3948376}, 4180 }, 4181 { 4182 {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, 4183 -8537131, -12833048, -30772034, -15486313}, 4184 {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, 4185 -31135347, -16049879, 10928917, 3011958}, 4186 {-6957757, -15594337, 31696059, 334240, 29576716, 14796075, 4187 -30831056, -12805180, 18008031, 10258577}, 4188 }, 4189 { 4190 {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, 4191 -1853465, 1367120, 25127874, 6671743}, 4192 {29701166, -14373934, -10878120, 9279288, -17568, 13127210, 4193 21382910, 11042292, 25838796, 4642684}, 4194 {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 4195 30468147, -13900640, 18423289, 4177476}, 4196 }, 4197 }, 4198 }; 4199 4200 static uint8_t negative(signed char b) 4201 { 4202 uint32_t x = b; 4203 4204 x >>= 31; /* 1: yes; 0: no */ 4205 return x; 4206 } 4207 4208 static void table_select(ge_precomp *t, int pos, signed char b) 4209 { 4210 ge_precomp minust; 4211 uint8_t bnegative = negative(b); 4212 uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1); 4213 4214 ge_precomp_0(t); 4215 cmov(t, &k25519Precomp[pos][0], equal(babs, 1)); 4216 cmov(t, &k25519Precomp[pos][1], equal(babs, 2)); 4217 cmov(t, &k25519Precomp[pos][2], equal(babs, 3)); 4218 cmov(t, &k25519Precomp[pos][3], equal(babs, 4)); 4219 cmov(t, &k25519Precomp[pos][4], equal(babs, 5)); 4220 cmov(t, &k25519Precomp[pos][5], equal(babs, 6)); 4221 cmov(t, &k25519Precomp[pos][6], equal(babs, 7)); 4222 cmov(t, &k25519Precomp[pos][7], equal(babs, 8)); 4223 fe_copy(minust.yplusx, t->yminusx); 4224 fe_copy(minust.yminusx, t->yplusx); 4225 fe_neg(minust.xy2d, t->xy2d); 4226 cmov(t, &minust, bnegative); 4227 } 4228 4229 /* 4230 * h = a * B 4231 * 4232 * where a = a[0]+256*a[1]+...+256^31 a[31] 4233 * B is the Ed25519 base point (x,4/5) with x positive. 4234 * 4235 * Preconditions: 4236 * a[31] <= 127 4237 */ 4238 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a) 4239 { 4240 signed char e[64]; 4241 signed char carry; 4242 ge_p1p1 r; 4243 ge_p2 s; 4244 ge_precomp t; 4245 int i; 4246 4247 for (i = 0; i < 32; ++i) { 4248 e[2 * i + 0] = (a[i] >> 0) & 15; 4249 e[2 * i + 1] = (a[i] >> 4) & 15; 4250 } 4251 /* each e[i] is between 0 and 15 */ 4252 /* e[63] is between 0 and 7 */ 4253 4254 carry = 0; 4255 for (i = 0; i < 63; ++i) { 4256 e[i] += carry; 4257 carry = e[i] + 8; 4258 carry >>= 4; 4259 e[i] -= carry << 4; 4260 } 4261 e[63] += carry; 4262 /* each e[i] is between -8 and 8 */ 4263 4264 ge_p3_0(h); 4265 for (i = 1; i < 64; i += 2) { 4266 table_select(&t, i / 2, e[i]); 4267 ge_madd(&r, h, &t); 4268 ge_p1p1_to_p3(h, &r); 4269 } 4270 4271 ge_p3_dbl(&r, h); 4272 ge_p1p1_to_p2(&s, &r); 4273 ge_p2_dbl(&r, &s); 4274 ge_p1p1_to_p2(&s, &r); 4275 ge_p2_dbl(&r, &s); 4276 ge_p1p1_to_p2(&s, &r); 4277 ge_p2_dbl(&r, &s); 4278 ge_p1p1_to_p3(h, &r); 4279 4280 for (i = 0; i < 64; i += 2) { 4281 table_select(&t, i / 2, e[i]); 4282 ge_madd(&r, h, &t); 4283 ge_p1p1_to_p3(h, &r); 4284 } 4285 4286 OPENSSL_cleanse(e, sizeof(e)); 4287 } 4288 4289 #if !defined(BASE_2_51_IMPLEMENTED) 4290 /* 4291 * Replace (f,g) with (g,f) if b == 1; 4292 * replace (f,g) with (f,g) if b == 0. 4293 * 4294 * Preconditions: b in {0,1}. 4295 */ 4296 static void fe_cswap(fe f, fe g, unsigned int b) 4297 { 4298 size_t i; 4299 4300 b = 0-b; 4301 for (i = 0; i < 10; i++) { 4302 int32_t x = f[i] ^ g[i]; 4303 x &= b; 4304 f[i] ^= x; 4305 g[i] ^= x; 4306 } 4307 } 4308 4309 /* 4310 * h = f * 121666 4311 * 4312 * Can overlap h with f. 4313 * 4314 * Preconditions: 4315 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 4316 * 4317 * Postconditions: 4318 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 4319 */ 4320 static void fe_mul121666(fe h, fe f) 4321 { 4322 int32_t f0 = f[0]; 4323 int32_t f1 = f[1]; 4324 int32_t f2 = f[2]; 4325 int32_t f3 = f[3]; 4326 int32_t f4 = f[4]; 4327 int32_t f5 = f[5]; 4328 int32_t f6 = f[6]; 4329 int32_t f7 = f[7]; 4330 int32_t f8 = f[8]; 4331 int32_t f9 = f[9]; 4332 int64_t h0 = f0 * (int64_t) 121666; 4333 int64_t h1 = f1 * (int64_t) 121666; 4334 int64_t h2 = f2 * (int64_t) 121666; 4335 int64_t h3 = f3 * (int64_t) 121666; 4336 int64_t h4 = f4 * (int64_t) 121666; 4337 int64_t h5 = f5 * (int64_t) 121666; 4338 int64_t h6 = f6 * (int64_t) 121666; 4339 int64_t h7 = f7 * (int64_t) 121666; 4340 int64_t h8 = f8 * (int64_t) 121666; 4341 int64_t h9 = f9 * (int64_t) 121666; 4342 int64_t carry0; 4343 int64_t carry1; 4344 int64_t carry2; 4345 int64_t carry3; 4346 int64_t carry4; 4347 int64_t carry5; 4348 int64_t carry6; 4349 int64_t carry7; 4350 int64_t carry8; 4351 int64_t carry9; 4352 4353 carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits; 4354 carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits; 4355 carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits; 4356 carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits; 4357 carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits; 4358 4359 carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits; 4360 carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits; 4361 carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits; 4362 carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits; 4363 carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits; 4364 4365 h[0] = (int32_t)h0; 4366 h[1] = (int32_t)h1; 4367 h[2] = (int32_t)h2; 4368 h[3] = (int32_t)h3; 4369 h[4] = (int32_t)h4; 4370 h[5] = (int32_t)h5; 4371 h[6] = (int32_t)h6; 4372 h[7] = (int32_t)h7; 4373 h[8] = (int32_t)h8; 4374 h[9] = (int32_t)h9; 4375 } 4376 4377 static void x25519_scalar_mult_generic(uint8_t out[32], 4378 const uint8_t scalar[32], 4379 const uint8_t point[32]) { 4380 fe x1, x2, z2, x3, z3, tmp0, tmp1; 4381 uint8_t e[32]; 4382 unsigned swap = 0; 4383 int pos; 4384 4385 memcpy(e, scalar, 32); 4386 e[0] &= 248; 4387 e[31] &= 127; 4388 e[31] |= 64; 4389 fe_frombytes(x1, point); 4390 fe_1(x2); 4391 fe_0(z2); 4392 fe_copy(x3, x1); 4393 fe_1(z3); 4394 4395 for (pos = 254; pos >= 0; --pos) { 4396 unsigned b = 1 & (e[pos / 8] >> (pos & 7)); 4397 swap ^= b; 4398 fe_cswap(x2, x3, swap); 4399 fe_cswap(z2, z3, swap); 4400 swap = b; 4401 fe_sub(tmp0, x3, z3); 4402 fe_sub(tmp1, x2, z2); 4403 fe_add(x2, x2, z2); 4404 fe_add(z2, x3, z3); 4405 fe_mul(z3, tmp0, x2); 4406 fe_mul(z2, z2, tmp1); 4407 fe_sq(tmp0, tmp1); 4408 fe_sq(tmp1, x2); 4409 fe_add(x3, z3, z2); 4410 fe_sub(z2, z3, z2); 4411 fe_mul(x2, tmp1, tmp0); 4412 fe_sub(tmp1, tmp1, tmp0); 4413 fe_sq(z2, z2); 4414 fe_mul121666(z3, tmp1); 4415 fe_sq(x3, x3); 4416 fe_add(tmp0, tmp0, z3); 4417 fe_mul(z3, x1, z2); 4418 fe_mul(z2, tmp1, tmp0); 4419 } 4420 4421 fe_invert(z2, z2); 4422 fe_mul(x2, x2, z2); 4423 fe_tobytes(out, x2); 4424 4425 OPENSSL_cleanse(e, sizeof(e)); 4426 } 4427 4428 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32], 4429 const uint8_t point[32]) { 4430 x25519_scalar_mult_generic(out, scalar, point); 4431 } 4432 #endif 4433 4434 static void slide(signed char *r, const uint8_t *a) 4435 { 4436 int i; 4437 int b; 4438 int k; 4439 4440 for (i = 0; i < 256; ++i) { 4441 r[i] = 1 & (a[i >> 3] >> (i & 7)); 4442 } 4443 4444 for (i = 0; i < 256; ++i) { 4445 if (r[i]) { 4446 for (b = 1; b <= 6 && i + b < 256; ++b) { 4447 if (r[i + b]) { 4448 if (r[i] + (r[i + b] << b) <= 15) { 4449 r[i] += r[i + b] << b; 4450 r[i + b] = 0; 4451 } else if (r[i] - (r[i + b] << b) >= -15) { 4452 r[i] -= r[i + b] << b; 4453 for (k = i + b; k < 256; ++k) { 4454 if (!r[k]) { 4455 r[k] = 1; 4456 break; 4457 } 4458 r[k] = 0; 4459 } 4460 } else { 4461 break; 4462 } 4463 } 4464 } 4465 } 4466 } 4467 } 4468 4469 static const ge_precomp Bi[8] = { 4470 { 4471 {25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, 4472 -11754271, -6079156, 2047605}, 4473 {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 4474 5043384, 19500929, -15469378}, 4475 {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 4476 11864899, -24514362, -4438546}, 4477 }, 4478 { 4479 {15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, 4480 -14772189, 28944400, -1550024}, 4481 {16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, 4482 -11775962, 7689662, 11199574}, 4483 {30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 4484 10017326, -17749093, -9920357}, 4485 }, 4486 { 4487 {10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 4488 14515107, -15438304, 10819380}, 4489 {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 4490 12483688, -12668491, 5581306}, 4491 {19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 4492 13850243, -23678021, -15815942}, 4493 }, 4494 { 4495 {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 4496 5230134, -23952439, -15175766}, 4497 {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 4498 16520125, 30598449, 7715701}, 4499 {28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 4500 1370708, 29794553, -1409300}, 4501 }, 4502 { 4503 {-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, 4504 -1361450, -13062696, 13821877}, 4505 {-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, 4506 -7212327, 18853322, -14220951}, 4507 {4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, 4508 -10431137, 2207753, -3209784}, 4509 }, 4510 { 4511 {-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, 4512 -663000, -31111463, -16132436}, 4513 {25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 4514 15725684, 171356, 6466918}, 4515 {23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, 4516 -14088058, -30714912, 16193877}, 4517 }, 4518 { 4519 {-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4520 4729455, -18074513, 9256800}, 4521 {-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 4522 9761698, -19827198, 630305}, 4523 {-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, 4524 -15960994, -2449256, -14291300}, 4525 }, 4526 { 4527 {-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 4528 15033784, 25105118, -7894876}, 4529 {-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 4530 1573892, -2625887, 2198790, -15804619}, 4531 {-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, 4532 -16236442, -32461234, -12290683}, 4533 }, 4534 }; 4535 4536 /* 4537 * r = a * A + b * B 4538 * 4539 * where a = a[0]+256*a[1]+...+256^31 a[31]. 4540 * and b = b[0]+256*b[1]+...+256^31 b[31]. 4541 * B is the Ed25519 base point (x,4/5) with x positive. 4542 */ 4543 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a, 4544 const ge_p3 *A, const uint8_t *b) 4545 { 4546 signed char aslide[256]; 4547 signed char bslide[256]; 4548 ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */ 4549 ge_p1p1 t; 4550 ge_p3 u; 4551 ge_p3 A2; 4552 int i; 4553 4554 slide(aslide, a); 4555 slide(bslide, b); 4556 4557 ge_p3_to_cached(&Ai[0], A); 4558 ge_p3_dbl(&t, A); 4559 ge_p1p1_to_p3(&A2, &t); 4560 ge_add(&t, &A2, &Ai[0]); 4561 ge_p1p1_to_p3(&u, &t); 4562 ge_p3_to_cached(&Ai[1], &u); 4563 ge_add(&t, &A2, &Ai[1]); 4564 ge_p1p1_to_p3(&u, &t); 4565 ge_p3_to_cached(&Ai[2], &u); 4566 ge_add(&t, &A2, &Ai[2]); 4567 ge_p1p1_to_p3(&u, &t); 4568 ge_p3_to_cached(&Ai[3], &u); 4569 ge_add(&t, &A2, &Ai[3]); 4570 ge_p1p1_to_p3(&u, &t); 4571 ge_p3_to_cached(&Ai[4], &u); 4572 ge_add(&t, &A2, &Ai[4]); 4573 ge_p1p1_to_p3(&u, &t); 4574 ge_p3_to_cached(&Ai[5], &u); 4575 ge_add(&t, &A2, &Ai[5]); 4576 ge_p1p1_to_p3(&u, &t); 4577 ge_p3_to_cached(&Ai[6], &u); 4578 ge_add(&t, &A2, &Ai[6]); 4579 ge_p1p1_to_p3(&u, &t); 4580 ge_p3_to_cached(&Ai[7], &u); 4581 4582 ge_p2_0(r); 4583 4584 for (i = 255; i >= 0; --i) { 4585 if (aslide[i] || bslide[i]) { 4586 break; 4587 } 4588 } 4589 4590 for (; i >= 0; --i) { 4591 ge_p2_dbl(&t, r); 4592 4593 if (aslide[i] > 0) { 4594 ge_p1p1_to_p3(&u, &t); 4595 ge_add(&t, &u, &Ai[aslide[i] / 2]); 4596 } else if (aslide[i] < 0) { 4597 ge_p1p1_to_p3(&u, &t); 4598 ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]); 4599 } 4600 4601 if (bslide[i] > 0) { 4602 ge_p1p1_to_p3(&u, &t); 4603 ge_madd(&t, &u, &Bi[bslide[i] / 2]); 4604 } else if (bslide[i] < 0) { 4605 ge_p1p1_to_p3(&u, &t); 4606 ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]); 4607 } 4608 4609 ge_p1p1_to_p2(r, &t); 4610 } 4611 } 4612 4613 /* 4614 * The set of scalars is \Z/l 4615 * where l = 2^252 + 27742317777372353535851937790883648493. 4616 * 4617 * Input: 4618 * s[0]+256*s[1]+...+256^63*s[63] = s 4619 * 4620 * Output: 4621 * s[0]+256*s[1]+...+256^31*s[31] = s mod l 4622 * where l = 2^252 + 27742317777372353535851937790883648493. 4623 * Overwrites s in place. 4624 */ 4625 static void x25519_sc_reduce(uint8_t *s) 4626 { 4627 int64_t s0 = kBottom21Bits & load_3(s); 4628 int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5); 4629 int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2); 4630 int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7); 4631 int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4); 4632 int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1); 4633 int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6); 4634 int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3); 4635 int64_t s8 = kBottom21Bits & load_3(s + 21); 4636 int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5); 4637 int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2); 4638 int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7); 4639 int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4); 4640 int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1); 4641 int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6); 4642 int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3); 4643 int64_t s16 = kBottom21Bits & load_3(s + 42); 4644 int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5); 4645 int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2); 4646 int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7); 4647 int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4); 4648 int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1); 4649 int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6); 4650 int64_t s23 = (load_4(s + 60) >> 3); 4651 int64_t carry0; 4652 int64_t carry1; 4653 int64_t carry2; 4654 int64_t carry3; 4655 int64_t carry4; 4656 int64_t carry5; 4657 int64_t carry6; 4658 int64_t carry7; 4659 int64_t carry8; 4660 int64_t carry9; 4661 int64_t carry10; 4662 int64_t carry11; 4663 int64_t carry12; 4664 int64_t carry13; 4665 int64_t carry14; 4666 int64_t carry15; 4667 int64_t carry16; 4668 4669 s11 += s23 * 666643; 4670 s12 += s23 * 470296; 4671 s13 += s23 * 654183; 4672 s14 -= s23 * 997805; 4673 s15 += s23 * 136657; 4674 s16 -= s23 * 683901; 4675 s23 = 0; 4676 4677 s10 += s22 * 666643; 4678 s11 += s22 * 470296; 4679 s12 += s22 * 654183; 4680 s13 -= s22 * 997805; 4681 s14 += s22 * 136657; 4682 s15 -= s22 * 683901; 4683 s22 = 0; 4684 4685 s9 += s21 * 666643; 4686 s10 += s21 * 470296; 4687 s11 += s21 * 654183; 4688 s12 -= s21 * 997805; 4689 s13 += s21 * 136657; 4690 s14 -= s21 * 683901; 4691 s21 = 0; 4692 4693 s8 += s20 * 666643; 4694 s9 += s20 * 470296; 4695 s10 += s20 * 654183; 4696 s11 -= s20 * 997805; 4697 s12 += s20 * 136657; 4698 s13 -= s20 * 683901; 4699 s20 = 0; 4700 4701 s7 += s19 * 666643; 4702 s8 += s19 * 470296; 4703 s9 += s19 * 654183; 4704 s10 -= s19 * 997805; 4705 s11 += s19 * 136657; 4706 s12 -= s19 * 683901; 4707 s19 = 0; 4708 4709 s6 += s18 * 666643; 4710 s7 += s18 * 470296; 4711 s8 += s18 * 654183; 4712 s9 -= s18 * 997805; 4713 s10 += s18 * 136657; 4714 s11 -= s18 * 683901; 4715 s18 = 0; 4716 4717 carry6 = (s6 + (1 << 20)) >> 21; 4718 s7 += carry6; 4719 s6 -= carry6 * (1 << 21); 4720 carry8 = (s8 + (1 << 20)) >> 21; 4721 s9 += carry8; 4722 s8 -= carry8 * (1 << 21); 4723 carry10 = (s10 + (1 << 20)) >> 21; 4724 s11 += carry10; 4725 s10 -= carry10 * (1 << 21); 4726 carry12 = (s12 + (1 << 20)) >> 21; 4727 s13 += carry12; 4728 s12 -= carry12 * (1 << 21); 4729 carry14 = (s14 + (1 << 20)) >> 21; 4730 s15 += carry14; 4731 s14 -= carry14 * (1 << 21); 4732 carry16 = (s16 + (1 << 20)) >> 21; 4733 s17 += carry16; 4734 s16 -= carry16 * (1 << 21); 4735 4736 carry7 = (s7 + (1 << 20)) >> 21; 4737 s8 += carry7; 4738 s7 -= carry7 * (1 << 21); 4739 carry9 = (s9 + (1 << 20)) >> 21; 4740 s10 += carry9; 4741 s9 -= carry9 * (1 << 21); 4742 carry11 = (s11 + (1 << 20)) >> 21; 4743 s12 += carry11; 4744 s11 -= carry11 * (1 << 21); 4745 carry13 = (s13 + (1 << 20)) >> 21; 4746 s14 += carry13; 4747 s13 -= carry13 * (1 << 21); 4748 carry15 = (s15 + (1 << 20)) >> 21; 4749 s16 += carry15; 4750 s15 -= carry15 * (1 << 21); 4751 4752 s5 += s17 * 666643; 4753 s6 += s17 * 470296; 4754 s7 += s17 * 654183; 4755 s8 -= s17 * 997805; 4756 s9 += s17 * 136657; 4757 s10 -= s17 * 683901; 4758 s17 = 0; 4759 4760 s4 += s16 * 666643; 4761 s5 += s16 * 470296; 4762 s6 += s16 * 654183; 4763 s7 -= s16 * 997805; 4764 s8 += s16 * 136657; 4765 s9 -= s16 * 683901; 4766 s16 = 0; 4767 4768 s3 += s15 * 666643; 4769 s4 += s15 * 470296; 4770 s5 += s15 * 654183; 4771 s6 -= s15 * 997805; 4772 s7 += s15 * 136657; 4773 s8 -= s15 * 683901; 4774 s15 = 0; 4775 4776 s2 += s14 * 666643; 4777 s3 += s14 * 470296; 4778 s4 += s14 * 654183; 4779 s5 -= s14 * 997805; 4780 s6 += s14 * 136657; 4781 s7 -= s14 * 683901; 4782 s14 = 0; 4783 4784 s1 += s13 * 666643; 4785 s2 += s13 * 470296; 4786 s3 += s13 * 654183; 4787 s4 -= s13 * 997805; 4788 s5 += s13 * 136657; 4789 s6 -= s13 * 683901; 4790 s13 = 0; 4791 4792 s0 += s12 * 666643; 4793 s1 += s12 * 470296; 4794 s2 += s12 * 654183; 4795 s3 -= s12 * 997805; 4796 s4 += s12 * 136657; 4797 s5 -= s12 * 683901; 4798 s12 = 0; 4799 4800 carry0 = (s0 + (1 << 20)) >> 21; 4801 s1 += carry0; 4802 s0 -= carry0 * (1 << 21); 4803 carry2 = (s2 + (1 << 20)) >> 21; 4804 s3 += carry2; 4805 s2 -= carry2 * (1 << 21); 4806 carry4 = (s4 + (1 << 20)) >> 21; 4807 s5 += carry4; 4808 s4 -= carry4 * (1 << 21); 4809 carry6 = (s6 + (1 << 20)) >> 21; 4810 s7 += carry6; 4811 s6 -= carry6 * (1 << 21); 4812 carry8 = (s8 + (1 << 20)) >> 21; 4813 s9 += carry8; 4814 s8 -= carry8 * (1 << 21); 4815 carry10 = (s10 + (1 << 20)) >> 21; 4816 s11 += carry10; 4817 s10 -= carry10 * (1 << 21); 4818 4819 carry1 = (s1 + (1 << 20)) >> 21; 4820 s2 += carry1; 4821 s1 -= carry1 * (1 << 21); 4822 carry3 = (s3 + (1 << 20)) >> 21; 4823 s4 += carry3; 4824 s3 -= carry3 * (1 << 21); 4825 carry5 = (s5 + (1 << 20)) >> 21; 4826 s6 += carry5; 4827 s5 -= carry5 * (1 << 21); 4828 carry7 = (s7 + (1 << 20)) >> 21; 4829 s8 += carry7; 4830 s7 -= carry7 * (1 << 21); 4831 carry9 = (s9 + (1 << 20)) >> 21; 4832 s10 += carry9; 4833 s9 -= carry9 * (1 << 21); 4834 carry11 = (s11 + (1 << 20)) >> 21; 4835 s12 += carry11; 4836 s11 -= carry11 * (1 << 21); 4837 4838 s0 += s12 * 666643; 4839 s1 += s12 * 470296; 4840 s2 += s12 * 654183; 4841 s3 -= s12 * 997805; 4842 s4 += s12 * 136657; 4843 s5 -= s12 * 683901; 4844 s12 = 0; 4845 4846 carry0 = s0 >> 21; 4847 s1 += carry0; 4848 s0 -= carry0 * (1 << 21); 4849 carry1 = s1 >> 21; 4850 s2 += carry1; 4851 s1 -= carry1 * (1 << 21); 4852 carry2 = s2 >> 21; 4853 s3 += carry2; 4854 s2 -= carry2 * (1 << 21); 4855 carry3 = s3 >> 21; 4856 s4 += carry3; 4857 s3 -= carry3 * (1 << 21); 4858 carry4 = s4 >> 21; 4859 s5 += carry4; 4860 s4 -= carry4 * (1 << 21); 4861 carry5 = s5 >> 21; 4862 s6 += carry5; 4863 s5 -= carry5 * (1 << 21); 4864 carry6 = s6 >> 21; 4865 s7 += carry6; 4866 s6 -= carry6 * (1 << 21); 4867 carry7 = s7 >> 21; 4868 s8 += carry7; 4869 s7 -= carry7 * (1 << 21); 4870 carry8 = s8 >> 21; 4871 s9 += carry8; 4872 s8 -= carry8 * (1 << 21); 4873 carry9 = s9 >> 21; 4874 s10 += carry9; 4875 s9 -= carry9 * (1 << 21); 4876 carry10 = s10 >> 21; 4877 s11 += carry10; 4878 s10 -= carry10 * (1 << 21); 4879 carry11 = s11 >> 21; 4880 s12 += carry11; 4881 s11 -= carry11 * (1 << 21); 4882 4883 s0 += s12 * 666643; 4884 s1 += s12 * 470296; 4885 s2 += s12 * 654183; 4886 s3 -= s12 * 997805; 4887 s4 += s12 * 136657; 4888 s5 -= s12 * 683901; 4889 s12 = 0; 4890 4891 carry0 = s0 >> 21; 4892 s1 += carry0; 4893 s0 -= carry0 * (1 << 21); 4894 carry1 = s1 >> 21; 4895 s2 += carry1; 4896 s1 -= carry1 * (1 << 21); 4897 carry2 = s2 >> 21; 4898 s3 += carry2; 4899 s2 -= carry2 * (1 << 21); 4900 carry3 = s3 >> 21; 4901 s4 += carry3; 4902 s3 -= carry3 * (1 << 21); 4903 carry4 = s4 >> 21; 4904 s5 += carry4; 4905 s4 -= carry4 * (1 << 21); 4906 carry5 = s5 >> 21; 4907 s6 += carry5; 4908 s5 -= carry5 * (1 << 21); 4909 carry6 = s6 >> 21; 4910 s7 += carry6; 4911 s6 -= carry6 * (1 << 21); 4912 carry7 = s7 >> 21; 4913 s8 += carry7; 4914 s7 -= carry7 * (1 << 21); 4915 carry8 = s8 >> 21; 4916 s9 += carry8; 4917 s8 -= carry8 * (1 << 21); 4918 carry9 = s9 >> 21; 4919 s10 += carry9; 4920 s9 -= carry9 * (1 << 21); 4921 carry10 = s10 >> 21; 4922 s11 += carry10; 4923 s10 -= carry10 * (1 << 21); 4924 4925 s[ 0] = (uint8_t) (s0 >> 0); 4926 s[ 1] = (uint8_t) (s0 >> 8); 4927 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5)); 4928 s[ 3] = (uint8_t) (s1 >> 3); 4929 s[ 4] = (uint8_t) (s1 >> 11); 4930 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2)); 4931 s[ 6] = (uint8_t) (s2 >> 6); 4932 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7)); 4933 s[ 8] = (uint8_t) (s3 >> 1); 4934 s[ 9] = (uint8_t) (s3 >> 9); 4935 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4)); 4936 s[11] = (uint8_t) (s4 >> 4); 4937 s[12] = (uint8_t) (s4 >> 12); 4938 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1)); 4939 s[14] = (uint8_t) (s5 >> 7); 4940 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6)); 4941 s[16] = (uint8_t) (s6 >> 2); 4942 s[17] = (uint8_t) (s6 >> 10); 4943 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3)); 4944 s[19] = (uint8_t) (s7 >> 5); 4945 s[20] = (uint8_t) (s7 >> 13); 4946 s[21] = (uint8_t) (s8 >> 0); 4947 s[22] = (uint8_t) (s8 >> 8); 4948 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5)); 4949 s[24] = (uint8_t) (s9 >> 3); 4950 s[25] = (uint8_t) (s9 >> 11); 4951 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2)); 4952 s[27] = (uint8_t) (s10 >> 6); 4953 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7)); 4954 s[29] = (uint8_t) (s11 >> 1); 4955 s[30] = (uint8_t) (s11 >> 9); 4956 s[31] = (uint8_t) (s11 >> 17); 4957 } 4958 4959 /* 4960 * Input: 4961 * a[0]+256*a[1]+...+256^31*a[31] = a 4962 * b[0]+256*b[1]+...+256^31*b[31] = b 4963 * c[0]+256*c[1]+...+256^31*c[31] = c 4964 * 4965 * Output: 4966 * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l 4967 * where l = 2^252 + 27742317777372353535851937790883648493. 4968 */ 4969 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b, 4970 const uint8_t *c) 4971 { 4972 int64_t a0 = kBottom21Bits & load_3(a); 4973 int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5); 4974 int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2); 4975 int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7); 4976 int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4); 4977 int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1); 4978 int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6); 4979 int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3); 4980 int64_t a8 = kBottom21Bits & load_3(a + 21); 4981 int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5); 4982 int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2); 4983 int64_t a11 = (load_4(a + 28) >> 7); 4984 int64_t b0 = kBottom21Bits & load_3(b); 4985 int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5); 4986 int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2); 4987 int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7); 4988 int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4); 4989 int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1); 4990 int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6); 4991 int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3); 4992 int64_t b8 = kBottom21Bits & load_3(b + 21); 4993 int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5); 4994 int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2); 4995 int64_t b11 = (load_4(b + 28) >> 7); 4996 int64_t c0 = kBottom21Bits & load_3(c); 4997 int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5); 4998 int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2); 4999 int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7); 5000 int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4); 5001 int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1); 5002 int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6); 5003 int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3); 5004 int64_t c8 = kBottom21Bits & load_3(c + 21); 5005 int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5); 5006 int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2); 5007 int64_t c11 = (load_4(c + 28) >> 7); 5008 int64_t s0; 5009 int64_t s1; 5010 int64_t s2; 5011 int64_t s3; 5012 int64_t s4; 5013 int64_t s5; 5014 int64_t s6; 5015 int64_t s7; 5016 int64_t s8; 5017 int64_t s9; 5018 int64_t s10; 5019 int64_t s11; 5020 int64_t s12; 5021 int64_t s13; 5022 int64_t s14; 5023 int64_t s15; 5024 int64_t s16; 5025 int64_t s17; 5026 int64_t s18; 5027 int64_t s19; 5028 int64_t s20; 5029 int64_t s21; 5030 int64_t s22; 5031 int64_t s23; 5032 int64_t carry0; 5033 int64_t carry1; 5034 int64_t carry2; 5035 int64_t carry3; 5036 int64_t carry4; 5037 int64_t carry5; 5038 int64_t carry6; 5039 int64_t carry7; 5040 int64_t carry8; 5041 int64_t carry9; 5042 int64_t carry10; 5043 int64_t carry11; 5044 int64_t carry12; 5045 int64_t carry13; 5046 int64_t carry14; 5047 int64_t carry15; 5048 int64_t carry16; 5049 int64_t carry17; 5050 int64_t carry18; 5051 int64_t carry19; 5052 int64_t carry20; 5053 int64_t carry21; 5054 int64_t carry22; 5055 5056 s0 = c0 + a0 * b0; 5057 s1 = c1 + a0 * b1 + a1 * b0; 5058 s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0; 5059 s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0; 5060 s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0; 5061 s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0; 5062 s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0; 5063 s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0; 5064 s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0; 5065 s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0; 5066 s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0; 5067 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; 5068 s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1; 5069 s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2; 5070 s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3; 5071 s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4; 5072 s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5; 5073 s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6; 5074 s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7; 5075 s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8; 5076 s20 = a9 * b11 + a10 * b10 + a11 * b9; 5077 s21 = a10 * b11 + a11 * b10; 5078 s22 = a11 * b11; 5079 s23 = 0; 5080 5081 carry0 = (s0 + (1 << 20)) >> 21; 5082 s1 += carry0; 5083 s0 -= carry0 * (1 << 21); 5084 carry2 = (s2 + (1 << 20)) >> 21; 5085 s3 += carry2; 5086 s2 -= carry2 * (1 << 21); 5087 carry4 = (s4 + (1 << 20)) >> 21; 5088 s5 += carry4; 5089 s4 -= carry4 * (1 << 21); 5090 carry6 = (s6 + (1 << 20)) >> 21; 5091 s7 += carry6; 5092 s6 -= carry6 * (1 << 21); 5093 carry8 = (s8 + (1 << 20)) >> 21; 5094 s9 += carry8; 5095 s8 -= carry8 * (1 << 21); 5096 carry10 = (s10 + (1 << 20)) >> 21; 5097 s11 += carry10; 5098 s10 -= carry10 * (1 << 21); 5099 carry12 = (s12 + (1 << 20)) >> 21; 5100 s13 += carry12; 5101 s12 -= carry12 * (1 << 21); 5102 carry14 = (s14 + (1 << 20)) >> 21; 5103 s15 += carry14; 5104 s14 -= carry14 * (1 << 21); 5105 carry16 = (s16 + (1 << 20)) >> 21; 5106 s17 += carry16; 5107 s16 -= carry16 * (1 << 21); 5108 carry18 = (s18 + (1 << 20)) >> 21; 5109 s19 += carry18; 5110 s18 -= carry18 * (1 << 21); 5111 carry20 = (s20 + (1 << 20)) >> 21; 5112 s21 += carry20; 5113 s20 -= carry20 * (1 << 21); 5114 carry22 = (s22 + (1 << 20)) >> 21; 5115 s23 += carry22; 5116 s22 -= carry22 * (1 << 21); 5117 5118 carry1 = (s1 + (1 << 20)) >> 21; 5119 s2 += carry1; 5120 s1 -= carry1 * (1 << 21); 5121 carry3 = (s3 + (1 << 20)) >> 21; 5122 s4 += carry3; 5123 s3 -= carry3 * (1 << 21); 5124 carry5 = (s5 + (1 << 20)) >> 21; 5125 s6 += carry5; 5126 s5 -= carry5 * (1 << 21); 5127 carry7 = (s7 + (1 << 20)) >> 21; 5128 s8 += carry7; 5129 s7 -= carry7 * (1 << 21); 5130 carry9 = (s9 + (1 << 20)) >> 21; 5131 s10 += carry9; 5132 s9 -= carry9 * (1 << 21); 5133 carry11 = (s11 + (1 << 20)) >> 21; 5134 s12 += carry11; 5135 s11 -= carry11 * (1 << 21); 5136 carry13 = (s13 + (1 << 20)) >> 21; 5137 s14 += carry13; 5138 s13 -= carry13 * (1 << 21); 5139 carry15 = (s15 + (1 << 20)) >> 21; 5140 s16 += carry15; 5141 s15 -= carry15 * (1 << 21); 5142 carry17 = (s17 + (1 << 20)) >> 21; 5143 s18 += carry17; 5144 s17 -= carry17 * (1 << 21); 5145 carry19 = (s19 + (1 << 20)) >> 21; 5146 s20 += carry19; 5147 s19 -= carry19 * (1 << 21); 5148 carry21 = (s21 + (1 << 20)) >> 21; 5149 s22 += carry21; 5150 s21 -= carry21 * (1 << 21); 5151 5152 s11 += s23 * 666643; 5153 s12 += s23 * 470296; 5154 s13 += s23 * 654183; 5155 s14 -= s23 * 997805; 5156 s15 += s23 * 136657; 5157 s16 -= s23 * 683901; 5158 s23 = 0; 5159 5160 s10 += s22 * 666643; 5161 s11 += s22 * 470296; 5162 s12 += s22 * 654183; 5163 s13 -= s22 * 997805; 5164 s14 += s22 * 136657; 5165 s15 -= s22 * 683901; 5166 s22 = 0; 5167 5168 s9 += s21 * 666643; 5169 s10 += s21 * 470296; 5170 s11 += s21 * 654183; 5171 s12 -= s21 * 997805; 5172 s13 += s21 * 136657; 5173 s14 -= s21 * 683901; 5174 s21 = 0; 5175 5176 s8 += s20 * 666643; 5177 s9 += s20 * 470296; 5178 s10 += s20 * 654183; 5179 s11 -= s20 * 997805; 5180 s12 += s20 * 136657; 5181 s13 -= s20 * 683901; 5182 s20 = 0; 5183 5184 s7 += s19 * 666643; 5185 s8 += s19 * 470296; 5186 s9 += s19 * 654183; 5187 s10 -= s19 * 997805; 5188 s11 += s19 * 136657; 5189 s12 -= s19 * 683901; 5190 s19 = 0; 5191 5192 s6 += s18 * 666643; 5193 s7 += s18 * 470296; 5194 s8 += s18 * 654183; 5195 s9 -= s18 * 997805; 5196 s10 += s18 * 136657; 5197 s11 -= s18 * 683901; 5198 s18 = 0; 5199 5200 carry6 = (s6 + (1 << 20)) >> 21; 5201 s7 += carry6; 5202 s6 -= carry6 * (1 << 21); 5203 carry8 = (s8 + (1 << 20)) >> 21; 5204 s9 += carry8; 5205 s8 -= carry8 * (1 << 21); 5206 carry10 = (s10 + (1 << 20)) >> 21; 5207 s11 += carry10; 5208 s10 -= carry10 * (1 << 21); 5209 carry12 = (s12 + (1 << 20)) >> 21; 5210 s13 += carry12; 5211 s12 -= carry12 * (1 << 21); 5212 carry14 = (s14 + (1 << 20)) >> 21; 5213 s15 += carry14; 5214 s14 -= carry14 * (1 << 21); 5215 carry16 = (s16 + (1 << 20)) >> 21; 5216 s17 += carry16; 5217 s16 -= carry16 * (1 << 21); 5218 5219 carry7 = (s7 + (1 << 20)) >> 21; 5220 s8 += carry7; 5221 s7 -= carry7 * (1 << 21); 5222 carry9 = (s9 + (1 << 20)) >> 21; 5223 s10 += carry9; 5224 s9 -= carry9 * (1 << 21); 5225 carry11 = (s11 + (1 << 20)) >> 21; 5226 s12 += carry11; 5227 s11 -= carry11 * (1 << 21); 5228 carry13 = (s13 + (1 << 20)) >> 21; 5229 s14 += carry13; 5230 s13 -= carry13 * (1 << 21); 5231 carry15 = (s15 + (1 << 20)) >> 21; 5232 s16 += carry15; 5233 s15 -= carry15 * (1 << 21); 5234 5235 s5 += s17 * 666643; 5236 s6 += s17 * 470296; 5237 s7 += s17 * 654183; 5238 s8 -= s17 * 997805; 5239 s9 += s17 * 136657; 5240 s10 -= s17 * 683901; 5241 s17 = 0; 5242 5243 s4 += s16 * 666643; 5244 s5 += s16 * 470296; 5245 s6 += s16 * 654183; 5246 s7 -= s16 * 997805; 5247 s8 += s16 * 136657; 5248 s9 -= s16 * 683901; 5249 s16 = 0; 5250 5251 s3 += s15 * 666643; 5252 s4 += s15 * 470296; 5253 s5 += s15 * 654183; 5254 s6 -= s15 * 997805; 5255 s7 += s15 * 136657; 5256 s8 -= s15 * 683901; 5257 s15 = 0; 5258 5259 s2 += s14 * 666643; 5260 s3 += s14 * 470296; 5261 s4 += s14 * 654183; 5262 s5 -= s14 * 997805; 5263 s6 += s14 * 136657; 5264 s7 -= s14 * 683901; 5265 s14 = 0; 5266 5267 s1 += s13 * 666643; 5268 s2 += s13 * 470296; 5269 s3 += s13 * 654183; 5270 s4 -= s13 * 997805; 5271 s5 += s13 * 136657; 5272 s6 -= s13 * 683901; 5273 s13 = 0; 5274 5275 s0 += s12 * 666643; 5276 s1 += s12 * 470296; 5277 s2 += s12 * 654183; 5278 s3 -= s12 * 997805; 5279 s4 += s12 * 136657; 5280 s5 -= s12 * 683901; 5281 s12 = 0; 5282 5283 carry0 = (s0 + (1 << 20)) >> 21; 5284 s1 += carry0; 5285 s0 -= carry0 * (1 << 21); 5286 carry2 = (s2 + (1 << 20)) >> 21; 5287 s3 += carry2; 5288 s2 -= carry2 * (1 << 21); 5289 carry4 = (s4 + (1 << 20)) >> 21; 5290 s5 += carry4; 5291 s4 -= carry4 * (1 << 21); 5292 carry6 = (s6 + (1 << 20)) >> 21; 5293 s7 += carry6; 5294 s6 -= carry6 * (1 << 21); 5295 carry8 = (s8 + (1 << 20)) >> 21; 5296 s9 += carry8; 5297 s8 -= carry8 * (1 << 21); 5298 carry10 = (s10 + (1 << 20)) >> 21; 5299 s11 += carry10; 5300 s10 -= carry10 * (1 << 21); 5301 5302 carry1 = (s1 + (1 << 20)) >> 21; 5303 s2 += carry1; 5304 s1 -= carry1 * (1 << 21); 5305 carry3 = (s3 + (1 << 20)) >> 21; 5306 s4 += carry3; 5307 s3 -= carry3 * (1 << 21); 5308 carry5 = (s5 + (1 << 20)) >> 21; 5309 s6 += carry5; 5310 s5 -= carry5 * (1 << 21); 5311 carry7 = (s7 + (1 << 20)) >> 21; 5312 s8 += carry7; 5313 s7 -= carry7 * (1 << 21); 5314 carry9 = (s9 + (1 << 20)) >> 21; 5315 s10 += carry9; 5316 s9 -= carry9 * (1 << 21); 5317 carry11 = (s11 + (1 << 20)) >> 21; 5318 s12 += carry11; 5319 s11 -= carry11 * (1 << 21); 5320 5321 s0 += s12 * 666643; 5322 s1 += s12 * 470296; 5323 s2 += s12 * 654183; 5324 s3 -= s12 * 997805; 5325 s4 += s12 * 136657; 5326 s5 -= s12 * 683901; 5327 s12 = 0; 5328 5329 carry0 = s0 >> 21; 5330 s1 += carry0; 5331 s0 -= carry0 * (1 << 21); 5332 carry1 = s1 >> 21; 5333 s2 += carry1; 5334 s1 -= carry1 * (1 << 21); 5335 carry2 = s2 >> 21; 5336 s3 += carry2; 5337 s2 -= carry2 * (1 << 21); 5338 carry3 = s3 >> 21; 5339 s4 += carry3; 5340 s3 -= carry3 * (1 << 21); 5341 carry4 = s4 >> 21; 5342 s5 += carry4; 5343 s4 -= carry4 * (1 << 21); 5344 carry5 = s5 >> 21; 5345 s6 += carry5; 5346 s5 -= carry5 * (1 << 21); 5347 carry6 = s6 >> 21; 5348 s7 += carry6; 5349 s6 -= carry6 * (1 << 21); 5350 carry7 = s7 >> 21; 5351 s8 += carry7; 5352 s7 -= carry7 * (1 << 21); 5353 carry8 = s8 >> 21; 5354 s9 += carry8; 5355 s8 -= carry8 * (1 << 21); 5356 carry9 = s9 >> 21; 5357 s10 += carry9; 5358 s9 -= carry9 * (1 << 21); 5359 carry10 = s10 >> 21; 5360 s11 += carry10; 5361 s10 -= carry10 * (1 << 21); 5362 carry11 = s11 >> 21; 5363 s12 += carry11; 5364 s11 -= carry11 * (1 << 21); 5365 5366 s0 += s12 * 666643; 5367 s1 += s12 * 470296; 5368 s2 += s12 * 654183; 5369 s3 -= s12 * 997805; 5370 s4 += s12 * 136657; 5371 s5 -= s12 * 683901; 5372 s12 = 0; 5373 5374 carry0 = s0 >> 21; 5375 s1 += carry0; 5376 s0 -= carry0 * (1 << 21); 5377 carry1 = s1 >> 21; 5378 s2 += carry1; 5379 s1 -= carry1 * (1 << 21); 5380 carry2 = s2 >> 21; 5381 s3 += carry2; 5382 s2 -= carry2 * (1 << 21); 5383 carry3 = s3 >> 21; 5384 s4 += carry3; 5385 s3 -= carry3 * (1 << 21); 5386 carry4 = s4 >> 21; 5387 s5 += carry4; 5388 s4 -= carry4 * (1 << 21); 5389 carry5 = s5 >> 21; 5390 s6 += carry5; 5391 s5 -= carry5 * (1 << 21); 5392 carry6 = s6 >> 21; 5393 s7 += carry6; 5394 s6 -= carry6 * (1 << 21); 5395 carry7 = s7 >> 21; 5396 s8 += carry7; 5397 s7 -= carry7 * (1 << 21); 5398 carry8 = s8 >> 21; 5399 s9 += carry8; 5400 s8 -= carry8 * (1 << 21); 5401 carry9 = s9 >> 21; 5402 s10 += carry9; 5403 s9 -= carry9 * (1 << 21); 5404 carry10 = s10 >> 21; 5405 s11 += carry10; 5406 s10 -= carry10 * (1 << 21); 5407 5408 s[ 0] = (uint8_t) (s0 >> 0); 5409 s[ 1] = (uint8_t) (s0 >> 8); 5410 s[ 2] = (uint8_t)((s0 >> 16) | (s1 << 5)); 5411 s[ 3] = (uint8_t) (s1 >> 3); 5412 s[ 4] = (uint8_t) (s1 >> 11); 5413 s[ 5] = (uint8_t)((s1 >> 19) | (s2 << 2)); 5414 s[ 6] = (uint8_t) (s2 >> 6); 5415 s[ 7] = (uint8_t)((s2 >> 14) | (s3 << 7)); 5416 s[ 8] = (uint8_t) (s3 >> 1); 5417 s[ 9] = (uint8_t) (s3 >> 9); 5418 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4)); 5419 s[11] = (uint8_t) (s4 >> 4); 5420 s[12] = (uint8_t) (s4 >> 12); 5421 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1)); 5422 s[14] = (uint8_t) (s5 >> 7); 5423 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6)); 5424 s[16] = (uint8_t) (s6 >> 2); 5425 s[17] = (uint8_t) (s6 >> 10); 5426 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3)); 5427 s[19] = (uint8_t) (s7 >> 5); 5428 s[20] = (uint8_t) (s7 >> 13); 5429 s[21] = (uint8_t) (s8 >> 0); 5430 s[22] = (uint8_t) (s8 >> 8); 5431 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5)); 5432 s[24] = (uint8_t) (s9 >> 3); 5433 s[25] = (uint8_t) (s9 >> 11); 5434 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2)); 5435 s[27] = (uint8_t) (s10 >> 6); 5436 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7)); 5437 s[29] = (uint8_t) (s11 >> 1); 5438 s[30] = (uint8_t) (s11 >> 9); 5439 s[31] = (uint8_t) (s11 >> 17); 5440 } 5441 5442 int 5443 ossl_ed25519_sign(uint8_t *out_sig, const uint8_t *message, size_t message_len, 5444 const uint8_t public_key[32], const uint8_t private_key[32], 5445 OSSL_LIB_CTX *libctx, const char *propq) 5446 { 5447 uint8_t az[SHA512_DIGEST_LENGTH]; 5448 uint8_t nonce[SHA512_DIGEST_LENGTH]; 5449 ge_p3 R; 5450 uint8_t hram[SHA512_DIGEST_LENGTH]; 5451 EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq); 5452 EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new(); 5453 unsigned int sz; 5454 int res = 0; 5455 5456 if (sha512 == NULL || hash_ctx == NULL) 5457 goto err; 5458 5459 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL) 5460 || !EVP_DigestUpdate(hash_ctx, private_key, 32) 5461 || !EVP_DigestFinal_ex(hash_ctx, az, &sz)) 5462 goto err; 5463 5464 az[0] &= 248; 5465 az[31] &= 63; 5466 az[31] |= 64; 5467 5468 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL) 5469 || !EVP_DigestUpdate(hash_ctx, az + 32, 32) 5470 || !EVP_DigestUpdate(hash_ctx, message, message_len) 5471 || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz)) 5472 goto err; 5473 5474 x25519_sc_reduce(nonce); 5475 ge_scalarmult_base(&R, nonce); 5476 ge_p3_tobytes(out_sig, &R); 5477 5478 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL) 5479 || !EVP_DigestUpdate(hash_ctx, out_sig, 32) 5480 || !EVP_DigestUpdate(hash_ctx, public_key, 32) 5481 || !EVP_DigestUpdate(hash_ctx, message, message_len) 5482 || !EVP_DigestFinal_ex(hash_ctx, hram, &sz)) 5483 goto err; 5484 5485 x25519_sc_reduce(hram); 5486 sc_muladd(out_sig + 32, hram, az, nonce); 5487 5488 res = 1; 5489 err: 5490 OPENSSL_cleanse(nonce, sizeof(nonce)); 5491 OPENSSL_cleanse(az, sizeof(az)); 5492 EVP_MD_free(sha512); 5493 EVP_MD_CTX_free(hash_ctx); 5494 return res; 5495 } 5496 5497 static const char allzeroes[15]; 5498 5499 int 5500 ossl_ed25519_verify(const uint8_t *message, size_t message_len, 5501 const uint8_t signature[64], const uint8_t public_key[32], 5502 OSSL_LIB_CTX *libctx, const char *propq) 5503 { 5504 int i; 5505 ge_p3 A; 5506 const uint8_t *r, *s; 5507 EVP_MD *sha512; 5508 EVP_MD_CTX *hash_ctx = NULL; 5509 unsigned int sz; 5510 int res = 0; 5511 ge_p2 R; 5512 uint8_t rcheck[32]; 5513 uint8_t h[SHA512_DIGEST_LENGTH]; 5514 /* 27742317777372353535851937790883648493 in little endian format */ 5515 const uint8_t l_low[16] = { 5516 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2, 5517 0xDE, 0xF9, 0xDE, 0x14 5518 }; 5519 5520 r = signature; 5521 s = signature + 32; 5522 5523 /* 5524 * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493 5525 * 5526 * If not the signature is publicly invalid. Since it's public we can do the 5527 * check in variable time. 5528 * 5529 * First check the most significant byte 5530 */ 5531 if (s[31] > 0x10) 5532 return 0; 5533 if (s[31] == 0x10) { 5534 /* 5535 * Most significant byte indicates a value close to 2^252 so check the 5536 * rest 5537 */ 5538 if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0) 5539 return 0; 5540 for (i = 15; i >= 0; i--) { 5541 if (s[i] < l_low[i]) 5542 break; 5543 if (s[i] > l_low[i]) 5544 return 0; 5545 } 5546 if (i < 0) 5547 return 0; 5548 } 5549 5550 if (ge_frombytes_vartime(&A, public_key) != 0) { 5551 return 0; 5552 } 5553 5554 fe_neg(A.X, A.X); 5555 fe_neg(A.T, A.T); 5556 5557 sha512 = EVP_MD_fetch(libctx, SN_sha512, propq); 5558 if (sha512 == NULL) 5559 return 0; 5560 hash_ctx = EVP_MD_CTX_new(); 5561 if (hash_ctx == NULL) 5562 goto err; 5563 5564 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL) 5565 || !EVP_DigestUpdate(hash_ctx, r, 32) 5566 || !EVP_DigestUpdate(hash_ctx, public_key, 32) 5567 || !EVP_DigestUpdate(hash_ctx, message, message_len) 5568 || !EVP_DigestFinal_ex(hash_ctx, h, &sz)) 5569 goto err; 5570 5571 x25519_sc_reduce(h); 5572 5573 ge_double_scalarmult_vartime(&R, h, &A, s); 5574 5575 ge_tobytes(rcheck, &R); 5576 5577 res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0; 5578 err: 5579 EVP_MD_free(sha512); 5580 EVP_MD_CTX_free(hash_ctx); 5581 return res; 5582 } 5583 5584 int 5585 ossl_ed25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32], 5586 const uint8_t private_key[32], 5587 const char *propq) 5588 { 5589 uint8_t az[SHA512_DIGEST_LENGTH]; 5590 ge_p3 A; 5591 int r; 5592 EVP_MD *sha512 = NULL; 5593 5594 sha512 = EVP_MD_fetch(ctx, SN_sha512, propq); 5595 if (sha512 == NULL) 5596 return 0; 5597 r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL); 5598 EVP_MD_free(sha512); 5599 if (!r) { 5600 OPENSSL_cleanse(az, sizeof(az)); 5601 return 0; 5602 } 5603 5604 az[0] &= 248; 5605 az[31] &= 63; 5606 az[31] |= 64; 5607 5608 ge_scalarmult_base(&A, az); 5609 ge_p3_tobytes(out_public_key, &A); 5610 5611 OPENSSL_cleanse(az, sizeof(az)); 5612 return 1; 5613 } 5614 5615 int 5616 ossl_x25519(uint8_t out_shared_key[32], const uint8_t private_key[32], 5617 const uint8_t peer_public_value[32]) 5618 { 5619 static const uint8_t kZeros[32] = {0}; 5620 x25519_scalar_mult(out_shared_key, private_key, peer_public_value); 5621 /* The all-zero output results when the input is a point of small order. */ 5622 return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0; 5623 } 5624 5625 void 5626 ossl_x25519_public_from_private(uint8_t out_public_value[32], 5627 const uint8_t private_key[32]) 5628 { 5629 uint8_t e[32]; 5630 ge_p3 A; 5631 fe zplusy, zminusy, zminusy_inv; 5632 5633 memcpy(e, private_key, 32); 5634 e[0] &= 248; 5635 e[31] &= 127; 5636 e[31] |= 64; 5637 5638 ge_scalarmult_base(&A, e); 5639 5640 /* 5641 * We only need the u-coordinate of the curve25519 point. 5642 * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives 5643 * u=(Z+Y)/(Z-Y). 5644 */ 5645 fe_add(zplusy, A.Z, A.Y); 5646 fe_sub(zminusy, A.Z, A.Y); 5647 fe_invert(zminusy_inv, zminusy); 5648 fe_mul(zplusy, zplusy, zminusy_inv); 5649 fe_tobytes(out_public_value, zplusy); 5650 5651 OPENSSL_cleanse(e, sizeof(e)); 5652 } 5653