1 /* 2 * Copyright 2016-2024 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__) || defined(_M_AMD64) || defined(_M_X64)) 25 26 #define BASE_2_64_IMPLEMENTED 27 28 typedef uint64_t fe64[4]; 29 30 int x25519_fe64_eligible(void); 31 32 /* 33 * Following subroutines perform corresponding operations modulo 34 * 2^256-38, i.e. double the curve modulus. However, inputs and 35 * outputs are permitted to be partially reduced, i.e. to remain 36 * in [0..2^256) range. It's all tied up in final fe64_tobytes 37 * that performs full reduction modulo 2^255-19. 38 * 39 * There are no reference C implementations for these. 40 */ 41 void x25519_fe64_mul(fe64 h, const fe64 f, const fe64 g); 42 void x25519_fe64_sqr(fe64 h, const fe64 f); 43 void x25519_fe64_mul121666(fe64 h, fe64 f); 44 void x25519_fe64_add(fe64 h, const fe64 f, const fe64 g); 45 void x25519_fe64_sub(fe64 h, const fe64 f, const fe64 g); 46 void x25519_fe64_tobytes(uint8_t *s, const fe64 f); 47 #define fe64_mul x25519_fe64_mul 48 #define fe64_sqr x25519_fe64_sqr 49 #define fe64_mul121666 x25519_fe64_mul121666 50 #define fe64_add x25519_fe64_add 51 #define fe64_sub x25519_fe64_sub 52 #define fe64_tobytes x25519_fe64_tobytes 53 54 static uint64_t load_8(const uint8_t *in) 55 { 56 uint64_t result; 57 58 result = in[0]; 59 result |= ((uint64_t)in[1]) << 8; 60 result |= ((uint64_t)in[2]) << 16; 61 result |= ((uint64_t)in[3]) << 24; 62 result |= ((uint64_t)in[4]) << 32; 63 result |= ((uint64_t)in[5]) << 40; 64 result |= ((uint64_t)in[6]) << 48; 65 result |= ((uint64_t)in[7]) << 56; 66 67 return result; 68 } 69 70 static void fe64_frombytes(fe64 h, const uint8_t *s) 71 { 72 h[0] = load_8(s); 73 h[1] = load_8(s + 8); 74 h[2] = load_8(s + 16); 75 h[3] = load_8(s + 24) & 0x7fffffffffffffff; 76 } 77 78 static void fe64_0(fe64 h) 79 { 80 h[0] = 0; 81 h[1] = 0; 82 h[2] = 0; 83 h[3] = 0; 84 } 85 86 static void fe64_1(fe64 h) 87 { 88 h[0] = 1; 89 h[1] = 0; 90 h[2] = 0; 91 h[3] = 0; 92 } 93 94 static void fe64_copy(fe64 h, const fe64 f) 95 { 96 h[0] = f[0]; 97 h[1] = f[1]; 98 h[2] = f[2]; 99 h[3] = f[3]; 100 } 101 102 static void fe64_cswap(fe64 f, fe64 g, unsigned int b) 103 { 104 int i; 105 uint64_t mask = 0 - (uint64_t)b; 106 107 for (i = 0; i < 4; i++) { 108 uint64_t x = f[i] ^ g[i]; 109 x &= mask; 110 f[i] ^= x; 111 g[i] ^= x; 112 } 113 } 114 115 static void fe64_invert(fe64 out, const fe64 z) 116 { 117 fe64 t0; 118 fe64 t1; 119 fe64 t2; 120 fe64 t3; 121 int i; 122 123 /* 124 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 125 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 126 */ 127 128 /* t0 = z ** 2 */ 129 fe64_sqr(t0, z); 130 131 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 132 fe64_sqr(t1, t0); 133 fe64_sqr(t1, t1); 134 135 /* t1 = z * t1 = z ** 9 */ 136 fe64_mul(t1, z, t1); 137 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 138 fe64_mul(t0, t0, t1); 139 140 /* t2 = t0 ** 2 = z ** 22 */ 141 fe64_sqr(t2, t0); 142 143 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 144 fe64_mul(t1, t1, t2); 145 146 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 147 fe64_sqr(t2, t1); 148 for (i = 1; i < 5; ++i) 149 fe64_sqr(t2, t2); 150 151 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 152 fe64_mul(t1, t2, t1); 153 154 /* Continuing similarly... */ 155 156 /* t2 = z ** (2 ** 20 - 1) */ 157 fe64_sqr(t2, t1); 158 for (i = 1; i < 10; ++i) 159 fe64_sqr(t2, t2); 160 161 fe64_mul(t2, t2, t1); 162 163 /* t2 = z ** (2 ** 40 - 1) */ 164 fe64_sqr(t3, t2); 165 for (i = 1; i < 20; ++i) 166 fe64_sqr(t3, t3); 167 168 fe64_mul(t2, t3, t2); 169 170 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 171 for (i = 0; i < 10; ++i) 172 fe64_sqr(t2, t2); 173 174 /* t1 = z ** (2 ** 50 - 1) */ 175 fe64_mul(t1, t2, t1); 176 177 /* t2 = z ** (2 ** 100 - 1) */ 178 fe64_sqr(t2, t1); 179 for (i = 1; i < 50; ++i) 180 fe64_sqr(t2, t2); 181 182 fe64_mul(t2, t2, t1); 183 184 /* t2 = z ** (2 ** 200 - 1) */ 185 fe64_sqr(t3, t2); 186 for (i = 1; i < 100; ++i) 187 fe64_sqr(t3, t3); 188 189 fe64_mul(t2, t3, t2); 190 191 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 192 for (i = 0; i < 50; ++i) 193 fe64_sqr(t2, t2); 194 195 /* t1 = z ** (2 ** 250 - 1) */ 196 fe64_mul(t1, t2, t1); 197 198 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 199 for (i = 0; i < 5; ++i) 200 fe64_sqr(t1, t1); 201 202 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 203 fe64_mul(out, t1, t0); 204 } 205 206 /* 207 * Duplicate of original x25519_scalar_mult_generic, but using 208 * fe64_* subroutines. 209 */ 210 static void x25519_scalar_mulx(uint8_t out[32], const uint8_t scalar[32], 211 const uint8_t point[32]) 212 { 213 fe64 x1, x2, z2, x3, z3, tmp0, tmp1; 214 uint8_t e[32]; 215 unsigned swap = 0; 216 int pos; 217 218 memcpy(e, scalar, 32); 219 e[0] &= 0xf8; 220 e[31] &= 0x7f; 221 e[31] |= 0x40; 222 fe64_frombytes(x1, point); 223 fe64_1(x2); 224 fe64_0(z2); 225 fe64_copy(x3, x1); 226 fe64_1(z3); 227 228 for (pos = 254; pos >= 0; --pos) { 229 unsigned int b = 1 & (e[pos / 8] >> (pos & 7)); 230 231 swap ^= b; 232 fe64_cswap(x2, x3, swap); 233 fe64_cswap(z2, z3, swap); 234 swap = b; 235 fe64_sub(tmp0, x3, z3); 236 fe64_sub(tmp1, x2, z2); 237 fe64_add(x2, x2, z2); 238 fe64_add(z2, x3, z3); 239 fe64_mul(z3, x2, tmp0); 240 fe64_mul(z2, z2, tmp1); 241 fe64_sqr(tmp0, tmp1); 242 fe64_sqr(tmp1, x2); 243 fe64_add(x3, z3, z2); 244 fe64_sub(z2, z3, z2); 245 fe64_mul(x2, tmp1, tmp0); 246 fe64_sub(tmp1, tmp1, tmp0); 247 fe64_sqr(z2, z2); 248 fe64_mul121666(z3, tmp1); 249 fe64_sqr(x3, x3); 250 fe64_add(tmp0, tmp0, z3); 251 fe64_mul(z3, x1, z2); 252 fe64_mul(z2, tmp1, tmp0); 253 } 254 255 fe64_invert(z2, z2); 256 fe64_mul(x2, x2, z2); 257 fe64_tobytes(out, x2); 258 259 OPENSSL_cleanse(e, sizeof(e)); 260 } 261 #endif 262 263 #if defined(X25519_ASM) \ 264 || (defined(INT128_MAX) \ 265 && !defined(__sparc__) \ 266 && (!defined(__SIZEOF_LONG__) || (__SIZEOF_LONG__ == 8)) \ 267 && !(defined(__ANDROID__) && !defined(__clang__))) 268 /* 269 * Base 2^51 implementation. It's virtually no different from reference 270 * base 2^25.5 implementation in respect to lax boundary conditions for 271 * intermediate values and even individual limbs. So that whatever you 272 * know about the reference, applies even here... 273 */ 274 #define BASE_2_51_IMPLEMENTED 275 276 typedef uint64_t fe51[5]; 277 278 static const uint64_t MASK51 = 0x7ffffffffffff; 279 280 static uint64_t load_7(const uint8_t *in) 281 { 282 uint64_t result; 283 284 result = in[0]; 285 result |= ((uint64_t)in[1]) << 8; 286 result |= ((uint64_t)in[2]) << 16; 287 result |= ((uint64_t)in[3]) << 24; 288 result |= ((uint64_t)in[4]) << 32; 289 result |= ((uint64_t)in[5]) << 40; 290 result |= ((uint64_t)in[6]) << 48; 291 292 return result; 293 } 294 295 static uint64_t load_6(const uint8_t *in) 296 { 297 uint64_t result; 298 299 result = in[0]; 300 result |= ((uint64_t)in[1]) << 8; 301 result |= ((uint64_t)in[2]) << 16; 302 result |= ((uint64_t)in[3]) << 24; 303 result |= ((uint64_t)in[4]) << 32; 304 result |= ((uint64_t)in[5]) << 40; 305 306 return result; 307 } 308 309 static void fe51_frombytes(fe51 h, const uint8_t *s) 310 { 311 uint64_t h0 = load_7(s); /* 56 bits */ 312 uint64_t h1 = load_6(s + 7) << 5; /* 53 bits */ 313 uint64_t h2 = load_7(s + 13) << 2; /* 58 bits */ 314 uint64_t h3 = load_6(s + 20) << 7; /* 55 bits */ 315 uint64_t h4 = (load_6(s + 26) & 0x7fffffffffff) << 4; /* 51 bits */ 316 317 h1 |= h0 >> 51; 318 h0 &= MASK51; 319 h2 |= h1 >> 51; 320 h1 &= MASK51; 321 h3 |= h2 >> 51; 322 h2 &= MASK51; 323 h4 |= h3 >> 51; 324 h3 &= MASK51; 325 326 h[0] = h0; 327 h[1] = h1; 328 h[2] = h2; 329 h[3] = h3; 330 h[4] = h4; 331 } 332 333 static void fe51_tobytes(uint8_t *s, const fe51 h) 334 { 335 uint64_t h0 = h[0]; 336 uint64_t h1 = h[1]; 337 uint64_t h2 = h[2]; 338 uint64_t h3 = h[3]; 339 uint64_t h4 = h[4]; 340 uint64_t q; 341 342 /* compare to modulus */ 343 q = (h0 + 19) >> 51; 344 q = (h1 + q) >> 51; 345 q = (h2 + q) >> 51; 346 q = (h3 + q) >> 51; 347 q = (h4 + q) >> 51; 348 349 /* full reduce */ 350 h0 += 19 * q; 351 h1 += h0 >> 51; 352 h0 &= MASK51; 353 h2 += h1 >> 51; 354 h1 &= MASK51; 355 h3 += h2 >> 51; 356 h2 &= MASK51; 357 h4 += h3 >> 51; 358 h3 &= MASK51; 359 h4 &= MASK51; 360 361 /* smash */ 362 s[0] = (uint8_t)(h0 >> 0); 363 s[1] = (uint8_t)(h0 >> 8); 364 s[2] = (uint8_t)(h0 >> 16); 365 s[3] = (uint8_t)(h0 >> 24); 366 s[4] = (uint8_t)(h0 >> 32); 367 s[5] = (uint8_t)(h0 >> 40); 368 s[6] = (uint8_t)((h0 >> 48) | ((uint32_t)h1 << 3)); 369 s[7] = (uint8_t)(h1 >> 5); 370 s[8] = (uint8_t)(h1 >> 13); 371 s[9] = (uint8_t)(h1 >> 21); 372 s[10] = (uint8_t)(h1 >> 29); 373 s[11] = (uint8_t)(h1 >> 37); 374 s[12] = (uint8_t)((h1 >> 45) | ((uint32_t)h2 << 6)); 375 s[13] = (uint8_t)(h2 >> 2); 376 s[14] = (uint8_t)(h2 >> 10); 377 s[15] = (uint8_t)(h2 >> 18); 378 s[16] = (uint8_t)(h2 >> 26); 379 s[17] = (uint8_t)(h2 >> 34); 380 s[18] = (uint8_t)(h2 >> 42); 381 s[19] = (uint8_t)((h2 >> 50) | ((uint32_t)h3 << 1)); 382 s[20] = (uint8_t)(h3 >> 7); 383 s[21] = (uint8_t)(h3 >> 15); 384 s[22] = (uint8_t)(h3 >> 23); 385 s[23] = (uint8_t)(h3 >> 31); 386 s[24] = (uint8_t)(h3 >> 39); 387 s[25] = (uint8_t)((h3 >> 47) | ((uint32_t)h4 << 4)); 388 s[26] = (uint8_t)(h4 >> 4); 389 s[27] = (uint8_t)(h4 >> 12); 390 s[28] = (uint8_t)(h4 >> 20); 391 s[29] = (uint8_t)(h4 >> 28); 392 s[30] = (uint8_t)(h4 >> 36); 393 s[31] = (uint8_t)(h4 >> 44); 394 } 395 396 #if defined(X25519_ASM) 397 void x25519_fe51_mul(fe51 h, const fe51 f, const fe51 g); 398 void x25519_fe51_sqr(fe51 h, const fe51 f); 399 void x25519_fe51_mul121666(fe51 h, fe51 f); 400 #define fe51_mul x25519_fe51_mul 401 #define fe51_sq x25519_fe51_sqr 402 #define fe51_mul121666 x25519_fe51_mul121666 403 #else 404 405 typedef uint128_t u128; 406 407 static void fe51_mul(fe51 h, const fe51 f, const fe51 g) 408 { 409 u128 h0, h1, h2, h3, h4; 410 uint64_t f_i, g0, g1, g2, g3, g4; 411 412 f_i = f[0]; 413 h0 = (u128)f_i * (g0 = g[0]); 414 h1 = (u128)f_i * (g1 = g[1]); 415 h2 = (u128)f_i * (g2 = g[2]); 416 h3 = (u128)f_i * (g3 = g[3]); 417 h4 = (u128)f_i * (g4 = g[4]); 418 419 f_i = f[1]; 420 h0 += (u128)f_i * (g4 *= 19); 421 h1 += (u128)f_i * g0; 422 h2 += (u128)f_i * g1; 423 h3 += (u128)f_i * g2; 424 h4 += (u128)f_i * g3; 425 426 f_i = f[2]; 427 h0 += (u128)f_i * (g3 *= 19); 428 h1 += (u128)f_i * g4; 429 h2 += (u128)f_i * g0; 430 h3 += (u128)f_i * g1; 431 h4 += (u128)f_i * g2; 432 433 f_i = f[3]; 434 h0 += (u128)f_i * (g2 *= 19); 435 h1 += (u128)f_i * g3; 436 h2 += (u128)f_i * g4; 437 h3 += (u128)f_i * g0; 438 h4 += (u128)f_i * g1; 439 440 f_i = f[4]; 441 h0 += (u128)f_i * (g1 *= 19); 442 h1 += (u128)f_i * g2; 443 h2 += (u128)f_i * g3; 444 h3 += (u128)f_i * g4; 445 h4 += (u128)f_i * g0; 446 447 /* partial [lazy] reduction */ 448 h3 += (uint64_t)(h2 >> 51); 449 g2 = (uint64_t)h2 & MASK51; 450 h1 += (uint64_t)(h0 >> 51); 451 g0 = (uint64_t)h0 & MASK51; 452 453 h4 += (uint64_t)(h3 >> 51); 454 g3 = (uint64_t)h3 & MASK51; 455 g2 += (uint64_t)(h1 >> 51); 456 g1 = (uint64_t)h1 & MASK51; 457 458 g0 += (uint64_t)(h4 >> 51) * 19; 459 g4 = (uint64_t)h4 & MASK51; 460 g3 += g2 >> 51; 461 g2 &= MASK51; 462 g1 += g0 >> 51; 463 g0 &= MASK51; 464 465 h[0] = g0; 466 h[1] = g1; 467 h[2] = g2; 468 h[3] = g3; 469 h[4] = g4; 470 } 471 472 static void fe51_sq(fe51 h, const fe51 f) 473 { 474 #if defined(OPENSSL_SMALL_FOOTPRINT) 475 fe51_mul(h, f, f); 476 #else 477 /* dedicated squaring gives 16-25% overall improvement */ 478 uint64_t g0 = f[0]; 479 uint64_t g1 = f[1]; 480 uint64_t g2 = f[2]; 481 uint64_t g3 = f[3]; 482 uint64_t g4 = f[4]; 483 u128 h0, h1, h2, h3, h4; 484 485 h0 = (u128)g0 * g0; 486 g0 *= 2; 487 h1 = (u128)g0 * g1; 488 h2 = (u128)g0 * g2; 489 h3 = (u128)g0 * g3; 490 h4 = (u128)g0 * g4; 491 492 g0 = g4; /* borrow g0 */ 493 h3 += (u128)g0 * (g4 *= 19); 494 495 h2 += (u128)g1 * g1; 496 g1 *= 2; 497 h3 += (u128)g1 * g2; 498 h4 += (u128)g1 * g3; 499 h0 += (u128)g1 * g4; 500 501 g0 = g3; /* borrow g0 */ 502 h1 += (u128)g0 * (g3 *= 19); 503 h2 += (u128)(g0 * 2) * g4; 504 505 h4 += (u128)g2 * g2; 506 g2 *= 2; 507 h0 += (u128)g2 * g3; 508 h1 += (u128)g2 * g4; 509 510 /* partial [lazy] reduction */ 511 h3 += (uint64_t)(h2 >> 51); 512 g2 = (uint64_t)h2 & MASK51; 513 h1 += (uint64_t)(h0 >> 51); 514 g0 = (uint64_t)h0 & MASK51; 515 516 h4 += (uint64_t)(h3 >> 51); 517 g3 = (uint64_t)h3 & MASK51; 518 g2 += (uint64_t)(h1 >> 51); 519 g1 = (uint64_t)h1 & MASK51; 520 521 g0 += (uint64_t)(h4 >> 51) * 19; 522 g4 = (uint64_t)h4 & MASK51; 523 g3 += g2 >> 51; 524 g2 &= MASK51; 525 g1 += g0 >> 51; 526 g0 &= MASK51; 527 528 h[0] = g0; 529 h[1] = g1; 530 h[2] = g2; 531 h[3] = g3; 532 h[4] = g4; 533 #endif 534 } 535 536 static void fe51_mul121666(fe51 h, fe51 f) 537 { 538 u128 h0 = f[0] * (u128)121666; 539 u128 h1 = f[1] * (u128)121666; 540 u128 h2 = f[2] * (u128)121666; 541 u128 h3 = f[3] * (u128)121666; 542 u128 h4 = f[4] * (u128)121666; 543 uint64_t g0, g1, g2, g3, g4; 544 545 h3 += (uint64_t)(h2 >> 51); 546 g2 = (uint64_t)h2 & MASK51; 547 h1 += (uint64_t)(h0 >> 51); 548 g0 = (uint64_t)h0 & MASK51; 549 550 h4 += (uint64_t)(h3 >> 51); 551 g3 = (uint64_t)h3 & MASK51; 552 g2 += (uint64_t)(h1 >> 51); 553 g1 = (uint64_t)h1 & MASK51; 554 555 g0 += (uint64_t)(h4 >> 51) * 19; 556 g4 = (uint64_t)h4 & MASK51; 557 g3 += g2 >> 51; 558 g2 &= MASK51; 559 g1 += g0 >> 51; 560 g0 &= MASK51; 561 562 h[0] = g0; 563 h[1] = g1; 564 h[2] = g2; 565 h[3] = g3; 566 h[4] = g4; 567 } 568 #endif 569 570 static void fe51_add(fe51 h, const fe51 f, const fe51 g) 571 { 572 h[0] = f[0] + g[0]; 573 h[1] = f[1] + g[1]; 574 h[2] = f[2] + g[2]; 575 h[3] = f[3] + g[3]; 576 h[4] = f[4] + g[4]; 577 } 578 579 static void fe51_sub(fe51 h, const fe51 f, const fe51 g) 580 { 581 /* 582 * Add 2*modulus to ensure that result remains positive 583 * even if subtrahend is partially reduced. 584 */ 585 h[0] = (f[0] + 0xfffffffffffda) - g[0]; 586 h[1] = (f[1] + 0xffffffffffffe) - g[1]; 587 h[2] = (f[2] + 0xffffffffffffe) - g[2]; 588 h[3] = (f[3] + 0xffffffffffffe) - g[3]; 589 h[4] = (f[4] + 0xffffffffffffe) - g[4]; 590 } 591 592 static void fe51_0(fe51 h) 593 { 594 h[0] = 0; 595 h[1] = 0; 596 h[2] = 0; 597 h[3] = 0; 598 h[4] = 0; 599 } 600 601 static void fe51_1(fe51 h) 602 { 603 h[0] = 1; 604 h[1] = 0; 605 h[2] = 0; 606 h[3] = 0; 607 h[4] = 0; 608 } 609 610 static void fe51_copy(fe51 h, const fe51 f) 611 { 612 h[0] = f[0]; 613 h[1] = f[1]; 614 h[2] = f[2]; 615 h[3] = f[3]; 616 h[4] = f[4]; 617 } 618 619 static void fe51_cswap(fe51 f, fe51 g, unsigned int b) 620 { 621 int i; 622 uint64_t mask = 0 - (uint64_t)b; 623 624 for (i = 0; i < 5; i++) { 625 int64_t x = f[i] ^ g[i]; 626 x &= mask; 627 f[i] ^= x; 628 g[i] ^= x; 629 } 630 } 631 632 static void fe51_invert(fe51 out, const fe51 z) 633 { 634 fe51 t0; 635 fe51 t1; 636 fe51 t2; 637 fe51 t3; 638 int i; 639 640 /* 641 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 642 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 643 */ 644 645 /* t0 = z ** 2 */ 646 fe51_sq(t0, z); 647 648 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 649 fe51_sq(t1, t0); 650 fe51_sq(t1, t1); 651 652 /* t1 = z * t1 = z ** 9 */ 653 fe51_mul(t1, z, t1); 654 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 655 fe51_mul(t0, t0, t1); 656 657 /* t2 = t0 ** 2 = z ** 22 */ 658 fe51_sq(t2, t0); 659 660 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 661 fe51_mul(t1, t1, t2); 662 663 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 664 fe51_sq(t2, t1); 665 for (i = 1; i < 5; ++i) 666 fe51_sq(t2, t2); 667 668 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 669 fe51_mul(t1, t2, t1); 670 671 /* Continuing similarly... */ 672 673 /* t2 = z ** (2 ** 20 - 1) */ 674 fe51_sq(t2, t1); 675 for (i = 1; i < 10; ++i) 676 fe51_sq(t2, t2); 677 678 fe51_mul(t2, t2, t1); 679 680 /* t2 = z ** (2 ** 40 - 1) */ 681 fe51_sq(t3, t2); 682 for (i = 1; i < 20; ++i) 683 fe51_sq(t3, t3); 684 685 fe51_mul(t2, t3, t2); 686 687 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 688 for (i = 0; i < 10; ++i) 689 fe51_sq(t2, t2); 690 691 /* t1 = z ** (2 ** 50 - 1) */ 692 fe51_mul(t1, t2, t1); 693 694 /* t2 = z ** (2 ** 100 - 1) */ 695 fe51_sq(t2, t1); 696 for (i = 1; i < 50; ++i) 697 fe51_sq(t2, t2); 698 699 fe51_mul(t2, t2, t1); 700 701 /* t2 = z ** (2 ** 200 - 1) */ 702 fe51_sq(t3, t2); 703 for (i = 1; i < 100; ++i) 704 fe51_sq(t3, t3); 705 706 fe51_mul(t2, t3, t2); 707 708 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 709 for (i = 0; i < 50; ++i) 710 fe51_sq(t2, t2); 711 712 /* t1 = z ** (2 ** 250 - 1) */ 713 fe51_mul(t1, t2, t1); 714 715 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 716 for (i = 0; i < 5; ++i) 717 fe51_sq(t1, t1); 718 719 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 720 fe51_mul(out, t1, t0); 721 } 722 723 /* 724 * Duplicate of original x25519_scalar_mult_generic, but using 725 * fe51_* subroutines. 726 */ 727 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32], 728 const uint8_t point[32]) 729 { 730 fe51 x1, x2, z2, x3, z3, tmp0, tmp1; 731 uint8_t e[32]; 732 unsigned swap = 0; 733 int pos; 734 735 #ifdef BASE_2_64_IMPLEMENTED 736 if (x25519_fe64_eligible()) { 737 x25519_scalar_mulx(out, scalar, point); 738 return; 739 } 740 #endif 741 742 memcpy(e, scalar, 32); 743 e[0] &= 0xf8; 744 e[31] &= 0x7f; 745 e[31] |= 0x40; 746 fe51_frombytes(x1, point); 747 fe51_1(x2); 748 fe51_0(z2); 749 fe51_copy(x3, x1); 750 fe51_1(z3); 751 752 for (pos = 254; pos >= 0; --pos) { 753 unsigned int b = 1 & (e[pos / 8] >> (pos & 7)); 754 755 swap ^= b; 756 fe51_cswap(x2, x3, swap); 757 fe51_cswap(z2, z3, swap); 758 swap = b; 759 fe51_sub(tmp0, x3, z3); 760 fe51_sub(tmp1, x2, z2); 761 fe51_add(x2, x2, z2); 762 fe51_add(z2, x3, z3); 763 fe51_mul(z3, tmp0, x2); 764 fe51_mul(z2, z2, tmp1); 765 fe51_sq(tmp0, tmp1); 766 fe51_sq(tmp1, x2); 767 fe51_add(x3, z3, z2); 768 fe51_sub(z2, z3, z2); 769 fe51_mul(x2, tmp1, tmp0); 770 fe51_sub(tmp1, tmp1, tmp0); 771 fe51_sq(z2, z2); 772 fe51_mul121666(z3, tmp1); 773 fe51_sq(x3, x3); 774 fe51_add(tmp0, tmp0, z3); 775 fe51_mul(z3, x1, z2); 776 fe51_mul(z2, tmp1, tmp0); 777 } 778 779 fe51_invert(z2, z2); 780 fe51_mul(x2, x2, z2); 781 fe51_tobytes(out, x2); 782 783 OPENSSL_cleanse(e, sizeof(e)); 784 } 785 #endif 786 787 /* 788 * Reference base 2^25.5 implementation. 789 * 790 * This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP 791 * 20141124 (http://bench.cr.yp.to/supercop.html). 792 * 793 * The field functions are shared by Ed25519 and X25519 where possible. 794 */ 795 796 /* 797 * fe means field element. Here the field is \Z/(2^255-19). An element t, 798 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 799 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on 800 * context. 801 */ 802 typedef int32_t fe[10]; 803 804 static const int64_t kBottom21Bits = 0x1fffffLL; 805 static const int64_t kBottom25Bits = 0x1ffffffLL; 806 static const int64_t kBottom26Bits = 0x3ffffffLL; 807 static const int64_t kTop39Bits = 0xfffffffffe000000LL; 808 static const int64_t kTop38Bits = 0xfffffffffc000000LL; 809 810 static uint64_t load_3(const uint8_t *in) 811 { 812 uint64_t result; 813 814 result = ((uint64_t)in[0]); 815 result |= ((uint64_t)in[1]) << 8; 816 result |= ((uint64_t)in[2]) << 16; 817 return result; 818 } 819 820 static uint64_t load_4(const uint8_t *in) 821 { 822 uint64_t result; 823 824 result = ((uint64_t)in[0]); 825 result |= ((uint64_t)in[1]) << 8; 826 result |= ((uint64_t)in[2]) << 16; 827 result |= ((uint64_t)in[3]) << 24; 828 return result; 829 } 830 831 static void fe_frombytes(fe h, const uint8_t *s) 832 { 833 /* Ignores top bit of h. */ 834 int64_t h0 = load_4(s); 835 int64_t h1 = load_3(s + 4) << 6; 836 int64_t h2 = load_3(s + 7) << 5; 837 int64_t h3 = load_3(s + 10) << 3; 838 int64_t h4 = load_3(s + 13) << 2; 839 int64_t h5 = load_4(s + 16); 840 int64_t h6 = load_3(s + 20) << 7; 841 int64_t h7 = load_3(s + 23) << 5; 842 int64_t h8 = load_3(s + 26) << 4; 843 int64_t h9 = (load_3(s + 29) & 0x7fffff) << 2; 844 int64_t carry0; 845 int64_t carry1; 846 int64_t carry2; 847 int64_t carry3; 848 int64_t carry4; 849 int64_t carry5; 850 int64_t carry6; 851 int64_t carry7; 852 int64_t carry8; 853 int64_t carry9; 854 855 carry9 = h9 + (1 << 24); 856 h0 += (carry9 >> 25) * 19; 857 h9 -= carry9 & kTop39Bits; 858 carry1 = h1 + (1 << 24); 859 h2 += carry1 >> 25; 860 h1 -= carry1 & kTop39Bits; 861 carry3 = h3 + (1 << 24); 862 h4 += carry3 >> 25; 863 h3 -= carry3 & kTop39Bits; 864 carry5 = h5 + (1 << 24); 865 h6 += carry5 >> 25; 866 h5 -= carry5 & kTop39Bits; 867 carry7 = h7 + (1 << 24); 868 h8 += carry7 >> 25; 869 h7 -= carry7 & kTop39Bits; 870 871 carry0 = h0 + (1 << 25); 872 h1 += carry0 >> 26; 873 h0 -= carry0 & kTop38Bits; 874 carry2 = h2 + (1 << 25); 875 h3 += carry2 >> 26; 876 h2 -= carry2 & kTop38Bits; 877 carry4 = h4 + (1 << 25); 878 h5 += carry4 >> 26; 879 h4 -= carry4 & kTop38Bits; 880 carry6 = h6 + (1 << 25); 881 h7 += carry6 >> 26; 882 h6 -= carry6 & kTop38Bits; 883 carry8 = h8 + (1 << 25); 884 h9 += carry8 >> 26; 885 h8 -= carry8 & kTop38Bits; 886 887 h[0] = (int32_t)h0; 888 h[1] = (int32_t)h1; 889 h[2] = (int32_t)h2; 890 h[3] = (int32_t)h3; 891 h[4] = (int32_t)h4; 892 h[5] = (int32_t)h5; 893 h[6] = (int32_t)h6; 894 h[7] = (int32_t)h7; 895 h[8] = (int32_t)h8; 896 h[9] = (int32_t)h9; 897 } 898 899 /* 900 * Preconditions: 901 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 902 * 903 * Write p=2^255-19; q=floor(h/p). 904 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). 905 * 906 * Proof: 907 * Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. 908 * Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4. 909 * 910 * Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). 911 * Then 0<y<1. 912 * 913 * Write r=h-pq. 914 * Have 0<=r<=p-1=2^255-20. 915 * Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1. 916 * 917 * Write x=r+19(2^-255)r+y. 918 * Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q. 919 * 920 * Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1)) 921 * so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q. 922 */ 923 static void fe_tobytes(uint8_t *s, const fe h) 924 { 925 int32_t h0 = h[0]; 926 int32_t h1 = h[1]; 927 int32_t h2 = h[2]; 928 int32_t h3 = h[3]; 929 int32_t h4 = h[4]; 930 int32_t h5 = h[5]; 931 int32_t h6 = h[6]; 932 int32_t h7 = h[7]; 933 int32_t h8 = h[8]; 934 int32_t h9 = h[9]; 935 int32_t q; 936 937 q = (19 * h9 + (((int32_t)1) << 24)) >> 25; 938 q = (h0 + q) >> 26; 939 q = (h1 + q) >> 25; 940 q = (h2 + q) >> 26; 941 q = (h3 + q) >> 25; 942 q = (h4 + q) >> 26; 943 q = (h5 + q) >> 25; 944 q = (h6 + q) >> 26; 945 q = (h7 + q) >> 25; 946 q = (h8 + q) >> 26; 947 q = (h9 + q) >> 25; 948 949 /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */ 950 h0 += 19 * q; 951 /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */ 952 953 h1 += h0 >> 26; 954 h0 &= kBottom26Bits; 955 h2 += h1 >> 25; 956 h1 &= kBottom25Bits; 957 h3 += h2 >> 26; 958 h2 &= kBottom26Bits; 959 h4 += h3 >> 25; 960 h3 &= kBottom25Bits; 961 h5 += h4 >> 26; 962 h4 &= kBottom26Bits; 963 h6 += h5 >> 25; 964 h5 &= kBottom25Bits; 965 h7 += h6 >> 26; 966 h6 &= kBottom26Bits; 967 h8 += h7 >> 25; 968 h7 &= kBottom25Bits; 969 h9 += h8 >> 26; 970 h8 &= kBottom26Bits; 971 h9 &= kBottom25Bits; 972 /* h10 = carry9 */ 973 974 /* 975 * Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. 976 * Have h0+...+2^230 h9 between 0 and 2^255-1; 977 * evidently 2^255 h10-2^255 q = 0. 978 * Goal: Output h0+...+2^230 h9. 979 */ 980 s[0] = (uint8_t)(h0 >> 0); 981 s[1] = (uint8_t)(h0 >> 8); 982 s[2] = (uint8_t)(h0 >> 16); 983 s[3] = (uint8_t)((h0 >> 24) | ((uint32_t)(h1) << 2)); 984 s[4] = (uint8_t)(h1 >> 6); 985 s[5] = (uint8_t)(h1 >> 14); 986 s[6] = (uint8_t)((h1 >> 22) | ((uint32_t)(h2) << 3)); 987 s[7] = (uint8_t)(h2 >> 5); 988 s[8] = (uint8_t)(h2 >> 13); 989 s[9] = (uint8_t)((h2 >> 21) | ((uint32_t)(h3) << 5)); 990 s[10] = (uint8_t)(h3 >> 3); 991 s[11] = (uint8_t)(h3 >> 11); 992 s[12] = (uint8_t)((h3 >> 19) | ((uint32_t)(h4) << 6)); 993 s[13] = (uint8_t)(h4 >> 2); 994 s[14] = (uint8_t)(h4 >> 10); 995 s[15] = (uint8_t)(h4 >> 18); 996 s[16] = (uint8_t)(h5 >> 0); 997 s[17] = (uint8_t)(h5 >> 8); 998 s[18] = (uint8_t)(h5 >> 16); 999 s[19] = (uint8_t)((h5 >> 24) | ((uint32_t)(h6) << 1)); 1000 s[20] = (uint8_t)(h6 >> 7); 1001 s[21] = (uint8_t)(h6 >> 15); 1002 s[22] = (uint8_t)((h6 >> 23) | ((uint32_t)(h7) << 3)); 1003 s[23] = (uint8_t)(h7 >> 5); 1004 s[24] = (uint8_t)(h7 >> 13); 1005 s[25] = (uint8_t)((h7 >> 21) | ((uint32_t)(h8) << 4)); 1006 s[26] = (uint8_t)(h8 >> 4); 1007 s[27] = (uint8_t)(h8 >> 12); 1008 s[28] = (uint8_t)((h8 >> 20) | ((uint32_t)(h9) << 6)); 1009 s[29] = (uint8_t)(h9 >> 2); 1010 s[30] = (uint8_t)(h9 >> 10); 1011 s[31] = (uint8_t)(h9 >> 18); 1012 } 1013 1014 /* h = f */ 1015 static void fe_copy(fe h, const fe f) 1016 { 1017 memmove(h, f, sizeof(int32_t) * 10); 1018 } 1019 1020 /* h = 0 */ 1021 static void fe_0(fe h) 1022 { 1023 memset(h, 0, sizeof(int32_t) * 10); 1024 } 1025 1026 /* h = 1 */ 1027 static void fe_1(fe h) 1028 { 1029 memset(h, 0, sizeof(int32_t) * 10); 1030 h[0] = 1; 1031 } 1032 1033 /* 1034 * h = f + g 1035 * 1036 * Can overlap h with f or g. 1037 * 1038 * Preconditions: 1039 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1040 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1041 * 1042 * Postconditions: 1043 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1044 */ 1045 static void fe_add(fe h, const fe f, const fe g) 1046 { 1047 unsigned i; 1048 1049 for (i = 0; i < 10; i++) { 1050 h[i] = f[i] + g[i]; 1051 } 1052 } 1053 1054 /* 1055 * h = f - g 1056 * 1057 * Can overlap h with f or g. 1058 * 1059 * Preconditions: 1060 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1061 * |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1062 * 1063 * Postconditions: 1064 * |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1065 */ 1066 static void fe_sub(fe h, const fe f, const fe g) 1067 { 1068 unsigned i; 1069 1070 for (i = 0; i < 10; i++) { 1071 h[i] = f[i] - g[i]; 1072 } 1073 } 1074 1075 /* 1076 * h = f * g 1077 * 1078 * Can overlap h with f or g. 1079 * 1080 * Preconditions: 1081 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1082 * |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1083 * 1084 * Postconditions: 1085 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1086 * 1087 * Notes on implementation strategy: 1088 * 1089 * Using schoolbook multiplication. 1090 * Karatsuba would save a little in some cost models. 1091 * 1092 * Most multiplications by 2 and 19 are 32-bit precomputations; 1093 * cheaper than 64-bit postcomputations. 1094 * 1095 * There is one remaining multiplication by 19 in the carry chain; 1096 * one *19 precomputation can be merged into this, 1097 * but the resulting data flow is considerably less clean. 1098 * 1099 * There are 12 carries below. 1100 * 10 of them are 2-way parallelizable and vectorizable. 1101 * Can get away with 11 carries, but then data flow is much deeper. 1102 * 1103 * With tighter constraints on inputs can squeeze carries into int32. 1104 */ 1105 static void fe_mul(fe h, const fe f, const fe g) 1106 { 1107 int32_t f0 = f[0]; 1108 int32_t f1 = f[1]; 1109 int32_t f2 = f[2]; 1110 int32_t f3 = f[3]; 1111 int32_t f4 = f[4]; 1112 int32_t f5 = f[5]; 1113 int32_t f6 = f[6]; 1114 int32_t f7 = f[7]; 1115 int32_t f8 = f[8]; 1116 int32_t f9 = f[9]; 1117 int32_t g0 = g[0]; 1118 int32_t g1 = g[1]; 1119 int32_t g2 = g[2]; 1120 int32_t g3 = g[3]; 1121 int32_t g4 = g[4]; 1122 int32_t g5 = g[5]; 1123 int32_t g6 = g[6]; 1124 int32_t g7 = g[7]; 1125 int32_t g8 = g[8]; 1126 int32_t g9 = g[9]; 1127 int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */ 1128 int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */ 1129 int32_t g3_19 = 19 * g3; 1130 int32_t g4_19 = 19 * g4; 1131 int32_t g5_19 = 19 * g5; 1132 int32_t g6_19 = 19 * g6; 1133 int32_t g7_19 = 19 * g7; 1134 int32_t g8_19 = 19 * g8; 1135 int32_t g9_19 = 19 * g9; 1136 int32_t f1_2 = 2 * f1; 1137 int32_t f3_2 = 2 * f3; 1138 int32_t f5_2 = 2 * f5; 1139 int32_t f7_2 = 2 * f7; 1140 int32_t f9_2 = 2 * f9; 1141 int64_t f0g0 = f0 * (int64_t)g0; 1142 int64_t f0g1 = f0 * (int64_t)g1; 1143 int64_t f0g2 = f0 * (int64_t)g2; 1144 int64_t f0g3 = f0 * (int64_t)g3; 1145 int64_t f0g4 = f0 * (int64_t)g4; 1146 int64_t f0g5 = f0 * (int64_t)g5; 1147 int64_t f0g6 = f0 * (int64_t)g6; 1148 int64_t f0g7 = f0 * (int64_t)g7; 1149 int64_t f0g8 = f0 * (int64_t)g8; 1150 int64_t f0g9 = f0 * (int64_t)g9; 1151 int64_t f1g0 = f1 * (int64_t)g0; 1152 int64_t f1g1_2 = f1_2 * (int64_t)g1; 1153 int64_t f1g2 = f1 * (int64_t)g2; 1154 int64_t f1g3_2 = f1_2 * (int64_t)g3; 1155 int64_t f1g4 = f1 * (int64_t)g4; 1156 int64_t f1g5_2 = f1_2 * (int64_t)g5; 1157 int64_t f1g6 = f1 * (int64_t)g6; 1158 int64_t f1g7_2 = f1_2 * (int64_t)g7; 1159 int64_t f1g8 = f1 * (int64_t)g8; 1160 int64_t f1g9_38 = f1_2 * (int64_t)g9_19; 1161 int64_t f2g0 = f2 * (int64_t)g0; 1162 int64_t f2g1 = f2 * (int64_t)g1; 1163 int64_t f2g2 = f2 * (int64_t)g2; 1164 int64_t f2g3 = f2 * (int64_t)g3; 1165 int64_t f2g4 = f2 * (int64_t)g4; 1166 int64_t f2g5 = f2 * (int64_t)g5; 1167 int64_t f2g6 = f2 * (int64_t)g6; 1168 int64_t f2g7 = f2 * (int64_t)g7; 1169 int64_t f2g8_19 = f2 * (int64_t)g8_19; 1170 int64_t f2g9_19 = f2 * (int64_t)g9_19; 1171 int64_t f3g0 = f3 * (int64_t)g0; 1172 int64_t f3g1_2 = f3_2 * (int64_t)g1; 1173 int64_t f3g2 = f3 * (int64_t)g2; 1174 int64_t f3g3_2 = f3_2 * (int64_t)g3; 1175 int64_t f3g4 = f3 * (int64_t)g4; 1176 int64_t f3g5_2 = f3_2 * (int64_t)g5; 1177 int64_t f3g6 = f3 * (int64_t)g6; 1178 int64_t f3g7_38 = f3_2 * (int64_t)g7_19; 1179 int64_t f3g8_19 = f3 * (int64_t)g8_19; 1180 int64_t f3g9_38 = f3_2 * (int64_t)g9_19; 1181 int64_t f4g0 = f4 * (int64_t)g0; 1182 int64_t f4g1 = f4 * (int64_t)g1; 1183 int64_t f4g2 = f4 * (int64_t)g2; 1184 int64_t f4g3 = f4 * (int64_t)g3; 1185 int64_t f4g4 = f4 * (int64_t)g4; 1186 int64_t f4g5 = f4 * (int64_t)g5; 1187 int64_t f4g6_19 = f4 * (int64_t)g6_19; 1188 int64_t f4g7_19 = f4 * (int64_t)g7_19; 1189 int64_t f4g8_19 = f4 * (int64_t)g8_19; 1190 int64_t f4g9_19 = f4 * (int64_t)g9_19; 1191 int64_t f5g0 = f5 * (int64_t)g0; 1192 int64_t f5g1_2 = f5_2 * (int64_t)g1; 1193 int64_t f5g2 = f5 * (int64_t)g2; 1194 int64_t f5g3_2 = f5_2 * (int64_t)g3; 1195 int64_t f5g4 = f5 * (int64_t)g4; 1196 int64_t f5g5_38 = f5_2 * (int64_t)g5_19; 1197 int64_t f5g6_19 = f5 * (int64_t)g6_19; 1198 int64_t f5g7_38 = f5_2 * (int64_t)g7_19; 1199 int64_t f5g8_19 = f5 * (int64_t)g8_19; 1200 int64_t f5g9_38 = f5_2 * (int64_t)g9_19; 1201 int64_t f6g0 = f6 * (int64_t)g0; 1202 int64_t f6g1 = f6 * (int64_t)g1; 1203 int64_t f6g2 = f6 * (int64_t)g2; 1204 int64_t f6g3 = f6 * (int64_t)g3; 1205 int64_t f6g4_19 = f6 * (int64_t)g4_19; 1206 int64_t f6g5_19 = f6 * (int64_t)g5_19; 1207 int64_t f6g6_19 = f6 * (int64_t)g6_19; 1208 int64_t f6g7_19 = f6 * (int64_t)g7_19; 1209 int64_t f6g8_19 = f6 * (int64_t)g8_19; 1210 int64_t f6g9_19 = f6 * (int64_t)g9_19; 1211 int64_t f7g0 = f7 * (int64_t)g0; 1212 int64_t f7g1_2 = f7_2 * (int64_t)g1; 1213 int64_t f7g2 = f7 * (int64_t)g2; 1214 int64_t f7g3_38 = f7_2 * (int64_t)g3_19; 1215 int64_t f7g4_19 = f7 * (int64_t)g4_19; 1216 int64_t f7g5_38 = f7_2 * (int64_t)g5_19; 1217 int64_t f7g6_19 = f7 * (int64_t)g6_19; 1218 int64_t f7g7_38 = f7_2 * (int64_t)g7_19; 1219 int64_t f7g8_19 = f7 * (int64_t)g8_19; 1220 int64_t f7g9_38 = f7_2 * (int64_t)g9_19; 1221 int64_t f8g0 = f8 * (int64_t)g0; 1222 int64_t f8g1 = f8 * (int64_t)g1; 1223 int64_t f8g2_19 = f8 * (int64_t)g2_19; 1224 int64_t f8g3_19 = f8 * (int64_t)g3_19; 1225 int64_t f8g4_19 = f8 * (int64_t)g4_19; 1226 int64_t f8g5_19 = f8 * (int64_t)g5_19; 1227 int64_t f8g6_19 = f8 * (int64_t)g6_19; 1228 int64_t f8g7_19 = f8 * (int64_t)g7_19; 1229 int64_t f8g8_19 = f8 * (int64_t)g8_19; 1230 int64_t f8g9_19 = f8 * (int64_t)g9_19; 1231 int64_t f9g0 = f9 * (int64_t)g0; 1232 int64_t f9g1_38 = f9_2 * (int64_t)g1_19; 1233 int64_t f9g2_19 = f9 * (int64_t)g2_19; 1234 int64_t f9g3_38 = f9_2 * (int64_t)g3_19; 1235 int64_t f9g4_19 = f9 * (int64_t)g4_19; 1236 int64_t f9g5_38 = f9_2 * (int64_t)g5_19; 1237 int64_t f9g6_19 = f9 * (int64_t)g6_19; 1238 int64_t f9g7_38 = f9_2 * (int64_t)g7_19; 1239 int64_t f9g8_19 = f9 * (int64_t)g8_19; 1240 int64_t f9g9_38 = f9_2 * (int64_t)g9_19; 1241 int64_t h0 = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38; 1242 int64_t h1 = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19; 1243 int64_t h2 = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38; 1244 int64_t h3 = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19; 1245 int64_t h4 = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38; 1246 int64_t h5 = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19; 1247 int64_t h6 = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38; 1248 int64_t h7 = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19; 1249 int64_t h8 = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38; 1250 int64_t h9 = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0; 1251 int64_t carry0; 1252 int64_t carry1; 1253 int64_t carry2; 1254 int64_t carry3; 1255 int64_t carry4; 1256 int64_t carry5; 1257 int64_t carry6; 1258 int64_t carry7; 1259 int64_t carry8; 1260 int64_t carry9; 1261 1262 /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38)) 1263 * i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8 1264 * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19)) 1265 * i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */ 1266 1267 carry0 = h0 + (1 << 25); 1268 h1 += carry0 >> 26; 1269 h0 -= carry0 & kTop38Bits; 1270 carry4 = h4 + (1 << 25); 1271 h5 += carry4 >> 26; 1272 h4 -= carry4 & kTop38Bits; 1273 /* |h0| <= 2^25 */ 1274 /* |h4| <= 2^25 */ 1275 /* |h1| <= 1.71*2^59 */ 1276 /* |h5| <= 1.71*2^59 */ 1277 1278 carry1 = h1 + (1 << 24); 1279 h2 += carry1 >> 25; 1280 h1 -= carry1 & kTop39Bits; 1281 carry5 = h5 + (1 << 24); 1282 h6 += carry5 >> 25; 1283 h5 -= carry5 & kTop39Bits; 1284 /* |h1| <= 2^24; from now on fits into int32 */ 1285 /* |h5| <= 2^24; from now on fits into int32 */ 1286 /* |h2| <= 1.41*2^60 */ 1287 /* |h6| <= 1.41*2^60 */ 1288 1289 carry2 = h2 + (1 << 25); 1290 h3 += carry2 >> 26; 1291 h2 -= carry2 & kTop38Bits; 1292 carry6 = h6 + (1 << 25); 1293 h7 += carry6 >> 26; 1294 h6 -= carry6 & kTop38Bits; 1295 /* |h2| <= 2^25; from now on fits into int32 unchanged */ 1296 /* |h6| <= 2^25; from now on fits into int32 unchanged */ 1297 /* |h3| <= 1.71*2^59 */ 1298 /* |h7| <= 1.71*2^59 */ 1299 1300 carry3 = h3 + (1 << 24); 1301 h4 += carry3 >> 25; 1302 h3 -= carry3 & kTop39Bits; 1303 carry7 = h7 + (1 << 24); 1304 h8 += carry7 >> 25; 1305 h7 -= carry7 & kTop39Bits; 1306 /* |h3| <= 2^24; from now on fits into int32 unchanged */ 1307 /* |h7| <= 2^24; from now on fits into int32 unchanged */ 1308 /* |h4| <= 1.72*2^34 */ 1309 /* |h8| <= 1.41*2^60 */ 1310 1311 carry4 = h4 + (1 << 25); 1312 h5 += carry4 >> 26; 1313 h4 -= carry4 & kTop38Bits; 1314 carry8 = h8 + (1 << 25); 1315 h9 += carry8 >> 26; 1316 h8 -= carry8 & kTop38Bits; 1317 /* |h4| <= 2^25; from now on fits into int32 unchanged */ 1318 /* |h8| <= 2^25; from now on fits into int32 unchanged */ 1319 /* |h5| <= 1.01*2^24 */ 1320 /* |h9| <= 1.71*2^59 */ 1321 1322 carry9 = h9 + (1 << 24); 1323 h0 += (carry9 >> 25) * 19; 1324 h9 -= carry9 & kTop39Bits; 1325 /* |h9| <= 2^24; from now on fits into int32 unchanged */ 1326 /* |h0| <= 1.1*2^39 */ 1327 1328 carry0 = h0 + (1 << 25); 1329 h1 += carry0 >> 26; 1330 h0 -= carry0 & kTop38Bits; 1331 /* |h0| <= 2^25; from now on fits into int32 unchanged */ 1332 /* |h1| <= 1.01*2^24 */ 1333 1334 h[0] = (int32_t)h0; 1335 h[1] = (int32_t)h1; 1336 h[2] = (int32_t)h2; 1337 h[3] = (int32_t)h3; 1338 h[4] = (int32_t)h4; 1339 h[5] = (int32_t)h5; 1340 h[6] = (int32_t)h6; 1341 h[7] = (int32_t)h7; 1342 h[8] = (int32_t)h8; 1343 h[9] = (int32_t)h9; 1344 } 1345 1346 /* 1347 * h = f * f 1348 * 1349 * Can overlap h with f. 1350 * 1351 * Preconditions: 1352 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1353 * 1354 * Postconditions: 1355 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1356 * 1357 * See fe_mul.c for discussion of implementation strategy. 1358 */ 1359 static void fe_sq(fe h, const fe f) 1360 { 1361 int32_t f0 = f[0]; 1362 int32_t f1 = f[1]; 1363 int32_t f2 = f[2]; 1364 int32_t f3 = f[3]; 1365 int32_t f4 = f[4]; 1366 int32_t f5 = f[5]; 1367 int32_t f6 = f[6]; 1368 int32_t f7 = f[7]; 1369 int32_t f8 = f[8]; 1370 int32_t f9 = f[9]; 1371 int32_t f0_2 = 2 * f0; 1372 int32_t f1_2 = 2 * f1; 1373 int32_t f2_2 = 2 * f2; 1374 int32_t f3_2 = 2 * f3; 1375 int32_t f4_2 = 2 * f4; 1376 int32_t f5_2 = 2 * f5; 1377 int32_t f6_2 = 2 * f6; 1378 int32_t f7_2 = 2 * f7; 1379 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */ 1380 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */ 1381 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */ 1382 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */ 1383 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */ 1384 int64_t f0f0 = f0 * (int64_t)f0; 1385 int64_t f0f1_2 = f0_2 * (int64_t)f1; 1386 int64_t f0f2_2 = f0_2 * (int64_t)f2; 1387 int64_t f0f3_2 = f0_2 * (int64_t)f3; 1388 int64_t f0f4_2 = f0_2 * (int64_t)f4; 1389 int64_t f0f5_2 = f0_2 * (int64_t)f5; 1390 int64_t f0f6_2 = f0_2 * (int64_t)f6; 1391 int64_t f0f7_2 = f0_2 * (int64_t)f7; 1392 int64_t f0f8_2 = f0_2 * (int64_t)f8; 1393 int64_t f0f9_2 = f0_2 * (int64_t)f9; 1394 int64_t f1f1_2 = f1_2 * (int64_t)f1; 1395 int64_t f1f2_2 = f1_2 * (int64_t)f2; 1396 int64_t f1f3_4 = f1_2 * (int64_t)f3_2; 1397 int64_t f1f4_2 = f1_2 * (int64_t)f4; 1398 int64_t f1f5_4 = f1_2 * (int64_t)f5_2; 1399 int64_t f1f6_2 = f1_2 * (int64_t)f6; 1400 int64_t f1f7_4 = f1_2 * (int64_t)f7_2; 1401 int64_t f1f8_2 = f1_2 * (int64_t)f8; 1402 int64_t f1f9_76 = f1_2 * (int64_t)f9_38; 1403 int64_t f2f2 = f2 * (int64_t)f2; 1404 int64_t f2f3_2 = f2_2 * (int64_t)f3; 1405 int64_t f2f4_2 = f2_2 * (int64_t)f4; 1406 int64_t f2f5_2 = f2_2 * (int64_t)f5; 1407 int64_t f2f6_2 = f2_2 * (int64_t)f6; 1408 int64_t f2f7_2 = f2_2 * (int64_t)f7; 1409 int64_t f2f8_38 = f2_2 * (int64_t)f8_19; 1410 int64_t f2f9_38 = f2 * (int64_t)f9_38; 1411 int64_t f3f3_2 = f3_2 * (int64_t)f3; 1412 int64_t f3f4_2 = f3_2 * (int64_t)f4; 1413 int64_t f3f5_4 = f3_2 * (int64_t)f5_2; 1414 int64_t f3f6_2 = f3_2 * (int64_t)f6; 1415 int64_t f3f7_76 = f3_2 * (int64_t)f7_38; 1416 int64_t f3f8_38 = f3_2 * (int64_t)f8_19; 1417 int64_t f3f9_76 = f3_2 * (int64_t)f9_38; 1418 int64_t f4f4 = f4 * (int64_t)f4; 1419 int64_t f4f5_2 = f4_2 * (int64_t)f5; 1420 int64_t f4f6_38 = f4_2 * (int64_t)f6_19; 1421 int64_t f4f7_38 = f4 * (int64_t)f7_38; 1422 int64_t f4f8_38 = f4_2 * (int64_t)f8_19; 1423 int64_t f4f9_38 = f4 * (int64_t)f9_38; 1424 int64_t f5f5_38 = f5 * (int64_t)f5_38; 1425 int64_t f5f6_38 = f5_2 * (int64_t)f6_19; 1426 int64_t f5f7_76 = f5_2 * (int64_t)f7_38; 1427 int64_t f5f8_38 = f5_2 * (int64_t)f8_19; 1428 int64_t f5f9_76 = f5_2 * (int64_t)f9_38; 1429 int64_t f6f6_19 = f6 * (int64_t)f6_19; 1430 int64_t f6f7_38 = f6 * (int64_t)f7_38; 1431 int64_t f6f8_38 = f6_2 * (int64_t)f8_19; 1432 int64_t f6f9_38 = f6 * (int64_t)f9_38; 1433 int64_t f7f7_38 = f7 * (int64_t)f7_38; 1434 int64_t f7f8_38 = f7_2 * (int64_t)f8_19; 1435 int64_t f7f9_76 = f7_2 * (int64_t)f9_38; 1436 int64_t f8f8_19 = f8 * (int64_t)f8_19; 1437 int64_t f8f9_38 = f8 * (int64_t)f9_38; 1438 int64_t f9f9_38 = f9 * (int64_t)f9_38; 1439 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38; 1440 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38; 1441 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19; 1442 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38; 1443 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38; 1444 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38; 1445 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19; 1446 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38; 1447 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38; 1448 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2; 1449 int64_t carry0; 1450 int64_t carry1; 1451 int64_t carry2; 1452 int64_t carry3; 1453 int64_t carry4; 1454 int64_t carry5; 1455 int64_t carry6; 1456 int64_t carry7; 1457 int64_t carry8; 1458 int64_t carry9; 1459 1460 carry0 = h0 + (1 << 25); 1461 h1 += carry0 >> 26; 1462 h0 -= carry0 & kTop38Bits; 1463 carry4 = h4 + (1 << 25); 1464 h5 += carry4 >> 26; 1465 h4 -= carry4 & kTop38Bits; 1466 1467 carry1 = h1 + (1 << 24); 1468 h2 += carry1 >> 25; 1469 h1 -= carry1 & kTop39Bits; 1470 carry5 = h5 + (1 << 24); 1471 h6 += carry5 >> 25; 1472 h5 -= carry5 & kTop39Bits; 1473 1474 carry2 = h2 + (1 << 25); 1475 h3 += carry2 >> 26; 1476 h2 -= carry2 & kTop38Bits; 1477 carry6 = h6 + (1 << 25); 1478 h7 += carry6 >> 26; 1479 h6 -= carry6 & kTop38Bits; 1480 1481 carry3 = h3 + (1 << 24); 1482 h4 += carry3 >> 25; 1483 h3 -= carry3 & kTop39Bits; 1484 carry7 = h7 + (1 << 24); 1485 h8 += carry7 >> 25; 1486 h7 -= carry7 & kTop39Bits; 1487 1488 carry4 = h4 + (1 << 25); 1489 h5 += carry4 >> 26; 1490 h4 -= carry4 & kTop38Bits; 1491 carry8 = h8 + (1 << 25); 1492 h9 += carry8 >> 26; 1493 h8 -= carry8 & kTop38Bits; 1494 1495 carry9 = h9 + (1 << 24); 1496 h0 += (carry9 >> 25) * 19; 1497 h9 -= carry9 & kTop39Bits; 1498 1499 carry0 = h0 + (1 << 25); 1500 h1 += carry0 >> 26; 1501 h0 -= carry0 & kTop38Bits; 1502 1503 h[0] = (int32_t)h0; 1504 h[1] = (int32_t)h1; 1505 h[2] = (int32_t)h2; 1506 h[3] = (int32_t)h3; 1507 h[4] = (int32_t)h4; 1508 h[5] = (int32_t)h5; 1509 h[6] = (int32_t)h6; 1510 h[7] = (int32_t)h7; 1511 h[8] = (int32_t)h8; 1512 h[9] = (int32_t)h9; 1513 } 1514 1515 static void fe_invert(fe out, const fe z) 1516 { 1517 fe t0; 1518 fe t1; 1519 fe t2; 1520 fe t3; 1521 int i; 1522 1523 /* 1524 * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as 1525 * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11. 1526 */ 1527 1528 /* t0 = z ** 2 */ 1529 fe_sq(t0, z); 1530 1531 /* t1 = t0 ** (2 ** 2) = z ** 8 */ 1532 fe_sq(t1, t0); 1533 fe_sq(t1, t1); 1534 1535 /* t1 = z * t1 = z ** 9 */ 1536 fe_mul(t1, z, t1); 1537 /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */ 1538 fe_mul(t0, t0, t1); 1539 1540 /* t2 = t0 ** 2 = z ** 22 */ 1541 fe_sq(t2, t0); 1542 1543 /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */ 1544 fe_mul(t1, t1, t2); 1545 1546 /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */ 1547 fe_sq(t2, t1); 1548 for (i = 1; i < 5; ++i) { 1549 fe_sq(t2, t2); 1550 } 1551 1552 /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */ 1553 fe_mul(t1, t2, t1); 1554 1555 /* Continuing similarly... */ 1556 1557 /* t2 = z ** (2 ** 20 - 1) */ 1558 fe_sq(t2, t1); 1559 for (i = 1; i < 10; ++i) { 1560 fe_sq(t2, t2); 1561 } 1562 fe_mul(t2, t2, t1); 1563 1564 /* t2 = z ** (2 ** 40 - 1) */ 1565 fe_sq(t3, t2); 1566 for (i = 1; i < 20; ++i) { 1567 fe_sq(t3, t3); 1568 } 1569 fe_mul(t2, t3, t2); 1570 1571 /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */ 1572 for (i = 0; i < 10; ++i) { 1573 fe_sq(t2, t2); 1574 } 1575 /* t1 = z ** (2 ** 50 - 1) */ 1576 fe_mul(t1, t2, t1); 1577 1578 /* t2 = z ** (2 ** 100 - 1) */ 1579 fe_sq(t2, t1); 1580 for (i = 1; i < 50; ++i) { 1581 fe_sq(t2, t2); 1582 } 1583 fe_mul(t2, t2, t1); 1584 1585 /* t2 = z ** (2 ** 200 - 1) */ 1586 fe_sq(t3, t2); 1587 for (i = 1; i < 100; ++i) { 1588 fe_sq(t3, t3); 1589 } 1590 fe_mul(t2, t3, t2); 1591 1592 /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */ 1593 fe_sq(t2, t2); 1594 for (i = 1; i < 50; ++i) { 1595 fe_sq(t2, t2); 1596 } 1597 1598 /* t1 = z ** (2 ** 250 - 1) */ 1599 fe_mul(t1, t2, t1); 1600 1601 /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */ 1602 fe_sq(t1, t1); 1603 for (i = 1; i < 5; ++i) { 1604 fe_sq(t1, t1); 1605 } 1606 1607 /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */ 1608 fe_mul(out, t1, t0); 1609 } 1610 1611 /* 1612 * h = -f 1613 * 1614 * Preconditions: 1615 * |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1616 * 1617 * Postconditions: 1618 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 1619 */ 1620 static void fe_neg(fe h, const fe f) 1621 { 1622 unsigned i; 1623 1624 for (i = 0; i < 10; i++) { 1625 h[i] = -f[i]; 1626 } 1627 } 1628 1629 /* 1630 * Replace (f,g) with (g,g) if b == 1; 1631 * replace (f,g) with (f,g) if b == 0. 1632 * 1633 * Preconditions: b in {0,1}. 1634 */ 1635 static void fe_cmov(fe f, const fe g, unsigned b) 1636 { 1637 size_t i; 1638 1639 b = 0 - b; 1640 for (i = 0; i < 10; i++) { 1641 int32_t x = f[i] ^ g[i]; 1642 x &= b; 1643 f[i] ^= x; 1644 } 1645 } 1646 1647 /* 1648 * return 0 if f == 0 1649 * return 1 if f != 0 1650 * 1651 * Preconditions: 1652 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1653 */ 1654 static int fe_isnonzero(const fe f) 1655 { 1656 uint8_t s[32]; 1657 static const uint8_t zero[32] = { 0 }; 1658 1659 fe_tobytes(s, f); 1660 1661 return CRYPTO_memcmp(s, zero, sizeof(zero)) != 0; 1662 } 1663 1664 /* 1665 * return 1 if f is in {1,3,5,...,q-2} 1666 * return 0 if f is in {0,2,4,...,q-1} 1667 * 1668 * Preconditions: 1669 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 1670 */ 1671 static int fe_isnegative(const fe f) 1672 { 1673 uint8_t s[32]; 1674 1675 fe_tobytes(s, f); 1676 return s[0] & 1; 1677 } 1678 1679 /* 1680 * h = 2 * f * f 1681 * 1682 * Can overlap h with f. 1683 * 1684 * Preconditions: 1685 * |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. 1686 * 1687 * Postconditions: 1688 * |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. 1689 * 1690 * See fe_mul.c for discussion of implementation strategy. 1691 */ 1692 static void fe_sq2(fe h, const fe f) 1693 { 1694 int32_t f0 = f[0]; 1695 int32_t f1 = f[1]; 1696 int32_t f2 = f[2]; 1697 int32_t f3 = f[3]; 1698 int32_t f4 = f[4]; 1699 int32_t f5 = f[5]; 1700 int32_t f6 = f[6]; 1701 int32_t f7 = f[7]; 1702 int32_t f8 = f[8]; 1703 int32_t f9 = f[9]; 1704 int32_t f0_2 = 2 * f0; 1705 int32_t f1_2 = 2 * f1; 1706 int32_t f2_2 = 2 * f2; 1707 int32_t f3_2 = 2 * f3; 1708 int32_t f4_2 = 2 * f4; 1709 int32_t f5_2 = 2 * f5; 1710 int32_t f6_2 = 2 * f6; 1711 int32_t f7_2 = 2 * f7; 1712 int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */ 1713 int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */ 1714 int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */ 1715 int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */ 1716 int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */ 1717 int64_t f0f0 = f0 * (int64_t)f0; 1718 int64_t f0f1_2 = f0_2 * (int64_t)f1; 1719 int64_t f0f2_2 = f0_2 * (int64_t)f2; 1720 int64_t f0f3_2 = f0_2 * (int64_t)f3; 1721 int64_t f0f4_2 = f0_2 * (int64_t)f4; 1722 int64_t f0f5_2 = f0_2 * (int64_t)f5; 1723 int64_t f0f6_2 = f0_2 * (int64_t)f6; 1724 int64_t f0f7_2 = f0_2 * (int64_t)f7; 1725 int64_t f0f8_2 = f0_2 * (int64_t)f8; 1726 int64_t f0f9_2 = f0_2 * (int64_t)f9; 1727 int64_t f1f1_2 = f1_2 * (int64_t)f1; 1728 int64_t f1f2_2 = f1_2 * (int64_t)f2; 1729 int64_t f1f3_4 = f1_2 * (int64_t)f3_2; 1730 int64_t f1f4_2 = f1_2 * (int64_t)f4; 1731 int64_t f1f5_4 = f1_2 * (int64_t)f5_2; 1732 int64_t f1f6_2 = f1_2 * (int64_t)f6; 1733 int64_t f1f7_4 = f1_2 * (int64_t)f7_2; 1734 int64_t f1f8_2 = f1_2 * (int64_t)f8; 1735 int64_t f1f9_76 = f1_2 * (int64_t)f9_38; 1736 int64_t f2f2 = f2 * (int64_t)f2; 1737 int64_t f2f3_2 = f2_2 * (int64_t)f3; 1738 int64_t f2f4_2 = f2_2 * (int64_t)f4; 1739 int64_t f2f5_2 = f2_2 * (int64_t)f5; 1740 int64_t f2f6_2 = f2_2 * (int64_t)f6; 1741 int64_t f2f7_2 = f2_2 * (int64_t)f7; 1742 int64_t f2f8_38 = f2_2 * (int64_t)f8_19; 1743 int64_t f2f9_38 = f2 * (int64_t)f9_38; 1744 int64_t f3f3_2 = f3_2 * (int64_t)f3; 1745 int64_t f3f4_2 = f3_2 * (int64_t)f4; 1746 int64_t f3f5_4 = f3_2 * (int64_t)f5_2; 1747 int64_t f3f6_2 = f3_2 * (int64_t)f6; 1748 int64_t f3f7_76 = f3_2 * (int64_t)f7_38; 1749 int64_t f3f8_38 = f3_2 * (int64_t)f8_19; 1750 int64_t f3f9_76 = f3_2 * (int64_t)f9_38; 1751 int64_t f4f4 = f4 * (int64_t)f4; 1752 int64_t f4f5_2 = f4_2 * (int64_t)f5; 1753 int64_t f4f6_38 = f4_2 * (int64_t)f6_19; 1754 int64_t f4f7_38 = f4 * (int64_t)f7_38; 1755 int64_t f4f8_38 = f4_2 * (int64_t)f8_19; 1756 int64_t f4f9_38 = f4 * (int64_t)f9_38; 1757 int64_t f5f5_38 = f5 * (int64_t)f5_38; 1758 int64_t f5f6_38 = f5_2 * (int64_t)f6_19; 1759 int64_t f5f7_76 = f5_2 * (int64_t)f7_38; 1760 int64_t f5f8_38 = f5_2 * (int64_t)f8_19; 1761 int64_t f5f9_76 = f5_2 * (int64_t)f9_38; 1762 int64_t f6f6_19 = f6 * (int64_t)f6_19; 1763 int64_t f6f7_38 = f6 * (int64_t)f7_38; 1764 int64_t f6f8_38 = f6_2 * (int64_t)f8_19; 1765 int64_t f6f9_38 = f6 * (int64_t)f9_38; 1766 int64_t f7f7_38 = f7 * (int64_t)f7_38; 1767 int64_t f7f8_38 = f7_2 * (int64_t)f8_19; 1768 int64_t f7f9_76 = f7_2 * (int64_t)f9_38; 1769 int64_t f8f8_19 = f8 * (int64_t)f8_19; 1770 int64_t f8f9_38 = f8 * (int64_t)f9_38; 1771 int64_t f9f9_38 = f9 * (int64_t)f9_38; 1772 int64_t h0 = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38; 1773 int64_t h1 = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38; 1774 int64_t h2 = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19; 1775 int64_t h3 = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38; 1776 int64_t h4 = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38; 1777 int64_t h5 = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38; 1778 int64_t h6 = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19; 1779 int64_t h7 = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38; 1780 int64_t h8 = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38; 1781 int64_t h9 = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2; 1782 int64_t carry0; 1783 int64_t carry1; 1784 int64_t carry2; 1785 int64_t carry3; 1786 int64_t carry4; 1787 int64_t carry5; 1788 int64_t carry6; 1789 int64_t carry7; 1790 int64_t carry8; 1791 int64_t carry9; 1792 1793 h0 += h0; 1794 h1 += h1; 1795 h2 += h2; 1796 h3 += h3; 1797 h4 += h4; 1798 h5 += h5; 1799 h6 += h6; 1800 h7 += h7; 1801 h8 += h8; 1802 h9 += h9; 1803 1804 carry0 = h0 + (1 << 25); 1805 h1 += carry0 >> 26; 1806 h0 -= carry0 & kTop38Bits; 1807 carry4 = h4 + (1 << 25); 1808 h5 += carry4 >> 26; 1809 h4 -= carry4 & kTop38Bits; 1810 1811 carry1 = h1 + (1 << 24); 1812 h2 += carry1 >> 25; 1813 h1 -= carry1 & kTop39Bits; 1814 carry5 = h5 + (1 << 24); 1815 h6 += carry5 >> 25; 1816 h5 -= carry5 & kTop39Bits; 1817 1818 carry2 = h2 + (1 << 25); 1819 h3 += carry2 >> 26; 1820 h2 -= carry2 & kTop38Bits; 1821 carry6 = h6 + (1 << 25); 1822 h7 += carry6 >> 26; 1823 h6 -= carry6 & kTop38Bits; 1824 1825 carry3 = h3 + (1 << 24); 1826 h4 += carry3 >> 25; 1827 h3 -= carry3 & kTop39Bits; 1828 carry7 = h7 + (1 << 24); 1829 h8 += carry7 >> 25; 1830 h7 -= carry7 & kTop39Bits; 1831 1832 carry4 = h4 + (1 << 25); 1833 h5 += carry4 >> 26; 1834 h4 -= carry4 & kTop38Bits; 1835 carry8 = h8 + (1 << 25); 1836 h9 += carry8 >> 26; 1837 h8 -= carry8 & kTop38Bits; 1838 1839 carry9 = h9 + (1 << 24); 1840 h0 += (carry9 >> 25) * 19; 1841 h9 -= carry9 & kTop39Bits; 1842 1843 carry0 = h0 + (1 << 25); 1844 h1 += carry0 >> 26; 1845 h0 -= carry0 & kTop38Bits; 1846 1847 h[0] = (int32_t)h0; 1848 h[1] = (int32_t)h1; 1849 h[2] = (int32_t)h2; 1850 h[3] = (int32_t)h3; 1851 h[4] = (int32_t)h4; 1852 h[5] = (int32_t)h5; 1853 h[6] = (int32_t)h6; 1854 h[7] = (int32_t)h7; 1855 h[8] = (int32_t)h8; 1856 h[9] = (int32_t)h9; 1857 } 1858 1859 static void fe_pow22523(fe out, const fe z) 1860 { 1861 fe t0; 1862 fe t1; 1863 fe t2; 1864 int i; 1865 1866 fe_sq(t0, z); 1867 fe_sq(t1, t0); 1868 for (i = 1; i < 2; ++i) { 1869 fe_sq(t1, t1); 1870 } 1871 fe_mul(t1, z, t1); 1872 fe_mul(t0, t0, t1); 1873 fe_sq(t0, t0); 1874 fe_mul(t0, t1, t0); 1875 fe_sq(t1, t0); 1876 for (i = 1; i < 5; ++i) { 1877 fe_sq(t1, t1); 1878 } 1879 fe_mul(t0, t1, t0); 1880 fe_sq(t1, t0); 1881 for (i = 1; i < 10; ++i) { 1882 fe_sq(t1, t1); 1883 } 1884 fe_mul(t1, t1, t0); 1885 fe_sq(t2, t1); 1886 for (i = 1; i < 20; ++i) { 1887 fe_sq(t2, t2); 1888 } 1889 fe_mul(t1, t2, t1); 1890 fe_sq(t1, t1); 1891 for (i = 1; i < 10; ++i) { 1892 fe_sq(t1, t1); 1893 } 1894 fe_mul(t0, t1, t0); 1895 fe_sq(t1, t0); 1896 for (i = 1; i < 50; ++i) { 1897 fe_sq(t1, t1); 1898 } 1899 fe_mul(t1, t1, t0); 1900 fe_sq(t2, t1); 1901 for (i = 1; i < 100; ++i) { 1902 fe_sq(t2, t2); 1903 } 1904 fe_mul(t1, t2, t1); 1905 fe_sq(t1, t1); 1906 for (i = 1; i < 50; ++i) { 1907 fe_sq(t1, t1); 1908 } 1909 fe_mul(t0, t1, t0); 1910 fe_sq(t0, t0); 1911 for (i = 1; i < 2; ++i) { 1912 fe_sq(t0, t0); 1913 } 1914 fe_mul(out, t0, z); 1915 } 1916 1917 /* 1918 * ge means group element. 1919 * 1920 * Here the group is the set of pairs (x,y) of field elements (see fe.h) 1921 * satisfying -x^2 + y^2 = 1 + d x^2y^2 1922 * where d = -121665/121666. 1923 * 1924 * Representations: 1925 * ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z 1926 * ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT 1927 * ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T 1928 * ge_precomp (Duif): (y+x,y-x,2dxy) 1929 */ 1930 typedef struct { 1931 fe X; 1932 fe Y; 1933 fe Z; 1934 } ge_p2; 1935 1936 typedef struct { 1937 fe X; 1938 fe Y; 1939 fe Z; 1940 fe T; 1941 } ge_p3; 1942 1943 typedef struct { 1944 fe X; 1945 fe Y; 1946 fe Z; 1947 fe T; 1948 } ge_p1p1; 1949 1950 typedef struct { 1951 fe yplusx; 1952 fe yminusx; 1953 fe xy2d; 1954 } ge_precomp; 1955 1956 typedef struct { 1957 fe YplusX; 1958 fe YminusX; 1959 fe Z; 1960 fe T2d; 1961 } ge_cached; 1962 1963 static void ge_tobytes(uint8_t *s, const ge_p2 *h) 1964 { 1965 fe recip; 1966 fe x; 1967 fe y; 1968 1969 fe_invert(recip, h->Z); 1970 fe_mul(x, h->X, recip); 1971 fe_mul(y, h->Y, recip); 1972 fe_tobytes(s, y); 1973 s[31] ^= fe_isnegative(x) << 7; 1974 } 1975 1976 static void ge_p3_tobytes(uint8_t *s, const ge_p3 *h) 1977 { 1978 fe recip; 1979 fe x; 1980 fe y; 1981 1982 fe_invert(recip, h->Z); 1983 fe_mul(x, h->X, recip); 1984 fe_mul(y, h->Y, recip); 1985 fe_tobytes(s, y); 1986 s[31] ^= fe_isnegative(x) << 7; 1987 } 1988 1989 static const fe d = { 1990 -10913610, 13857413, -15372611, 6949391, 114729, 1991 -8787816, -6275908, -3247719, -18696448, -12055116 1992 }; 1993 1994 static const fe sqrtm1 = { 1995 -32595792, -7943725, 9377950, 3500415, 12389472, 1996 -272473, -25146209, -2005654, 326686, 11406482 1997 }; 1998 1999 static int ge_frombytes_vartime(ge_p3 *h, const uint8_t *s) 2000 { 2001 fe u; 2002 fe v; 2003 fe w; 2004 fe vxx; 2005 fe check; 2006 2007 fe_frombytes(h->Y, s); 2008 fe_1(h->Z); 2009 fe_sq(u, h->Y); 2010 fe_mul(v, u, d); 2011 fe_sub(u, u, h->Z); /* u = y^2-1 */ 2012 fe_add(v, v, h->Z); /* v = dy^2+1 */ 2013 2014 fe_mul(w, u, v); /* w = u*v */ 2015 2016 fe_pow22523(h->X, w); /* x = w^((q-5)/8) */ 2017 fe_mul(h->X, h->X, u); /* x = u * w^((q-5)/8) */ 2018 2019 fe_sq(vxx, h->X); 2020 fe_mul(vxx, vxx, v); 2021 fe_sub(check, vxx, u); /* vx^2-u */ 2022 if (fe_isnonzero(check)) { 2023 fe_add(check, vxx, u); /* vx^2+u */ 2024 if (fe_isnonzero(check)) { 2025 return -1; 2026 } 2027 fe_mul(h->X, h->X, sqrtm1); 2028 } 2029 2030 if (fe_isnegative(h->X) != (s[31] >> 7)) { 2031 fe_neg(h->X, h->X); 2032 } 2033 2034 fe_mul(h->T, h->X, h->Y); 2035 return 0; 2036 } 2037 2038 static void ge_p2_0(ge_p2 *h) 2039 { 2040 fe_0(h->X); 2041 fe_1(h->Y); 2042 fe_1(h->Z); 2043 } 2044 2045 static void ge_p3_0(ge_p3 *h) 2046 { 2047 fe_0(h->X); 2048 fe_1(h->Y); 2049 fe_1(h->Z); 2050 fe_0(h->T); 2051 } 2052 2053 static void ge_precomp_0(ge_precomp *h) 2054 { 2055 fe_1(h->yplusx); 2056 fe_1(h->yminusx); 2057 fe_0(h->xy2d); 2058 } 2059 2060 /* r = p */ 2061 static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) 2062 { 2063 fe_copy(r->X, p->X); 2064 fe_copy(r->Y, p->Y); 2065 fe_copy(r->Z, p->Z); 2066 } 2067 2068 static const fe d2 = { 2069 -21827239, -5839606, -30745221, 13898782, 229458, 2070 15978800, -12551817, -6495438, 29715968, 9444199 2071 }; 2072 2073 /* r = p */ 2074 static void ge_p3_to_cached(ge_cached *r, const ge_p3 *p) 2075 { 2076 fe_add(r->YplusX, p->Y, p->X); 2077 fe_sub(r->YminusX, p->Y, p->X); 2078 fe_copy(r->Z, p->Z); 2079 fe_mul(r->T2d, p->T, d2); 2080 } 2081 2082 /* r = p */ 2083 static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) 2084 { 2085 fe_mul(r->X, p->X, p->T); 2086 fe_mul(r->Y, p->Y, p->Z); 2087 fe_mul(r->Z, p->Z, p->T); 2088 } 2089 2090 /* r = p */ 2091 static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p) 2092 { 2093 fe_mul(r->X, p->X, p->T); 2094 fe_mul(r->Y, p->Y, p->Z); 2095 fe_mul(r->Z, p->Z, p->T); 2096 fe_mul(r->T, p->X, p->Y); 2097 } 2098 2099 /* r = 2 * p */ 2100 static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p) 2101 { 2102 fe t0; 2103 2104 fe_sq(r->X, p->X); 2105 fe_sq(r->Z, p->Y); 2106 fe_sq2(r->T, p->Z); 2107 fe_add(r->Y, p->X, p->Y); 2108 fe_sq(t0, r->Y); 2109 fe_add(r->Y, r->Z, r->X); 2110 fe_sub(r->Z, r->Z, r->X); 2111 fe_sub(r->X, t0, r->Y); 2112 fe_sub(r->T, r->T, r->Z); 2113 } 2114 2115 /* r = 2 * p */ 2116 static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p) 2117 { 2118 ge_p2 q; 2119 ge_p3_to_p2(&q, p); 2120 ge_p2_dbl(r, &q); 2121 } 2122 2123 /* r = p + q */ 2124 static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) 2125 { 2126 fe t0; 2127 2128 fe_add(r->X, p->Y, p->X); 2129 fe_sub(r->Y, p->Y, p->X); 2130 fe_mul(r->Z, r->X, q->yplusx); 2131 fe_mul(r->Y, r->Y, q->yminusx); 2132 fe_mul(r->T, q->xy2d, p->T); 2133 fe_add(t0, p->Z, p->Z); 2134 fe_sub(r->X, r->Z, r->Y); 2135 fe_add(r->Y, r->Z, r->Y); 2136 fe_add(r->Z, t0, r->T); 2137 fe_sub(r->T, t0, r->T); 2138 } 2139 2140 /* r = p - q */ 2141 static void ge_msub(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) 2142 { 2143 fe t0; 2144 2145 fe_add(r->X, p->Y, p->X); 2146 fe_sub(r->Y, p->Y, p->X); 2147 fe_mul(r->Z, r->X, q->yminusx); 2148 fe_mul(r->Y, r->Y, q->yplusx); 2149 fe_mul(r->T, q->xy2d, p->T); 2150 fe_add(t0, p->Z, p->Z); 2151 fe_sub(r->X, r->Z, r->Y); 2152 fe_add(r->Y, r->Z, r->Y); 2153 fe_sub(r->Z, t0, r->T); 2154 fe_add(r->T, t0, r->T); 2155 } 2156 2157 /* r = p + q */ 2158 static void ge_add(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) 2159 { 2160 fe t0; 2161 2162 fe_add(r->X, p->Y, p->X); 2163 fe_sub(r->Y, p->Y, p->X); 2164 fe_mul(r->Z, r->X, q->YplusX); 2165 fe_mul(r->Y, r->Y, q->YminusX); 2166 fe_mul(r->T, q->T2d, p->T); 2167 fe_mul(r->X, p->Z, q->Z); 2168 fe_add(t0, r->X, r->X); 2169 fe_sub(r->X, r->Z, r->Y); 2170 fe_add(r->Y, r->Z, r->Y); 2171 fe_add(r->Z, t0, r->T); 2172 fe_sub(r->T, t0, r->T); 2173 } 2174 2175 /* r = p - q */ 2176 static void ge_sub(ge_p1p1 *r, const ge_p3 *p, const ge_cached *q) 2177 { 2178 fe t0; 2179 2180 fe_add(r->X, p->Y, p->X); 2181 fe_sub(r->Y, p->Y, p->X); 2182 fe_mul(r->Z, r->X, q->YminusX); 2183 fe_mul(r->Y, r->Y, q->YplusX); 2184 fe_mul(r->T, q->T2d, p->T); 2185 fe_mul(r->X, p->Z, q->Z); 2186 fe_add(t0, r->X, r->X); 2187 fe_sub(r->X, r->Z, r->Y); 2188 fe_add(r->Y, r->Z, r->Y); 2189 fe_sub(r->Z, t0, r->T); 2190 fe_add(r->T, t0, r->T); 2191 } 2192 2193 static uint8_t equal(signed char b, signed char c) 2194 { 2195 uint8_t ub = b; 2196 uint8_t uc = c; 2197 uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */ 2198 uint32_t y = x; /* 0: yes; 1..255: no */ 2199 y -= 1; /* 4294967295: yes; 0..254: no */ 2200 y >>= 31; /* 1: yes; 0: no */ 2201 return y; 2202 } 2203 2204 static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b) 2205 { 2206 fe_cmov(t->yplusx, u->yplusx, b); 2207 fe_cmov(t->yminusx, u->yminusx, b); 2208 fe_cmov(t->xy2d, u->xy2d, b); 2209 } 2210 2211 /* k25519Precomp[i][j] = (j+1)*256^i*B */ 2212 static const ge_precomp k25519Precomp[32][8] = { 2213 { 2214 { 2215 { 25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 2216 27544626, -11754271, -6079156, 2047605 }, 2217 { -12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 2218 5043384, 19500929, -15469378 }, 2219 { -8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 2220 29287919, 11864899, -24514362, -4438546 }, 2221 }, 2222 { 2223 { -12815894, -12976347, -21581243, 11784320, -25355658, -2750717, 2224 -11717903, -3814571, -358445, -10211303 }, 2225 { -21703237, 6903825, 27185491, 6451973, -29577724, -9554005, 2226 -15616551, 11189268, -26829678, -5319081 }, 2227 { 26966642, 11152617, 32442495, 15396054, 14353839, -12752335, 2228 -3128826, -9541118, -15472047, -4166697 }, 2229 }, 2230 { 2231 { 15636291, -9688557, 24204773, -7912398, 616977, -16685262, 2232 27787600, -14772189, 28944400, -1550024 }, 2233 { 16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 2234 16354577, -11775962, 7689662, 11199574 }, 2235 { 30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 2236 7512774, 10017326, -17749093, -9920357 }, 2237 }, 2238 { 2239 { -17036878, 13921892, 10945806, -6033431, 27105052, -16084379, 2240 -28926210, 15006023, 3284568, -6276540 }, 2241 { 23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 2242 7464579, 9656445, 13059162, 10374397 }, 2243 { 7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, 2244 -3839045, -641708, -101325 }, 2245 }, 2246 { 2247 { 10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 2248 32867885, 14515107, -15438304, 10819380 }, 2249 { 4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 2250 12483688, -12668491, 5581306 }, 2251 { 19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 2252 28542350, 13850243, -23678021, -15815942 }, 2253 }, 2254 { 2255 { -15371964, -12862754, 32573250, 4720197, -26436522, 5875511, 2256 -19188627, -15224819, -9818940, -12085777 }, 2257 { -8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, 2258 -15689887, 1762328, 14866737 }, 2259 { -18199695, -15951423, -10473290, 1707278, -17185920, 3916101, 2260 -28236412, 3959421, 27914454, 4383652 }, 2261 }, 2262 { 2263 { 5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 2264 5230134, -23952439, -15175766 }, 2265 { -30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 2266 20654025, 16520125, 30598449, 7715701 }, 2267 { 28881845, 14381568, 9657904, 3680757, -20181635, 7843316, 2268 -31400660, 1370708, 29794553, -1409300 }, 2269 }, 2270 { 2271 { 14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 2272 30290735, 10876454, -33154098, 2381726 }, 2273 { -7195431, -2655363, -14730155, 462251, -27724326, 3941372, 2274 -6236617, 3696005, -32300832, 15351955 }, 2275 { 27431194, 8222322, 16448760, -3907995, -18707002, 11938355, 2276 -32961401, -2970515, 29551813, 10109425 }, 2277 }, 2278 }, 2279 { 2280 { 2281 { -13657040, -13155431, -31283750, 11777098, 21447386, 6519384, 2282 -2378284, -1627556, 10092783, -4764171 }, 2283 { 27939166, 14210322, 4677035, 16277044, -22964462, -12398139, 2284 -32508754, 12005538, -17810127, 12803510 }, 2285 { 17228999, -15661624, -1233527, 300140, -1224870, -11714777, 2286 30364213, -9038194, 18016357, 4397660 }, 2287 }, 2288 { 2289 { -10958843, -7690207, 4776341, -14954238, 27850028, -15602212, 2290 -26619106, 14544525, -17477504, 982639 }, 2291 { 29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, 2292 -4120128, -21047696, 9934963 }, 2293 { 5793303, 16271923, -24131614, -10116404, 29188560, 1206517, 2294 -14747930, 4559895, -30123922, -10897950 }, 2295 }, 2296 { 2297 { -27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 2298 24191034, 4541697, -13338309, 5500568 }, 2299 { 12650548, -1497113, 9052871, 11355358, -17680037, -8400164, 2300 -17430592, 12264343, 10874051, 13524335 }, 2301 { 25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 2302 5080568, -22528059, 5376628 }, 2303 }, 2304 { 2305 { -26088264, -4011052, -17013699, -3537628, -6726793, 1920897, 2306 -22321305, -9447443, 4535768, 1569007 }, 2307 { -2255422, 14606630, -21692440, -8039818, 28430649, 8775819, 2308 -30494562, 3044290, 31848280, 12543772 }, 2309 { -22028579, 2943893, -31857513, 6777306, 13784462, -4292203, 2310 -27377195, -2062731, 7718482, 14474653 }, 2311 }, 2312 { 2313 { 2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, 2314 -7236665, 24316168, -5253567 }, 2315 { 13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 2316 33040651, -13424532, -20729456, 8321686 }, 2317 { 21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 2318 23845965, -11874838, -9984458, 608372 }, 2319 }, 2320 { 2321 { -13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 2322 1123968, -6780577, 27229399, 23887 }, 2323 { -23244140, -294205, -11744728, 14712571, -29465699, -2029617, 2324 12797024, -6440308, -1633405, 16678954 }, 2325 { -29500620, 4770662, -16054387, 14001338, 7830047, 9564805, 2326 -1508144, -4795045, -17169265, 4904953 }, 2327 }, 2328 { 2329 { 24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 2330 5169211, 16191880, 2128236, -4326833 }, 2331 { -16981152, 4124966, -8540610, -10653797, 30336522, -14105247, 2332 -29806336, 916033, -6882542, -2986532 }, 2333 { -22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 2334 285431, 2763829, 15736322, 4143876 }, 2335 }, 2336 { 2337 { 2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, 2338 -14594663, 23527084, -16458268 }, 2339 { 33431127, -11130478, -17838966, -15626900, 8909499, 8376530, 2340 -32625340, 4087881, -15188911, -14416214 }, 2341 { 1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055, 2342 4357868, -4774191, -16323038 }, 2343 }, 2344 }, 2345 { 2346 { 2347 { 6721966, 13833823, -23523388, -1551314, 26354293, -11863321, 2348 23365147, -3949732, 7390890, 2759800 }, 2349 { 4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353, 2350 -4264057, 1244380, -12919645 }, 2351 { -4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 2352 9208236, 15886429, 16489664 }, 2353 }, 2354 { 2355 { 1996075, 10375649, 14346367, 13311202, -6874135, -16438411, 2356 -13693198, 398369, -30606455, -712933 }, 2357 { -25307465, 9795880, -2777414, 14878809, -33531835, 14780363, 2358 13348553, 12076947, -30836462, 5113182 }, 2359 { -17770784, 11797796, 31950843, 13929123, -25888302, 12288344, 2360 -30341101, -7336386, 13847711, 5387222 }, 2361 }, 2362 { 2363 { -18582163, -3416217, 17824843, -2340966, 22744343, -10442611, 2364 8763061, 3617786, -19600662, 10370991 }, 2365 { 20246567, -14369378, 22358229, -543712, 18507283, -10413996, 2366 14554437, -8746092, 32232924, 16763880 }, 2367 { 9648505, 10094563, 26416693, 14745928, -30374318, -6472621, 2368 11094161, 15689506, 3140038, -16510092 }, 2369 }, 2370 { 2371 { -16160072, 5472695, 31895588, 4744994, 8823515, 10365685, 2372 -27224800, 9448613, -28774454, 366295 }, 2373 { 19153450, 11523972, -11096490, -6503142, -24647631, 5420647, 2374 28344573, 8041113, 719605, 11671788 }, 2375 { 8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916, 2376 -15266516, 27000813, -10195553 }, 2377 }, 2378 { 2379 { -15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065, 2380 5336097, 6750977, -14521026 }, 2381 { 11836410, -3979488, 26297894, 16080799, 23455045, 15735944, 2382 1695823, -8819122, 8169720, 16220347 }, 2383 { -18115838, 8653647, 17578566, -6092619, -8025777, -16012763, 2384 -11144307, -2627664, -5990708, -14166033 }, 2385 }, 2386 { 2387 { -23308498, -10968312, 15213228, -10081214, -30853605, -11050004, 2388 27884329, 2847284, 2655861, 1738395 }, 2389 { -27537433, -14253021, -25336301, -8002780, -9370762, 8129821, 2390 21651608, -3239336, -19087449, -11005278 }, 2391 { 1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092, 2392 5821408, 10478196, 8544890 }, 2393 }, 2394 { 2395 { 32173121, -16129311, 24896207, 3921497, 22579056, -3410854, 2396 19270449, 12217473, 17789017, -3395995 }, 2397 { -30552961, -2228401, -15578829, -10147201, 13243889, 517024, 2398 15479401, -3853233, 30460520, 1052596 }, 2399 { -11614875, 13323618, 32618793, 8175907, -15230173, 12596687, 2400 27491595, -4612359, 3179268, -9478891 }, 2401 }, 2402 { 2403 { 31947069, -14366651, -4640583, -15339921, -15125977, -6039709, 2404 -14756777, -16411740, 19072640, -9511060 }, 2405 { 11685058, 11822410, 3158003, -13952594, 33402194, -4165066, 2406 5977896, -5215017, 473099, 5040608 }, 2407 { -20290863, 8198642, -27410132, 11602123, 1290375, -2799760, 2408 28326862, 1721092, -19558642, -3131606 }, 2409 }, 2410 }, 2411 { 2412 { 2413 { 7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786, 2414 8076149, -27868496, 11538389 }, 2415 { -19935666, 3899861, 18283497, -6801568, -15728660, -11249211, 2416 8754525, 7446702, -5676054, 5797016 }, 2417 { -11295600, -3793569, -15782110, -7964573, 12708869, -8456199, 2418 2014099, -9050574, -2369172, -5877341 }, 2419 }, 2420 { 2421 { -22472376, -11568741, -27682020, 1146375, 18956691, 16640559, 2422 1192730, -3714199, 15123619, 10811505 }, 2423 { 14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363, 2424 15776356, -28886779, -11974553 }, 2425 { -28241164, -8072475, -4978962, -5315317, 29416931, 1847569, 2426 -20654173, -16484855, 4714547, -9600655 }, 2427 }, 2428 { 2429 { 15200332, 8368572, 19679101, 15970074, -31872674, 1959451, 2430 24611599, -4543832, -11745876, 12340220 }, 2431 { 12876937, -10480056, 33134381, 6590940, -6307776, 14872440, 2432 9613953, 8241152, 15370987, 9608631 }, 2433 { -4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868, 2434 15866074, -28210621, -8814099 }, 2435 }, 2436 { 2437 { 26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233, 2438 858697, 20571223, 8420556 }, 2439 { 14620715, 13067227, -15447274, 8264467, 14106269, 15080814, 2440 33531827, 12516406, -21574435, -12476749 }, 2441 { 236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519, 2442 7256740, 8791136, 15069930 }, 2443 }, 2444 { 2445 { 1276410, -9371918, 22949635, -16322807, -23493039, -5702186, 2446 14711875, 4874229, -30663140, -2331391 }, 2447 { 5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175, 2448 -7912378, -33069337, 9234253 }, 2449 { 20590503, -9018988, 31529744, -7352666, -2706834, 10650548, 2450 31559055, -11609587, 18979186, 13396066 }, 2451 }, 2452 { 2453 { 24474287, 4968103, 22267082, 4407354, 24063882, -8325180, 2454 -18816887, 13594782, 33514650, 7021958 }, 2455 { -11566906, -6565505, -21365085, 15928892, -26158305, 4315421, 2456 -25948728, -3916677, -21480480, 12868082 }, 2457 { -28635013, 13504661, 19988037, -2132761, 21078225, 6443208, 2458 -21446107, 2244500, -12455797, -8089383 }, 2459 }, 2460 { 2461 { -30595528, 13793479, -5852820, 319136, -25723172, -6263899, 2462 33086546, 8957937, -15233648, 5540521 }, 2463 { -11630176, -11503902, -8119500, -7643073, 2620056, 1022908, 2464 -23710744, -1568984, -16128528, -14962807 }, 2465 { 23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819, 2466 892185, -11513277, -15205948 }, 2467 }, 2468 { 2469 { 9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819, 2470 4763127, -19179614, 5867134 }, 2471 { -32765025, 1927590, 31726409, -4753295, 23962434, -16019500, 2472 27846559, 5931263, -29749703, -16108455 }, 2473 { 27461885, -2977536, 22380810, 1815854, -23033753, -3031938, 2474 7283490, -15148073, -19526700, 7734629 }, 2475 }, 2476 }, 2477 { 2478 { 2479 { -8010264, -9590817, -11120403, 6196038, 29344158, -13430885, 2480 7585295, -3176626, 18549497, 15302069 }, 2481 { -32658337, -6171222, -7672793, -11051681, 6258878, 13504381, 2482 10458790, -6418461, -8872242, 8424746 }, 2483 { 24687205, 8613276, -30667046, -3233545, 1863892, -1830544, 2484 19206234, 7134917, -11284482, -828919 }, 2485 }, 2486 { 2487 { 11334899, -9218022, 8025293, 12707519, 17523892, -10476071, 2488 10243738, -14685461, -5066034, 16498837 }, 2489 { 8911542, 6887158, -9584260, -6958590, 11145641, -9543680, 17303925, 2490 -14124238, 6536641, 10543906 }, 2491 { -28946384, 15479763, -17466835, 568876, -1497683, 11223454, 2492 -2669190, -16625574, -27235709, 8876771 }, 2493 }, 2494 { 2495 { -25742899, -12566864, -15649966, -846607, -33026686, -796288, 2496 -33481822, 15824474, -604426, -9039817 }, 2497 { 10330056, 70051, 7957388, -9002667, 9764902, 15609756, 27698697, 2498 -4890037, 1657394, 3084098 }, 2499 { 10477963, -7470260, 12119566, -13250805, 29016247, -5365589, 2500 31280319, 14396151, -30233575, 15272409 }, 2501 }, 2502 { 2503 { -12288309, 3169463, 28813183, 16658753, 25116432, -5630466, 2504 -25173957, -12636138, -25014757, 1950504 }, 2505 { -26180358, 9489187, 11053416, -14746161, -31053720, 5825630, 2506 -8384306, -8767532, 15341279, 8373727 }, 2507 { 28685821, 7759505, -14378516, -12002860, -31971820, 4079242, 2508 298136, -10232602, -2878207, 15190420 }, 2509 }, 2510 { 2511 { -32932876, 13806336, -14337485, -15794431, -24004620, 10940928, 2512 8669718, 2742393, -26033313, -6875003 }, 2513 { -1580388, -11729417, -25979658, -11445023, -17411874, -10912854, 2514 9291594, -16247779, -12154742, 6048605 }, 2515 { -30305315, 14843444, 1539301, 11864366, 20201677, 1900163, 2516 13934231, 5128323, 11213262, 9168384 }, 2517 }, 2518 { 2519 { -26280513, 11007847, 19408960, -940758, -18592965, -4328580, 2520 -5088060, -11105150, 20470157, -16398701 }, 2521 { -23136053, 9282192, 14855179, -15390078, -7362815, -14408560, 2522 -22783952, 14461608, 14042978, 5230683 }, 2523 { 29969567, -2741594, -16711867, -8552442, 9175486, -2468974, 2524 21556951, 3506042, -5933891, -12449708 }, 2525 }, 2526 { 2527 { -3144746, 8744661, 19704003, 4581278, -20430686, 6830683, 2528 -21284170, 8971513, -28539189, 15326563 }, 2529 { -19464629, 10110288, -17262528, -3503892, -23500387, 1355669, 2530 -15523050, 15300988, -20514118, 9168260 }, 2531 { -5353335, 4488613, -23803248, 16314347, 7780487, -15638939, 2532 -28948358, 9601605, 33087103, -9011387 }, 2533 }, 2534 { 2535 { -19443170, -15512900, -20797467, -12445323, -29824447, 10229461, 2536 -27444329, -15000531, -5996870, 15664672 }, 2537 { 23294591, -16632613, -22650781, -8470978, 27844204, 11461195, 2538 13099750, -2460356, 18151676, 13417686 }, 2539 { -24722913, -4176517, -31150679, 5988919, -26858785, 6685065, 2540 1661597, -12551441, 15271676, -15452665 }, 2541 }, 2542 }, 2543 { 2544 { 2545 { 11433042, -13228665, 8239631, -5279517, -1985436, -725718, 2546 -18698764, 2167544, -6921301, -13440182 }, 2547 { -31436171, 15575146, 30436815, 12192228, -22463353, 9395379, 2548 -9917708, -8638997, 12215110, 12028277 }, 2549 { 14098400, 6555944, 23007258, 5757252, -15427832, -12950502, 2550 30123440, 4617780, -16900089, -655628 }, 2551 }, 2552 { 2553 { -4026201, -15240835, 11893168, 13718664, -14809462, 1847385, 2554 -15819999, 10154009, 23973261, -12684474 }, 2555 { -26531820, -3695990, -1908898, 2534301, -31870557, -16550355, 2556 18341390, -11419951, 32013174, -10103539 }, 2557 { -25479301, 10876443, -11771086, -14625140, -12369567, 1838104, 2558 21911214, 6354752, 4425632, -837822 }, 2559 }, 2560 { 2561 { -10433389, -14612966, 22229858, -3091047, -13191166, 776729, 2562 -17415375, -12020462, 4725005, 14044970 }, 2563 { 19268650, -7304421, 1555349, 8692754, -21474059, -9910664, 6347390, 2564 -1411784, -19522291, -16109756 }, 2565 { -24864089, 12986008, -10898878, -5558584, -11312371, -148526, 2566 19541418, 8180106, 9282262, 10282508 }, 2567 }, 2568 { 2569 { -26205082, 4428547, -8661196, -13194263, 4098402, -14165257, 2570 15522535, 8372215, 5542595, -10702683 }, 2571 { -10562541, 14895633, 26814552, -16673850, -17480754, -2489360, 2572 -2781891, 6993761, -18093885, 10114655 }, 2573 { -20107055, -929418, 31422704, 10427861, -7110749, 6150669, 2574 -29091755, -11529146, 25953725, -106158 }, 2575 }, 2576 { 2577 { -4234397, -8039292, -9119125, 3046000, 2101609, -12607294, 2578 19390020, 6094296, -3315279, 12831125 }, 2579 { -15998678, 7578152, 5310217, 14408357, -33548620, -224739, 2580 31575954, 6326196, 7381791, -2421839 }, 2581 { -20902779, 3296811, 24736065, -16328389, 18374254, 7318640, 2582 6295303, 8082724, -15362489, 12339664 }, 2583 }, 2584 { 2585 { 27724736, 2291157, 6088201, -14184798, 1792727, 5857634, 13848414, 2586 15768922, 25091167, 14856294 }, 2587 { -18866652, 8331043, 24373479, 8541013, -701998, -9269457, 12927300, 2588 -12695493, -22182473, -9012899 }, 2589 { -11423429, -5421590, 11632845, 3405020, 30536730, -11674039, 2590 -27260765, 13866390, 30146206, 9142070 }, 2591 }, 2592 { 2593 { 3924129, -15307516, -13817122, -10054960, 12291820, -668366, 2594 -27702774, 9326384, -8237858, 4171294 }, 2595 { -15921940, 16037937, 6713787, 16606682, -21612135, 2790944, 2596 26396185, 3731949, 345228, -5462949 }, 2597 { -21327538, 13448259, 25284571, 1143661, 20614966, -8849387, 2598 2031539, -12391231, -16253183, -13582083 }, 2599 }, 2600 { 2601 { 31016211, -16722429, 26371392, -14451233, -5027349, 14854137, 2602 17477601, 3842657, 28012650, -16405420 }, 2603 { -5075835, 9368966, -8562079, -4600902, -15249953, 6970560, 2604 -9189873, 16292057, -8867157, 3507940 }, 2605 { 29439664, 3537914, 23333589, 6997794, -17555561, -11018068, 2606 -15209202, -15051267, -9164929, 6580396 }, 2607 }, 2608 }, 2609 { 2610 { 2611 { -12185861, -7679788, 16438269, 10826160, -8696817, -6235611, 2612 17860444, -9273846, -2095802, 9304567 }, 2613 { 20714564, -4336911, 29088195, 7406487, 11426967, -5095705, 2614 14792667, -14608617, 5289421, -477127 }, 2615 { -16665533, -10650790, -6160345, -13305760, 9192020, -1802462, 2616 17271490, 12349094, 26939669, -3752294 }, 2617 }, 2618 { 2619 { -12889898, 9373458, 31595848, 16374215, 21471720, 13221525, 2620 -27283495, -12348559, -3698806, 117887 }, 2621 { 22263325, -6560050, 3984570, -11174646, -15114008, -566785, 2622 28311253, 5358056, -23319780, 541964 }, 2623 { 16259219, 3261970, 2309254, -15534474, -16885711, -4581916, 2624 24134070, -16705829, -13337066, -13552195 }, 2625 }, 2626 { 2627 { 9378160, -13140186, -22845982, -12745264, 28198281, -7244098, 2628 -2399684, -717351, 690426, 14876244 }, 2629 { 24977353, -314384, -8223969, -13465086, 28432343, -1176353, 2630 -13068804, -12297348, -22380984, 6618999 }, 2631 { -1538174, 11685646, 12944378, 13682314, -24389511, -14413193, 2632 8044829, -13817328, 32239829, -5652762 }, 2633 }, 2634 { 2635 { -18603066, 4762990, -926250, 8885304, -28412480, -3187315, 9781647, 2636 -10350059, 32779359, 5095274 }, 2637 { -33008130, -5214506, -32264887, -3685216, 9460461, -9327423, 2638 -24601656, 14506724, 21639561, -2630236 }, 2639 { -16400943, -13112215, 25239338, 15531969, 3987758, -4499318, 2640 -1289502, -6863535, 17874574, 558605 }, 2641 }, 2642 { 2643 { -13600129, 10240081, 9171883, 16131053, -20869254, 9599700, 2644 33499487, 5080151, 2085892, 5119761 }, 2645 { -22205145, -2519528, -16381601, 414691, -25019550, 2170430, 2646 30634760, -8363614, -31999993, -5759884 }, 2647 { -6845704, 15791202, 8550074, -1312654, 29928809, -12092256, 2648 27534430, -7192145, -22351378, 12961482 }, 2649 }, 2650 { 2651 { -24492060, -9570771, 10368194, 11582341, -23397293, -2245287, 2652 16533930, 8206996, -30194652, -5159638 }, 2653 { -11121496, -3382234, 2307366, 6362031, -135455, 8868177, -16835630, 2654 7031275, 7589640, 8945490 }, 2655 { -32152748, 8917967, 6661220, -11677616, -1192060, -15793393, 2656 7251489, -11182180, 24099109, -14456170 }, 2657 }, 2658 { 2659 { 5019558, -7907470, 4244127, -14714356, -26933272, 6453165, 2660 -19118182, -13289025, -6231896, -10280736 }, 2661 { 10853594, 10721687, 26480089, 5861829, -22995819, 1972175, 2662 -1866647, -10557898, -3363451, -6441124 }, 2663 { -17002408, 5906790, 221599, -6563147, 7828208, -13248918, 24362661, 2664 -2008168, -13866408, 7421392 }, 2665 }, 2666 { 2667 { 8139927, -6546497, 32257646, -5890546, 30375719, 1886181, 2668 -21175108, 15441252, 28826358, -4123029 }, 2669 { 6267086, 9695052, 7709135, -16603597, -32869068, -1886135, 2670 14795160, -7840124, 13746021, -1742048 }, 2671 { 28584902, 7787108, -6732942, -15050729, 22846041, -7571236, 2672 -3181936, -363524, 4771362, -8419958 }, 2673 }, 2674 }, 2675 { 2676 { 2677 { 24949256, 6376279, -27466481, -8174608, -18646154, -9930606, 2678 33543569, -12141695, 3569627, 11342593 }, 2679 { 26514989, 4740088, 27912651, 3697550, 19331575, -11472339, 6809886, 2680 4608608, 7325975, -14801071 }, 2681 { -11618399, -14554430, -24321212, 7655128, -1369274, 5214312, 2682 -27400540, 10258390, -17646694, -8186692 }, 2683 }, 2684 { 2685 { 11431204, 15823007, 26570245, 14329124, 18029990, 4796082, 2686 -31446179, 15580664, 9280358, -3973687 }, 2687 { -160783, -10326257, -22855316, -4304997, -20861367, -13621002, 2688 -32810901, -11181622, -15545091, 4387441 }, 2689 { -20799378, 12194512, 3937617, -5805892, -27154820, 9340370, 2690 -24513992, 8548137, 20617071, -7482001 }, 2691 }, 2692 { 2693 { -938825, -3930586, -8714311, 16124718, 24603125, -6225393, 2694 -13775352, -11875822, 24345683, 10325460 }, 2695 { -19855277, -1568885, -22202708, 8714034, 14007766, 6928528, 2696 16318175, -1010689, 4766743, 3552007 }, 2697 { -21751364, -16730916, 1351763, -803421, -4009670, 3950935, 3217514, 2698 14481909, 10988822, -3994762 }, 2699 }, 2700 { 2701 { 15564307, -14311570, 3101243, 5684148, 30446780, -8051356, 2702 12677127, -6505343, -8295852, 13296005 }, 2703 { -9442290, 6624296, -30298964, -11913677, -4670981, -2057379, 2704 31521204, 9614054, -30000824, 12074674 }, 2705 { 4771191, -135239, 14290749, -13089852, 27992298, 14998318, 2706 -1413936, -1556716, 29832613, -16391035 }, 2707 }, 2708 { 2709 { 7064884, -7541174, -19161962, -5067537, -18891269, -2912736, 2710 25825242, 5293297, -27122660, 13101590 }, 2711 { -2298563, 2439670, -7466610, 1719965, -27267541, -16328445, 2712 32512469, -5317593, -30356070, -4190957 }, 2713 { -30006540, 10162316, -33180176, 3981723, -16482138, -13070044, 2714 14413974, 9515896, 19568978, 9628812 }, 2715 }, 2716 { 2717 { 33053803, 199357, 15894591, 1583059, 27380243, -4580435, -17838894, 2718 -6106839, -6291786, 3437740 }, 2719 { -18978877, 3884493, 19469877, 12726490, 15913552, 13614290, 2720 -22961733, 70104, 7463304, 4176122 }, 2721 { -27124001, 10659917, 11482427, -16070381, 12771467, -6635117, 2722 -32719404, -5322751, 24216882, 5944158 }, 2723 }, 2724 { 2725 { 8894125, 7450974, -2664149, -9765752, -28080517, -12389115, 2726 19345746, 14680796, 11632993, 5847885 }, 2727 { 26942781, -2315317, 9129564, -4906607, 26024105, 11769399, 2728 -11518837, 6367194, -9727230, 4782140 }, 2729 { 19916461, -4828410, -22910704, -11414391, 25606324, -5972441, 2730 33253853, 8220911, 6358847, -1873857 }, 2731 }, 2732 { 2733 { 801428, -2081702, 16569428, 11065167, 29875704, 96627, 7908388, 2734 -4480480, -13538503, 1387155 }, 2735 { 19646058, 5720633, -11416706, 12814209, 11607948, 12749789, 2736 14147075, 15156355, -21866831, 11835260 }, 2737 { 19299512, 1155910, 28703737, 14890794, 2925026, 7269399, 26121523, 2738 15467869, -26560550, 5052483 }, 2739 }, 2740 }, 2741 { 2742 { 2743 { -3017432, 10058206, 1980837, 3964243, 22160966, 12322533, -6431123, 2744 -12618185, 12228557, -7003677 }, 2745 { 32944382, 14922211, -22844894, 5188528, 21913450, -8719943, 2746 4001465, 13238564, -6114803, 8653815 }, 2747 { 22865569, -4652735, 27603668, -12545395, 14348958, 8234005, 2748 24808405, 5719875, 28483275, 2841751 }, 2749 }, 2750 { 2751 { -16420968, -1113305, -327719, -12107856, 21886282, -15552774, 2752 -1887966, -315658, 19932058, -12739203 }, 2753 { -11656086, 10087521, -8864888, -5536143, -19278573, -3055912, 2754 3999228, 13239134, -4777469, -13910208 }, 2755 { 1382174, -11694719, 17266790, 9194690, -13324356, 9720081, 2756 20403944, 11284705, -14013818, 3093230 }, 2757 }, 2758 { 2759 { 16650921, -11037932, -1064178, 1570629, -8329746, 7352753, -302424, 2760 16271225, -24049421, -6691850 }, 2761 { -21911077, -5927941, -4611316, -5560156, -31744103, -10785293, 2762 24123614, 15193618, -21652117, -16739389 }, 2763 { -9935934, -4289447, -25279823, 4372842, 2087473, 10399484, 2764 31870908, 14690798, 17361620, 11864968 }, 2765 }, 2766 { 2767 { -11307610, 6210372, 13206574, 5806320, -29017692, -13967200, 2768 -12331205, -7486601, -25578460, -16240689 }, 2769 { 14668462, -12270235, 26039039, 15305210, 25515617, 4542480, 2770 10453892, 6577524, 9145645, -6443880 }, 2771 { 5974874, 3053895, -9433049, -10385191, -31865124, 3225009, 2772 -7972642, 3936128, -5652273, -3050304 }, 2773 }, 2774 { 2775 { 30625386, -4729400, -25555961, -12792866, -20484575, 7695099, 2776 17097188, -16303496, -27999779, 1803632 }, 2777 { -3553091, 9865099, -5228566, 4272701, -5673832, -16689700, 2778 14911344, 12196514, -21405489, 7047412 }, 2779 { 20093277, 9920966, -11138194, -5343857, 13161587, 12044805, 2780 -32856851, 4124601, -32343828, -10257566 }, 2781 }, 2782 { 2783 { -20788824, 14084654, -13531713, 7842147, 19119038, -13822605, 2784 4752377, -8714640, -21679658, 2288038 }, 2785 { -26819236, -3283715, 29965059, 3039786, -14473765, 2540457, 2786 29457502, 14625692, -24819617, 12570232 }, 2787 { -1063558, -11551823, 16920318, 12494842, 1278292, -5869109, 2788 -21159943, -3498680, -11974704, 4724943 }, 2789 }, 2790 { 2791 { 17960970, -11775534, -4140968, -9702530, -8876562, -1410617, 2792 -12907383, -8659932, -29576300, 1903856 }, 2793 { 23134274, -14279132, -10681997, -1611936, 20684485, 15770816, 2794 -12989750, 3190296, 26955097, 14109738 }, 2795 { 15308788, 5320727, -30113809, -14318877, 22902008, 7767164, 2796 29425325, -11277562, 31960942, 11934971 }, 2797 }, 2798 { 2799 { -27395711, 8435796, 4109644, 12222639, -24627868, 14818669, 2800 20638173, 4875028, 10491392, 1379718 }, 2801 { -13159415, 9197841, 3875503, -8936108, -1383712, -5879801, 2802 33518459, 16176658, 21432314, 12180697 }, 2803 { -11787308, 11500838, 13787581, -13832590, -22430679, 10140205, 2804 1465425, 12689540, -10301319, -13872883 }, 2805 }, 2806 }, 2807 { 2808 { 2809 { 5414091, -15386041, -21007664, 9643570, 12834970, 1186149, 2810 -2622916, -1342231, 26128231, 6032912 }, 2811 { -26337395, -13766162, 32496025, -13653919, 17847801, -12669156, 2812 3604025, 8316894, -25875034, -10437358 }, 2813 { 3296484, 6223048, 24680646, -12246460, -23052020, 5903205, 2814 -8862297, -4639164, 12376617, 3188849 }, 2815 }, 2816 { 2817 { 29190488, -14659046, 27549113, -1183516, 3520066, -10697301, 2818 32049515, -7309113, -16109234, -9852307 }, 2819 { -14744486, -9309156, 735818, -598978, -20407687, -5057904, 2820 25246078, -15795669, 18640741, -960977 }, 2821 { -6928835, -16430795, 10361374, 5642961, 4910474, 12345252, 2822 -31638386, -494430, 10530747, 1053335 }, 2823 }, 2824 { 2825 { -29265967, -14186805, -13538216, -12117373, -19457059, -10655384, 2826 -31462369, -2948985, 24018831, 15026644 }, 2827 { -22592535, -3145277, -2289276, 5953843, -13440189, 9425631, 2828 25310643, 13003497, -2314791, -15145616 }, 2829 { -27419985, -603321, -8043984, -1669117, -26092265, 13987819, 2830 -27297622, 187899, -23166419, -2531735 }, 2831 }, 2832 { 2833 { -21744398, -13810475, 1844840, 5021428, -10434399, -15911473, 2834 9716667, 16266922, -5070217, 726099 }, 2835 { 29370922, -6053998, 7334071, -15342259, 9385287, 2247707, 2836 -13661962, -4839461, 30007388, -15823341 }, 2837 { -936379, 16086691, 23751945, -543318, -1167538, -5189036, 9137109, 2838 730663, 9835848, 4555336 }, 2839 }, 2840 { 2841 { -23376435, 1410446, -22253753, -12899614, 30867635, 15826977, 2842 17693930, 544696, -11985298, 12422646 }, 2843 { 31117226, -12215734, -13502838, 6561947, -9876867, -12757670, 2844 -5118685, -4096706, 29120153, 13924425 }, 2845 { -17400879, -14233209, 19675799, -2734756, -11006962, -5858820, 2846 -9383939, -11317700, 7240931, -237388 }, 2847 }, 2848 { 2849 { -31361739, -11346780, -15007447, -5856218, -22453340, -12152771, 2850 1222336, 4389483, 3293637, -15551743 }, 2851 { -16684801, -14444245, 11038544, 11054958, -13801175, -3338533, 2852 -24319580, 7733547, 12796905, -6335822 }, 2853 { -8759414, -10817836, -25418864, 10783769, -30615557, -9746811, 2854 -28253339, 3647836, 3222231, -11160462 }, 2855 }, 2856 { 2857 { 18606113, 1693100, -25448386, -15170272, 4112353, 10045021, 2858 23603893, -2048234, -7550776, 2484985 }, 2859 { 9255317, -3131197, -12156162, -1004256, 13098013, -9214866, 2860 16377220, -2102812, -19802075, -3034702 }, 2861 { -22729289, 7496160, -5742199, 11329249, 19991973, -3347502, 2862 -31718148, 9936966, -30097688, -10618797 }, 2863 }, 2864 { 2865 { 21878590, -5001297, 4338336, 13643897, -3036865, 13160960, 2866 19708896, 5415497, -7360503, -4109293 }, 2867 { 27736861, 10103576, 12500508, 8502413, -3413016, -9633558, 2868 10436918, -1550276, -23659143, -8132100 }, 2869 { 19492550, -12104365, -29681976, -852630, -3208171, 12403437, 2870 30066266, 8367329, 13243957, 8709688 }, 2871 }, 2872 }, 2873 { 2874 { 2875 { 12015105, 2801261, 28198131, 10151021, 24818120, -4743133, 2876 -11194191, -5645734, 5150968, 7274186 }, 2877 { 2831366, -12492146, 1478975, 6122054, 23825128, -12733586, 2878 31097299, 6083058, 31021603, -9793610 }, 2879 { -2529932, -2229646, 445613, 10720828, -13849527, -11505937, 2880 -23507731, 16354465, 15067285, -14147707 }, 2881 }, 2882 { 2883 { 7840942, 14037873, -33364863, 15934016, -728213, -3642706, 2884 21403988, 1057586, -19379462, -12403220 }, 2885 { 915865, -16469274, 15608285, -8789130, -24357026, 6060030, 2886 -17371319, 8410997, -7220461, 16527025 }, 2887 { 32922597, -556987, 20336074, -16184568, 10903705, -5384487, 2888 16957574, 52992, 23834301, 6588044 }, 2889 }, 2890 { 2891 { 32752030, 11232950, 3381995, -8714866, 22652988, -10744103, 2892 17159699, 16689107, -20314580, -1305992 }, 2893 { -4689649, 9166776, -25710296, -10847306, 11576752, 12733943, 2894 7924251, -2752281, 1976123, -7249027 }, 2895 { 21251222, 16309901, -2983015, -6783122, 30810597, 12967303, 156041, 2896 -3371252, 12331345, -8237197 }, 2897 }, 2898 { 2899 { 8651614, -4477032, -16085636, -4996994, 13002507, 2950805, 2900 29054427, -5106970, 10008136, -4667901 }, 2901 { 31486080, 15114593, -14261250, 12951354, 14369431, -7387845, 2902 16347321, -13662089, 8684155, -10532952 }, 2903 { 19443825, 11385320, 24468943, -9659068, -23919258, 2187569, 2904 -26263207, -6086921, 31316348, 14219878 }, 2905 }, 2906 { 2907 { -28594490, 1193785, 32245219, 11392485, 31092169, 15722801, 2908 27146014, 6992409, 29126555, 9207390 }, 2909 { 32382935, 1110093, 18477781, 11028262, -27411763, -7548111, 2910 -4980517, 10843782, -7957600, -14435730 }, 2911 { 2814918, 7836403, 27519878, -7868156, -20894015, -11553689, 2912 -21494559, 8550130, 28346258, 1994730 }, 2913 }, 2914 { 2915 { -19578299, 8085545, -14000519, -3948622, 2785838, -16231307, 2916 -19516951, 7174894, 22628102, 8115180 }, 2917 { -30405132, 955511, -11133838, -15078069, -32447087, -13278079, 2918 -25651578, 3317160, -9943017, 930272 }, 2919 { -15303681, -6833769, 28856490, 1357446, 23421993, 1057177, 2920 24091212, -1388970, -22765376, -10650715 }, 2921 }, 2922 { 2923 { -22751231, -5303997, -12907607, -12768866, -15811511, -7797053, 2924 -14839018, -16554220, -1867018, 8398970 }, 2925 { -31969310, 2106403, -4736360, 1362501, 12813763, 16200670, 2926 22981545, -6291273, 18009408, -15772772 }, 2927 { -17220923, -9545221, -27784654, 14166835, 29815394, 7444469, 2928 29551787, -3727419, 19288549, 1325865 }, 2929 }, 2930 { 2931 { 15100157, -15835752, -23923978, -1005098, -26450192, 15509408, 2932 12376730, -3479146, 33166107, -8042750 }, 2933 { 20909231, 13023121, -9209752, 16251778, -5778415, -8094914, 2934 12412151, 10018715, 2213263, -13878373 }, 2935 { 32529814, -11074689, 30361439, -16689753, -9135940, 1513226, 2936 22922121, 6382134, -5766928, 8371348 }, 2937 }, 2938 }, 2939 { 2940 { 2941 { 9923462, 11271500, 12616794, 3544722, -29998368, -1721626, 2942 12891687, -8193132, -26442943, 10486144 }, 2943 { -22597207, -7012665, 8587003, -8257861, 4084309, -12970062, 361726, 2944 2610596, -23921530, -11455195 }, 2945 { 5408411, -1136691, -4969122, 10561668, 24145918, 14240566, 2946 31319731, -4235541, 19985175, -3436086 }, 2947 }, 2948 { 2949 { -13994457, 16616821, 14549246, 3341099, 32155958, 13648976, 2950 -17577068, 8849297, 65030, 8370684 }, 2951 { -8320926, -12049626, 31204563, 5839400, -20627288, -1057277, 2952 -19442942, 6922164, 12743482, -9800518 }, 2953 { -2361371, 12678785, 28815050, 4759974, -23893047, 4884717, 2954 23783145, 11038569, 18800704, 255233 }, 2955 }, 2956 { 2957 { -5269658, -1773886, 13957886, 7990715, 23132995, 728773, 13393847, 2958 9066957, 19258688, -14753793 }, 2959 { -2936654, -10827535, -10432089, 14516793, -3640786, 4372541, 2960 -31934921, 2209390, -1524053, 2055794 }, 2961 { 580882, 16705327, 5468415, -2683018, -30926419, -14696000, 2962 -7203346, -8994389, -30021019, 7394435 }, 2963 }, 2964 { 2965 { 23838809, 1822728, -15738443, 15242727, 8318092, -3733104, 2966 -21672180, -3492205, -4821741, 14799921 }, 2967 { 13345610, 9759151, 3371034, -16137791, 16353039, 8577942, 31129804, 2968 13496856, -9056018, 7402518 }, 2969 { 2286874, -4435931, -20042458, -2008336, -13696227, 5038122, 2970 11006906, -15760352, 8205061, 1607563 }, 2971 }, 2972 { 2973 { 14414086, -8002132, 3331830, -3208217, 22249151, -5594188, 2974 18364661, -2906958, 30019587, -9029278 }, 2975 { -27688051, 1585953, -10775053, 931069, -29120221, -11002319, 2976 -14410829, 12029093, 9944378, 8024 }, 2977 { 4368715, -3709630, 29874200, -15022983, -20230386, -11410704, 2978 -16114594, -999085, -8142388, 5640030 }, 2979 }, 2980 { 2981 { 10299610, 13746483, 11661824, 16234854, 7630238, 5998374, 9809887, 2982 -16694564, 15219798, -14327783 }, 2983 { 27425505, -5719081, 3055006, 10660664, 23458024, 595578, -15398605, 2984 -1173195, -18342183, 9742717 }, 2985 { 6744077, 2427284, 26042789, 2720740, -847906, 1118974, 32324614, 2986 7406442, 12420155, 1994844 }, 2987 }, 2988 { 2989 { 14012521, -5024720, -18384453, -9578469, -26485342, -3936439, 2990 -13033478, -10909803, 24319929, -6446333 }, 2991 { 16412690, -4507367, 10772641, 15929391, -17068788, -4658621, 2992 10555945, -10484049, -30102368, -4739048 }, 2993 { 22397382, -7767684, -9293161, -12792868, 17166287, -9755136, 2994 -27333065, 6199366, 21880021, -12250760 }, 2995 }, 2996 { 2997 { -4283307, 5368523, -31117018, 8163389, -30323063, 3209128, 2998 16557151, 8890729, 8840445, 4957760 }, 2999 { -15447727, 709327, -6919446, -10870178, -29777922, 6522332, 3000 -21720181, 12130072, -14796503, 5005757 }, 3001 { -2114751, -14308128, 23019042, 15765735, -25269683, 6002752, 3002 10183197, -13239326, -16395286, -2176112 }, 3003 }, 3004 }, 3005 { 3006 { 3007 { -19025756, 1632005, 13466291, -7995100, -23640451, 16573537, 3008 -32013908, -3057104, 22208662, 2000468 }, 3009 { 3065073, -1412761, -25598674, -361432, -17683065, -5703415, 3010 -8164212, 11248527, -3691214, -7414184 }, 3011 { 10379208, -6045554, 8877319, 1473647, -29291284, -12507580, 3012 16690915, 2553332, -3132688, 16400289 }, 3013 }, 3014 { 3015 { 15716668, 1254266, -18472690, 7446274, -8448918, 6344164, 3016 -22097271, -7285580, 26894937, 9132066 }, 3017 { 24158887, 12938817, 11085297, -8177598, -28063478, -4457083, 3018 -30576463, 64452, -6817084, -2692882 }, 3019 { 13488534, 7794716, 22236231, 5989356, 25426474, -12578208, 2350710, 3020 -3418511, -4688006, 2364226 }, 3021 }, 3022 { 3023 { 16335052, 9132434, 25640582, 6678888, 1725628, 8517937, -11807024, 3024 -11697457, 15445875, -7798101 }, 3025 { 29004207, -7867081, 28661402, -640412, -12794003, -7943086, 3026 31863255, -4135540, -278050, -15759279 }, 3027 { -6122061, -14866665, -28614905, 14569919, -10857999, -3591829, 3028 10343412, -6976290, -29828287, -10815811 }, 3029 }, 3030 { 3031 { 27081650, 3463984, 14099042, -4517604, 1616303, -6205604, 29542636, 3032 15372179, 17293797, 960709 }, 3033 { 20263915, 11434237, -5765435, 11236810, 13505955, -10857102, 3034 -16111345, 6493122, -19384511, 7639714 }, 3035 { -2830798, -14839232, 25403038, -8215196, -8317012, -16173699, 3036 18006287, -16043750, 29994677, -15808121 }, 3037 }, 3038 { 3039 { 9769828, 5202651, -24157398, -13631392, -28051003, -11561624, 3040 -24613141, -13860782, -31184575, 709464 }, 3041 { 12286395, 13076066, -21775189, -1176622, -25003198, 4057652, 3042 -32018128, -8890874, 16102007, 13205847 }, 3043 { 13733362, 5599946, 10557076, 3195751, -5557991, 8536970, -25540170, 3044 8525972, 10151379, 10394400 }, 3045 }, 3046 { 3047 { 4024660, -16137551, 22436262, 12276534, -9099015, -2686099, 3048 19698229, 11743039, -33302334, 8934414 }, 3049 { -15879800, -4525240, -8580747, -2934061, 14634845, -698278, 3050 -9449077, 3137094, -11536886, 11721158 }, 3051 { 17555939, -5013938, 8268606, 2331751, -22738815, 9761013, 9319229, 3052 8835153, -9205489, -1280045 }, 3053 }, 3054 { 3055 { -461409, -7830014, 20614118, 16688288, -7514766, -4807119, 3056 22300304, 505429, 6108462, -6183415 }, 3057 { -5070281, 12367917, -30663534, 3234473, 32617080, -8422642, 3058 29880583, -13483331, -26898490, -7867459 }, 3059 { -31975283, 5726539, 26934134, 10237677, -3173717, -605053, 3060 24199304, 3795095, 7592688, -14992079 }, 3061 }, 3062 { 3063 { 21594432, -14964228, 17466408, -4077222, 32537084, 2739898, 3064 6407723, 12018833, -28256052, 4298412 }, 3065 { -20650503, -11961496, -27236275, 570498, 3767144, -1717540, 3066 13891942, -1569194, 13717174, 10805743 }, 3067 { -14676630, -15644296, 15287174, 11927123, 24177847, -8175568, 3068 -796431, 14860609, -26938930, -5863836 }, 3069 }, 3070 }, 3071 { 3072 { 3073 { 12962541, 5311799, -10060768, 11658280, 18855286, -7954201, 3074 13286263, -12808704, -4381056, 9882022 }, 3075 { 18512079, 11319350, -20123124, 15090309, 18818594, 5271736, 3076 -22727904, 3666879, -23967430, -3299429 }, 3077 { -6789020, -3146043, 16192429, 13241070, 15898607, -14206114, 3078 -10084880, -6661110, -2403099, 5276065 }, 3079 }, 3080 { 3081 { 30169808, -5317648, 26306206, -11750859, 27814964, 7069267, 3082 7152851, 3684982, 1449224, 13082861 }, 3083 { 10342826, 3098505, 2119311, 193222, 25702612, 12233820, 23697382, 3084 15056736, -21016438, -8202000 }, 3085 { -33150110, 3261608, 22745853, 7948688, 19370557, -15177665, 3086 -26171976, 6482814, -10300080, -11060101 }, 3087 }, 3088 { 3089 { 32869458, -5408545, 25609743, 15678670, -10687769, -15471071, 3090 26112421, 2521008, -22664288, 6904815 }, 3091 { 29506923, 4457497, 3377935, -9796444, -30510046, 12935080, 1561737, 3092 3841096, -29003639, -6657642 }, 3093 { 10340844, -6630377, -18656632, -2278430, 12621151, -13339055, 3094 30878497, -11824370, -25584551, 5181966 }, 3095 }, 3096 { 3097 { 25940115, -12658025, 17324188, -10307374, -8671468, 15029094, 3098 24396252, -16450922, -2322852, -12388574 }, 3099 { -21765684, 9916823, -1300409, 4079498, -1028346, 11909559, 1782390, 3100 12641087, 20603771, -6561742 }, 3101 { -18882287, -11673380, 24849422, 11501709, 13161720, -4768874, 3102 1925523, 11914390, 4662781, 7820689 }, 3103 }, 3104 { 3105 { 12241050, -425982, 8132691, 9393934, 32846760, -1599620, 29749456, 3106 12172924, 16136752, 15264020 }, 3107 { -10349955, -14680563, -8211979, 2330220, -17662549, -14545780, 3108 10658213, 6671822, 19012087, 3772772 }, 3109 { 3753511, -3421066, 10617074, 2028709, 14841030, -6721664, 28718732, 3110 -15762884, 20527771, 12988982 }, 3111 }, 3112 { 3113 { -14822485, -5797269, -3707987, 12689773, -898983, -10914866, 3114 -24183046, -10564943, 3299665, -12424953 }, 3115 { -16777703, -15253301, -9642417, 4978983, 3308785, 8755439, 6943197, 3116 6461331, -25583147, 8991218 }, 3117 { -17226263, 1816362, -1673288, -6086439, 31783888, -8175991, 3118 -32948145, 7417950, -30242287, 1507265 }, 3119 }, 3120 { 3121 { 29692663, 6829891, -10498800, 4334896, 20945975, -11906496, 3122 -28887608, 8209391, 14606362, -10647073 }, 3123 { -3481570, 8707081, 32188102, 5672294, 22096700, 1711240, -33020695, 3124 9761487, 4170404, -2085325 }, 3125 { -11587470, 14855945, -4127778, -1531857, -26649089, 15084046, 3126 22186522, 16002000, -14276837, -8400798 }, 3127 }, 3128 { 3129 { -4811456, 13761029, -31703877, -2483919, -3312471, 7869047, 3130 -7113572, -9620092, 13240845, 10965870 }, 3131 { -7742563, -8256762, -14768334, -13656260, -23232383, 12387166, 3132 4498947, 14147411, 29514390, 4302863 }, 3133 { -13413405, -12407859, 20757302, -13801832, 14785143, 8976368, 3134 -5061276, -2144373, 17846988, -13971927 }, 3135 }, 3136 }, 3137 { 3138 { 3139 { -2244452, -754728, -4597030, -1066309, -6247172, 1455299, 3140 -21647728, -9214789, -5222701, 12650267 }, 3141 { -9906797, -16070310, 21134160, 12198166, -27064575, 708126, 387813, 3142 13770293, -19134326, 10958663 }, 3143 { 22470984, 12369526, 23446014, -5441109, -21520802, -9698723, 3144 -11772496, -11574455, -25083830, 4271862 }, 3145 }, 3146 { 3147 { -25169565, -10053642, -19909332, 15361595, -5984358, 2159192, 3148 75375, -4278529, -32526221, 8469673 }, 3149 { 15854970, 4148314, -8893890, 7259002, 11666551, 13824734, 3150 -30531198, 2697372, 24154791, -9460943 }, 3151 { 15446137, -15806644, 29759747, 14019369, 30811221, -9610191, 3152 -31582008, 12840104, 24913809, 9815020 }, 3153 }, 3154 { 3155 { -4709286, -5614269, -31841498, -12288893, -14443537, 10799414, 3156 -9103676, 13438769, 18735128, 9466238 }, 3157 { 11933045, 9281483, 5081055, -5183824, -2628162, -4905629, -7727821, 3158 -10896103, -22728655, 16199064 }, 3159 { 14576810, 379472, -26786533, -8317236, -29426508, -10812974, 3160 -102766, 1876699, 30801119, 2164795 }, 3161 }, 3162 { 3163 { 15995086, 3199873, 13672555, 13712240, -19378835, -4647646, 3164 -13081610, -15496269, -13492807, 1268052 }, 3165 { -10290614, -3659039, -3286592, 10948818, 23037027, 3794475, 3166 -3470338, -12600221, -17055369, 3565904 }, 3167 { 29210088, -9419337, -5919792, -4952785, 10834811, -13327726, 3168 -16512102, -10820713, -27162222, -14030531 }, 3169 }, 3170 { 3171 { -13161890, 15508588, 16663704, -8156150, -28349942, 9019123, 3172 -29183421, -3769423, 2244111, -14001979 }, 3173 { -5152875, -3800936, -9306475, -6071583, 16243069, 14684434, 3174 -25673088, -16180800, 13491506, 4641841 }, 3175 { 10813417, 643330, -19188515, -728916, 30292062, -16600078, 3176 27548447, -7721242, 14476989, -12767431 }, 3177 }, 3178 { 3179 { 10292079, 9984945, 6481436, 8279905, -7251514, 7032743, 27282937, 3180 -1644259, -27912810, 12651324 }, 3181 { -31185513, -813383, 22271204, 11835308, 10201545, 15351028, 3182 17099662, 3988035, 21721536, -3148940 }, 3183 { 10202177, -6545839, -31373232, -9574638, -32150642, -8119683, 3184 -12906320, 3852694, 13216206, 14842320 }, 3185 }, 3186 { 3187 { -15815640, -10601066, -6538952, -7258995, -6984659, -6581778, 3188 -31500847, 13765824, -27434397, 9900184 }, 3189 { 14465505, -13833331, -32133984, -14738873, -27443187, 12990492, 3190 33046193, 15796406, -7051866, -8040114 }, 3191 { 30924417, -8279620, 6359016, -12816335, 16508377, 9071735, 3192 -25488601, 15413635, 9524356, -7018878 }, 3193 }, 3194 { 3195 { 12274201, -13175547, 32627641, -1785326, 6736625, 13267305, 3196 5237659, -5109483, 15663516, 4035784 }, 3197 { -2951309, 8903985, 17349946, 601635, -16432815, -4612556, 3198 -13732739, -15889334, -22258478, 4659091 }, 3199 { -16916263, -4952973, -30393711, -15158821, 20774812, 15897498, 3200 5736189, 15026997, -2178256, -13455585 }, 3201 }, 3202 }, 3203 { 3204 { 3205 { -8858980, -2219056, 28571666, -10155518, -474467, -10105698, 3206 -3801496, 278095, 23440562, -290208 }, 3207 { 10226241, -5928702, 15139956, 120818, -14867693, 5218603, 32937275, 3208 11551483, -16571960, -7442864 }, 3209 { 17932739, -12437276, -24039557, 10749060, 11316803, 7535897, 3210 22503767, 5561594, -3646624, 3898661 }, 3211 }, 3212 { 3213 { 7749907, -969567, -16339731, -16464, -25018111, 15122143, -1573531, 3214 7152530, 21831162, 1245233 }, 3215 { 26958459, -14658026, 4314586, 8346991, -5677764, 11960072, 3216 -32589295, -620035, -30402091, -16716212 }, 3217 { -12165896, 9166947, 33491384, 13673479, 29787085, 13096535, 3218 6280834, 14587357, -22338025, 13987525 }, 3219 }, 3220 { 3221 { -24349909, 7778775, 21116000, 15572597, -4833266, -5357778, 3222 -4300898, -5124639, -7469781, -2858068 }, 3223 { 9681908, -6737123, -31951644, 13591838, -6883821, 386950, 31622781, 3224 6439245, -14581012, 4091397 }, 3225 { -8426427, 1470727, -28109679, -1596990, 3978627, -5123623, 3226 -19622683, 12092163, 29077877, -14741988 }, 3227 }, 3228 { 3229 { 5269168, -6859726, -13230211, -8020715, 25932563, 1763552, 3230 -5606110, -5505881, -20017847, 2357889 }, 3231 { 32264008, -15407652, -5387735, -1160093, -2091322, -3946900, 3232 23104804, -12869908, 5727338, 189038 }, 3233 { 14609123, -8954470, -6000566, -16622781, -14577387, -7743898, 3234 -26745169, 10942115, -25888931, -14884697 }, 3235 }, 3236 { 3237 { 20513500, 5557931, -15604613, 7829531, 26413943, -2019404, 3238 -21378968, 7471781, 13913677, -5137875 }, 3239 { -25574376, 11967826, 29233242, 12948236, -6754465, 4713227, 3240 -8940970, 14059180, 12878652, 8511905 }, 3241 { -25656801, 3393631, -2955415, -7075526, -2250709, 9366908, 3242 -30223418, 6812974, 5568676, -3127656 }, 3243 }, 3244 { 3245 { 11630004, 12144454, 2116339, 13606037, 27378885, 15676917, 3246 -17408753, -13504373, -14395196, 8070818 }, 3247 { 27117696, -10007378, -31282771, -5570088, 1127282, 12772488, 3248 -29845906, 10483306, -11552749, -1028714 }, 3249 { 10637467, -5688064, 5674781, 1072708, -26343588, -6982302, 3250 -1683975, 9177853, -27493162, 15431203 }, 3251 }, 3252 { 3253 { 20525145, 10892566, -12742472, 12779443, -29493034, 16150075, 3254 -28240519, 14943142, -15056790, -7935931 }, 3255 { -30024462, 5626926, -551567, -9981087, 753598, 11981191, 25244767, 3256 -3239766, -3356550, 9594024 }, 3257 { -23752644, 2636870, -5163910, -10103818, 585134, 7877383, 11345683, 3258 -6492290, 13352335, -10977084 }, 3259 }, 3260 { 3261 { -1931799, -5407458, 3304649, -12884869, 17015806, -4877091, 3262 -29783850, -7752482, -13215537, -319204 }, 3263 { 20239939, 6607058, 6203985, 3483793, -18386976, -779229, -20723742, 3264 15077870, -22750759, 14523817 }, 3265 { 27406042, -6041657, 27423596, -4497394, 4996214, 10002360, 3266 -28842031, -4545494, -30172742, -4805667 }, 3267 }, 3268 }, 3269 { 3270 { 3271 { 11374242, 12660715, 17861383, -12540833, 10935568, 1099227, 3272 -13886076, -9091740, -27727044, 11358504 }, 3273 { -12730809, 10311867, 1510375, 10778093, -2119455, -9145702, 3274 32676003, 11149336, -26123651, 4985768 }, 3275 { -19096303, 341147, -6197485, -239033, 15756973, -8796662, -983043, 3276 13794114, -19414307, -15621255 }, 3277 }, 3278 { 3279 { 6490081, 11940286, 25495923, -7726360, 8668373, -8751316, 3367603, 3280 6970005, -1691065, -9004790 }, 3281 { 1656497, 13457317, 15370807, 6364910, 13605745, 8362338, -19174622, 3282 -5475723, -16796596, -5031438 }, 3283 { -22273315, -13524424, -64685, -4334223, -18605636, -10921968, 3284 -20571065, -7007978, -99853, -10237333 }, 3285 }, 3286 { 3287 { 17747465, 10039260, 19368299, -4050591, -20630635, -16041286, 3288 31992683, -15857976, -29260363, -5511971 }, 3289 { 31932027, -4986141, -19612382, 16366580, 22023614, 88450, 11371999, 3290 -3744247, 4882242, -10626905 }, 3291 { 29796507, 37186, 19818052, 10115756, -11829032, 3352736, 18551198, 3292 3272828, -5190932, -4162409 }, 3293 }, 3294 { 3295 { 12501286, 4044383, -8612957, -13392385, -32430052, 5136599, 3296 -19230378, -3529697, 330070, -3659409 }, 3297 { 6384877, 2899513, 17807477, 7663917, -2358888, 12363165, 25366522, 3298 -8573892, -271295, 12071499 }, 3299 { -8365515, -4042521, 25133448, -4517355, -6211027, 2265927, 3300 -32769618, 1936675, -5159697, 3829363 }, 3301 }, 3302 { 3303 { 28425966, -5835433, -577090, -4697198, -14217555, 6870930, 7921550, 3304 -6567787, 26333140, 14267664 }, 3305 { -11067219, 11871231, 27385719, -10559544, -4585914, -11189312, 3306 10004786, -8709488, -21761224, 8930324 }, 3307 { -21197785, -16396035, 25654216, -1725397, 12282012, 11008919, 3308 1541940, 4757911, -26491501, -16408940 }, 3309 }, 3310 { 3311 { 13537262, -7759490, -20604840, 10961927, -5922820, -13218065, 3312 -13156584, 6217254, -15943699, 13814990 }, 3313 { -17422573, 15157790, 18705543, 29619, 24409717, -260476, 27361681, 3314 9257833, -1956526, -1776914 }, 3315 { -25045300, -10191966, 15366585, 15166509, -13105086, 8423556, 3316 -29171540, 12361135, -18685978, 4578290 }, 3317 }, 3318 { 3319 { 24579768, 3711570, 1342322, -11180126, -27005135, 14124956, 3320 -22544529, 14074919, 21964432, 8235257 }, 3321 { -6528613, -2411497, 9442966, -5925588, 12025640, -1487420, 3322 -2981514, -1669206, 13006806, 2355433 }, 3323 { -16304899, -13605259, -6632427, -5142349, 16974359, -10911083, 3324 27202044, 1719366, 1141648, -12796236 }, 3325 }, 3326 { 3327 { -12863944, -13219986, -8318266, -11018091, -6810145, -4843894, 3328 13475066, -3133972, 32674895, 13715045 }, 3329 { 11423335, -5468059, 32344216, 8962751, 24989809, 9241752, 3330 -13265253, 16086212, -28740881, -15642093 }, 3331 { -1409668, 12530728, -6368726, 10847387, 19531186, -14132160, 3332 -11709148, 7791794, -27245943, 4383347 }, 3333 }, 3334 }, 3335 { 3336 { 3337 { -28970898, 5271447, -1266009, -9736989, -12455236, 16732599, 3338 -4862407, -4906449, 27193557, 6245191 }, 3339 { -15193956, 5362278, -1783893, 2695834, 4960227, 12840725, 23061898, 3340 3260492, 22510453, 8577507 }, 3341 { -12632451, 11257346, -32692994, 13548177, -721004, 10879011, 3342 31168030, 13952092, -29571492, -3635906 }, 3343 }, 3344 { 3345 { 3877321, -9572739, 32416692, 5405324, -11004407, -13656635, 3346 3759769, 11935320, 5611860, 8164018 }, 3347 { -16275802, 14667797, 15906460, 12155291, -22111149, -9039718, 3348 32003002, -8832289, 5773085, -8422109 }, 3349 { -23788118, -8254300, 1950875, 8937633, 18686727, 16459170, -905725, 3350 12376320, 31632953, 190926 }, 3351 }, 3352 { 3353 { -24593607, -16138885, -8423991, 13378746, 14162407, 6901328, 3354 -8288749, 4508564, -25341555, -3627528 }, 3355 { 8884438, -5884009, 6023974, 10104341, -6881569, -4941533, 18722941, 3356 -14786005, -1672488, 827625 }, 3357 { -32720583, -16289296, -32503547, 7101210, 13354605, 2659080, 3358 -1800575, -14108036, -24878478, 1541286 }, 3359 }, 3360 { 3361 { 2901347, -1117687, 3880376, -10059388, -17620940, -3612781, 3362 -21802117, -3567481, 20456845, -1885033 }, 3363 { 27019610, 12299467, -13658288, -1603234, -12861660, -4861471, 3364 -19540150, -5016058, 29439641, 15138866 }, 3365 { 21536104, -6626420, -32447818, -10690208, -22408077, 5175814, 3366 -5420040, -16361163, 7779328, 109896 }, 3367 }, 3368 { 3369 { 30279744, 14648750, -8044871, 6425558, 13639621, -743509, 28698390, 3370 12180118, 23177719, -554075 }, 3371 { 26572847, 3405927, -31701700, 12890905, -19265668, 5335866, 3372 -6493768, 2378492, 4439158, -13279347 }, 3373 { -22716706, 3489070, -9225266, -332753, 18875722, -1140095, 3374 14819434, -12731527, -17717757, -5461437 }, 3375 }, 3376 { 3377 { -5056483, 16566551, 15953661, 3767752, -10436499, 15627060, 3378 -820954, 2177225, 8550082, -15114165 }, 3379 { -18473302, 16596775, -381660, 15663611, 22860960, 15585581, 3380 -27844109, -3582739, -23260460, -8428588 }, 3381 { -32480551, 15707275, -8205912, -5652081, 29464558, 2713815, 3382 -22725137, 15860482, -21902570, 1494193 }, 3383 }, 3384 { 3385 { -19562091, -14087393, -25583872, -9299552, 13127842, 759709, 3386 21923482, 16529112, 8742704, 12967017 }, 3387 { -28464899, 1553205, 32536856, -10473729, -24691605, -406174, 3388 -8914625, -2933896, -29903758, 15553883 }, 3389 { 21877909, 3230008, 9881174, 10539357, -4797115, 2841332, 11543572, 3390 14513274, 19375923, -12647961 }, 3391 }, 3392 { 3393 { 8832269, -14495485, 13253511, 5137575, 5037871, 4078777, 24880818, 3394 -6222716, 2862653, 9455043 }, 3395 { 29306751, 5123106, 20245049, -14149889, 9592566, 8447059, -2077124, 3396 -2990080, 15511449, 4789663 }, 3397 { -20679756, 7004547, 8824831, -9434977, -4045704, -3750736, 3398 -5754762, 108893, 23513200, 16652362 }, 3399 }, 3400 }, 3401 { 3402 { 3403 { -33256173, 4144782, -4476029, -6579123, 10770039, -7155542, 3404 -6650416, -12936300, -18319198, 10212860 }, 3405 { 2756081, 8598110, 7383731, -6859892, 22312759, -1105012, 21179801, 3406 2600940, -9988298, -12506466 }, 3407 { -24645692, 13317462, -30449259, -15653928, 21365574, -10869657, 3408 11344424, 864440, -2499677, -16710063 }, 3409 }, 3410 { 3411 { -26432803, 6148329, -17184412, -14474154, 18782929, -275997, 3412 -22561534, 211300, 2719757, 4940997 }, 3413 { -1323882, 3911313, -6948744, 14759765, -30027150, 7851207, 3414 21690126, 8518463, 26699843, 5276295 }, 3415 { -13149873, -6429067, 9396249, 365013, 24703301, -10488939, 1321586, 3416 149635, -15452774, 7159369 }, 3417 }, 3418 { 3419 { 9987780, -3404759, 17507962, 9505530, 9731535, -2165514, 22356009, 3420 8312176, 22477218, -8403385 }, 3421 { 18155857, -16504990, 19744716, 9006923, 15154154, -10538976, 3422 24256460, -4864995, -22548173, 9334109 }, 3423 { 2986088, -4911893, 10776628, -3473844, 10620590, -7083203, 3424 -21413845, 14253545, -22587149, 536906 }, 3425 }, 3426 { 3427 { 4377756, 8115836, 24567078, 15495314, 11625074, 13064599, 7390551, 3428 10589625, 10838060, -15420424 }, 3429 { -19342404, 867880, 9277171, -3218459, -14431572, -1986443, 3430 19295826, -15796950, 6378260, 699185 }, 3431 { 7895026, 4057113, -7081772, -13077756, -17886831, -323126, -716039, 3432 15693155, -5045064, -13373962 }, 3433 }, 3434 { 3435 { -7737563, -5869402, -14566319, -7406919, 11385654, 13201616, 3436 31730678, -10962840, -3918636, -9669325 }, 3437 { 10188286, -15770834, -7336361, 13427543, 22223443, 14896287, 3438 30743455, 7116568, -21786507, 5427593 }, 3439 { 696102, 13206899, 27047647, -10632082, 15285305, -9853179, 3440 10798490, -4578720, 19236243, 12477404 }, 3441 }, 3442 { 3443 { -11229439, 11243796, -17054270, -8040865, -788228, -8167967, 3444 -3897669, 11180504, -23169516, 7733644 }, 3445 { 17800790, -14036179, -27000429, -11766671, 23887827, 3149671, 3446 23466177, -10538171, 10322027, 15313801 }, 3447 { 26246234, 11968874, 32263343, -5468728, 6830755, -13323031, 3448 -15794704, -101982, -24449242, 10890804 }, 3449 }, 3450 { 3451 { -31365647, 10271363, -12660625, -6267268, 16690207, -13062544, 3452 -14982212, 16484931, 25180797, -5334884 }, 3453 { -586574, 10376444, -32586414, -11286356, 19801893, 10997610, 3454 2276632, 9482883, 316878, 13820577 }, 3455 { -9882808, -4510367, -2115506, 16457136, -11100081, 11674996, 3456 30756178, -7515054, 30696930, -3712849 }, 3457 }, 3458 { 3459 { 32988917, -9603412, 12499366, 7910787, -10617257, -11931514, 3460 -7342816, -9985397, -32349517, 7392473 }, 3461 { -8855661, 15927861, 9866406, -3649411, -2396914, -16655781, 3462 -30409476, -9134995, 25112947, -2926644 }, 3463 { -2504044, -436966, 25621774, -5678772, 15085042, -5479877, 3464 -24884878, -13526194, 5537438, -13914319 }, 3465 }, 3466 }, 3467 { 3468 { 3469 { -11225584, 2320285, -9584280, 10149187, -33444663, 5808648, 3470 -14876251, -1729667, 31234590, 6090599 }, 3471 { -9633316, 116426, 26083934, 2897444, -6364437, -2688086, 609721, 3472 15878753, -6970405, -9034768 }, 3473 { -27757857, 247744, -15194774, -9002551, 23288161, -10011936, 3474 -23869595, 6503646, 20650474, 1804084 }, 3475 }, 3476 { 3477 { -27589786, 15456424, 8972517, 8469608, 15640622, 4439847, 3121995, 3478 -10329713, 27842616, -202328 }, 3479 { -15306973, 2839644, 22530074, 10026331, 4602058, 5048462, 28248656, 3480 5031932, -11375082, 12714369 }, 3481 { 20807691, -7270825, 29286141, 11421711, -27876523, -13868230, 3482 -21227475, 1035546, -19733229, 12796920 }, 3483 }, 3484 { 3485 { 12076899, -14301286, -8785001, -11848922, -25012791, 16400684, 3486 -17591495, -12899438, 3480665, -15182815 }, 3487 { -32361549, 5457597, 28548107, 7833186, 7303070, -11953545, 3488 -24363064, -15921875, -33374054, 2771025 }, 3489 { -21389266, 421932, 26597266, 6860826, 22486084, -6737172, 3490 -17137485, -4210226, -24552282, 15673397 }, 3491 }, 3492 { 3493 { -20184622, 2338216, 19788685, -9620956, -4001265, -8740893, 3494 -20271184, 4733254, 3727144, -12934448 }, 3495 { 6120119, 814863, -11794402, -622716, 6812205, -15747771, 2019594, 3496 7975683, 31123697, -10958981 }, 3497 { 30069250, -11435332, 30434654, 2958439, 18399564, -976289, 3498 12296869, 9204260, -16432438, 9648165 }, 3499 }, 3500 { 3501 { 32705432, -1550977, 30705658, 7451065, -11805606, 9631813, 3305266, 3502 5248604, -26008332, -11377501 }, 3503 { 17219865, 2375039, -31570947, -5575615, -19459679, 9219903, 294711, 3504 15298639, 2662509, -16297073 }, 3505 { -1172927, -7558695, -4366770, -4287744, -21346413, -8434326, 3506 32087529, -1222777, 32247248, -14389861 }, 3507 }, 3508 { 3509 { 14312628, 1221556, 17395390, -8700143, -4945741, -8684635, 3510 -28197744, -9637817, -16027623, -13378845 }, 3511 { -1428825, -9678990, -9235681, 6549687, -7383069, -468664, 23046502, 3512 9803137, 17597934, 2346211 }, 3513 { 18510800, 15337574, 26171504, 981392, -22241552, 7827556, 3514 -23491134, -11323352, 3059833, -11782870 }, 3515 }, 3516 { 3517 { 10141598, 6082907, 17829293, -1947643, 9830092, 13613136, 3518 -25556636, -5544586, -33502212, 3592096 }, 3519 { 33114168, -15889352, -26525686, -13343397, 33076705, 8716171, 3520 1151462, 1521897, -982665, -6837803 }, 3521 { -32939165, -4255815, 23947181, -324178, -33072974, -12305637, 3522 -16637686, 3891704, 26353178, 693168 }, 3523 }, 3524 { 3525 { 30374239, 1595580, -16884039, 13186931, 4600344, 406904, 9585294, 3526 -400668, 31375464, 14369965 }, 3527 { -14370654, -7772529, 1510301, 6434173, -18784789, -6262728, 3528 32732230, -13108839, 17901441, 16011505 }, 3529 { 18171223, -11934626, -12500402, 15197122, -11038147, -15230035, 3530 -19172240, -16046376, 8764035, 12309598 }, 3531 }, 3532 }, 3533 { 3534 { 3535 { 5975908, -5243188, -19459362, -9681747, -11541277, 14015782, 3536 -23665757, 1228319, 17544096, -10593782 }, 3537 { 5811932, -1715293, 3442887, -2269310, -18367348, -8359541, 3538 -18044043, -15410127, -5565381, 12348900 }, 3539 { -31399660, 11407555, 25755363, 6891399, -3256938, 14872274, 3540 -24849353, 8141295, -10632534, -585479 }, 3541 }, 3542 { 3543 { -12675304, 694026, -5076145, 13300344, 14015258, -14451394, 3544 -9698672, -11329050, 30944593, 1130208 }, 3545 { 8247766, -6710942, -26562381, -7709309, -14401939, -14648910, 3546 4652152, 2488540, 23550156, -271232 }, 3547 { 17294316, -3788438, 7026748, 15626851, 22990044, 113481, 2267737, 3548 -5908146, -408818, -137719 }, 3549 }, 3550 { 3551 { 16091085, -16253926, 18599252, 7340678, 2137637, -1221657, 3552 -3364161, 14550936, 3260525, -7166271 }, 3553 { -4910104, -13332887, 18550887, 10864893, -16459325, -7291596, 3554 -23028869, -13204905, -12748722, 2701326 }, 3555 { -8574695, 16099415, 4629974, -16340524, -20786213, -6005432, 3556 -10018363, 9276971, 11329923, 1862132 }, 3557 }, 3558 { 3559 { 14763076, -15903608, -30918270, 3689867, 3511892, 10313526, 3560 -21951088, 12219231, -9037963, -940300 }, 3561 { 8894987, -3446094, 6150753, 3013931, 301220, 15693451, -31981216, 3562 -2909717, -15438168, 11595570 }, 3563 { 15214962, 3537601, -26238722, -14058872, 4418657, -15230761, 3564 13947276, 10730794, -13489462, -4363670 }, 3565 }, 3566 { 3567 { -2538306, 7682793, 32759013, 263109, -29984731, -7955452, 3568 -22332124, -10188635, 977108, 699994 }, 3569 { -12466472, 4195084, -9211532, 550904, -15565337, 12917920, 3570 19118110, -439841, -30534533, -14337913 }, 3571 { 31788461, -14507657, 4799989, 7372237, 8808585, -14747943, 9408237, 3572 -10051775, 12493932, -5409317 }, 3573 }, 3574 { 3575 { -25680606, 5260744, -19235809, -6284470, -3695942, 16566087, 3576 27218280, 2607121, 29375955, 6024730 }, 3577 { 842132, -2794693, -4763381, -8722815, 26332018, -12405641, 3578 11831880, 6985184, -9940361, 2854096 }, 3579 { -4847262, -7969331, 2516242, -5847713, 9695691, -7221186, 16512645, 3580 960770, 12121869, 16648078 }, 3581 }, 3582 { 3583 { -15218652, 14667096, -13336229, 2013717, 30598287, -464137, 3584 -31504922, -7882064, 20237806, 2838411 }, 3585 { -19288047, 4453152, 15298546, -16178388, 22115043, -15972604, 3586 12544294, -13470457, 1068881, -12499905 }, 3587 { -9558883, -16518835, 33238498, 13506958, 30505848, -1114596, 3588 -8486907, -2630053, 12521378, 4845654 }, 3589 }, 3590 { 3591 { -28198521, 10744108, -2958380, 10199664, 7759311, -13088600, 3592 3409348, -873400, -6482306, -12885870 }, 3593 { -23561822, 6230156, -20382013, 10655314, -24040585, -11621172, 3594 10477734, -1240216, -3113227, 13974498 }, 3595 { 12966261, 15550616, -32038948, -1615346, 21025980, -629444, 3596 5642325, 7188737, 18895762, 12629579 }, 3597 }, 3598 }, 3599 { 3600 { 3601 { 14741879, -14946887, 22177208, -11721237, 1279741, 8058600, 3602 11758140, 789443, 32195181, 3895677 }, 3603 { 10758205, 15755439, -4509950, 9243698, -4879422, 6879879, -2204575, 3604 -3566119, -8982069, 4429647 }, 3605 { -2453894, 15725973, -20436342, -10410672, -5803908, -11040220, 3606 -7135870, -11642895, 18047436, -15281743 }, 3607 }, 3608 { 3609 { -25173001, -11307165, 29759956, 11776784, -22262383, -15820455, 3610 10993114, -12850837, -17620701, -9408468 }, 3611 { 21987233, 700364, -24505048, 14972008, -7774265, -5718395, 3612 32155026, 2581431, -29958985, 8773375 }, 3613 { -25568350, 454463, -13211935, 16126715, 25240068, 8594567, 3614 20656846, 12017935, -7874389, -13920155 }, 3615 }, 3616 { 3617 { 6028182, 6263078, -31011806, -11301710, -818919, 2461772, 3618 -31841174, -5468042, -1721788, -2776725 }, 3619 { -12278994, 16624277, 987579, -5922598, 32908203, 1248608, 7719845, 3620 -4166698, 28408820, 6816612 }, 3621 { -10358094, -8237829, 19549651, -12169222, 22082623, 16147817, 3622 20613181, 13982702, -10339570, 5067943 }, 3623 }, 3624 { 3625 { -30505967, -3821767, 12074681, 13582412, -19877972, 2443951, 3626 -19719286, 12746132, 5331210, -10105944 }, 3627 { 30528811, 3601899, -1957090, 4619785, -27361822, -15436388, 3628 24180793, -12570394, 27679908, -1648928 }, 3629 { 9402404, -13957065, 32834043, 10838634, -26580150, -13237195, 3630 26653274, -8685565, 22611444, -12715406 }, 3631 }, 3632 { 3633 { 22190590, 1118029, 22736441, 15130463, -30460692, -5991321, 3634 19189625, -4648942, 4854859, 6622139 }, 3635 { -8310738, -2953450, -8262579, -3388049, -10401731, -271929, 3636 13424426, -3567227, 26404409, 13001963 }, 3637 { -31241838, -15415700, -2994250, 8939346, 11562230, -12840670, 3638 -26064365, -11621720, -15405155, 11020693 }, 3639 }, 3640 { 3641 { 1866042, -7949489, -7898649, -10301010, 12483315, 13477547, 3642 3175636, -12424163, 28761762, 1406734 }, 3643 { -448555, -1777666, 13018551, 3194501, -9580420, -11161737, 3644 24760585, -4347088, 25577411, -13378680 }, 3645 { -24290378, 4759345, -690653, -1852816, 2066747, 10693769, 3646 -29595790, 9884936, -9368926, 4745410 }, 3647 }, 3648 { 3649 { -9141284, 6049714, -19531061, -4341411, -31260798, 9944276, 3650 -15462008, -11311852, 10931924, -11931931 }, 3651 { -16561513, 14112680, -8012645, 4817318, -8040464, -11414606, 3652 -22853429, 10856641, -20470770, 13434654 }, 3653 { 22759489, -10073434, -16766264, -1871422, 13637442, -10168091, 3654 1765144, -12654326, 28445307, -5364710 }, 3655 }, 3656 { 3657 { 29875063, 12493613, 2795536, -3786330, 1710620, 15181182, 3658 -10195717, -8788675, 9074234, 1167180 }, 3659 { -26205683, 11014233, -9842651, -2635485, -26908120, 7532294, 3660 -18716888, -9535498, 3843903, 9367684 }, 3661 { -10969595, -6403711, 9591134, 9582310, 11349256, 108879, 16235123, 3662 8601684, -139197, 4242895 }, 3663 }, 3664 }, 3665 { 3666 { 3667 { 22092954, -13191123, -2042793, -11968512, 32186753, -11517388, 3668 -6574341, 2470660, -27417366, 16625501 }, 3669 { -11057722, 3042016, 13770083, -9257922, 584236, -544855, -7770857, 3670 2602725, -27351616, 14247413 }, 3671 { 6314175, -10264892, -32772502, 15957557, -10157730, 168750, 3672 -8618807, 14290061, 27108877, -1180880 }, 3673 }, 3674 { 3675 { -8586597, -7170966, 13241782, 10960156, -32991015, -13794596, 3676 33547976, -11058889, -27148451, 981874 }, 3677 { 22833440, 9293594, -32649448, -13618667, -9136966, 14756819, 3678 -22928859, -13970780, -10479804, -16197962 }, 3679 { -7768587, 3326786, -28111797, 10783824, 19178761, 14905060, 3680 22680049, 13906969, -15933690, 3797899 }, 3681 }, 3682 { 3683 { 21721356, -4212746, -12206123, 9310182, -3882239, -13653110, 3684 23740224, -2709232, 20491983, -8042152 }, 3685 { 9209270, -15135055, -13256557, -6167798, -731016, 15289673, 3686 25947805, 15286587, 30997318, -6703063 }, 3687 { 7392032, 16618386, 23946583, -8039892, -13265164, -1533858, 3688 -14197445, -2321576, 17649998, -250080 }, 3689 }, 3690 { 3691 { -9301088, -14193827, 30609526, -3049543, -25175069, -1283752, 3692 -15241566, -9525724, -2233253, 7662146 }, 3693 { -17558673, 1763594, -33114336, 15908610, -30040870, -12174295, 3694 7335080, -8472199, -3174674, 3440183 }, 3695 { -19889700, -5977008, -24111293, -9688870, 10799743, -16571957, 3696 40450, -4431835, 4862400, 1133 }, 3697 }, 3698 { 3699 { -32856209, -7873957, -5422389, 14860950, -16319031, 7956142, 3700 7258061, 311861, -30594991, -7379421 }, 3701 { -3773428, -1565936, 28985340, 7499440, 24445838, 9325937, 29727763, 3702 16527196, 18278453, 15405622 }, 3703 { -4381906, 8508652, -19898366, -3674424, -5984453, 15149970, 3704 -13313598, 843523, -21875062, 13626197 }, 3705 }, 3706 { 3707 { 2281448, -13487055, -10915418, -2609910, 1879358, 16164207, 3708 -10783882, 3953792, 13340839, 15928663 }, 3709 { 31727126, -7179855, -18437503, -8283652, 2875793, -16390330, 3710 -25269894, -7014826, -23452306, 5964753 }, 3711 { 4100420, -5959452, -17179337, 6017714, -18705837, 12227141, 3712 -26684835, 11344144, 2538215, -7570755 }, 3713 }, 3714 { 3715 { -9433605, 6123113, 11159803, -2156608, 30016280, 14966241, 3716 -20474983, 1485421, -629256, -15958862 }, 3717 { -26804558, 4260919, 11851389, 9658551, -32017107, 16367492, 3718 -20205425, -13191288, 11659922, -11115118 }, 3719 { 26180396, 10015009, -30844224, -8581293, 5418197, 9480663, 2231568, 3720 -10170080, 33100372, -1306171 }, 3721 }, 3722 { 3723 { 15121113, -5201871, -10389905, 15427821, -27509937, -15992507, 3724 21670947, 4486675, -5931810, -14466380 }, 3725 { 16166486, -9483733, -11104130, 6023908, -31926798, -1364923, 3726 2340060, -16254968, -10735770, -10039824 }, 3727 { 28042865, -3557089, -12126526, 12259706, -3717498, -6945899, 3728 6766453, -8689599, 18036436, 5803270 }, 3729 }, 3730 }, 3731 { 3732 { 3733 { -817581, 6763912, 11803561, 1585585, 10958447, -2671165, 23855391, 3734 4598332, -6159431, -14117438 }, 3735 { -31031306, -14256194, 17332029, -2383520, 31312682, -5967183, 3736 696309, 50292, -20095739, 11763584 }, 3737 { -594563, -2514283, -32234153, 12643980, 12650761, 14811489, 665117, 3738 -12613632, -19773211, -10713562 }, 3739 }, 3740 { 3741 { 30464590, -11262872, -4127476, -12734478, 19835327, -7105613, 3742 -24396175, 2075773, -17020157, 992471 }, 3743 { 18357185, -6994433, 7766382, 16342475, -29324918, 411174, 14578841, 3744 8080033, -11574335, -10601610 }, 3745 { 19598397, 10334610, 12555054, 2555664, 18821899, -10339780, 3746 21873263, 16014234, 26224780, 16452269 }, 3747 }, 3748 { 3749 { -30223925, 5145196, 5944548, 16385966, 3976735, 2009897, -11377804, 3750 -7618186, -20533829, 3698650 }, 3751 { 14187449, 3448569, -10636236, -10810935, -22663880, -3433596, 3752 7268410, -10890444, 27394301, 12015369 }, 3753 { 19695761, 16087646, 28032085, 12999827, 6817792, 11427614, 3754 20244189, -1312777, -13259127, -3402461 }, 3755 }, 3756 { 3757 { 30860103, 12735208, -1888245, -4699734, -16974906, 2256940, 3758 -8166013, 12298312, -8550524, -10393462 }, 3759 { -5719826, -11245325, -1910649, 15569035, 26642876, -7587760, 3760 -5789354, -15118654, -4976164, 12651793 }, 3761 { -2848395, 9953421, 11531313, -5282879, 26895123, -12697089, 3762 -13118820, -16517902, 9768698, -2533218 }, 3763 }, 3764 { 3765 { -24719459, 1894651, -287698, -4704085, 15348719, -8156530, 3766 32767513, 12765450, 4940095, 10678226 }, 3767 { 18860224, 15980149, -18987240, -1562570, -26233012, -11071856, 3768 -7843882, 13944024, -24372348, 16582019 }, 3769 { -15504260, 4970268, -29893044, 4175593, -20993212, -2199756, 3770 -11704054, 15444560, -11003761, 7989037 }, 3771 }, 3772 { 3773 { 31490452, 5568061, -2412803, 2182383, -32336847, 4531686, 3774 -32078269, 6200206, -19686113, -14800171 }, 3775 { -17308668, -15879940, -31522777, -2831, -32887382, 16375549, 3776 8680158, -16371713, 28550068, -6857132 }, 3777 { -28126887, -5688091, 16837845, -1820458, -6850681, 12700016, 3778 -30039981, 4364038, 1155602, 5988841 }, 3779 }, 3780 { 3781 { 21890435, -13272907, -12624011, 12154349, -7831873, 15300496, 3782 23148983, -4470481, 24618407, 8283181 }, 3783 { -33136107, -10512751, 9975416, 6841041, -31559793, 16356536, 3784 3070187, -7025928, 1466169, 10740210 }, 3785 { -1509399, -15488185, -13503385, -10655916, 32799044, 909394, 3786 -13938903, -5779719, -32164649, -15327040 }, 3787 }, 3788 { 3789 { 3960823, -14267803, -28026090, -15918051, -19404858, 13146868, 3790 15567327, 951507, -3260321, -573935 }, 3791 { 24740841, 5052253, -30094131, 8961361, 25877428, 6165135, 3792 -24368180, 14397372, -7380369, -6144105 }, 3793 { -28888365, 3510803, -28103278, -1158478, -11238128, -10631454, 3794 -15441463, -14453128, -1625486, -6494814 }, 3795 }, 3796 }, 3797 { 3798 { 3799 { 793299, -9230478, 8836302, -6235707, -27360908, -2369593, 33152843, 3800 -4885251, -9906200, -621852 }, 3801 { 5666233, 525582, 20782575, -8038419, -24538499, 14657740, 16099374, 3802 1468826, -6171428, -15186581 }, 3803 { -4859255, -3779343, -2917758, -6748019, 7778750, 11688288, 3804 -30404353, -9871238, -1558923, -9863646 }, 3805 }, 3806 { 3807 { 10896332, -7719704, 824275, 472601, -19460308, 3009587, 25248958, 3808 14783338, -30581476, -15757844 }, 3809 { 10566929, 12612572, -31944212, 11118703, -12633376, 12362879, 3810 21752402, 8822496, 24003793, 14264025 }, 3811 { 27713862, -7355973, -11008240, 9227530, 27050101, 2504721, 3812 23886875, -13117525, 13958495, -5732453 }, 3813 }, 3814 { 3815 { -23481610, 4867226, -27247128, 3900521, 29838369, -8212291, 3816 -31889399, -10041781, 7340521, -15410068 }, 3817 { 4646514, -8011124, -22766023, -11532654, 23184553, 8566613, 3818 31366726, -1381061, -15066784, -10375192 }, 3819 { -17270517, 12723032, -16993061, 14878794, 21619651, -6197576, 3820 27584817, 3093888, -8843694, 3849921 }, 3821 }, 3822 { 3823 { -9064912, 2103172, 25561640, -15125738, -5239824, 9582958, 3824 32477045, -9017955, 5002294, -15550259 }, 3825 { -12057553, -11177906, 21115585, -13365155, 8808712, -12030708, 3826 16489530, 13378448, -25845716, 12741426 }, 3827 { -5946367, 10645103, -30911586, 15390284, -3286982, -7118677, 3828 24306472, 15852464, 28834118, -7646072 }, 3829 }, 3830 { 3831 { -17335748, -9107057, -24531279, 9434953, -8472084, -583362, 3832 -13090771, 455841, 20461858, 5491305 }, 3833 { 13669248, -16095482, -12481974, -10203039, -14569770, -11893198, 3834 -24995986, 11293807, -28588204, -9421832 }, 3835 { 28497928, 6272777, -33022994, 14470570, 8906179, -1225630, 3836 18504674, -14165166, 29867745, -8795943 }, 3837 }, 3838 { 3839 { -16207023, 13517196, -27799630, -13697798, 24009064, -6373891, 3840 -6367600, -13175392, 22853429, -4012011 }, 3841 { 24191378, 16712145, -13931797, 15217831, 14542237, 1646131, 3842 18603514, -11037887, 12876623, -2112447 }, 3843 { 17902668, 4518229, -411702, -2829247, 26878217, 5258055, -12860753, 3844 608397, 16031844, 3723494 }, 3845 }, 3846 { 3847 { -28632773, 12763728, -20446446, 7577504, 33001348, -13017745, 3848 17558842, -7872890, 23896954, -4314245 }, 3849 { -20005381, -12011952, 31520464, 605201, 2543521, 5991821, -2945064, 3850 7229064, -9919646, -8826859 }, 3851 { 28816045, 298879, -28165016, -15920938, 19000928, -1665890, 3852 -12680833, -2949325, -18051778, -2082915 }, 3853 }, 3854 { 3855 { 16000882, -344896, 3493092, -11447198, -29504595, -13159789, 3856 12577740, 16041268, -19715240, 7847707 }, 3857 { 10151868, 10572098, 27312476, 7922682, 14825339, 4723128, 3858 -32855931, -6519018, -10020567, 3852848 }, 3859 { -11430470, 15697596, -21121557, -4420647, 5386314, 15063598, 3860 16514493, -15932110, 29330899, -15076224 }, 3861 }, 3862 }, 3863 { 3864 { 3865 { -25499735, -4378794, -15222908, -6901211, 16615731, 2051784, 3866 3303702, 15490, -27548796, 12314391 }, 3867 { 15683520, -6003043, 18109120, -9980648, 15337968, -5997823, 3868 -16717435, 15921866, 16103996, -3731215 }, 3869 { -23169824, -10781249, 13588192, -1628807, -3798557, -1074929, 3870 -19273607, 5402699, -29815713, -9841101 }, 3871 }, 3872 { 3873 { 23190676, 2384583, -32714340, 3462154, -29903655, -1529132, 3874 -11266856, 8911517, -25205859, 2739713 }, 3875 { 21374101, -3554250, -33524649, 9874411, 15377179, 11831242, 3876 -33529904, 6134907, 4931255, 11987849 }, 3877 { -7732, -2978858, -16223486, 7277597, 105524, -322051, -31480539, 3878 13861388, -30076310, 10117930 }, 3879 }, 3880 { 3881 { -29501170, -10744872, -26163768, 13051539, -25625564, 5089643, 3882 -6325503, 6704079, 12890019, 15728940 }, 3883 { -21972360, -11771379, -951059, -4418840, 14704840, 2695116, 903376, 3884 -10428139, 12885167, 8311031 }, 3885 { -17516482, 5352194, 10384213, -13811658, 7506451, 13453191, 3886 26423267, 4384730, 1888765, -5435404 }, 3887 }, 3888 { 3889 { -25817338, -3107312, -13494599, -3182506, 30896459, -13921729, 3890 -32251644, -12707869, -19464434, -3340243 }, 3891 { -23607977, -2665774, -526091, 4651136, 5765089, 4618330, 6092245, 3892 14845197, 17151279, -9854116 }, 3893 { -24830458, -12733720, -15165978, 10367250, -29530908, -265356, 3894 22825805, -7087279, -16866484, 16176525 }, 3895 }, 3896 { 3897 { -23583256, 6564961, 20063689, 3798228, -4740178, 7359225, 2006182, 3898 -10363426, -28746253, -10197509 }, 3899 { -10626600, -4486402, -13320562, -5125317, 3432136, -6393229, 3900 23632037, -1940610, 32808310, 1099883 }, 3901 { 15030977, 5768825, -27451236, -2887299, -6427378, -15361371, 3902 -15277896, -6809350, 2051441, -15225865 }, 3903 }, 3904 { 3905 { -3362323, -7239372, 7517890, 9824992, 23555850, 295369, 5148398, 3906 -14154188, -22686354, 16633660 }, 3907 { 4577086, -16752288, 13249841, -15304328, 19958763, -14537274, 3908 18559670, -10759549, 8402478, -9864273 }, 3909 { -28406330, -1051581, -26790155, -907698, -17212414, -11030789, 3910 9453451, -14980072, 17983010, 9967138 }, 3911 }, 3912 { 3913 { -25762494, 6524722, 26585488, 9969270, 24709298, 1220360, -1677990, 3914 7806337, 17507396, 3651560 }, 3915 { -10420457, -4118111, 14584639, 15971087, -15768321, 8861010, 3916 26556809, -5574557, -18553322, -11357135 }, 3917 { 2839101, 14284142, 4029895, 3472686, 14402957, 12689363, -26642121, 3918 8459447, -5605463, -7621941 }, 3919 }, 3920 { 3921 { -4839289, -3535444, 9744961, 2871048, 25113978, 3187018, -25110813, 3922 -849066, 17258084, -7977739 }, 3923 { 18164541, -10595176, -17154882, -1542417, 19237078, -9745295, 3924 23357533, -15217008, 26908270, 12150756 }, 3925 { -30264870, -7647865, 5112249, -7036672, -1499807, -6974257, 43168, 3926 -5537701, -32302074, 16215819 }, 3927 }, 3928 }, 3929 { 3930 { 3931 { -6898905, 9824394, -12304779, -4401089, -31397141, -6276835, 3932 32574489, 12532905, -7503072, -8675347 }, 3933 { -27343522, -16515468, -27151524, -10722951, 946346, 16291093, 3934 254968, 7168080, 21676107, -1943028 }, 3935 { 21260961, -8424752, -16831886, -11920822, -23677961, 3968121, 3936 -3651949, -6215466, -3556191, -7913075 }, 3937 }, 3938 { 3939 { 16544754, 13250366, -16804428, 15546242, -4583003, 12757258, 3940 -2462308, -8680336, -18907032, -9662799 }, 3941 { -2415239, -15577728, 18312303, 4964443, -15272530, -12653564, 3942 26820651, 16690659, 25459437, -4564609 }, 3943 { -25144690, 11425020, 28423002, -11020557, -6144921, -15826224, 3944 9142795, -2391602, -6432418, -1644817 }, 3945 }, 3946 { 3947 { -23104652, 6253476, 16964147, -3768872, -25113972, -12296437, 3948 -27457225, -16344658, 6335692, 7249989 }, 3949 { -30333227, 13979675, 7503222, -12368314, -11956721, -4621693, 3950 -30272269, 2682242, 25993170, -12478523 }, 3951 { 4364628, 5930691, 32304656, -10044554, -8054781, 15091131, 3952 22857016, -10598955, 31820368, 15075278 }, 3953 }, 3954 { 3955 { 31879134, -8918693, 17258761, 90626, -8041836, -4917709, 24162788, 3956 -9650886, -17970238, 12833045 }, 3957 { 19073683, 14851414, -24403169, -11860168, 7625278, 11091125, 3958 -19619190, 2074449, -9413939, 14905377 }, 3959 { 24483667, -11935567, -2518866, -11547418, -1553130, 15355506, 3960 -25282080, 9253129, 27628530, -7555480 }, 3961 }, 3962 { 3963 { 17597607, 8340603, 19355617, 552187, 26198470, -3176583, 4593324, 3964 -9157582, -14110875, 15297016 }, 3965 { 510886, 14337390, -31785257, 16638632, 6328095, 2713355, -20217417, 3966 -11864220, 8683221, 2921426 }, 3967 { 18606791, 11874196, 27155355, -5281482, -24031742, 6265446, 3968 -25178240, -1278924, 4674690, 13890525 }, 3969 }, 3970 { 3971 { 13609624, 13069022, -27372361, -13055908, 24360586, 9592974, 3972 14977157, 9835105, 4389687, 288396 }, 3973 { 9922506, -519394, 13613107, 5883594, -18758345, -434263, -12304062, 3974 8317628, 23388070, 16052080 }, 3975 { 12720016, 11937594, -31970060, -5028689, 26900120, 8561328, 3976 -20155687, -11632979, -14754271, -10812892 }, 3977 }, 3978 { 3979 { 15961858, 14150409, 26716931, -665832, -22794328, 13603569, 3980 11829573, 7467844, -28822128, 929275 }, 3981 { 11038231, -11582396, -27310482, -7316562, -10498527, -16307831, 3982 -23479533, -9371869, -21393143, 2465074 }, 3983 { 20017163, -4323226, 27915242, 1529148, 12396362, 15675764, 3984 13817261, -9658066, 2463391, -4622140 }, 3985 }, 3986 { 3987 { -16358878, -12663911, -12065183, 4996454, -1256422, 1073572, 3988 9583558, 12851107, 4003896, 12673717 }, 3989 { -1731589, -15155870, -3262930, 16143082, 19294135, 13385325, 3990 14741514, -9103726, 7903886, 2348101 }, 3991 { 24536016, -16515207, 12715592, -3862155, 1511293, 10047386, 3992 -3842346, -7129159, -28377538, 10048127 }, 3993 }, 3994 }, 3995 { 3996 { 3997 { -12622226, -6204820, 30718825, 2591312, -10617028, 12192840, 3998 18873298, -7297090, -32297756, 15221632 }, 3999 { -26478122, -11103864, 11546244, -1852483, 9180880, 7656409, 4000 -21343950, 2095755, 29769758, 6593415 }, 4001 { -31994208, -2907461, 4176912, 3264766, 12538965, -868111, 26312345, 4002 -6118678, 30958054, 8292160 }, 4003 }, 4004 { 4005 { 31429822, -13959116, 29173532, 15632448, 12174511, -2760094, 4006 32808831, 3977186, 26143136, -3148876 }, 4007 { 22648901, 1402143, -22799984, 13746059, 7936347, 365344, -8668633, 4008 -1674433, -3758243, -2304625 }, 4009 { -15491917, 8012313, -2514730, -12702462, -23965846, -10254029, 4010 -1612713, -1535569, -16664475, 8194478 }, 4011 }, 4012 { 4013 { 27338066, -7507420, -7414224, 10140405, -19026427, -6589889, 4014 27277191, 8855376, 28572286, 3005164 }, 4015 { 26287124, 4821776, 25476601, -4145903, -3764513, -15788984, 4016 -18008582, 1182479, -26094821, -13079595 }, 4017 { -7171154, 3178080, 23970071, 6201893, -17195577, -4489192, 4018 -21876275, -13982627, 32208683, -1198248 }, 4019 }, 4020 { 4021 { -16657702, 2817643, -10286362, 14811298, 6024667, 13349505, 4022 -27315504, -10497842, -27672585, -11539858 }, 4023 { 15941029, -9405932, -21367050, 8062055, 31876073, -238629, 4024 -15278393, -1444429, 15397331, -4130193 }, 4025 { 8934485, -13485467, -23286397, -13423241, -32446090, 14047986, 4026 31170398, -1441021, -27505566, 15087184 }, 4027 }, 4028 { 4029 { -18357243, -2156491, 24524913, -16677868, 15520427, -6360776, 4030 -15502406, 11461896, 16788528, -5868942 }, 4031 { -1947386, 16013773, 21750665, 3714552, -17401782, -16055433, 4032 -3770287, -10323320, 31322514, -11615635 }, 4033 { 21426655, -5650218, -13648287, -5347537, -28812189, -4920970, 4034 -18275391, -14621414, 13040862, -12112948 }, 4035 }, 4036 { 4037 { 11293895, 12478086, -27136401, 15083750, -29307421, 14748872, 4038 14555558, -13417103, 1613711, 4896935 }, 4039 { -25894883, 15323294, -8489791, -8057900, 25967126, -13425460, 4040 2825960, -4897045, -23971776, -11267415 }, 4041 { -15924766, -5229880, -17443532, 6410664, 3622847, 10243618, 4042 20615400, 12405433, -23753030, -8436416 }, 4043 }, 4044 { 4045 { -7091295, 12556208, -20191352, 9025187, -17072479, 4333801, 4046 4378436, 2432030, 23097949, -566018 }, 4047 { 4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264, 4048 10103221, -18512313, 2424778 }, 4049 { 366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678, 4050 1344109, -3642553, 12412659 }, 4051 }, 4052 { 4053 { -24001791, 7690286, 14929416, -168257, -32210835, -13412986, 4054 24162697, -15326504, -3141501, 11179385 }, 4055 { 18289522, -14724954, 8056945, 16430056, -21729724, 7842514, 4056 -6001441, -1486897, -18684645, -11443503 }, 4057 { 476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959, 4058 13403813, 11052904, 5219329 }, 4059 }, 4060 }, 4061 { 4062 { 4063 { 20678546, -8375738, -32671898, 8849123, -5009758, 14574752, 4064 31186971, -3973730, 9014762, -8579056 }, 4065 { -13644050, -10350239, -15962508, 5075808, -1514661, -11534600, 4066 -33102500, 9160280, 8473550, -3256838 }, 4067 { 24900749, 14435722, 17209120, -15292541, -22592275, 9878983, 4068 -7689309, -16335821, -24568481, 11788948 }, 4069 }, 4070 { 4071 { -3118155, -11395194, -13802089, 14797441, 9652448, -6845904, 4072 -20037437, 10410733, -24568470, -1458691 }, 4073 { -15659161, 16736706, -22467150, 10215878, -9097177, 7563911, 4074 11871841, -12505194, -18513325, 8464118 }, 4075 { -23400612, 8348507, -14585951, -861714, -3950205, -6373419, 4076 14325289, 8628612, 33313881, -8370517 }, 4077 }, 4078 { 4079 { -20186973, -4967935, 22367356, 5271547, -1097117, -4788838, 4080 -24805667, -10236854, -8940735, -5818269 }, 4081 { -6948785, -1795212, -32625683, -16021179, 32635414, -7374245, 4082 15989197, -12838188, 28358192, -4253904 }, 4083 { -23561781, -2799059, -32351682, -1661963, -9147719, 10429267, 4084 -16637684, 4072016, -5351664, 5596589 }, 4085 }, 4086 { 4087 { -28236598, -3390048, 12312896, 6213178, 3117142, 16078565, 4088 29266239, 2557221, 1768301, 15373193 }, 4089 { -7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902, 4090 -4504991, -24660491, 3442910 }, 4091 { -30210571, 5124043, 14181784, 8197961, 18964734, -11939093, 4092 22597931, 7176455, -18585478, 13365930 }, 4093 }, 4094 { 4095 { -7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107, 4096 -8570186, -9689599, -3031667 }, 4097 { 25008904, -10771599, -4305031, -9638010, 16265036, 15721635, 4098 683793, -11823784, 15723479, -15163481 }, 4099 { -9660625, 12374379, -27006999, -7026148, -7724114, -12314514, 4100 11879682, 5400171, 519526, -1235876 }, 4101 }, 4102 { 4103 { 22258397, -16332233, -7869817, 14613016, -22520255, -2950923, 4104 -20353881, 7315967, 16648397, 7605640 }, 4105 { -8081308, -8464597, -8223311, 9719710, 19259459, -15348212, 4106 23994942, -5281555, -9468848, 4763278 }, 4107 { -21699244, 9220969, -15730624, 1084137, -25476107, -2852390, 4108 31088447, -7764523, -11356529, 728112 }, 4109 }, 4110 { 4111 { 26047220, -11751471, -6900323, -16521798, 24092068, 9158119, 4112 -4273545, -12555558, -29365436, -5498272 }, 4113 { 17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007, 4114 12327945, 10750447, 10014012 }, 4115 { -10312768, 3936952, 9156313, -8897683, 16498692, -994647, 4116 -27481051, -666732, 3424691, 7540221 }, 4117 }, 4118 { 4119 { 30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422, 4120 -16317219, -9244265, 15258046 }, 4121 { 13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406, 4122 2711395, 1062915, -5136345 }, 4123 { -19240248, -11254599, -29509029, -7499965, -5835763, 13005411, 4124 -6066489, 12194497, 32960380, 1459310 }, 4125 }, 4126 }, 4127 { 4128 { 4129 { 19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197, 4130 -6101885, 18638003, -11174937 }, 4131 { 31395534, 15098109, 26581030, 8030562, -16527914, -5007134, 4132 9012486, -7584354, -6643087, -5442636 }, 4133 { -9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222, 4134 9677543, -32294889, -6456008 }, 4135 }, 4136 { 4137 { -2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579, 4138 -7839692, -7852844, -8138429 }, 4139 { -15236356, -15433509, 7766470, 746860, 26346930, -10221762, 4140 -27333451, 10754588, -9431476, 5203576 }, 4141 { 31834314, 14135496, -770007, 5159118, 20917671, -16768096, 4142 -7467973, -7337524, 31809243, 7347066 }, 4143 }, 4144 { 4145 { -9606723, -11874240, 20414459, 13033986, 13716524, -11691881, 4146 19797970, -12211255, 15192876, -2087490 }, 4147 { -12663563, -2181719, 1168162, -3804809, 26747877, -14138091, 4148 10609330, 12694420, 33473243, -13382104 }, 4149 { 33184999, 11180355, 15832085, -11385430, -1633671, 225884, 4150 15089336, -11023903, -6135662, 14480053 }, 4151 }, 4152 { 4153 { 31308717, -5619998, 31030840, -1897099, 15674547, -6582883, 4154 5496208, 13685227, 27595050, 8737275 }, 4155 { -20318852, -15150239, 10933843, -16178022, 8335352, -7546022, 4156 -31008351, -12610604, 26498114, 66511 }, 4157 { 22644454, -8761729, -16671776, 4884562, -3105614, -13559366, 4158 30540766, -4286747, -13327787, -7515095 }, 4159 }, 4160 { 4161 { -28017847, 9834845, 18617207, -2681312, -3401956, -13307506, 4162 8205540, 13585437, -17127465, 15115439 }, 4163 { 23711543, -672915, 31206561, -8362711, 6164647, -9709987, 4164 -33535882, -1426096, 8236921, 16492939 }, 4165 { -23910559, -13515526, -26299483, -4503841, 25005590, -7687270, 4166 19574902, 10071562, 6708380, -6222424 }, 4167 }, 4168 { 4169 { 2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017, 4170 9328700, 29955601, -11678310 }, 4171 { 3096359, 9271816, -21620864, -15521844, -14847996, -7592937, 4172 -25892142, -12635595, -9917575, 6216608 }, 4173 { -32615849, 338663, -25195611, 2510422, -29213566, -13820213, 4174 24822830, -6146567, -26767480, 7525079 }, 4175 }, 4176 { 4177 { -23066649, -13985623, 16133487, -7896178, -3389565, 778788, 4178 -910336, -2782495, -19386633, 11994101 }, 4179 { 21691500, -13624626, -641331, -14367021, 3285881, -3483596, 4180 -25064666, 9718258, -7477437, 13381418 }, 4181 { 18445390, -4202236, 14979846, 11622458, -1727110, -3582980, 4182 23111648, -6375247, 28535282, 15779576 }, 4183 }, 4184 { 4185 { 30098053, 3089662, -9234387, 16662135, -21306940, 11308411, 4186 -14068454, 12021730, 9955285, -16303356 }, 4187 { 9734894, -14576830, -7473633, -9138735, 2060392, 11313496, 4188 -18426029, 9924399, 20194861, 13380996 }, 4189 { -26378102, -7965207, -22167821, 15789297, -18055342, -6168792, 4190 -1984914, 15707771, 26342023, 10146099 }, 4191 }, 4192 }, 4193 { 4194 { 4195 { -26016874, -219943, 21339191, -41388, 19745256, -2878700, 4196 -29637280, 2227040, 21612326, -545728 }, 4197 { -13077387, 1184228, 23562814, -5970442, -20351244, -6348714, 4198 25764461, 12243797, -20856566, 11649658 }, 4199 { -10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944, 4200 6114064, 33514190, 2333242 }, 4201 }, 4202 { 4203 { -21433588, -12421821, 8119782, 7219913, -21830522, -9016134, 4204 -6679750, -12670638, 24350578, -13450001 }, 4205 { -4116307, -11271533, -23886186, 4843615, -30088339, 690623, 4206 -31536088, -10406836, 8317860, 12352766 }, 4207 { 18200138, -14475911, -33087759, -2696619, -23702521, -9102511, 4208 -23552096, -2287550, 20712163, 6719373 }, 4209 }, 4210 { 4211 { 26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, 4212 -3763210, 26224235, -3297458 }, 4213 { -17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 4214 21728352, 9493610, 18620611, -16428628 }, 4215 { -13323321, 13325349, 11432106, 5964811, 18609221, 6062965, 4216 -5269471, -9725556, -30701573, -16479657 }, 4217 }, 4218 { 4219 { -23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 4220 12248509, -5240639, 13735342, 1934062 }, 4221 { 25089769, 6742589, 17081145, -13406266, 21909293, -16067981, 4222 -15136294, -3765346, -21277997, 5473616 }, 4223 { 31883677, -7961101, 1083432, -11572403, 22828471, 13290673, 4224 -7125085, 12469656, 29111212, -5451014 }, 4225 }, 4226 { 4227 { 24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 4228 24367466, 6388839, -10295587, 452383 }, 4229 { -25640782, -3417841, 5217916, 16224624, 19987036, -4082269, 4230 -24236251, -5915248, 15766062, 8407814 }, 4231 { -20406999, 13990231, 15495425, 16395525, 5377168, 15166495, 4232 -8917023, -4388953, -8067909, 2276718 }, 4233 }, 4234 { 4235 { 30157918, 12924066, -17712050, 9245753, 19895028, 3368142, 4236 -23827587, 5096219, 22740376, -7303417 }, 4237 { 2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 4238 24051124, 13742383, -15637599, 13295222 }, 4239 { 33338237, -8505733, 12532113, 7977527, 9106186, -1715251, 4240 -17720195, -4612972, -4451357, -14669444 }, 4241 }, 4242 { 4243 { -20045281, 5454097, -14346548, 6447146, 28862071, 1883651, 4244 -2469266, -4141880, 7770569, 9620597 }, 4245 { 23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, 4246 -1694323, -33502340, -14767970 }, 4247 { 1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 4248 1220118, 30494170, -11440799 }, 4249 }, 4250 { 4251 { -5037580, -13028295, -2970559, -3061767, 15640974, -6701666, 4252 -26739026, 926050, -1684339, -13333647 }, 4253 { 13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 4254 9021034, 9078865, 3353509, 4033511 }, 4255 { -29663431, -15113610, 32259991, -344482, 24295849, -12912123, 4256 23161163, 8839127, 27485041, 7356032 }, 4257 }, 4258 }, 4259 { 4260 { 4261 { 9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 4262 2625015, 28431036, -16771834 }, 4263 { -23839233, -8311415, -25945511, 7480958, -17681669, -8354183, 4264 -22545972, 14150565, 15970762, 4099461 }, 4265 { 29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 4266 13617293, -9937143, 11465739, 8317062 }, 4267 }, 4268 { 4269 { -25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 4270 14898637, 3848455, 20969334, -5157516 }, 4271 { -20384450, -14347713, -18336405, 13884722, -33039454, 2842114, 4272 -21610826, -3649888, 11177095, 14989547 }, 4273 { -24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 4274 13515641, 2581286, -28487508, 9930240 }, 4275 }, 4276 { 4277 { -17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 4278 18345767, -13403753, 16291481, -5314038 }, 4279 { -33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 4280 6957617, 4368891, 9788741 }, 4281 { 16660756, 7281060, -10830758, 12911820, 20108584, -8101676, 4282 -21722536, -8613148, 16250552, -11111103 }, 4283 }, 4284 { 4285 { -19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 4286 10604807, -30190403, 4782747 }, 4287 { -1354539, 14736941, -7367442, -13292886, 7710542, -14155590, 4288 -9981571, 4383045, 22546403, 437323 }, 4289 { 31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 4290 27343084, 2786261, -30633590, -14097016 }, 4291 }, 4292 { 4293 { -14467279, -683715, -33374107, 7448552, 19294360, 14334329, 4294 -19690631, 2355319, -19284671, -6114373 }, 4295 { 15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 4296 18952177, 15496498, -29380133, 11754228 }, 4297 { -2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 4298 7141596, 11724556, 22761615, -10134141 }, 4299 }, 4300 { 4301 { 16918416, 11729663, -18083579, 3022987, -31015732, -13339659, 4302 -28741185, -12227393, 32851222, 11717399 }, 4303 { 11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 4304 31474879, 3483633, -1193175, -4030831 }, 4305 { -185635, 9921305, 31456609, -13536438, -12013818, 13348923, 4306 33142652, 6546660, -19985279, -3948376 }, 4307 }, 4308 { 4309 { -32460596, 11266712, -11197107, -7899103, 31703694, 3855903, 4310 -8537131, -12833048, -30772034, -15486313 }, 4311 { -18006477, 12709068, 3991746, -6479188, -21491523, -10550425, 4312 -31135347, -16049879, 10928917, 3011958 }, 4313 { -6957757, -15594337, 31696059, 334240, 29576716, 14796075, 4314 -30831056, -12805180, 18008031, 10258577 }, 4315 }, 4316 { 4317 { -22448644, 15655569, 7018479, -4410003, -30314266, -1201591, 4318 -1853465, 1367120, 25127874, 6671743 }, 4319 { 29701166, -14373934, -10878120, 9279288, -17568, 13127210, 4320 21382910, 11042292, 25838796, 4642684 }, 4321 { -20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 4322 30468147, -13900640, 18423289, 4177476 }, 4323 }, 4324 }, 4325 }; 4326 4327 static uint8_t negative(signed char b) 4328 { 4329 uint32_t x = b; 4330 4331 x >>= 31; /* 1: yes; 0: no */ 4332 return x; 4333 } 4334 4335 static void table_select(ge_precomp *t, int pos, signed char b) 4336 { 4337 ge_precomp minust; 4338 uint8_t bnegative = negative(b); 4339 uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1); 4340 4341 ge_precomp_0(t); 4342 cmov(t, &k25519Precomp[pos][0], equal(babs, 1)); 4343 cmov(t, &k25519Precomp[pos][1], equal(babs, 2)); 4344 cmov(t, &k25519Precomp[pos][2], equal(babs, 3)); 4345 cmov(t, &k25519Precomp[pos][3], equal(babs, 4)); 4346 cmov(t, &k25519Precomp[pos][4], equal(babs, 5)); 4347 cmov(t, &k25519Precomp[pos][5], equal(babs, 6)); 4348 cmov(t, &k25519Precomp[pos][6], equal(babs, 7)); 4349 cmov(t, &k25519Precomp[pos][7], equal(babs, 8)); 4350 fe_copy(minust.yplusx, t->yminusx); 4351 fe_copy(minust.yminusx, t->yplusx); 4352 fe_neg(minust.xy2d, t->xy2d); 4353 cmov(t, &minust, bnegative); 4354 } 4355 4356 /* 4357 * h = a * B 4358 * 4359 * where a = a[0]+256*a[1]+...+256^31 a[31] 4360 * B is the Ed25519 base point (x,4/5) with x positive. 4361 * 4362 * Preconditions: 4363 * a[31] <= 127 4364 */ 4365 static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a) 4366 { 4367 signed char e[64]; 4368 signed char carry; 4369 ge_p1p1 r; 4370 ge_p2 s; 4371 ge_precomp t; 4372 int i; 4373 4374 for (i = 0; i < 32; ++i) { 4375 e[2 * i + 0] = (a[i] >> 0) & 15; 4376 e[2 * i + 1] = (a[i] >> 4) & 15; 4377 } 4378 /* each e[i] is between 0 and 15 */ 4379 /* e[63] is between 0 and 7 */ 4380 4381 carry = 0; 4382 for (i = 0; i < 63; ++i) { 4383 e[i] += carry; 4384 carry = e[i] + 8; 4385 carry >>= 4; 4386 e[i] -= carry << 4; 4387 } 4388 e[63] += carry; 4389 /* each e[i] is between -8 and 8 */ 4390 4391 ge_p3_0(h); 4392 for (i = 1; i < 64; i += 2) { 4393 table_select(&t, i / 2, e[i]); 4394 ge_madd(&r, h, &t); 4395 ge_p1p1_to_p3(h, &r); 4396 } 4397 4398 ge_p3_dbl(&r, h); 4399 ge_p1p1_to_p2(&s, &r); 4400 ge_p2_dbl(&r, &s); 4401 ge_p1p1_to_p2(&s, &r); 4402 ge_p2_dbl(&r, &s); 4403 ge_p1p1_to_p2(&s, &r); 4404 ge_p2_dbl(&r, &s); 4405 ge_p1p1_to_p3(h, &r); 4406 4407 for (i = 0; i < 64; i += 2) { 4408 table_select(&t, i / 2, e[i]); 4409 ge_madd(&r, h, &t); 4410 ge_p1p1_to_p3(h, &r); 4411 } 4412 4413 OPENSSL_cleanse(e, sizeof(e)); 4414 } 4415 4416 #if !defined(BASE_2_51_IMPLEMENTED) 4417 /* 4418 * Replace (f,g) with (g,f) if b == 1; 4419 * replace (f,g) with (f,g) if b == 0. 4420 * 4421 * Preconditions: b in {0,1}. 4422 */ 4423 static void fe_cswap(fe f, fe g, unsigned int b) 4424 { 4425 size_t i; 4426 4427 b = 0 - b; 4428 for (i = 0; i < 10; i++) { 4429 int32_t x = f[i] ^ g[i]; 4430 x &= b; 4431 f[i] ^= x; 4432 g[i] ^= x; 4433 } 4434 } 4435 4436 /* 4437 * h = f * 121666 4438 * 4439 * Can overlap h with f. 4440 * 4441 * Preconditions: 4442 * |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. 4443 * 4444 * Postconditions: 4445 * |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. 4446 */ 4447 static void fe_mul121666(fe h, fe f) 4448 { 4449 int32_t f0 = f[0]; 4450 int32_t f1 = f[1]; 4451 int32_t f2 = f[2]; 4452 int32_t f3 = f[3]; 4453 int32_t f4 = f[4]; 4454 int32_t f5 = f[5]; 4455 int32_t f6 = f[6]; 4456 int32_t f7 = f[7]; 4457 int32_t f8 = f[8]; 4458 int32_t f9 = f[9]; 4459 int64_t h0 = f0 * (int64_t)121666; 4460 int64_t h1 = f1 * (int64_t)121666; 4461 int64_t h2 = f2 * (int64_t)121666; 4462 int64_t h3 = f3 * (int64_t)121666; 4463 int64_t h4 = f4 * (int64_t)121666; 4464 int64_t h5 = f5 * (int64_t)121666; 4465 int64_t h6 = f6 * (int64_t)121666; 4466 int64_t h7 = f7 * (int64_t)121666; 4467 int64_t h8 = f8 * (int64_t)121666; 4468 int64_t h9 = f9 * (int64_t)121666; 4469 int64_t carry0; 4470 int64_t carry1; 4471 int64_t carry2; 4472 int64_t carry3; 4473 int64_t carry4; 4474 int64_t carry5; 4475 int64_t carry6; 4476 int64_t carry7; 4477 int64_t carry8; 4478 int64_t carry9; 4479 4480 carry9 = h9 + (1 << 24); 4481 h0 += (carry9 >> 25) * 19; 4482 h9 -= carry9 & kTop39Bits; 4483 carry1 = h1 + (1 << 24); 4484 h2 += carry1 >> 25; 4485 h1 -= carry1 & kTop39Bits; 4486 carry3 = h3 + (1 << 24); 4487 h4 += carry3 >> 25; 4488 h3 -= carry3 & kTop39Bits; 4489 carry5 = h5 + (1 << 24); 4490 h6 += carry5 >> 25; 4491 h5 -= carry5 & kTop39Bits; 4492 carry7 = h7 + (1 << 24); 4493 h8 += carry7 >> 25; 4494 h7 -= carry7 & kTop39Bits; 4495 4496 carry0 = h0 + (1 << 25); 4497 h1 += carry0 >> 26; 4498 h0 -= carry0 & kTop38Bits; 4499 carry2 = h2 + (1 << 25); 4500 h3 += carry2 >> 26; 4501 h2 -= carry2 & kTop38Bits; 4502 carry4 = h4 + (1 << 25); 4503 h5 += carry4 >> 26; 4504 h4 -= carry4 & kTop38Bits; 4505 carry6 = h6 + (1 << 25); 4506 h7 += carry6 >> 26; 4507 h6 -= carry6 & kTop38Bits; 4508 carry8 = h8 + (1 << 25); 4509 h9 += carry8 >> 26; 4510 h8 -= carry8 & kTop38Bits; 4511 4512 h[0] = (int32_t)h0; 4513 h[1] = (int32_t)h1; 4514 h[2] = (int32_t)h2; 4515 h[3] = (int32_t)h3; 4516 h[4] = (int32_t)h4; 4517 h[5] = (int32_t)h5; 4518 h[6] = (int32_t)h6; 4519 h[7] = (int32_t)h7; 4520 h[8] = (int32_t)h8; 4521 h[9] = (int32_t)h9; 4522 } 4523 4524 static void x25519_scalar_mult_generic(uint8_t out[32], 4525 const uint8_t scalar[32], 4526 const uint8_t point[32]) 4527 { 4528 fe x1, x2, z2, x3, z3, tmp0, tmp1; 4529 uint8_t e[32]; 4530 unsigned swap = 0; 4531 int pos; 4532 4533 memcpy(e, scalar, 32); 4534 e[0] &= 248; 4535 e[31] &= 127; 4536 e[31] |= 64; 4537 fe_frombytes(x1, point); 4538 fe_1(x2); 4539 fe_0(z2); 4540 fe_copy(x3, x1); 4541 fe_1(z3); 4542 4543 for (pos = 254; pos >= 0; --pos) { 4544 unsigned b = 1 & (e[pos / 8] >> (pos & 7)); 4545 swap ^= b; 4546 fe_cswap(x2, x3, swap); 4547 fe_cswap(z2, z3, swap); 4548 swap = b; 4549 fe_sub(tmp0, x3, z3); 4550 fe_sub(tmp1, x2, z2); 4551 fe_add(x2, x2, z2); 4552 fe_add(z2, x3, z3); 4553 fe_mul(z3, tmp0, x2); 4554 fe_mul(z2, z2, tmp1); 4555 fe_sq(tmp0, tmp1); 4556 fe_sq(tmp1, x2); 4557 fe_add(x3, z3, z2); 4558 fe_sub(z2, z3, z2); 4559 fe_mul(x2, tmp1, tmp0); 4560 fe_sub(tmp1, tmp1, tmp0); 4561 fe_sq(z2, z2); 4562 fe_mul121666(z3, tmp1); 4563 fe_sq(x3, x3); 4564 fe_add(tmp0, tmp0, z3); 4565 fe_mul(z3, x1, z2); 4566 fe_mul(z2, tmp1, tmp0); 4567 } 4568 4569 fe_invert(z2, z2); 4570 fe_mul(x2, x2, z2); 4571 fe_tobytes(out, x2); 4572 4573 OPENSSL_cleanse(e, sizeof(e)); 4574 } 4575 4576 static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32], 4577 const uint8_t point[32]) 4578 { 4579 x25519_scalar_mult_generic(out, scalar, point); 4580 } 4581 #endif 4582 4583 static void slide(signed char *r, const uint8_t *a) 4584 { 4585 int i; 4586 int b; 4587 int k; 4588 4589 for (i = 0; i < 256; ++i) { 4590 r[i] = 1 & (a[i >> 3] >> (i & 7)); 4591 } 4592 4593 for (i = 0; i < 256; ++i) { 4594 if (r[i]) { 4595 for (b = 1; b <= 6 && i + b < 256; ++b) { 4596 if (r[i + b]) { 4597 if (r[i] + (r[i + b] << b) <= 15) { 4598 r[i] += r[i + b] << b; 4599 r[i + b] = 0; 4600 } else if (r[i] - (r[i + b] << b) >= -15) { 4601 r[i] -= r[i + b] << b; 4602 for (k = i + b; k < 256; ++k) { 4603 if (!r[k]) { 4604 r[k] = 1; 4605 break; 4606 } 4607 r[k] = 0; 4608 } 4609 } else { 4610 break; 4611 } 4612 } 4613 } 4614 } 4615 } 4616 } 4617 4618 static const ge_precomp Bi[8] = { 4619 { 4620 { 25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, 4621 -11754271, -6079156, 2047605 }, 4622 { -12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 4623 5043384, 19500929, -15469378 }, 4624 { -8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 4625 11864899, -24514362, -4438546 }, 4626 }, 4627 { 4628 { 15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, 4629 -14772189, 28944400, -1550024 }, 4630 { 16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, 4631 -11775962, 7689662, 11199574 }, 4632 { 30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 4633 10017326, -17749093, -9920357 }, 4634 }, 4635 { 4636 { 10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 4637 14515107, -15438304, 10819380 }, 4638 { 4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 4639 12483688, -12668491, 5581306 }, 4640 { 19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 4641 13850243, -23678021, -15815942 }, 4642 }, 4643 { 4644 { 5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 4645 5230134, -23952439, -15175766 }, 4646 { -30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 4647 16520125, 30598449, 7715701 }, 4648 { 28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 4649 1370708, 29794553, -1409300 }, 4650 }, 4651 { 4652 { -22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, 4653 -1361450, -13062696, 13821877 }, 4654 { -6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, 4655 -7212327, 18853322, -14220951 }, 4656 { 4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, 4657 -10431137, 2207753, -3209784 }, 4658 }, 4659 { 4660 { -25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, 4661 -663000, -31111463, -16132436 }, 4662 { 25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 4663 15725684, 171356, 6466918 }, 4664 { 23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, 4665 -14088058, -30714912, 16193877 }, 4666 }, 4667 { 4668 { -33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4669 4729455, -18074513, 9256800 }, 4670 { -25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 4671 9761698, -19827198, 630305 }, 4672 { -13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, 4673 -15960994, -2449256, -14291300 }, 4674 }, 4675 { 4676 { -3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 4677 15033784, 25105118, -7894876 }, 4678 { -24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 4679 1573892, -2625887, 2198790, -15804619 }, 4680 { -3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, 4681 -16236442, -32461234, -12290683 }, 4682 }, 4683 }; 4684 4685 /* 4686 * r = a * A + b * B 4687 * 4688 * where a = a[0]+256*a[1]+...+256^31 a[31]. 4689 * and b = b[0]+256*b[1]+...+256^31 b[31]. 4690 * B is the Ed25519 base point (x,4/5) with x positive. 4691 */ 4692 static void ge_double_scalarmult_vartime(ge_p2 *r, const uint8_t *a, 4693 const ge_p3 *A, const uint8_t *b) 4694 { 4695 signed char aslide[256]; 4696 signed char bslide[256]; 4697 ge_cached Ai[8]; /* A,3A,5A,7A,9A,11A,13A,15A */ 4698 ge_p1p1 t; 4699 ge_p3 u; 4700 ge_p3 A2; 4701 int i; 4702 4703 slide(aslide, a); 4704 slide(bslide, b); 4705 4706 ge_p3_to_cached(&Ai[0], A); 4707 ge_p3_dbl(&t, A); 4708 ge_p1p1_to_p3(&A2, &t); 4709 ge_add(&t, &A2, &Ai[0]); 4710 ge_p1p1_to_p3(&u, &t); 4711 ge_p3_to_cached(&Ai[1], &u); 4712 ge_add(&t, &A2, &Ai[1]); 4713 ge_p1p1_to_p3(&u, &t); 4714 ge_p3_to_cached(&Ai[2], &u); 4715 ge_add(&t, &A2, &Ai[2]); 4716 ge_p1p1_to_p3(&u, &t); 4717 ge_p3_to_cached(&Ai[3], &u); 4718 ge_add(&t, &A2, &Ai[3]); 4719 ge_p1p1_to_p3(&u, &t); 4720 ge_p3_to_cached(&Ai[4], &u); 4721 ge_add(&t, &A2, &Ai[4]); 4722 ge_p1p1_to_p3(&u, &t); 4723 ge_p3_to_cached(&Ai[5], &u); 4724 ge_add(&t, &A2, &Ai[5]); 4725 ge_p1p1_to_p3(&u, &t); 4726 ge_p3_to_cached(&Ai[6], &u); 4727 ge_add(&t, &A2, &Ai[6]); 4728 ge_p1p1_to_p3(&u, &t); 4729 ge_p3_to_cached(&Ai[7], &u); 4730 4731 ge_p2_0(r); 4732 4733 for (i = 255; i >= 0; --i) { 4734 if (aslide[i] || bslide[i]) { 4735 break; 4736 } 4737 } 4738 4739 for (; i >= 0; --i) { 4740 ge_p2_dbl(&t, r); 4741 4742 if (aslide[i] > 0) { 4743 ge_p1p1_to_p3(&u, &t); 4744 ge_add(&t, &u, &Ai[aslide[i] / 2]); 4745 } else if (aslide[i] < 0) { 4746 ge_p1p1_to_p3(&u, &t); 4747 ge_sub(&t, &u, &Ai[(-aslide[i]) / 2]); 4748 } 4749 4750 if (bslide[i] > 0) { 4751 ge_p1p1_to_p3(&u, &t); 4752 ge_madd(&t, &u, &Bi[bslide[i] / 2]); 4753 } else if (bslide[i] < 0) { 4754 ge_p1p1_to_p3(&u, &t); 4755 ge_msub(&t, &u, &Bi[(-bslide[i]) / 2]); 4756 } 4757 4758 ge_p1p1_to_p2(r, &t); 4759 } 4760 } 4761 4762 /* 4763 * The set of scalars is \Z/l 4764 * where l = 2^252 + 27742317777372353535851937790883648493. 4765 * 4766 * Input: 4767 * s[0]+256*s[1]+...+256^63*s[63] = s 4768 * 4769 * Output: 4770 * s[0]+256*s[1]+...+256^31*s[31] = s mod l 4771 * where l = 2^252 + 27742317777372353535851937790883648493. 4772 * Overwrites s in place. 4773 */ 4774 static void x25519_sc_reduce(uint8_t *s) 4775 { 4776 int64_t s0 = kBottom21Bits & load_3(s); 4777 int64_t s1 = kBottom21Bits & (load_4(s + 2) >> 5); 4778 int64_t s2 = kBottom21Bits & (load_3(s + 5) >> 2); 4779 int64_t s3 = kBottom21Bits & (load_4(s + 7) >> 7); 4780 int64_t s4 = kBottom21Bits & (load_4(s + 10) >> 4); 4781 int64_t s5 = kBottom21Bits & (load_3(s + 13) >> 1); 4782 int64_t s6 = kBottom21Bits & (load_4(s + 15) >> 6); 4783 int64_t s7 = kBottom21Bits & (load_3(s + 18) >> 3); 4784 int64_t s8 = kBottom21Bits & load_3(s + 21); 4785 int64_t s9 = kBottom21Bits & (load_4(s + 23) >> 5); 4786 int64_t s10 = kBottom21Bits & (load_3(s + 26) >> 2); 4787 int64_t s11 = kBottom21Bits & (load_4(s + 28) >> 7); 4788 int64_t s12 = kBottom21Bits & (load_4(s + 31) >> 4); 4789 int64_t s13 = kBottom21Bits & (load_3(s + 34) >> 1); 4790 int64_t s14 = kBottom21Bits & (load_4(s + 36) >> 6); 4791 int64_t s15 = kBottom21Bits & (load_3(s + 39) >> 3); 4792 int64_t s16 = kBottom21Bits & load_3(s + 42); 4793 int64_t s17 = kBottom21Bits & (load_4(s + 44) >> 5); 4794 int64_t s18 = kBottom21Bits & (load_3(s + 47) >> 2); 4795 int64_t s19 = kBottom21Bits & (load_4(s + 49) >> 7); 4796 int64_t s20 = kBottom21Bits & (load_4(s + 52) >> 4); 4797 int64_t s21 = kBottom21Bits & (load_3(s + 55) >> 1); 4798 int64_t s22 = kBottom21Bits & (load_4(s + 57) >> 6); 4799 int64_t s23 = (load_4(s + 60) >> 3); 4800 int64_t carry0; 4801 int64_t carry1; 4802 int64_t carry2; 4803 int64_t carry3; 4804 int64_t carry4; 4805 int64_t carry5; 4806 int64_t carry6; 4807 int64_t carry7; 4808 int64_t carry8; 4809 int64_t carry9; 4810 int64_t carry10; 4811 int64_t carry11; 4812 int64_t carry12; 4813 int64_t carry13; 4814 int64_t carry14; 4815 int64_t carry15; 4816 int64_t carry16; 4817 4818 s11 += s23 * 666643; 4819 s12 += s23 * 470296; 4820 s13 += s23 * 654183; 4821 s14 -= s23 * 997805; 4822 s15 += s23 * 136657; 4823 s16 -= s23 * 683901; 4824 s23 = 0; 4825 4826 s10 += s22 * 666643; 4827 s11 += s22 * 470296; 4828 s12 += s22 * 654183; 4829 s13 -= s22 * 997805; 4830 s14 += s22 * 136657; 4831 s15 -= s22 * 683901; 4832 s22 = 0; 4833 4834 s9 += s21 * 666643; 4835 s10 += s21 * 470296; 4836 s11 += s21 * 654183; 4837 s12 -= s21 * 997805; 4838 s13 += s21 * 136657; 4839 s14 -= s21 * 683901; 4840 s21 = 0; 4841 4842 s8 += s20 * 666643; 4843 s9 += s20 * 470296; 4844 s10 += s20 * 654183; 4845 s11 -= s20 * 997805; 4846 s12 += s20 * 136657; 4847 s13 -= s20 * 683901; 4848 s20 = 0; 4849 4850 s7 += s19 * 666643; 4851 s8 += s19 * 470296; 4852 s9 += s19 * 654183; 4853 s10 -= s19 * 997805; 4854 s11 += s19 * 136657; 4855 s12 -= s19 * 683901; 4856 s19 = 0; 4857 4858 s6 += s18 * 666643; 4859 s7 += s18 * 470296; 4860 s8 += s18 * 654183; 4861 s9 -= s18 * 997805; 4862 s10 += s18 * 136657; 4863 s11 -= s18 * 683901; 4864 s18 = 0; 4865 4866 carry6 = (s6 + (1 << 20)) >> 21; 4867 s7 += carry6; 4868 s6 -= carry6 * (1 << 21); 4869 carry8 = (s8 + (1 << 20)) >> 21; 4870 s9 += carry8; 4871 s8 -= carry8 * (1 << 21); 4872 carry10 = (s10 + (1 << 20)) >> 21; 4873 s11 += carry10; 4874 s10 -= carry10 * (1 << 21); 4875 carry12 = (s12 + (1 << 20)) >> 21; 4876 s13 += carry12; 4877 s12 -= carry12 * (1 << 21); 4878 carry14 = (s14 + (1 << 20)) >> 21; 4879 s15 += carry14; 4880 s14 -= carry14 * (1 << 21); 4881 carry16 = (s16 + (1 << 20)) >> 21; 4882 s17 += carry16; 4883 s16 -= carry16 * (1 << 21); 4884 4885 carry7 = (s7 + (1 << 20)) >> 21; 4886 s8 += carry7; 4887 s7 -= carry7 * (1 << 21); 4888 carry9 = (s9 + (1 << 20)) >> 21; 4889 s10 += carry9; 4890 s9 -= carry9 * (1 << 21); 4891 carry11 = (s11 + (1 << 20)) >> 21; 4892 s12 += carry11; 4893 s11 -= carry11 * (1 << 21); 4894 carry13 = (s13 + (1 << 20)) >> 21; 4895 s14 += carry13; 4896 s13 -= carry13 * (1 << 21); 4897 carry15 = (s15 + (1 << 20)) >> 21; 4898 s16 += carry15; 4899 s15 -= carry15 * (1 << 21); 4900 4901 s5 += s17 * 666643; 4902 s6 += s17 * 470296; 4903 s7 += s17 * 654183; 4904 s8 -= s17 * 997805; 4905 s9 += s17 * 136657; 4906 s10 -= s17 * 683901; 4907 s17 = 0; 4908 4909 s4 += s16 * 666643; 4910 s5 += s16 * 470296; 4911 s6 += s16 * 654183; 4912 s7 -= s16 * 997805; 4913 s8 += s16 * 136657; 4914 s9 -= s16 * 683901; 4915 s16 = 0; 4916 4917 s3 += s15 * 666643; 4918 s4 += s15 * 470296; 4919 s5 += s15 * 654183; 4920 s6 -= s15 * 997805; 4921 s7 += s15 * 136657; 4922 s8 -= s15 * 683901; 4923 s15 = 0; 4924 4925 s2 += s14 * 666643; 4926 s3 += s14 * 470296; 4927 s4 += s14 * 654183; 4928 s5 -= s14 * 997805; 4929 s6 += s14 * 136657; 4930 s7 -= s14 * 683901; 4931 s14 = 0; 4932 4933 s1 += s13 * 666643; 4934 s2 += s13 * 470296; 4935 s3 += s13 * 654183; 4936 s4 -= s13 * 997805; 4937 s5 += s13 * 136657; 4938 s6 -= s13 * 683901; 4939 s13 = 0; 4940 4941 s0 += s12 * 666643; 4942 s1 += s12 * 470296; 4943 s2 += s12 * 654183; 4944 s3 -= s12 * 997805; 4945 s4 += s12 * 136657; 4946 s5 -= s12 * 683901; 4947 s12 = 0; 4948 4949 carry0 = (s0 + (1 << 20)) >> 21; 4950 s1 += carry0; 4951 s0 -= carry0 * (1 << 21); 4952 carry2 = (s2 + (1 << 20)) >> 21; 4953 s3 += carry2; 4954 s2 -= carry2 * (1 << 21); 4955 carry4 = (s4 + (1 << 20)) >> 21; 4956 s5 += carry4; 4957 s4 -= carry4 * (1 << 21); 4958 carry6 = (s6 + (1 << 20)) >> 21; 4959 s7 += carry6; 4960 s6 -= carry6 * (1 << 21); 4961 carry8 = (s8 + (1 << 20)) >> 21; 4962 s9 += carry8; 4963 s8 -= carry8 * (1 << 21); 4964 carry10 = (s10 + (1 << 20)) >> 21; 4965 s11 += carry10; 4966 s10 -= carry10 * (1 << 21); 4967 4968 carry1 = (s1 + (1 << 20)) >> 21; 4969 s2 += carry1; 4970 s1 -= carry1 * (1 << 21); 4971 carry3 = (s3 + (1 << 20)) >> 21; 4972 s4 += carry3; 4973 s3 -= carry3 * (1 << 21); 4974 carry5 = (s5 + (1 << 20)) >> 21; 4975 s6 += carry5; 4976 s5 -= carry5 * (1 << 21); 4977 carry7 = (s7 + (1 << 20)) >> 21; 4978 s8 += carry7; 4979 s7 -= carry7 * (1 << 21); 4980 carry9 = (s9 + (1 << 20)) >> 21; 4981 s10 += carry9; 4982 s9 -= carry9 * (1 << 21); 4983 carry11 = (s11 + (1 << 20)) >> 21; 4984 s12 += carry11; 4985 s11 -= carry11 * (1 << 21); 4986 4987 s0 += s12 * 666643; 4988 s1 += s12 * 470296; 4989 s2 += s12 * 654183; 4990 s3 -= s12 * 997805; 4991 s4 += s12 * 136657; 4992 s5 -= s12 * 683901; 4993 s12 = 0; 4994 4995 carry0 = s0 >> 21; 4996 s1 += carry0; 4997 s0 -= carry0 * (1 << 21); 4998 carry1 = s1 >> 21; 4999 s2 += carry1; 5000 s1 -= carry1 * (1 << 21); 5001 carry2 = s2 >> 21; 5002 s3 += carry2; 5003 s2 -= carry2 * (1 << 21); 5004 carry3 = s3 >> 21; 5005 s4 += carry3; 5006 s3 -= carry3 * (1 << 21); 5007 carry4 = s4 >> 21; 5008 s5 += carry4; 5009 s4 -= carry4 * (1 << 21); 5010 carry5 = s5 >> 21; 5011 s6 += carry5; 5012 s5 -= carry5 * (1 << 21); 5013 carry6 = s6 >> 21; 5014 s7 += carry6; 5015 s6 -= carry6 * (1 << 21); 5016 carry7 = s7 >> 21; 5017 s8 += carry7; 5018 s7 -= carry7 * (1 << 21); 5019 carry8 = s8 >> 21; 5020 s9 += carry8; 5021 s8 -= carry8 * (1 << 21); 5022 carry9 = s9 >> 21; 5023 s10 += carry9; 5024 s9 -= carry9 * (1 << 21); 5025 carry10 = s10 >> 21; 5026 s11 += carry10; 5027 s10 -= carry10 * (1 << 21); 5028 carry11 = s11 >> 21; 5029 s12 += carry11; 5030 s11 -= carry11 * (1 << 21); 5031 5032 s0 += s12 * 666643; 5033 s1 += s12 * 470296; 5034 s2 += s12 * 654183; 5035 s3 -= s12 * 997805; 5036 s4 += s12 * 136657; 5037 s5 -= s12 * 683901; 5038 s12 = 0; 5039 5040 carry0 = s0 >> 21; 5041 s1 += carry0; 5042 s0 -= carry0 * (1 << 21); 5043 carry1 = s1 >> 21; 5044 s2 += carry1; 5045 s1 -= carry1 * (1 << 21); 5046 carry2 = s2 >> 21; 5047 s3 += carry2; 5048 s2 -= carry2 * (1 << 21); 5049 carry3 = s3 >> 21; 5050 s4 += carry3; 5051 s3 -= carry3 * (1 << 21); 5052 carry4 = s4 >> 21; 5053 s5 += carry4; 5054 s4 -= carry4 * (1 << 21); 5055 carry5 = s5 >> 21; 5056 s6 += carry5; 5057 s5 -= carry5 * (1 << 21); 5058 carry6 = s6 >> 21; 5059 s7 += carry6; 5060 s6 -= carry6 * (1 << 21); 5061 carry7 = s7 >> 21; 5062 s8 += carry7; 5063 s7 -= carry7 * (1 << 21); 5064 carry8 = s8 >> 21; 5065 s9 += carry8; 5066 s8 -= carry8 * (1 << 21); 5067 carry9 = s9 >> 21; 5068 s10 += carry9; 5069 s9 -= carry9 * (1 << 21); 5070 carry10 = s10 >> 21; 5071 s11 += carry10; 5072 s10 -= carry10 * (1 << 21); 5073 5074 s[0] = (uint8_t)(s0 >> 0); 5075 s[1] = (uint8_t)(s0 >> 8); 5076 s[2] = (uint8_t)((s0 >> 16) | (s1 << 5)); 5077 s[3] = (uint8_t)(s1 >> 3); 5078 s[4] = (uint8_t)(s1 >> 11); 5079 s[5] = (uint8_t)((s1 >> 19) | (s2 << 2)); 5080 s[6] = (uint8_t)(s2 >> 6); 5081 s[7] = (uint8_t)((s2 >> 14) | (s3 << 7)); 5082 s[8] = (uint8_t)(s3 >> 1); 5083 s[9] = (uint8_t)(s3 >> 9); 5084 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4)); 5085 s[11] = (uint8_t)(s4 >> 4); 5086 s[12] = (uint8_t)(s4 >> 12); 5087 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1)); 5088 s[14] = (uint8_t)(s5 >> 7); 5089 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6)); 5090 s[16] = (uint8_t)(s6 >> 2); 5091 s[17] = (uint8_t)(s6 >> 10); 5092 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3)); 5093 s[19] = (uint8_t)(s7 >> 5); 5094 s[20] = (uint8_t)(s7 >> 13); 5095 s[21] = (uint8_t)(s8 >> 0); 5096 s[22] = (uint8_t)(s8 >> 8); 5097 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5)); 5098 s[24] = (uint8_t)(s9 >> 3); 5099 s[25] = (uint8_t)(s9 >> 11); 5100 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2)); 5101 s[27] = (uint8_t)(s10 >> 6); 5102 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7)); 5103 s[29] = (uint8_t)(s11 >> 1); 5104 s[30] = (uint8_t)(s11 >> 9); 5105 s[31] = (uint8_t)(s11 >> 17); 5106 } 5107 5108 /* 5109 * Input: 5110 * a[0]+256*a[1]+...+256^31*a[31] = a 5111 * b[0]+256*b[1]+...+256^31*b[31] = b 5112 * c[0]+256*c[1]+...+256^31*c[31] = c 5113 * 5114 * Output: 5115 * s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l 5116 * where l = 2^252 + 27742317777372353535851937790883648493. 5117 */ 5118 static void sc_muladd(uint8_t *s, const uint8_t *a, const uint8_t *b, 5119 const uint8_t *c) 5120 { 5121 int64_t a0 = kBottom21Bits & load_3(a); 5122 int64_t a1 = kBottom21Bits & (load_4(a + 2) >> 5); 5123 int64_t a2 = kBottom21Bits & (load_3(a + 5) >> 2); 5124 int64_t a3 = kBottom21Bits & (load_4(a + 7) >> 7); 5125 int64_t a4 = kBottom21Bits & (load_4(a + 10) >> 4); 5126 int64_t a5 = kBottom21Bits & (load_3(a + 13) >> 1); 5127 int64_t a6 = kBottom21Bits & (load_4(a + 15) >> 6); 5128 int64_t a7 = kBottom21Bits & (load_3(a + 18) >> 3); 5129 int64_t a8 = kBottom21Bits & load_3(a + 21); 5130 int64_t a9 = kBottom21Bits & (load_4(a + 23) >> 5); 5131 int64_t a10 = kBottom21Bits & (load_3(a + 26) >> 2); 5132 int64_t a11 = (load_4(a + 28) >> 7); 5133 int64_t b0 = kBottom21Bits & load_3(b); 5134 int64_t b1 = kBottom21Bits & (load_4(b + 2) >> 5); 5135 int64_t b2 = kBottom21Bits & (load_3(b + 5) >> 2); 5136 int64_t b3 = kBottom21Bits & (load_4(b + 7) >> 7); 5137 int64_t b4 = kBottom21Bits & (load_4(b + 10) >> 4); 5138 int64_t b5 = kBottom21Bits & (load_3(b + 13) >> 1); 5139 int64_t b6 = kBottom21Bits & (load_4(b + 15) >> 6); 5140 int64_t b7 = kBottom21Bits & (load_3(b + 18) >> 3); 5141 int64_t b8 = kBottom21Bits & load_3(b + 21); 5142 int64_t b9 = kBottom21Bits & (load_4(b + 23) >> 5); 5143 int64_t b10 = kBottom21Bits & (load_3(b + 26) >> 2); 5144 int64_t b11 = (load_4(b + 28) >> 7); 5145 int64_t c0 = kBottom21Bits & load_3(c); 5146 int64_t c1 = kBottom21Bits & (load_4(c + 2) >> 5); 5147 int64_t c2 = kBottom21Bits & (load_3(c + 5) >> 2); 5148 int64_t c3 = kBottom21Bits & (load_4(c + 7) >> 7); 5149 int64_t c4 = kBottom21Bits & (load_4(c + 10) >> 4); 5150 int64_t c5 = kBottom21Bits & (load_3(c + 13) >> 1); 5151 int64_t c6 = kBottom21Bits & (load_4(c + 15) >> 6); 5152 int64_t c7 = kBottom21Bits & (load_3(c + 18) >> 3); 5153 int64_t c8 = kBottom21Bits & load_3(c + 21); 5154 int64_t c9 = kBottom21Bits & (load_4(c + 23) >> 5); 5155 int64_t c10 = kBottom21Bits & (load_3(c + 26) >> 2); 5156 int64_t c11 = (load_4(c + 28) >> 7); 5157 int64_t s0; 5158 int64_t s1; 5159 int64_t s2; 5160 int64_t s3; 5161 int64_t s4; 5162 int64_t s5; 5163 int64_t s6; 5164 int64_t s7; 5165 int64_t s8; 5166 int64_t s9; 5167 int64_t s10; 5168 int64_t s11; 5169 int64_t s12; 5170 int64_t s13; 5171 int64_t s14; 5172 int64_t s15; 5173 int64_t s16; 5174 int64_t s17; 5175 int64_t s18; 5176 int64_t s19; 5177 int64_t s20; 5178 int64_t s21; 5179 int64_t s22; 5180 int64_t s23; 5181 int64_t carry0; 5182 int64_t carry1; 5183 int64_t carry2; 5184 int64_t carry3; 5185 int64_t carry4; 5186 int64_t carry5; 5187 int64_t carry6; 5188 int64_t carry7; 5189 int64_t carry8; 5190 int64_t carry9; 5191 int64_t carry10; 5192 int64_t carry11; 5193 int64_t carry12; 5194 int64_t carry13; 5195 int64_t carry14; 5196 int64_t carry15; 5197 int64_t carry16; 5198 int64_t carry17; 5199 int64_t carry18; 5200 int64_t carry19; 5201 int64_t carry20; 5202 int64_t carry21; 5203 int64_t carry22; 5204 5205 s0 = c0 + a0 * b0; 5206 s1 = c1 + a0 * b1 + a1 * b0; 5207 s2 = c2 + a0 * b2 + a1 * b1 + a2 * b0; 5208 s3 = c3 + a0 * b3 + a1 * b2 + a2 * b1 + a3 * b0; 5209 s4 = c4 + a0 * b4 + a1 * b3 + a2 * b2 + a3 * b1 + a4 * b0; 5210 s5 = c5 + a0 * b5 + a1 * b4 + a2 * b3 + a3 * b2 + a4 * b1 + a5 * b0; 5211 s6 = c6 + a0 * b6 + a1 * b5 + a2 * b4 + a3 * b3 + a4 * b2 + a5 * b1 + a6 * b0; 5212 s7 = c7 + a0 * b7 + a1 * b6 + a2 * b5 + a3 * b4 + a4 * b3 + a5 * b2 + a6 * b1 + a7 * b0; 5213 s8 = c8 + a0 * b8 + a1 * b7 + a2 * b6 + a3 * b5 + a4 * b4 + a5 * b3 + a6 * b2 + a7 * b1 + a8 * b0; 5214 s9 = c9 + a0 * b9 + a1 * b8 + a2 * b7 + a3 * b6 + a4 * b5 + a5 * b4 + a6 * b3 + a7 * b2 + a8 * b1 + a9 * b0; 5215 s10 = c10 + a0 * b10 + a1 * b9 + a2 * b8 + a3 * b7 + a4 * b6 + a5 * b5 + a6 * b4 + a7 * b3 + a8 * b2 + a9 * b1 + a10 * b0; 5216 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; 5217 s12 = a1 * b11 + a2 * b10 + a3 * b9 + a4 * b8 + a5 * b7 + a6 * b6 + a7 * b5 + a8 * b4 + a9 * b3 + a10 * b2 + a11 * b1; 5218 s13 = a2 * b11 + a3 * b10 + a4 * b9 + a5 * b8 + a6 * b7 + a7 * b6 + a8 * b5 + a9 * b4 + a10 * b3 + a11 * b2; 5219 s14 = a3 * b11 + a4 * b10 + a5 * b9 + a6 * b8 + a7 * b7 + a8 * b6 + a9 * b5 + a10 * b4 + a11 * b3; 5220 s15 = a4 * b11 + a5 * b10 + a6 * b9 + a7 * b8 + a8 * b7 + a9 * b6 + a10 * b5 + a11 * b4; 5221 s16 = a5 * b11 + a6 * b10 + a7 * b9 + a8 * b8 + a9 * b7 + a10 * b6 + a11 * b5; 5222 s17 = a6 * b11 + a7 * b10 + a8 * b9 + a9 * b8 + a10 * b7 + a11 * b6; 5223 s18 = a7 * b11 + a8 * b10 + a9 * b9 + a10 * b8 + a11 * b7; 5224 s19 = a8 * b11 + a9 * b10 + a10 * b9 + a11 * b8; 5225 s20 = a9 * b11 + a10 * b10 + a11 * b9; 5226 s21 = a10 * b11 + a11 * b10; 5227 s22 = a11 * b11; 5228 s23 = 0; 5229 5230 carry0 = (s0 + (1 << 20)) >> 21; 5231 s1 += carry0; 5232 s0 -= carry0 * (1 << 21); 5233 carry2 = (s2 + (1 << 20)) >> 21; 5234 s3 += carry2; 5235 s2 -= carry2 * (1 << 21); 5236 carry4 = (s4 + (1 << 20)) >> 21; 5237 s5 += carry4; 5238 s4 -= carry4 * (1 << 21); 5239 carry6 = (s6 + (1 << 20)) >> 21; 5240 s7 += carry6; 5241 s6 -= carry6 * (1 << 21); 5242 carry8 = (s8 + (1 << 20)) >> 21; 5243 s9 += carry8; 5244 s8 -= carry8 * (1 << 21); 5245 carry10 = (s10 + (1 << 20)) >> 21; 5246 s11 += carry10; 5247 s10 -= carry10 * (1 << 21); 5248 carry12 = (s12 + (1 << 20)) >> 21; 5249 s13 += carry12; 5250 s12 -= carry12 * (1 << 21); 5251 carry14 = (s14 + (1 << 20)) >> 21; 5252 s15 += carry14; 5253 s14 -= carry14 * (1 << 21); 5254 carry16 = (s16 + (1 << 20)) >> 21; 5255 s17 += carry16; 5256 s16 -= carry16 * (1 << 21); 5257 carry18 = (s18 + (1 << 20)) >> 21; 5258 s19 += carry18; 5259 s18 -= carry18 * (1 << 21); 5260 carry20 = (s20 + (1 << 20)) >> 21; 5261 s21 += carry20; 5262 s20 -= carry20 * (1 << 21); 5263 carry22 = (s22 + (1 << 20)) >> 21; 5264 s23 += carry22; 5265 s22 -= carry22 * (1 << 21); 5266 5267 carry1 = (s1 + (1 << 20)) >> 21; 5268 s2 += carry1; 5269 s1 -= carry1 * (1 << 21); 5270 carry3 = (s3 + (1 << 20)) >> 21; 5271 s4 += carry3; 5272 s3 -= carry3 * (1 << 21); 5273 carry5 = (s5 + (1 << 20)) >> 21; 5274 s6 += carry5; 5275 s5 -= carry5 * (1 << 21); 5276 carry7 = (s7 + (1 << 20)) >> 21; 5277 s8 += carry7; 5278 s7 -= carry7 * (1 << 21); 5279 carry9 = (s9 + (1 << 20)) >> 21; 5280 s10 += carry9; 5281 s9 -= carry9 * (1 << 21); 5282 carry11 = (s11 + (1 << 20)) >> 21; 5283 s12 += carry11; 5284 s11 -= carry11 * (1 << 21); 5285 carry13 = (s13 + (1 << 20)) >> 21; 5286 s14 += carry13; 5287 s13 -= carry13 * (1 << 21); 5288 carry15 = (s15 + (1 << 20)) >> 21; 5289 s16 += carry15; 5290 s15 -= carry15 * (1 << 21); 5291 carry17 = (s17 + (1 << 20)) >> 21; 5292 s18 += carry17; 5293 s17 -= carry17 * (1 << 21); 5294 carry19 = (s19 + (1 << 20)) >> 21; 5295 s20 += carry19; 5296 s19 -= carry19 * (1 << 21); 5297 carry21 = (s21 + (1 << 20)) >> 21; 5298 s22 += carry21; 5299 s21 -= carry21 * (1 << 21); 5300 5301 s11 += s23 * 666643; 5302 s12 += s23 * 470296; 5303 s13 += s23 * 654183; 5304 s14 -= s23 * 997805; 5305 s15 += s23 * 136657; 5306 s16 -= s23 * 683901; 5307 s23 = 0; 5308 5309 s10 += s22 * 666643; 5310 s11 += s22 * 470296; 5311 s12 += s22 * 654183; 5312 s13 -= s22 * 997805; 5313 s14 += s22 * 136657; 5314 s15 -= s22 * 683901; 5315 s22 = 0; 5316 5317 s9 += s21 * 666643; 5318 s10 += s21 * 470296; 5319 s11 += s21 * 654183; 5320 s12 -= s21 * 997805; 5321 s13 += s21 * 136657; 5322 s14 -= s21 * 683901; 5323 s21 = 0; 5324 5325 s8 += s20 * 666643; 5326 s9 += s20 * 470296; 5327 s10 += s20 * 654183; 5328 s11 -= s20 * 997805; 5329 s12 += s20 * 136657; 5330 s13 -= s20 * 683901; 5331 s20 = 0; 5332 5333 s7 += s19 * 666643; 5334 s8 += s19 * 470296; 5335 s9 += s19 * 654183; 5336 s10 -= s19 * 997805; 5337 s11 += s19 * 136657; 5338 s12 -= s19 * 683901; 5339 s19 = 0; 5340 5341 s6 += s18 * 666643; 5342 s7 += s18 * 470296; 5343 s8 += s18 * 654183; 5344 s9 -= s18 * 997805; 5345 s10 += s18 * 136657; 5346 s11 -= s18 * 683901; 5347 s18 = 0; 5348 5349 carry6 = (s6 + (1 << 20)) >> 21; 5350 s7 += carry6; 5351 s6 -= carry6 * (1 << 21); 5352 carry8 = (s8 + (1 << 20)) >> 21; 5353 s9 += carry8; 5354 s8 -= carry8 * (1 << 21); 5355 carry10 = (s10 + (1 << 20)) >> 21; 5356 s11 += carry10; 5357 s10 -= carry10 * (1 << 21); 5358 carry12 = (s12 + (1 << 20)) >> 21; 5359 s13 += carry12; 5360 s12 -= carry12 * (1 << 21); 5361 carry14 = (s14 + (1 << 20)) >> 21; 5362 s15 += carry14; 5363 s14 -= carry14 * (1 << 21); 5364 carry16 = (s16 + (1 << 20)) >> 21; 5365 s17 += carry16; 5366 s16 -= carry16 * (1 << 21); 5367 5368 carry7 = (s7 + (1 << 20)) >> 21; 5369 s8 += carry7; 5370 s7 -= carry7 * (1 << 21); 5371 carry9 = (s9 + (1 << 20)) >> 21; 5372 s10 += carry9; 5373 s9 -= carry9 * (1 << 21); 5374 carry11 = (s11 + (1 << 20)) >> 21; 5375 s12 += carry11; 5376 s11 -= carry11 * (1 << 21); 5377 carry13 = (s13 + (1 << 20)) >> 21; 5378 s14 += carry13; 5379 s13 -= carry13 * (1 << 21); 5380 carry15 = (s15 + (1 << 20)) >> 21; 5381 s16 += carry15; 5382 s15 -= carry15 * (1 << 21); 5383 5384 s5 += s17 * 666643; 5385 s6 += s17 * 470296; 5386 s7 += s17 * 654183; 5387 s8 -= s17 * 997805; 5388 s9 += s17 * 136657; 5389 s10 -= s17 * 683901; 5390 s17 = 0; 5391 5392 s4 += s16 * 666643; 5393 s5 += s16 * 470296; 5394 s6 += s16 * 654183; 5395 s7 -= s16 * 997805; 5396 s8 += s16 * 136657; 5397 s9 -= s16 * 683901; 5398 s16 = 0; 5399 5400 s3 += s15 * 666643; 5401 s4 += s15 * 470296; 5402 s5 += s15 * 654183; 5403 s6 -= s15 * 997805; 5404 s7 += s15 * 136657; 5405 s8 -= s15 * 683901; 5406 s15 = 0; 5407 5408 s2 += s14 * 666643; 5409 s3 += s14 * 470296; 5410 s4 += s14 * 654183; 5411 s5 -= s14 * 997805; 5412 s6 += s14 * 136657; 5413 s7 -= s14 * 683901; 5414 s14 = 0; 5415 5416 s1 += s13 * 666643; 5417 s2 += s13 * 470296; 5418 s3 += s13 * 654183; 5419 s4 -= s13 * 997805; 5420 s5 += s13 * 136657; 5421 s6 -= s13 * 683901; 5422 s13 = 0; 5423 5424 s0 += s12 * 666643; 5425 s1 += s12 * 470296; 5426 s2 += s12 * 654183; 5427 s3 -= s12 * 997805; 5428 s4 += s12 * 136657; 5429 s5 -= s12 * 683901; 5430 s12 = 0; 5431 5432 carry0 = (s0 + (1 << 20)) >> 21; 5433 s1 += carry0; 5434 s0 -= carry0 * (1 << 21); 5435 carry2 = (s2 + (1 << 20)) >> 21; 5436 s3 += carry2; 5437 s2 -= carry2 * (1 << 21); 5438 carry4 = (s4 + (1 << 20)) >> 21; 5439 s5 += carry4; 5440 s4 -= carry4 * (1 << 21); 5441 carry6 = (s6 + (1 << 20)) >> 21; 5442 s7 += carry6; 5443 s6 -= carry6 * (1 << 21); 5444 carry8 = (s8 + (1 << 20)) >> 21; 5445 s9 += carry8; 5446 s8 -= carry8 * (1 << 21); 5447 carry10 = (s10 + (1 << 20)) >> 21; 5448 s11 += carry10; 5449 s10 -= carry10 * (1 << 21); 5450 5451 carry1 = (s1 + (1 << 20)) >> 21; 5452 s2 += carry1; 5453 s1 -= carry1 * (1 << 21); 5454 carry3 = (s3 + (1 << 20)) >> 21; 5455 s4 += carry3; 5456 s3 -= carry3 * (1 << 21); 5457 carry5 = (s5 + (1 << 20)) >> 21; 5458 s6 += carry5; 5459 s5 -= carry5 * (1 << 21); 5460 carry7 = (s7 + (1 << 20)) >> 21; 5461 s8 += carry7; 5462 s7 -= carry7 * (1 << 21); 5463 carry9 = (s9 + (1 << 20)) >> 21; 5464 s10 += carry9; 5465 s9 -= carry9 * (1 << 21); 5466 carry11 = (s11 + (1 << 20)) >> 21; 5467 s12 += carry11; 5468 s11 -= carry11 * (1 << 21); 5469 5470 s0 += s12 * 666643; 5471 s1 += s12 * 470296; 5472 s2 += s12 * 654183; 5473 s3 -= s12 * 997805; 5474 s4 += s12 * 136657; 5475 s5 -= s12 * 683901; 5476 s12 = 0; 5477 5478 carry0 = s0 >> 21; 5479 s1 += carry0; 5480 s0 -= carry0 * (1 << 21); 5481 carry1 = s1 >> 21; 5482 s2 += carry1; 5483 s1 -= carry1 * (1 << 21); 5484 carry2 = s2 >> 21; 5485 s3 += carry2; 5486 s2 -= carry2 * (1 << 21); 5487 carry3 = s3 >> 21; 5488 s4 += carry3; 5489 s3 -= carry3 * (1 << 21); 5490 carry4 = s4 >> 21; 5491 s5 += carry4; 5492 s4 -= carry4 * (1 << 21); 5493 carry5 = s5 >> 21; 5494 s6 += carry5; 5495 s5 -= carry5 * (1 << 21); 5496 carry6 = s6 >> 21; 5497 s7 += carry6; 5498 s6 -= carry6 * (1 << 21); 5499 carry7 = s7 >> 21; 5500 s8 += carry7; 5501 s7 -= carry7 * (1 << 21); 5502 carry8 = s8 >> 21; 5503 s9 += carry8; 5504 s8 -= carry8 * (1 << 21); 5505 carry9 = s9 >> 21; 5506 s10 += carry9; 5507 s9 -= carry9 * (1 << 21); 5508 carry10 = s10 >> 21; 5509 s11 += carry10; 5510 s10 -= carry10 * (1 << 21); 5511 carry11 = s11 >> 21; 5512 s12 += carry11; 5513 s11 -= carry11 * (1 << 21); 5514 5515 s0 += s12 * 666643; 5516 s1 += s12 * 470296; 5517 s2 += s12 * 654183; 5518 s3 -= s12 * 997805; 5519 s4 += s12 * 136657; 5520 s5 -= s12 * 683901; 5521 s12 = 0; 5522 5523 carry0 = s0 >> 21; 5524 s1 += carry0; 5525 s0 -= carry0 * (1 << 21); 5526 carry1 = s1 >> 21; 5527 s2 += carry1; 5528 s1 -= carry1 * (1 << 21); 5529 carry2 = s2 >> 21; 5530 s3 += carry2; 5531 s2 -= carry2 * (1 << 21); 5532 carry3 = s3 >> 21; 5533 s4 += carry3; 5534 s3 -= carry3 * (1 << 21); 5535 carry4 = s4 >> 21; 5536 s5 += carry4; 5537 s4 -= carry4 * (1 << 21); 5538 carry5 = s5 >> 21; 5539 s6 += carry5; 5540 s5 -= carry5 * (1 << 21); 5541 carry6 = s6 >> 21; 5542 s7 += carry6; 5543 s6 -= carry6 * (1 << 21); 5544 carry7 = s7 >> 21; 5545 s8 += carry7; 5546 s7 -= carry7 * (1 << 21); 5547 carry8 = s8 >> 21; 5548 s9 += carry8; 5549 s8 -= carry8 * (1 << 21); 5550 carry9 = s9 >> 21; 5551 s10 += carry9; 5552 s9 -= carry9 * (1 << 21); 5553 carry10 = s10 >> 21; 5554 s11 += carry10; 5555 s10 -= carry10 * (1 << 21); 5556 5557 s[0] = (uint8_t)(s0 >> 0); 5558 s[1] = (uint8_t)(s0 >> 8); 5559 s[2] = (uint8_t)((s0 >> 16) | (s1 << 5)); 5560 s[3] = (uint8_t)(s1 >> 3); 5561 s[4] = (uint8_t)(s1 >> 11); 5562 s[5] = (uint8_t)((s1 >> 19) | (s2 << 2)); 5563 s[6] = (uint8_t)(s2 >> 6); 5564 s[7] = (uint8_t)((s2 >> 14) | (s3 << 7)); 5565 s[8] = (uint8_t)(s3 >> 1); 5566 s[9] = (uint8_t)(s3 >> 9); 5567 s[10] = (uint8_t)((s3 >> 17) | (s4 << 4)); 5568 s[11] = (uint8_t)(s4 >> 4); 5569 s[12] = (uint8_t)(s4 >> 12); 5570 s[13] = (uint8_t)((s4 >> 20) | (s5 << 1)); 5571 s[14] = (uint8_t)(s5 >> 7); 5572 s[15] = (uint8_t)((s5 >> 15) | (s6 << 6)); 5573 s[16] = (uint8_t)(s6 >> 2); 5574 s[17] = (uint8_t)(s6 >> 10); 5575 s[18] = (uint8_t)((s6 >> 18) | (s7 << 3)); 5576 s[19] = (uint8_t)(s7 >> 5); 5577 s[20] = (uint8_t)(s7 >> 13); 5578 s[21] = (uint8_t)(s8 >> 0); 5579 s[22] = (uint8_t)(s8 >> 8); 5580 s[23] = (uint8_t)((s8 >> 16) | (s9 << 5)); 5581 s[24] = (uint8_t)(s9 >> 3); 5582 s[25] = (uint8_t)(s9 >> 11); 5583 s[26] = (uint8_t)((s9 >> 19) | (s10 << 2)); 5584 s[27] = (uint8_t)(s10 >> 6); 5585 s[28] = (uint8_t)((s10 >> 14) | (s11 << 7)); 5586 s[29] = (uint8_t)(s11 >> 1); 5587 s[30] = (uint8_t)(s11 >> 9); 5588 s[31] = (uint8_t)(s11 >> 17); 5589 } 5590 5591 static int hash_init_with_dom(EVP_MD_CTX *hash_ctx, 5592 EVP_MD *sha512, 5593 const uint8_t dom2flag, 5594 const uint8_t phflag, 5595 const uint8_t *context, 5596 const size_t context_len) 5597 { 5598 /* ASCII: "SigEd25519 no Ed25519 collisions", in hex for EBCDIC compatibility */ 5599 const char dom_s[] = "\x53\x69\x67\x45\x64\x32\x35\x35\x31\x39\x20\x6e" 5600 "\x6f\x20\x45\x64\x32\x35\x35\x31\x39\x20\x63\x6f" 5601 "\x6c\x6c\x69\x73\x69\x6f\x6e\x73"; 5602 uint8_t dom[2]; 5603 5604 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL)) 5605 return 0; 5606 5607 /* return early if dom2flag is not set */ 5608 if (!dom2flag) 5609 return 1; 5610 5611 if (context_len > UINT8_MAX) 5612 return 0; 5613 5614 dom[0] = (uint8_t)(phflag >= 1 ? 1 : 0); 5615 dom[1] = (uint8_t)context_len; 5616 5617 if (!EVP_DigestUpdate(hash_ctx, dom_s, sizeof(dom_s) - 1) 5618 || !EVP_DigestUpdate(hash_ctx, dom, sizeof(dom)) 5619 || !EVP_DigestUpdate(hash_ctx, context, context_len)) { 5620 return 0; 5621 } 5622 5623 return 1; 5624 } 5625 5626 int ossl_ed25519_sign(uint8_t *out_sig, const uint8_t *tbs, size_t tbs_len, 5627 const uint8_t public_key[32], const uint8_t private_key[32], 5628 const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag, 5629 const uint8_t *context, size_t context_len, 5630 OSSL_LIB_CTX *libctx, const char *propq) 5631 { 5632 uint8_t az[SHA512_DIGEST_LENGTH]; 5633 uint8_t nonce[SHA512_DIGEST_LENGTH]; 5634 ge_p3 R; 5635 uint8_t hram[SHA512_DIGEST_LENGTH]; 5636 EVP_MD *sha512 = EVP_MD_fetch(libctx, SN_sha512, propq); 5637 EVP_MD_CTX *hash_ctx = EVP_MD_CTX_new(); 5638 unsigned int sz; 5639 int res = 0; 5640 5641 if (context == NULL) 5642 context_len = 0; 5643 5644 /* if csflag is set, then a non-empty context-string is required */ 5645 if (csflag && context_len == 0) 5646 goto err; 5647 5648 /* if dom2flag is not set, then an empty context-string is required */ 5649 if (!dom2flag && context_len > 0) 5650 goto err; 5651 5652 if (sha512 == NULL || hash_ctx == NULL) 5653 goto err; 5654 5655 if (!EVP_DigestInit_ex(hash_ctx, sha512, NULL) 5656 || !EVP_DigestUpdate(hash_ctx, private_key, 32) 5657 || !EVP_DigestFinal_ex(hash_ctx, az, &sz)) 5658 goto err; 5659 5660 az[0] &= 248; 5661 az[31] &= 63; 5662 az[31] |= 64; 5663 5664 if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len) 5665 || !EVP_DigestUpdate(hash_ctx, az + 32, 32) 5666 || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len) 5667 || !EVP_DigestFinal_ex(hash_ctx, nonce, &sz)) 5668 goto err; 5669 5670 x25519_sc_reduce(nonce); 5671 ge_scalarmult_base(&R, nonce); 5672 ge_p3_tobytes(out_sig, &R); 5673 5674 if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len) 5675 || !EVP_DigestUpdate(hash_ctx, out_sig, 32) 5676 || !EVP_DigestUpdate(hash_ctx, public_key, 32) 5677 || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len) 5678 || !EVP_DigestFinal_ex(hash_ctx, hram, &sz)) 5679 goto err; 5680 5681 x25519_sc_reduce(hram); 5682 sc_muladd(out_sig + 32, hram, az, nonce); 5683 5684 res = 1; 5685 err: 5686 OPENSSL_cleanse(nonce, sizeof(nonce)); 5687 OPENSSL_cleanse(az, sizeof(az)); 5688 EVP_MD_free(sha512); 5689 EVP_MD_CTX_free(hash_ctx); 5690 return res; 5691 } 5692 5693 /* 5694 * This function should not be necessary since ossl_ed25519_verify() already 5695 * does this check internally. 5696 * For some reason the FIPS ACVP requires a EDDSA KeyVer test. 5697 */ 5698 int ossl_ed25519_pubkey_verify(const uint8_t *pub, size_t pub_len) 5699 { 5700 ge_p3 A; 5701 5702 if (pub_len != ED25519_KEYLEN) 5703 return 0; 5704 return (ge_frombytes_vartime(&A, pub) == 0); 5705 } 5706 5707 static const char allzeroes[15]; 5708 5709 int ossl_ed25519_verify(const uint8_t *tbs, size_t tbs_len, 5710 const uint8_t signature[64], const uint8_t public_key[32], 5711 const uint8_t dom2flag, const uint8_t phflag, const uint8_t csflag, 5712 const uint8_t *context, size_t context_len, 5713 OSSL_LIB_CTX *libctx, const char *propq) 5714 { 5715 int i; 5716 ge_p3 A; 5717 const uint8_t *r, *s; 5718 EVP_MD *sha512; 5719 EVP_MD_CTX *hash_ctx = NULL; 5720 unsigned int sz; 5721 int res = 0; 5722 ge_p2 R; 5723 uint8_t rcheck[32]; 5724 uint8_t h[SHA512_DIGEST_LENGTH]; 5725 /* 27742317777372353535851937790883648493 in little endian format */ 5726 const uint8_t l_low[16] = { 5727 0xED, 0xD3, 0xF5, 0x5C, 0x1A, 0x63, 0x12, 0x58, 0xD6, 0x9C, 0xF7, 0xA2, 5728 0xDE, 0xF9, 0xDE, 0x14 5729 }; 5730 5731 if (context == NULL) 5732 context_len = 0; 5733 5734 /* if csflag is set, then a non-empty context-string is required */ 5735 if (csflag && context_len == 0) 5736 return 0; 5737 5738 /* if dom2flag is not set, then an empty context-string is required */ 5739 if (!dom2flag && context_len > 0) 5740 return 0; 5741 5742 r = signature; 5743 s = signature + 32; 5744 5745 /* 5746 * Check 0 <= s < L where L = 2^252 + 27742317777372353535851937790883648493 5747 * 5748 * If not the signature is publicly invalid. Since it's public we can do the 5749 * check in variable time. 5750 * 5751 * First check the most significant byte 5752 */ 5753 if (s[31] > 0x10) 5754 return 0; 5755 if (s[31] == 0x10) { 5756 /* 5757 * Most significant byte indicates a value close to 2^252 so check the 5758 * rest 5759 */ 5760 if (memcmp(s + 16, allzeroes, sizeof(allzeroes)) != 0) 5761 return 0; 5762 for (i = 15; i >= 0; i--) { 5763 if (s[i] < l_low[i]) 5764 break; 5765 if (s[i] > l_low[i]) 5766 return 0; 5767 } 5768 if (i < 0) 5769 return 0; 5770 } 5771 5772 if (ge_frombytes_vartime(&A, public_key) != 0) { 5773 return 0; 5774 } 5775 5776 fe_neg(A.X, A.X); 5777 fe_neg(A.T, A.T); 5778 5779 sha512 = EVP_MD_fetch(libctx, SN_sha512, propq); 5780 if (sha512 == NULL) 5781 return 0; 5782 hash_ctx = EVP_MD_CTX_new(); 5783 if (hash_ctx == NULL) 5784 goto err; 5785 5786 if (!hash_init_with_dom(hash_ctx, sha512, dom2flag, phflag, context, context_len) 5787 || !EVP_DigestUpdate(hash_ctx, r, 32) 5788 || !EVP_DigestUpdate(hash_ctx, public_key, 32) 5789 || !EVP_DigestUpdate(hash_ctx, tbs, tbs_len) 5790 || !EVP_DigestFinal_ex(hash_ctx, h, &sz)) 5791 goto err; 5792 5793 x25519_sc_reduce(h); 5794 5795 ge_double_scalarmult_vartime(&R, h, &A, s); 5796 5797 ge_tobytes(rcheck, &R); 5798 5799 res = CRYPTO_memcmp(rcheck, r, sizeof(rcheck)) == 0; 5800 5801 /* note that we have used the strict verification equation here. 5802 * we checked that ENC( [h](-A) + [s]B ) == r 5803 * B is the base point. 5804 * 5805 * the less strict verification equation uses the curve cofactor: 5806 * [h*8](-A) + [s*8]B == [8]R 5807 */ 5808 err: 5809 EVP_MD_free(sha512); 5810 EVP_MD_CTX_free(hash_ctx); 5811 return res; 5812 } 5813 5814 int ossl_ed25519_public_from_private(OSSL_LIB_CTX *ctx, uint8_t out_public_key[32], 5815 const uint8_t private_key[32], 5816 const char *propq) 5817 { 5818 uint8_t az[SHA512_DIGEST_LENGTH]; 5819 ge_p3 A; 5820 int r; 5821 EVP_MD *sha512 = NULL; 5822 5823 sha512 = EVP_MD_fetch(ctx, SN_sha512, propq); 5824 if (sha512 == NULL) 5825 return 0; 5826 r = EVP_Digest(private_key, 32, az, NULL, sha512, NULL); 5827 EVP_MD_free(sha512); 5828 if (!r) { 5829 OPENSSL_cleanse(az, sizeof(az)); 5830 return 0; 5831 } 5832 5833 az[0] &= 248; 5834 az[31] &= 63; 5835 az[31] |= 64; 5836 5837 ge_scalarmult_base(&A, az); 5838 ge_p3_tobytes(out_public_key, &A); 5839 5840 OPENSSL_cleanse(az, sizeof(az)); 5841 return 1; 5842 } 5843 5844 int ossl_x25519(uint8_t out_shared_key[32], const uint8_t private_key[32], 5845 const uint8_t peer_public_value[32]) 5846 { 5847 static const uint8_t kZeros[32] = { 0 }; 5848 x25519_scalar_mult(out_shared_key, private_key, peer_public_value); 5849 /* The all-zero output results when the input is a point of small order. */ 5850 return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0; 5851 } 5852 5853 void ossl_x25519_public_from_private(uint8_t out_public_value[32], 5854 const uint8_t private_key[32]) 5855 { 5856 uint8_t e[32]; 5857 ge_p3 A; 5858 fe zplusy, zminusy, zminusy_inv; 5859 5860 memcpy(e, private_key, 32); 5861 e[0] &= 248; 5862 e[31] &= 127; 5863 e[31] |= 64; 5864 5865 ge_scalarmult_base(&A, e); 5866 5867 /* 5868 * We only need the u-coordinate of the curve25519 point. 5869 * The map is u=(y+1)/(1-y). Since y=Y/Z, this gives 5870 * u=(Z+Y)/(Z-Y). 5871 */ 5872 fe_add(zplusy, A.Z, A.Y); 5873 fe_sub(zminusy, A.Z, A.Y); 5874 fe_invert(zminusy_inv, zminusy); 5875 fe_mul(zplusy, zplusy, zminusy_inv); 5876 fe_tobytes(out_public_value, zplusy); 5877 5878 OPENSSL_cleanse(e, sizeof(e)); 5879 } 5880