1 /* 2 * Copyright 2002-2021 The OpenSSL Project Authors. All Rights Reserved. 3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved 4 * 5 * Licensed under the Apache License 2.0 (the "License"). You may not use 6 * this file except in compliance with the License. You can obtain a copy 7 * in the file LICENSE in the source distribution or at 8 * https://www.openssl.org/source/license.html 9 */ 10 11 #include <assert.h> 12 #include <limits.h> 13 #include <stdio.h> 14 #include "internal/cryptlib.h" 15 #include "bn_local.h" 16 17 #ifndef OPENSSL_NO_EC2M 18 19 /* 20 * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should 21 * fail. 22 */ 23 # define MAX_ITERATIONS 50 24 25 # define SQR_nibble(w) ((((w) & 8) << 3) \ 26 | (((w) & 4) << 2) \ 27 | (((w) & 2) << 1) \ 28 | ((w) & 1)) 29 30 31 /* Platform-specific macros to accelerate squaring. */ 32 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 33 # define SQR1(w) \ 34 SQR_nibble((w) >> 60) << 56 | SQR_nibble((w) >> 56) << 48 | \ 35 SQR_nibble((w) >> 52) << 40 | SQR_nibble((w) >> 48) << 32 | \ 36 SQR_nibble((w) >> 44) << 24 | SQR_nibble((w) >> 40) << 16 | \ 37 SQR_nibble((w) >> 36) << 8 | SQR_nibble((w) >> 32) 38 # define SQR0(w) \ 39 SQR_nibble((w) >> 28) << 56 | SQR_nibble((w) >> 24) << 48 | \ 40 SQR_nibble((w) >> 20) << 40 | SQR_nibble((w) >> 16) << 32 | \ 41 SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >> 8) << 16 | \ 42 SQR_nibble((w) >> 4) << 8 | SQR_nibble((w) ) 43 # endif 44 # ifdef THIRTY_TWO_BIT 45 # define SQR1(w) \ 46 SQR_nibble((w) >> 28) << 24 | SQR_nibble((w) >> 24) << 16 | \ 47 SQR_nibble((w) >> 20) << 8 | SQR_nibble((w) >> 16) 48 # define SQR0(w) \ 49 SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >> 8) << 16 | \ 50 SQR_nibble((w) >> 4) << 8 | SQR_nibble((w) ) 51 # endif 52 53 # if !defined(OPENSSL_BN_ASM_GF2m) 54 /* 55 * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is 56 * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that 57 * the variables have the right amount of space allocated. 58 */ 59 # ifdef THIRTY_TWO_BIT 60 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, 61 const BN_ULONG b) 62 { 63 register BN_ULONG h, l, s; 64 BN_ULONG tab[8], top2b = a >> 30; 65 register BN_ULONG a1, a2, a4; 66 67 a1 = a & (0x3FFFFFFF); 68 a2 = a1 << 1; 69 a4 = a2 << 1; 70 71 tab[0] = 0; 72 tab[1] = a1; 73 tab[2] = a2; 74 tab[3] = a1 ^ a2; 75 tab[4] = a4; 76 tab[5] = a1 ^ a4; 77 tab[6] = a2 ^ a4; 78 tab[7] = a1 ^ a2 ^ a4; 79 80 s = tab[b & 0x7]; 81 l = s; 82 s = tab[b >> 3 & 0x7]; 83 l ^= s << 3; 84 h = s >> 29; 85 s = tab[b >> 6 & 0x7]; 86 l ^= s << 6; 87 h ^= s >> 26; 88 s = tab[b >> 9 & 0x7]; 89 l ^= s << 9; 90 h ^= s >> 23; 91 s = tab[b >> 12 & 0x7]; 92 l ^= s << 12; 93 h ^= s >> 20; 94 s = tab[b >> 15 & 0x7]; 95 l ^= s << 15; 96 h ^= s >> 17; 97 s = tab[b >> 18 & 0x7]; 98 l ^= s << 18; 99 h ^= s >> 14; 100 s = tab[b >> 21 & 0x7]; 101 l ^= s << 21; 102 h ^= s >> 11; 103 s = tab[b >> 24 & 0x7]; 104 l ^= s << 24; 105 h ^= s >> 8; 106 s = tab[b >> 27 & 0x7]; 107 l ^= s << 27; 108 h ^= s >> 5; 109 s = tab[b >> 30]; 110 l ^= s << 30; 111 h ^= s >> 2; 112 113 /* compensate for the top two bits of a */ 114 115 if (top2b & 01) { 116 l ^= b << 30; 117 h ^= b >> 2; 118 } 119 if (top2b & 02) { 120 l ^= b << 31; 121 h ^= b >> 1; 122 } 123 124 *r1 = h; 125 *r0 = l; 126 } 127 # endif 128 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 129 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, 130 const BN_ULONG b) 131 { 132 register BN_ULONG h, l, s; 133 BN_ULONG tab[16], top3b = a >> 61; 134 register BN_ULONG a1, a2, a4, a8; 135 136 a1 = a & (0x1FFFFFFFFFFFFFFFULL); 137 a2 = a1 << 1; 138 a4 = a2 << 1; 139 a8 = a4 << 1; 140 141 tab[0] = 0; 142 tab[1] = a1; 143 tab[2] = a2; 144 tab[3] = a1 ^ a2; 145 tab[4] = a4; 146 tab[5] = a1 ^ a4; 147 tab[6] = a2 ^ a4; 148 tab[7] = a1 ^ a2 ^ a4; 149 tab[8] = a8; 150 tab[9] = a1 ^ a8; 151 tab[10] = a2 ^ a8; 152 tab[11] = a1 ^ a2 ^ a8; 153 tab[12] = a4 ^ a8; 154 tab[13] = a1 ^ a4 ^ a8; 155 tab[14] = a2 ^ a4 ^ a8; 156 tab[15] = a1 ^ a2 ^ a4 ^ a8; 157 158 s = tab[b & 0xF]; 159 l = s; 160 s = tab[b >> 4 & 0xF]; 161 l ^= s << 4; 162 h = s >> 60; 163 s = tab[b >> 8 & 0xF]; 164 l ^= s << 8; 165 h ^= s >> 56; 166 s = tab[b >> 12 & 0xF]; 167 l ^= s << 12; 168 h ^= s >> 52; 169 s = tab[b >> 16 & 0xF]; 170 l ^= s << 16; 171 h ^= s >> 48; 172 s = tab[b >> 20 & 0xF]; 173 l ^= s << 20; 174 h ^= s >> 44; 175 s = tab[b >> 24 & 0xF]; 176 l ^= s << 24; 177 h ^= s >> 40; 178 s = tab[b >> 28 & 0xF]; 179 l ^= s << 28; 180 h ^= s >> 36; 181 s = tab[b >> 32 & 0xF]; 182 l ^= s << 32; 183 h ^= s >> 32; 184 s = tab[b >> 36 & 0xF]; 185 l ^= s << 36; 186 h ^= s >> 28; 187 s = tab[b >> 40 & 0xF]; 188 l ^= s << 40; 189 h ^= s >> 24; 190 s = tab[b >> 44 & 0xF]; 191 l ^= s << 44; 192 h ^= s >> 20; 193 s = tab[b >> 48 & 0xF]; 194 l ^= s << 48; 195 h ^= s >> 16; 196 s = tab[b >> 52 & 0xF]; 197 l ^= s << 52; 198 h ^= s >> 12; 199 s = tab[b >> 56 & 0xF]; 200 l ^= s << 56; 201 h ^= s >> 8; 202 s = tab[b >> 60]; 203 l ^= s << 60; 204 h ^= s >> 4; 205 206 /* compensate for the top three bits of a */ 207 208 if (top3b & 01) { 209 l ^= b << 61; 210 h ^= b >> 3; 211 } 212 if (top3b & 02) { 213 l ^= b << 62; 214 h ^= b >> 2; 215 } 216 if (top3b & 04) { 217 l ^= b << 63; 218 h ^= b >> 1; 219 } 220 221 *r1 = h; 222 *r0 = l; 223 } 224 # endif 225 226 /* 227 * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, 228 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST 229 * ensure that the variables have the right amount of space allocated. 230 */ 231 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, 232 const BN_ULONG b1, const BN_ULONG b0) 233 { 234 BN_ULONG m1, m0; 235 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ 236 bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1); 237 bn_GF2m_mul_1x1(r + 1, r, a0, b0); 238 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); 239 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ 240 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ 241 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ 242 } 243 # else 244 void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1, 245 BN_ULONG b0); 246 # endif 247 248 /* 249 * Add polynomials a and b and store result in r; r could be a or b, a and b 250 * could be equal; r is the bitwise XOR of a and b. 251 */ 252 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) 253 { 254 int i; 255 const BIGNUM *at, *bt; 256 257 bn_check_top(a); 258 bn_check_top(b); 259 260 if (a->top < b->top) { 261 at = b; 262 bt = a; 263 } else { 264 at = a; 265 bt = b; 266 } 267 268 if (bn_wexpand(r, at->top) == NULL) 269 return 0; 270 271 for (i = 0; i < bt->top; i++) { 272 r->d[i] = at->d[i] ^ bt->d[i]; 273 } 274 for (; i < at->top; i++) { 275 r->d[i] = at->d[i]; 276 } 277 278 r->top = at->top; 279 bn_correct_top(r); 280 281 return 1; 282 } 283 284 /*- 285 * Some functions allow for representation of the irreducible polynomials 286 * as an int[], say p. The irreducible f(t) is then of the form: 287 * t^p[0] + t^p[1] + ... + t^p[k] 288 * where m = p[0] > p[1] > ... > p[k] = 0. 289 */ 290 291 /* Performs modular reduction of a and store result in r. r could be a. */ 292 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]) 293 { 294 int j, k; 295 int n, dN, d0, d1; 296 BN_ULONG zz, *z; 297 298 bn_check_top(a); 299 300 if (p[0] == 0) { 301 /* reduction mod 1 => return 0 */ 302 BN_zero(r); 303 return 1; 304 } 305 306 /* 307 * Since the algorithm does reduction in the r value, if a != r, copy the 308 * contents of a into r so we can do reduction in r. 309 */ 310 if (a != r) { 311 if (!bn_wexpand(r, a->top)) 312 return 0; 313 for (j = 0; j < a->top; j++) { 314 r->d[j] = a->d[j]; 315 } 316 r->top = a->top; 317 } 318 z = r->d; 319 320 /* start reduction */ 321 dN = p[0] / BN_BITS2; 322 for (j = r->top - 1; j > dN;) { 323 zz = z[j]; 324 if (z[j] == 0) { 325 j--; 326 continue; 327 } 328 z[j] = 0; 329 330 for (k = 1; p[k] != 0; k++) { 331 /* reducing component t^p[k] */ 332 n = p[0] - p[k]; 333 d0 = n % BN_BITS2; 334 d1 = BN_BITS2 - d0; 335 n /= BN_BITS2; 336 z[j - n] ^= (zz >> d0); 337 if (d0) 338 z[j - n - 1] ^= (zz << d1); 339 } 340 341 /* reducing component t^0 */ 342 n = dN; 343 d0 = p[0] % BN_BITS2; 344 d1 = BN_BITS2 - d0; 345 z[j - n] ^= (zz >> d0); 346 if (d0) 347 z[j - n - 1] ^= (zz << d1); 348 } 349 350 /* final round of reduction */ 351 while (j == dN) { 352 353 d0 = p[0] % BN_BITS2; 354 zz = z[dN] >> d0; 355 if (zz == 0) 356 break; 357 d1 = BN_BITS2 - d0; 358 359 /* clear up the top d1 bits */ 360 if (d0) 361 z[dN] = (z[dN] << d1) >> d1; 362 else 363 z[dN] = 0; 364 z[0] ^= zz; /* reduction t^0 component */ 365 366 for (k = 1; p[k] != 0; k++) { 367 BN_ULONG tmp_ulong; 368 369 /* reducing component t^p[k] */ 370 n = p[k] / BN_BITS2; 371 d0 = p[k] % BN_BITS2; 372 d1 = BN_BITS2 - d0; 373 z[n] ^= (zz << d0); 374 if (d0 && (tmp_ulong = zz >> d1)) 375 z[n + 1] ^= tmp_ulong; 376 } 377 378 } 379 380 bn_correct_top(r); 381 return 1; 382 } 383 384 /* 385 * Performs modular reduction of a by p and store result in r. r could be a. 386 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 387 * function is only provided for convenience; for best performance, use the 388 * BN_GF2m_mod_arr function. 389 */ 390 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 391 { 392 int ret = 0; 393 int arr[6]; 394 bn_check_top(a); 395 bn_check_top(p); 396 ret = BN_GF2m_poly2arr(p, arr, OSSL_NELEM(arr)); 397 if (!ret || ret > (int)OSSL_NELEM(arr)) { 398 ERR_raise(ERR_LIB_BN, BN_R_INVALID_LENGTH); 399 return 0; 400 } 401 ret = BN_GF2m_mod_arr(r, a, arr); 402 bn_check_top(r); 403 return ret; 404 } 405 406 /* 407 * Compute the product of two polynomials a and b, reduce modulo p, and store 408 * the result in r. r could be a or b; a could be b. 409 */ 410 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 411 const int p[], BN_CTX *ctx) 412 { 413 int zlen, i, j, k, ret = 0; 414 BIGNUM *s; 415 BN_ULONG x1, x0, y1, y0, zz[4]; 416 417 bn_check_top(a); 418 bn_check_top(b); 419 420 if (a == b) { 421 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 422 } 423 424 BN_CTX_start(ctx); 425 if ((s = BN_CTX_get(ctx)) == NULL) 426 goto err; 427 428 zlen = a->top + b->top + 4; 429 if (!bn_wexpand(s, zlen)) 430 goto err; 431 s->top = zlen; 432 433 for (i = 0; i < zlen; i++) 434 s->d[i] = 0; 435 436 for (j = 0; j < b->top; j += 2) { 437 y0 = b->d[j]; 438 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1]; 439 for (i = 0; i < a->top; i += 2) { 440 x0 = a->d[i]; 441 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1]; 442 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 443 for (k = 0; k < 4; k++) 444 s->d[i + j + k] ^= zz[k]; 445 } 446 } 447 448 bn_correct_top(s); 449 if (BN_GF2m_mod_arr(r, s, p)) 450 ret = 1; 451 bn_check_top(r); 452 453 err: 454 BN_CTX_end(ctx); 455 return ret; 456 } 457 458 /* 459 * Compute the product of two polynomials a and b, reduce modulo p, and store 460 * the result in r. r could be a or b; a could equal b. This function calls 461 * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is 462 * only provided for convenience; for best performance, use the 463 * BN_GF2m_mod_mul_arr function. 464 */ 465 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 466 const BIGNUM *p, BN_CTX *ctx) 467 { 468 int ret = 0; 469 const int max = BN_num_bits(p) + 1; 470 int *arr; 471 472 bn_check_top(a); 473 bn_check_top(b); 474 bn_check_top(p); 475 476 arr = OPENSSL_malloc(sizeof(*arr) * max); 477 if (arr == NULL) { 478 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); 479 return 0; 480 } 481 ret = BN_GF2m_poly2arr(p, arr, max); 482 if (!ret || ret > max) { 483 ERR_raise(ERR_LIB_BN, BN_R_INVALID_LENGTH); 484 goto err; 485 } 486 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 487 bn_check_top(r); 488 err: 489 OPENSSL_free(arr); 490 return ret; 491 } 492 493 /* Square a, reduce the result mod p, and store it in a. r could be a. */ 494 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], 495 BN_CTX *ctx) 496 { 497 int i, ret = 0; 498 BIGNUM *s; 499 500 bn_check_top(a); 501 BN_CTX_start(ctx); 502 if ((s = BN_CTX_get(ctx)) == NULL) 503 goto err; 504 if (!bn_wexpand(s, 2 * a->top)) 505 goto err; 506 507 for (i = a->top - 1; i >= 0; i--) { 508 s->d[2 * i + 1] = SQR1(a->d[i]); 509 s->d[2 * i] = SQR0(a->d[i]); 510 } 511 512 s->top = 2 * a->top; 513 bn_correct_top(s); 514 if (!BN_GF2m_mod_arr(r, s, p)) 515 goto err; 516 bn_check_top(r); 517 ret = 1; 518 err: 519 BN_CTX_end(ctx); 520 return ret; 521 } 522 523 /* 524 * Square a, reduce the result mod p, and store it in a. r could be a. This 525 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this 526 * wrapper function is only provided for convenience; for best performance, 527 * use the BN_GF2m_mod_sqr_arr function. 528 */ 529 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 530 { 531 int ret = 0; 532 const int max = BN_num_bits(p) + 1; 533 int *arr; 534 535 bn_check_top(a); 536 bn_check_top(p); 537 538 arr = OPENSSL_malloc(sizeof(*arr) * max); 539 if (arr == NULL) { 540 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); 541 return 0; 542 } 543 ret = BN_GF2m_poly2arr(p, arr, max); 544 if (!ret || ret > max) { 545 ERR_raise(ERR_LIB_BN, BN_R_INVALID_LENGTH); 546 goto err; 547 } 548 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 549 bn_check_top(r); 550 err: 551 OPENSSL_free(arr); 552 return ret; 553 } 554 555 /* 556 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses 557 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D., 558 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic 559 * Curve Cryptography Over Binary Fields". 560 */ 561 static int BN_GF2m_mod_inv_vartime(BIGNUM *r, const BIGNUM *a, 562 const BIGNUM *p, BN_CTX *ctx) 563 { 564 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; 565 int ret = 0; 566 567 bn_check_top(a); 568 bn_check_top(p); 569 570 BN_CTX_start(ctx); 571 572 b = BN_CTX_get(ctx); 573 c = BN_CTX_get(ctx); 574 u = BN_CTX_get(ctx); 575 v = BN_CTX_get(ctx); 576 if (v == NULL) 577 goto err; 578 579 if (!BN_GF2m_mod(u, a, p)) 580 goto err; 581 if (BN_is_zero(u)) 582 goto err; 583 584 if (!BN_copy(v, p)) 585 goto err; 586 # if 0 587 if (!BN_one(b)) 588 goto err; 589 590 while (1) { 591 while (!BN_is_odd(u)) { 592 if (BN_is_zero(u)) 593 goto err; 594 if (!BN_rshift1(u, u)) 595 goto err; 596 if (BN_is_odd(b)) { 597 if (!BN_GF2m_add(b, b, p)) 598 goto err; 599 } 600 if (!BN_rshift1(b, b)) 601 goto err; 602 } 603 604 if (BN_abs_is_word(u, 1)) 605 break; 606 607 if (BN_num_bits(u) < BN_num_bits(v)) { 608 tmp = u; 609 u = v; 610 v = tmp; 611 tmp = b; 612 b = c; 613 c = tmp; 614 } 615 616 if (!BN_GF2m_add(u, u, v)) 617 goto err; 618 if (!BN_GF2m_add(b, b, c)) 619 goto err; 620 } 621 # else 622 { 623 int i; 624 int ubits = BN_num_bits(u); 625 int vbits = BN_num_bits(v); /* v is copy of p */ 626 int top = p->top; 627 BN_ULONG *udp, *bdp, *vdp, *cdp; 628 629 if (!bn_wexpand(u, top)) 630 goto err; 631 udp = u->d; 632 for (i = u->top; i < top; i++) 633 udp[i] = 0; 634 u->top = top; 635 if (!bn_wexpand(b, top)) 636 goto err; 637 bdp = b->d; 638 bdp[0] = 1; 639 for (i = 1; i < top; i++) 640 bdp[i] = 0; 641 b->top = top; 642 if (!bn_wexpand(c, top)) 643 goto err; 644 cdp = c->d; 645 for (i = 0; i < top; i++) 646 cdp[i] = 0; 647 c->top = top; 648 vdp = v->d; /* It pays off to "cache" *->d pointers, 649 * because it allows optimizer to be more 650 * aggressive. But we don't have to "cache" 651 * p->d, because *p is declared 'const'... */ 652 while (1) { 653 while (ubits && !(udp[0] & 1)) { 654 BN_ULONG u0, u1, b0, b1, mask; 655 656 u0 = udp[0]; 657 b0 = bdp[0]; 658 mask = (BN_ULONG)0 - (b0 & 1); 659 b0 ^= p->d[0] & mask; 660 for (i = 0; i < top - 1; i++) { 661 u1 = udp[i + 1]; 662 udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2; 663 u0 = u1; 664 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask); 665 bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2; 666 b0 = b1; 667 } 668 udp[i] = u0 >> 1; 669 bdp[i] = b0 >> 1; 670 ubits--; 671 } 672 673 if (ubits <= BN_BITS2) { 674 if (udp[0] == 0) /* poly was reducible */ 675 goto err; 676 if (udp[0] == 1) 677 break; 678 } 679 680 if (ubits < vbits) { 681 i = ubits; 682 ubits = vbits; 683 vbits = i; 684 tmp = u; 685 u = v; 686 v = tmp; 687 tmp = b; 688 b = c; 689 c = tmp; 690 udp = vdp; 691 vdp = v->d; 692 bdp = cdp; 693 cdp = c->d; 694 } 695 for (i = 0; i < top; i++) { 696 udp[i] ^= vdp[i]; 697 bdp[i] ^= cdp[i]; 698 } 699 if (ubits == vbits) { 700 BN_ULONG ul; 701 int utop = (ubits - 1) / BN_BITS2; 702 703 while ((ul = udp[utop]) == 0 && utop) 704 utop--; 705 ubits = utop * BN_BITS2 + BN_num_bits_word(ul); 706 } 707 } 708 bn_correct_top(b); 709 } 710 # endif 711 712 if (!BN_copy(r, b)) 713 goto err; 714 bn_check_top(r); 715 ret = 1; 716 717 err: 718 # ifdef BN_DEBUG 719 /* BN_CTX_end would complain about the expanded form */ 720 bn_correct_top(c); 721 bn_correct_top(u); 722 bn_correct_top(v); 723 # endif 724 BN_CTX_end(ctx); 725 return ret; 726 } 727 728 /*- 729 * Wrapper for BN_GF2m_mod_inv_vartime that blinds the input before calling. 730 * This is not constant time. 731 * But it does eliminate first order deduction on the input. 732 */ 733 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 734 { 735 BIGNUM *b = NULL; 736 int ret = 0; 737 int numbits; 738 739 BN_CTX_start(ctx); 740 if ((b = BN_CTX_get(ctx)) == NULL) 741 goto err; 742 743 /* Fail on a non-sensical input p value */ 744 numbits = BN_num_bits(p); 745 if (numbits <= 1) 746 goto err; 747 748 /* generate blinding value */ 749 do { 750 if (!BN_priv_rand_ex(b, numbits - 1, 751 BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY, 0, ctx)) 752 goto err; 753 } while (BN_is_zero(b)); 754 755 /* r := a * b */ 756 if (!BN_GF2m_mod_mul(r, a, b, p, ctx)) 757 goto err; 758 759 /* r := 1/(a * b) */ 760 if (!BN_GF2m_mod_inv_vartime(r, r, p, ctx)) 761 goto err; 762 763 /* r := b/(a * b) = 1/a */ 764 if (!BN_GF2m_mod_mul(r, r, b, p, ctx)) 765 goto err; 766 767 ret = 1; 768 769 err: 770 BN_CTX_end(ctx); 771 return ret; 772 } 773 774 /* 775 * Invert xx, reduce modulo p, and store the result in r. r could be xx. 776 * This function calls down to the BN_GF2m_mod_inv implementation; this 777 * wrapper function is only provided for convenience; for best performance, 778 * use the BN_GF2m_mod_inv function. 779 */ 780 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], 781 BN_CTX *ctx) 782 { 783 BIGNUM *field; 784 int ret = 0; 785 786 bn_check_top(xx); 787 BN_CTX_start(ctx); 788 if ((field = BN_CTX_get(ctx)) == NULL) 789 goto err; 790 if (!BN_GF2m_arr2poly(p, field)) 791 goto err; 792 793 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 794 bn_check_top(r); 795 796 err: 797 BN_CTX_end(ctx); 798 return ret; 799 } 800 801 /* 802 * Divide y by x, reduce modulo p, and store the result in r. r could be x 803 * or y, x could equal y. 804 */ 805 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 806 const BIGNUM *p, BN_CTX *ctx) 807 { 808 BIGNUM *xinv = NULL; 809 int ret = 0; 810 811 bn_check_top(y); 812 bn_check_top(x); 813 bn_check_top(p); 814 815 BN_CTX_start(ctx); 816 xinv = BN_CTX_get(ctx); 817 if (xinv == NULL) 818 goto err; 819 820 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) 821 goto err; 822 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) 823 goto err; 824 bn_check_top(r); 825 ret = 1; 826 827 err: 828 BN_CTX_end(ctx); 829 return ret; 830 } 831 832 /* 833 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 834 * * or yy, xx could equal yy. This function calls down to the 835 * BN_GF2m_mod_div implementation; this wrapper function is only provided for 836 * convenience; for best performance, use the BN_GF2m_mod_div function. 837 */ 838 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, 839 const int p[], BN_CTX *ctx) 840 { 841 BIGNUM *field; 842 int ret = 0; 843 844 bn_check_top(yy); 845 bn_check_top(xx); 846 847 BN_CTX_start(ctx); 848 if ((field = BN_CTX_get(ctx)) == NULL) 849 goto err; 850 if (!BN_GF2m_arr2poly(p, field)) 851 goto err; 852 853 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 854 bn_check_top(r); 855 856 err: 857 BN_CTX_end(ctx); 858 return ret; 859 } 860 861 /* 862 * Compute the bth power of a, reduce modulo p, and store the result in r. r 863 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE 864 * P1363. 865 */ 866 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 867 const int p[], BN_CTX *ctx) 868 { 869 int ret = 0, i, n; 870 BIGNUM *u; 871 872 bn_check_top(a); 873 bn_check_top(b); 874 875 if (BN_is_zero(b)) 876 return BN_one(r); 877 878 if (BN_abs_is_word(b, 1)) 879 return (BN_copy(r, a) != NULL); 880 881 BN_CTX_start(ctx); 882 if ((u = BN_CTX_get(ctx)) == NULL) 883 goto err; 884 885 if (!BN_GF2m_mod_arr(u, a, p)) 886 goto err; 887 888 n = BN_num_bits(b) - 1; 889 for (i = n - 1; i >= 0; i--) { 890 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) 891 goto err; 892 if (BN_is_bit_set(b, i)) { 893 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) 894 goto err; 895 } 896 } 897 if (!BN_copy(r, u)) 898 goto err; 899 bn_check_top(r); 900 ret = 1; 901 err: 902 BN_CTX_end(ctx); 903 return ret; 904 } 905 906 /* 907 * Compute the bth power of a, reduce modulo p, and store the result in r. r 908 * could be a. This function calls down to the BN_GF2m_mod_exp_arr 909 * implementation; this wrapper function is only provided for convenience; 910 * for best performance, use the BN_GF2m_mod_exp_arr function. 911 */ 912 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 913 const BIGNUM *p, BN_CTX *ctx) 914 { 915 int ret = 0; 916 const int max = BN_num_bits(p) + 1; 917 int *arr; 918 919 bn_check_top(a); 920 bn_check_top(b); 921 bn_check_top(p); 922 923 arr = OPENSSL_malloc(sizeof(*arr) * max); 924 if (arr == NULL) { 925 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); 926 return 0; 927 } 928 ret = BN_GF2m_poly2arr(p, arr, max); 929 if (!ret || ret > max) { 930 ERR_raise(ERR_LIB_BN, BN_R_INVALID_LENGTH); 931 goto err; 932 } 933 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 934 bn_check_top(r); 935 err: 936 OPENSSL_free(arr); 937 return ret; 938 } 939 940 /* 941 * Compute the square root of a, reduce modulo p, and store the result in r. 942 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 943 */ 944 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], 945 BN_CTX *ctx) 946 { 947 int ret = 0; 948 BIGNUM *u; 949 950 bn_check_top(a); 951 952 if (p[0] == 0) { 953 /* reduction mod 1 => return 0 */ 954 BN_zero(r); 955 return 1; 956 } 957 958 BN_CTX_start(ctx); 959 if ((u = BN_CTX_get(ctx)) == NULL) 960 goto err; 961 962 if (!BN_set_bit(u, p[0] - 1)) 963 goto err; 964 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 965 bn_check_top(r); 966 967 err: 968 BN_CTX_end(ctx); 969 return ret; 970 } 971 972 /* 973 * Compute the square root of a, reduce modulo p, and store the result in r. 974 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr 975 * implementation; this wrapper function is only provided for convenience; 976 * for best performance, use the BN_GF2m_mod_sqrt_arr function. 977 */ 978 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 979 { 980 int ret = 0; 981 const int max = BN_num_bits(p) + 1; 982 int *arr; 983 984 bn_check_top(a); 985 bn_check_top(p); 986 987 arr = OPENSSL_malloc(sizeof(*arr) * max); 988 if (arr == NULL) { 989 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); 990 return 0; 991 } 992 ret = BN_GF2m_poly2arr(p, arr, max); 993 if (!ret || ret > max) { 994 ERR_raise(ERR_LIB_BN, BN_R_INVALID_LENGTH); 995 goto err; 996 } 997 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 998 bn_check_top(r); 999 err: 1000 OPENSSL_free(arr); 1001 return ret; 1002 } 1003 1004 /* 1005 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1006 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 1007 */ 1008 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], 1009 BN_CTX *ctx) 1010 { 1011 int ret = 0, count = 0, j; 1012 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 1013 1014 bn_check_top(a_); 1015 1016 if (p[0] == 0) { 1017 /* reduction mod 1 => return 0 */ 1018 BN_zero(r); 1019 return 1; 1020 } 1021 1022 BN_CTX_start(ctx); 1023 a = BN_CTX_get(ctx); 1024 z = BN_CTX_get(ctx); 1025 w = BN_CTX_get(ctx); 1026 if (w == NULL) 1027 goto err; 1028 1029 if (!BN_GF2m_mod_arr(a, a_, p)) 1030 goto err; 1031 1032 if (BN_is_zero(a)) { 1033 BN_zero(r); 1034 ret = 1; 1035 goto err; 1036 } 1037 1038 if (p[0] & 0x1) { /* m is odd */ 1039 /* compute half-trace of a */ 1040 if (!BN_copy(z, a)) 1041 goto err; 1042 for (j = 1; j <= (p[0] - 1) / 2; j++) { 1043 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1044 goto err; 1045 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1046 goto err; 1047 if (!BN_GF2m_add(z, z, a)) 1048 goto err; 1049 } 1050 1051 } else { /* m is even */ 1052 1053 rho = BN_CTX_get(ctx); 1054 w2 = BN_CTX_get(ctx); 1055 tmp = BN_CTX_get(ctx); 1056 if (tmp == NULL) 1057 goto err; 1058 do { 1059 if (!BN_priv_rand_ex(rho, p[0], BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ANY, 1060 0, ctx)) 1061 goto err; 1062 if (!BN_GF2m_mod_arr(rho, rho, p)) 1063 goto err; 1064 BN_zero(z); 1065 if (!BN_copy(w, rho)) 1066 goto err; 1067 for (j = 1; j <= p[0] - 1; j++) { 1068 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1069 goto err; 1070 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) 1071 goto err; 1072 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) 1073 goto err; 1074 if (!BN_GF2m_add(z, z, tmp)) 1075 goto err; 1076 if (!BN_GF2m_add(w, w2, rho)) 1077 goto err; 1078 } 1079 count++; 1080 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1081 if (BN_is_zero(w)) { 1082 ERR_raise(ERR_LIB_BN, BN_R_TOO_MANY_ITERATIONS); 1083 goto err; 1084 } 1085 } 1086 1087 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) 1088 goto err; 1089 if (!BN_GF2m_add(w, z, w)) 1090 goto err; 1091 if (BN_GF2m_cmp(w, a)) { 1092 ERR_raise(ERR_LIB_BN, BN_R_NO_SOLUTION); 1093 goto err; 1094 } 1095 1096 if (!BN_copy(r, z)) 1097 goto err; 1098 bn_check_top(r); 1099 1100 ret = 1; 1101 1102 err: 1103 BN_CTX_end(ctx); 1104 return ret; 1105 } 1106 1107 /* 1108 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1109 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr 1110 * implementation; this wrapper function is only provided for convenience; 1111 * for best performance, use the BN_GF2m_mod_solve_quad_arr function. 1112 */ 1113 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, 1114 BN_CTX *ctx) 1115 { 1116 int ret = 0; 1117 const int max = BN_num_bits(p) + 1; 1118 int *arr; 1119 1120 bn_check_top(a); 1121 bn_check_top(p); 1122 1123 arr = OPENSSL_malloc(sizeof(*arr) * max); 1124 if (arr == NULL) { 1125 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); 1126 goto err; 1127 } 1128 ret = BN_GF2m_poly2arr(p, arr, max); 1129 if (!ret || ret > max) { 1130 ERR_raise(ERR_LIB_BN, BN_R_INVALID_LENGTH); 1131 goto err; 1132 } 1133 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1134 bn_check_top(r); 1135 err: 1136 OPENSSL_free(arr); 1137 return ret; 1138 } 1139 1140 /* 1141 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * 1142 * x^i) into an array of integers corresponding to the bits with non-zero 1143 * coefficient. Array is terminated with -1. Up to max elements of the array 1144 * will be filled. Return value is total number of array elements that would 1145 * be filled if array was large enough. 1146 */ 1147 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1148 { 1149 int i, j, k = 0; 1150 BN_ULONG mask; 1151 1152 if (BN_is_zero(a)) 1153 return 0; 1154 1155 for (i = a->top - 1; i >= 0; i--) { 1156 if (!a->d[i]) 1157 /* skip word if a->d[i] == 0 */ 1158 continue; 1159 mask = BN_TBIT; 1160 for (j = BN_BITS2 - 1; j >= 0; j--) { 1161 if (a->d[i] & mask) { 1162 if (k < max) 1163 p[k] = BN_BITS2 * i + j; 1164 k++; 1165 } 1166 mask >>= 1; 1167 } 1168 } 1169 1170 if (k < max) { 1171 p[k] = -1; 1172 k++; 1173 } 1174 1175 return k; 1176 } 1177 1178 /* 1179 * Convert the coefficient array representation of a polynomial to a 1180 * bit-string. The array must be terminated by -1. 1181 */ 1182 int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1183 { 1184 int i; 1185 1186 bn_check_top(a); 1187 BN_zero(a); 1188 for (i = 0; p[i] != -1; i++) { 1189 if (BN_set_bit(a, p[i]) == 0) 1190 return 0; 1191 } 1192 bn_check_top(a); 1193 1194 return 1; 1195 } 1196 1197 #endif 1198