1 /* 2 * Copyright 2002-2018 The OpenSSL Project Authors. All Rights Reserved. 3 * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved 4 * 5 * Licensed under the OpenSSL license (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]) { 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 BNerr(BN_F_BN_GF2M_MOD, 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 = NULL; 471 bn_check_top(a); 472 bn_check_top(b); 473 bn_check_top(p); 474 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) 475 goto err; 476 ret = BN_GF2m_poly2arr(p, arr, max); 477 if (!ret || ret > max) { 478 BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH); 479 goto err; 480 } 481 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 482 bn_check_top(r); 483 err: 484 OPENSSL_free(arr); 485 return ret; 486 } 487 488 /* Square a, reduce the result mod p, and store it in a. r could be a. */ 489 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], 490 BN_CTX *ctx) 491 { 492 int i, ret = 0; 493 BIGNUM *s; 494 495 bn_check_top(a); 496 BN_CTX_start(ctx); 497 if ((s = BN_CTX_get(ctx)) == NULL) 498 goto err; 499 if (!bn_wexpand(s, 2 * a->top)) 500 goto err; 501 502 for (i = a->top - 1; i >= 0; i--) { 503 s->d[2 * i + 1] = SQR1(a->d[i]); 504 s->d[2 * i] = SQR0(a->d[i]); 505 } 506 507 s->top = 2 * a->top; 508 bn_correct_top(s); 509 if (!BN_GF2m_mod_arr(r, s, p)) 510 goto err; 511 bn_check_top(r); 512 ret = 1; 513 err: 514 BN_CTX_end(ctx); 515 return ret; 516 } 517 518 /* 519 * Square a, reduce the result mod p, and store it in a. r could be a. This 520 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this 521 * wrapper function is only provided for convenience; for best performance, 522 * use the BN_GF2m_mod_sqr_arr function. 523 */ 524 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 525 { 526 int ret = 0; 527 const int max = BN_num_bits(p) + 1; 528 int *arr = NULL; 529 530 bn_check_top(a); 531 bn_check_top(p); 532 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) 533 goto err; 534 ret = BN_GF2m_poly2arr(p, arr, max); 535 if (!ret || ret > max) { 536 BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH); 537 goto err; 538 } 539 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 540 bn_check_top(r); 541 err: 542 OPENSSL_free(arr); 543 return ret; 544 } 545 546 /* 547 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses 548 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D., 549 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic 550 * Curve Cryptography Over Binary Fields". 551 */ 552 static int BN_GF2m_mod_inv_vartime(BIGNUM *r, const BIGNUM *a, 553 const BIGNUM *p, BN_CTX *ctx) 554 { 555 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; 556 int ret = 0; 557 558 bn_check_top(a); 559 bn_check_top(p); 560 561 BN_CTX_start(ctx); 562 563 b = BN_CTX_get(ctx); 564 c = BN_CTX_get(ctx); 565 u = BN_CTX_get(ctx); 566 v = BN_CTX_get(ctx); 567 if (v == NULL) 568 goto err; 569 570 if (!BN_GF2m_mod(u, a, p)) 571 goto err; 572 if (BN_is_zero(u)) 573 goto err; 574 575 if (!BN_copy(v, p)) 576 goto err; 577 # if 0 578 if (!BN_one(b)) 579 goto err; 580 581 while (1) { 582 while (!BN_is_odd(u)) { 583 if (BN_is_zero(u)) 584 goto err; 585 if (!BN_rshift1(u, u)) 586 goto err; 587 if (BN_is_odd(b)) { 588 if (!BN_GF2m_add(b, b, p)) 589 goto err; 590 } 591 if (!BN_rshift1(b, b)) 592 goto err; 593 } 594 595 if (BN_abs_is_word(u, 1)) 596 break; 597 598 if (BN_num_bits(u) < BN_num_bits(v)) { 599 tmp = u; 600 u = v; 601 v = tmp; 602 tmp = b; 603 b = c; 604 c = tmp; 605 } 606 607 if (!BN_GF2m_add(u, u, v)) 608 goto err; 609 if (!BN_GF2m_add(b, b, c)) 610 goto err; 611 } 612 # else 613 { 614 int i; 615 int ubits = BN_num_bits(u); 616 int vbits = BN_num_bits(v); /* v is copy of p */ 617 int top = p->top; 618 BN_ULONG *udp, *bdp, *vdp, *cdp; 619 620 if (!bn_wexpand(u, top)) 621 goto err; 622 udp = u->d; 623 for (i = u->top; i < top; i++) 624 udp[i] = 0; 625 u->top = top; 626 if (!bn_wexpand(b, top)) 627 goto err; 628 bdp = b->d; 629 bdp[0] = 1; 630 for (i = 1; i < top; i++) 631 bdp[i] = 0; 632 b->top = top; 633 if (!bn_wexpand(c, top)) 634 goto err; 635 cdp = c->d; 636 for (i = 0; i < top; i++) 637 cdp[i] = 0; 638 c->top = top; 639 vdp = v->d; /* It pays off to "cache" *->d pointers, 640 * because it allows optimizer to be more 641 * aggressive. But we don't have to "cache" 642 * p->d, because *p is declared 'const'... */ 643 while (1) { 644 while (ubits && !(udp[0] & 1)) { 645 BN_ULONG u0, u1, b0, b1, mask; 646 647 u0 = udp[0]; 648 b0 = bdp[0]; 649 mask = (BN_ULONG)0 - (b0 & 1); 650 b0 ^= p->d[0] & mask; 651 for (i = 0; i < top - 1; i++) { 652 u1 = udp[i + 1]; 653 udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2; 654 u0 = u1; 655 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask); 656 bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2; 657 b0 = b1; 658 } 659 udp[i] = u0 >> 1; 660 bdp[i] = b0 >> 1; 661 ubits--; 662 } 663 664 if (ubits <= BN_BITS2) { 665 if (udp[0] == 0) /* poly was reducible */ 666 goto err; 667 if (udp[0] == 1) 668 break; 669 } 670 671 if (ubits < vbits) { 672 i = ubits; 673 ubits = vbits; 674 vbits = i; 675 tmp = u; 676 u = v; 677 v = tmp; 678 tmp = b; 679 b = c; 680 c = tmp; 681 udp = vdp; 682 vdp = v->d; 683 bdp = cdp; 684 cdp = c->d; 685 } 686 for (i = 0; i < top; i++) { 687 udp[i] ^= vdp[i]; 688 bdp[i] ^= cdp[i]; 689 } 690 if (ubits == vbits) { 691 BN_ULONG ul; 692 int utop = (ubits - 1) / BN_BITS2; 693 694 while ((ul = udp[utop]) == 0 && utop) 695 utop--; 696 ubits = utop * BN_BITS2 + BN_num_bits_word(ul); 697 } 698 } 699 bn_correct_top(b); 700 } 701 # endif 702 703 if (!BN_copy(r, b)) 704 goto err; 705 bn_check_top(r); 706 ret = 1; 707 708 err: 709 # ifdef BN_DEBUG /* BN_CTX_end would complain about the 710 * expanded form */ 711 bn_correct_top(c); 712 bn_correct_top(u); 713 bn_correct_top(v); 714 # endif 715 BN_CTX_end(ctx); 716 return ret; 717 } 718 719 /*- 720 * Wrapper for BN_GF2m_mod_inv_vartime that blinds the input before calling. 721 * This is not constant time. 722 * But it does eliminate first order deduction on the input. 723 */ 724 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 725 { 726 BIGNUM *b = NULL; 727 int ret = 0; 728 729 BN_CTX_start(ctx); 730 if ((b = BN_CTX_get(ctx)) == NULL) 731 goto err; 732 733 /* generate blinding value */ 734 do { 735 if (!BN_priv_rand(b, BN_num_bits(p) - 1, 736 BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY)) 737 goto err; 738 } while (BN_is_zero(b)); 739 740 /* r := a * b */ 741 if (!BN_GF2m_mod_mul(r, a, b, p, ctx)) 742 goto err; 743 744 /* r := 1/(a * b) */ 745 if (!BN_GF2m_mod_inv_vartime(r, r, p, ctx)) 746 goto err; 747 748 /* r := b/(a * b) = 1/a */ 749 if (!BN_GF2m_mod_mul(r, r, b, p, ctx)) 750 goto err; 751 752 ret = 1; 753 754 err: 755 BN_CTX_end(ctx); 756 return ret; 757 } 758 759 /* 760 * Invert xx, reduce modulo p, and store the result in r. r could be xx. 761 * This function calls down to the BN_GF2m_mod_inv implementation; this 762 * wrapper function is only provided for convenience; for best performance, 763 * use the BN_GF2m_mod_inv function. 764 */ 765 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], 766 BN_CTX *ctx) 767 { 768 BIGNUM *field; 769 int ret = 0; 770 771 bn_check_top(xx); 772 BN_CTX_start(ctx); 773 if ((field = BN_CTX_get(ctx)) == NULL) 774 goto err; 775 if (!BN_GF2m_arr2poly(p, field)) 776 goto err; 777 778 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 779 bn_check_top(r); 780 781 err: 782 BN_CTX_end(ctx); 783 return ret; 784 } 785 786 /* 787 * Divide y by x, reduce modulo p, and store the result in r. r could be x 788 * or y, x could equal y. 789 */ 790 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 791 const BIGNUM *p, BN_CTX *ctx) 792 { 793 BIGNUM *xinv = NULL; 794 int ret = 0; 795 796 bn_check_top(y); 797 bn_check_top(x); 798 bn_check_top(p); 799 800 BN_CTX_start(ctx); 801 xinv = BN_CTX_get(ctx); 802 if (xinv == NULL) 803 goto err; 804 805 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) 806 goto err; 807 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) 808 goto err; 809 bn_check_top(r); 810 ret = 1; 811 812 err: 813 BN_CTX_end(ctx); 814 return ret; 815 } 816 817 /* 818 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 819 * * or yy, xx could equal yy. This function calls down to the 820 * BN_GF2m_mod_div implementation; this wrapper function is only provided for 821 * convenience; for best performance, use the BN_GF2m_mod_div function. 822 */ 823 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, 824 const int p[], BN_CTX *ctx) 825 { 826 BIGNUM *field; 827 int ret = 0; 828 829 bn_check_top(yy); 830 bn_check_top(xx); 831 832 BN_CTX_start(ctx); 833 if ((field = BN_CTX_get(ctx)) == NULL) 834 goto err; 835 if (!BN_GF2m_arr2poly(p, field)) 836 goto err; 837 838 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 839 bn_check_top(r); 840 841 err: 842 BN_CTX_end(ctx); 843 return ret; 844 } 845 846 /* 847 * Compute the bth power of a, reduce modulo p, and store the result in r. r 848 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE 849 * P1363. 850 */ 851 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 852 const int p[], BN_CTX *ctx) 853 { 854 int ret = 0, i, n; 855 BIGNUM *u; 856 857 bn_check_top(a); 858 bn_check_top(b); 859 860 if (BN_is_zero(b)) 861 return BN_one(r); 862 863 if (BN_abs_is_word(b, 1)) 864 return (BN_copy(r, a) != NULL); 865 866 BN_CTX_start(ctx); 867 if ((u = BN_CTX_get(ctx)) == NULL) 868 goto err; 869 870 if (!BN_GF2m_mod_arr(u, a, p)) 871 goto err; 872 873 n = BN_num_bits(b) - 1; 874 for (i = n - 1; i >= 0; i--) { 875 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) 876 goto err; 877 if (BN_is_bit_set(b, i)) { 878 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) 879 goto err; 880 } 881 } 882 if (!BN_copy(r, u)) 883 goto err; 884 bn_check_top(r); 885 ret = 1; 886 err: 887 BN_CTX_end(ctx); 888 return ret; 889 } 890 891 /* 892 * Compute the bth power of a, reduce modulo p, and store the result in r. r 893 * could be a. This function calls down to the BN_GF2m_mod_exp_arr 894 * implementation; this wrapper function is only provided for convenience; 895 * for best performance, use the BN_GF2m_mod_exp_arr function. 896 */ 897 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 898 const BIGNUM *p, BN_CTX *ctx) 899 { 900 int ret = 0; 901 const int max = BN_num_bits(p) + 1; 902 int *arr = NULL; 903 bn_check_top(a); 904 bn_check_top(b); 905 bn_check_top(p); 906 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) 907 goto err; 908 ret = BN_GF2m_poly2arr(p, arr, max); 909 if (!ret || ret > max) { 910 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH); 911 goto err; 912 } 913 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 914 bn_check_top(r); 915 err: 916 OPENSSL_free(arr); 917 return ret; 918 } 919 920 /* 921 * Compute the square root of a, reduce modulo p, and store the result in r. 922 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 923 */ 924 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], 925 BN_CTX *ctx) 926 { 927 int ret = 0; 928 BIGNUM *u; 929 930 bn_check_top(a); 931 932 if (!p[0]) { 933 /* reduction mod 1 => return 0 */ 934 BN_zero(r); 935 return 1; 936 } 937 938 BN_CTX_start(ctx); 939 if ((u = BN_CTX_get(ctx)) == NULL) 940 goto err; 941 942 if (!BN_set_bit(u, p[0] - 1)) 943 goto err; 944 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 945 bn_check_top(r); 946 947 err: 948 BN_CTX_end(ctx); 949 return ret; 950 } 951 952 /* 953 * Compute the square root of a, reduce modulo p, and store the result in r. 954 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr 955 * implementation; this wrapper function is only provided for convenience; 956 * for best performance, use the BN_GF2m_mod_sqrt_arr function. 957 */ 958 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 959 { 960 int ret = 0; 961 const int max = BN_num_bits(p) + 1; 962 int *arr = NULL; 963 bn_check_top(a); 964 bn_check_top(p); 965 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) 966 goto err; 967 ret = BN_GF2m_poly2arr(p, arr, max); 968 if (!ret || ret > max) { 969 BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH); 970 goto err; 971 } 972 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 973 bn_check_top(r); 974 err: 975 OPENSSL_free(arr); 976 return ret; 977 } 978 979 /* 980 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 981 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 982 */ 983 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], 984 BN_CTX *ctx) 985 { 986 int ret = 0, count = 0, j; 987 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 988 989 bn_check_top(a_); 990 991 if (!p[0]) { 992 /* reduction mod 1 => return 0 */ 993 BN_zero(r); 994 return 1; 995 } 996 997 BN_CTX_start(ctx); 998 a = BN_CTX_get(ctx); 999 z = BN_CTX_get(ctx); 1000 w = BN_CTX_get(ctx); 1001 if (w == NULL) 1002 goto err; 1003 1004 if (!BN_GF2m_mod_arr(a, a_, p)) 1005 goto err; 1006 1007 if (BN_is_zero(a)) { 1008 BN_zero(r); 1009 ret = 1; 1010 goto err; 1011 } 1012 1013 if (p[0] & 0x1) { /* m is odd */ 1014 /* compute half-trace of a */ 1015 if (!BN_copy(z, a)) 1016 goto err; 1017 for (j = 1; j <= (p[0] - 1) / 2; j++) { 1018 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1019 goto err; 1020 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1021 goto err; 1022 if (!BN_GF2m_add(z, z, a)) 1023 goto err; 1024 } 1025 1026 } else { /* m is even */ 1027 1028 rho = BN_CTX_get(ctx); 1029 w2 = BN_CTX_get(ctx); 1030 tmp = BN_CTX_get(ctx); 1031 if (tmp == NULL) 1032 goto err; 1033 do { 1034 if (!BN_priv_rand(rho, p[0], BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ANY)) 1035 goto err; 1036 if (!BN_GF2m_mod_arr(rho, rho, p)) 1037 goto err; 1038 BN_zero(z); 1039 if (!BN_copy(w, rho)) 1040 goto err; 1041 for (j = 1; j <= p[0] - 1; j++) { 1042 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1043 goto err; 1044 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) 1045 goto err; 1046 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) 1047 goto err; 1048 if (!BN_GF2m_add(z, z, tmp)) 1049 goto err; 1050 if (!BN_GF2m_add(w, w2, rho)) 1051 goto err; 1052 } 1053 count++; 1054 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1055 if (BN_is_zero(w)) { 1056 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS); 1057 goto err; 1058 } 1059 } 1060 1061 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) 1062 goto err; 1063 if (!BN_GF2m_add(w, z, w)) 1064 goto err; 1065 if (BN_GF2m_cmp(w, a)) { 1066 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1067 goto err; 1068 } 1069 1070 if (!BN_copy(r, z)) 1071 goto err; 1072 bn_check_top(r); 1073 1074 ret = 1; 1075 1076 err: 1077 BN_CTX_end(ctx); 1078 return ret; 1079 } 1080 1081 /* 1082 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1083 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr 1084 * implementation; this wrapper function is only provided for convenience; 1085 * for best performance, use the BN_GF2m_mod_solve_quad_arr function. 1086 */ 1087 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, 1088 BN_CTX *ctx) 1089 { 1090 int ret = 0; 1091 const int max = BN_num_bits(p) + 1; 1092 int *arr = NULL; 1093 bn_check_top(a); 1094 bn_check_top(p); 1095 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) 1096 goto err; 1097 ret = BN_GF2m_poly2arr(p, arr, max); 1098 if (!ret || ret > max) { 1099 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH); 1100 goto err; 1101 } 1102 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1103 bn_check_top(r); 1104 err: 1105 OPENSSL_free(arr); 1106 return ret; 1107 } 1108 1109 /* 1110 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * 1111 * x^i) into an array of integers corresponding to the bits with non-zero 1112 * coefficient. Array is terminated with -1. Up to max elements of the array 1113 * will be filled. Return value is total number of array elements that would 1114 * be filled if array was large enough. 1115 */ 1116 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1117 { 1118 int i, j, k = 0; 1119 BN_ULONG mask; 1120 1121 if (BN_is_zero(a)) 1122 return 0; 1123 1124 for (i = a->top - 1; i >= 0; i--) { 1125 if (!a->d[i]) 1126 /* skip word if a->d[i] == 0 */ 1127 continue; 1128 mask = BN_TBIT; 1129 for (j = BN_BITS2 - 1; j >= 0; j--) { 1130 if (a->d[i] & mask) { 1131 if (k < max) 1132 p[k] = BN_BITS2 * i + j; 1133 k++; 1134 } 1135 mask >>= 1; 1136 } 1137 } 1138 1139 if (k < max) { 1140 p[k] = -1; 1141 k++; 1142 } 1143 1144 return k; 1145 } 1146 1147 /* 1148 * Convert the coefficient array representation of a polynomial to a 1149 * bit-string. The array must be terminated by -1. 1150 */ 1151 int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1152 { 1153 int i; 1154 1155 bn_check_top(a); 1156 BN_zero(a); 1157 for (i = 0; p[i] != -1; i++) { 1158 if (BN_set_bit(a, p[i]) == 0) 1159 return 0; 1160 } 1161 bn_check_top(a); 1162 1163 return 1; 1164 } 1165 1166 #endif 1167