1/* crypto/bn/bn_gf2m.c */ 2/* ==================================================================== 3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. 4 * 5 * The Elliptic Curve Public-Key Crypto Library (ECC Code) included 6 * herein is developed by SUN MICROSYSTEMS, INC., and is contributed 7 * to the OpenSSL project. 8 * 9 * The ECC Code is licensed pursuant to the OpenSSL open source 10 * license provided below. 11 * 12 * In addition, Sun covenants to all licensees who provide a reciprocal 13 * covenant with respect to their own patents if any, not to sue under 14 * current and future patent claims necessarily infringed by the making, 15 * using, practicing, selling, offering for sale and/or otherwise 16 * disposing of the ECC Code as delivered hereunder (or portions thereof), 17 * provided that such covenant shall not apply: 18 * 1) for code that a licensee deletes from the ECC Code; 19 * 2) separates from the ECC Code; or 20 * 3) for infringements caused by: 21 * i) the modification of the ECC Code or 22 * ii) the combination of the ECC Code with other software or 23 * devices where such combination causes the infringement. 24 * 25 * The software is originally written by Sheueling Chang Shantz and 26 * Douglas Stebila of Sun Microsystems Laboratories. 27 * 28 */ 29 30/* 31 * NOTE: This file is licensed pursuant to the OpenSSL license below and may 32 * be modified; but after modifications, the above covenant may no longer 33 * apply! In such cases, the corresponding paragraph ["In addition, Sun 34 * covenants ... causes the infringement."] and this note can be edited out; 35 * but please keep the Sun copyright notice and attribution. 36 */ 37 38/* ==================================================================== 39 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved. 40 * 41 * Redistribution and use in source and binary forms, with or without 42 * modification, are permitted provided that the following conditions 43 * are met: 44 * 45 * 1. Redistributions of source code must retain the above copyright 46 * notice, this list of conditions and the following disclaimer. 47 * 48 * 2. Redistributions in binary form must reproduce the above copyright 49 * notice, this list of conditions and the following disclaimer in 50 * the documentation and/or other materials provided with the 51 * distribution. 52 * 53 * 3. All advertising materials mentioning features or use of this 54 * software must display the following acknowledgment: 55 * "This product includes software developed by the OpenSSL Project 56 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 57 * 58 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 59 * endorse or promote products derived from this software without 60 * prior written permission. For written permission, please contact 61 * openssl-core@openssl.org. 62 * 63 * 5. Products derived from this software may not be called "OpenSSL" 64 * nor may "OpenSSL" appear in their names without prior written 65 * permission of the OpenSSL Project. 66 * 67 * 6. Redistributions of any form whatsoever must retain the following 68 * acknowledgment: 69 * "This product includes software developed by the OpenSSL Project 70 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 71 * 72 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 73 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 74 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 75 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 76 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 77 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 78 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 79 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 80 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 81 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 82 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 83 * OF THE POSSIBILITY OF SUCH DAMAGE. 84 * ==================================================================== 85 * 86 * This product includes cryptographic software written by Eric Young 87 * (eay@cryptsoft.com). This product includes software written by Tim 88 * Hudson (tjh@cryptsoft.com). 89 * 90 */ 91 92#include <assert.h> 93#include <limits.h> 94#include <stdio.h> 95#include "cryptlib.h" 96#include "bn_lcl.h" 97 98#ifndef OPENSSL_NO_EC2M 99 100/* 101 * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should 102 * fail. 103 */ 104# define MAX_ITERATIONS 50 105 106static const BN_ULONG SQR_tb[16] = { 0, 1, 4, 5, 16, 17, 20, 21, 107 64, 65, 68, 69, 80, 81, 84, 85 108}; 109 110/* Platform-specific macros to accelerate squaring. */ 111# if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 112# define SQR1(w) \ 113 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \ 114 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \ 115 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \ 116 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF] 117# define SQR0(w) \ 118 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \ 119 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \ 120 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ 121 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 122# endif 123# ifdef THIRTY_TWO_BIT 124# define SQR1(w) \ 125 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \ 126 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF] 127# define SQR0(w) \ 128 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ 129 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 130# endif 131 132# if !defined(OPENSSL_BN_ASM_GF2m) 133/* 134 * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is 135 * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that 136 * the variables have the right amount of space allocated. 137 */ 138# ifdef THIRTY_TWO_BIT 139static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, 140 const BN_ULONG b) 141{ 142 register BN_ULONG h, l, s; 143 BN_ULONG tab[8], top2b = a >> 30; 144 register BN_ULONG a1, a2, a4; 145 146 a1 = a & (0x3FFFFFFF); 147 a2 = a1 << 1; 148 a4 = a2 << 1; 149 150 tab[0] = 0; 151 tab[1] = a1; 152 tab[2] = a2; 153 tab[3] = a1 ^ a2; 154 tab[4] = a4; 155 tab[5] = a1 ^ a4; 156 tab[6] = a2 ^ a4; 157 tab[7] = a1 ^ a2 ^ a4; 158 159 s = tab[b & 0x7]; 160 l = s; 161 s = tab[b >> 3 & 0x7]; 162 l ^= s << 3; 163 h = s >> 29; 164 s = tab[b >> 6 & 0x7]; 165 l ^= s << 6; 166 h ^= s >> 26; 167 s = tab[b >> 9 & 0x7]; 168 l ^= s << 9; 169 h ^= s >> 23; 170 s = tab[b >> 12 & 0x7]; 171 l ^= s << 12; 172 h ^= s >> 20; 173 s = tab[b >> 15 & 0x7]; 174 l ^= s << 15; 175 h ^= s >> 17; 176 s = tab[b >> 18 & 0x7]; 177 l ^= s << 18; 178 h ^= s >> 14; 179 s = tab[b >> 21 & 0x7]; 180 l ^= s << 21; 181 h ^= s >> 11; 182 s = tab[b >> 24 & 0x7]; 183 l ^= s << 24; 184 h ^= s >> 8; 185 s = tab[b >> 27 & 0x7]; 186 l ^= s << 27; 187 h ^= s >> 5; 188 s = tab[b >> 30]; 189 l ^= s << 30; 190 h ^= s >> 2; 191 192 /* compensate for the top two bits of a */ 193 194 if (top2b & 01) { 195 l ^= b << 30; 196 h ^= b >> 2; 197 } 198 if (top2b & 02) { 199 l ^= b << 31; 200 h ^= b >> 1; 201 } 202 203 *r1 = h; 204 *r0 = l; 205} 206# endif 207# if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 208static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, 209 const BN_ULONG b) 210{ 211 register BN_ULONG h, l, s; 212 BN_ULONG tab[16], top3b = a >> 61; 213 register BN_ULONG a1, a2, a4, a8; 214 215 a1 = a & (0x1FFFFFFFFFFFFFFFULL); 216 a2 = a1 << 1; 217 a4 = a2 << 1; 218 a8 = a4 << 1; 219 220 tab[0] = 0; 221 tab[1] = a1; 222 tab[2] = a2; 223 tab[3] = a1 ^ a2; 224 tab[4] = a4; 225 tab[5] = a1 ^ a4; 226 tab[6] = a2 ^ a4; 227 tab[7] = a1 ^ a2 ^ a4; 228 tab[8] = a8; 229 tab[9] = a1 ^ a8; 230 tab[10] = a2 ^ a8; 231 tab[11] = a1 ^ a2 ^ a8; 232 tab[12] = a4 ^ a8; 233 tab[13] = a1 ^ a4 ^ a8; 234 tab[14] = a2 ^ a4 ^ a8; 235 tab[15] = a1 ^ a2 ^ a4 ^ a8; 236 237 s = tab[b & 0xF]; 238 l = s; 239 s = tab[b >> 4 & 0xF]; 240 l ^= s << 4; 241 h = s >> 60; 242 s = tab[b >> 8 & 0xF]; 243 l ^= s << 8; 244 h ^= s >> 56; 245 s = tab[b >> 12 & 0xF]; 246 l ^= s << 12; 247 h ^= s >> 52; 248 s = tab[b >> 16 & 0xF]; 249 l ^= s << 16; 250 h ^= s >> 48; 251 s = tab[b >> 20 & 0xF]; 252 l ^= s << 20; 253 h ^= s >> 44; 254 s = tab[b >> 24 & 0xF]; 255 l ^= s << 24; 256 h ^= s >> 40; 257 s = tab[b >> 28 & 0xF]; 258 l ^= s << 28; 259 h ^= s >> 36; 260 s = tab[b >> 32 & 0xF]; 261 l ^= s << 32; 262 h ^= s >> 32; 263 s = tab[b >> 36 & 0xF]; 264 l ^= s << 36; 265 h ^= s >> 28; 266 s = tab[b >> 40 & 0xF]; 267 l ^= s << 40; 268 h ^= s >> 24; 269 s = tab[b >> 44 & 0xF]; 270 l ^= s << 44; 271 h ^= s >> 20; 272 s = tab[b >> 48 & 0xF]; 273 l ^= s << 48; 274 h ^= s >> 16; 275 s = tab[b >> 52 & 0xF]; 276 l ^= s << 52; 277 h ^= s >> 12; 278 s = tab[b >> 56 & 0xF]; 279 l ^= s << 56; 280 h ^= s >> 8; 281 s = tab[b >> 60]; 282 l ^= s << 60; 283 h ^= s >> 4; 284 285 /* compensate for the top three bits of a */ 286 287 if (top3b & 01) { 288 l ^= b << 61; 289 h ^= b >> 3; 290 } 291 if (top3b & 02) { 292 l ^= b << 62; 293 h ^= b >> 2; 294 } 295 if (top3b & 04) { 296 l ^= b << 63; 297 h ^= b >> 1; 298 } 299 300 *r1 = h; 301 *r0 = l; 302} 303# endif 304 305/* 306 * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, 307 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST 308 * ensure that the variables have the right amount of space allocated. 309 */ 310static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, 311 const BN_ULONG b1, const BN_ULONG b0) 312{ 313 BN_ULONG m1, m0; 314 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ 315 bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1); 316 bn_GF2m_mul_1x1(r + 1, r, a0, b0); 317 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); 318 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ 319 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ 320 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ 321} 322# else 323void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1, 324 BN_ULONG b0); 325# endif 326 327/* 328 * Add polynomials a and b and store result in r; r could be a or b, a and b 329 * could be equal; r is the bitwise XOR of a and b. 330 */ 331int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) 332{ 333 int i; 334 const BIGNUM *at, *bt; 335 336 bn_check_top(a); 337 bn_check_top(b); 338 339 if (a->top < b->top) { 340 at = b; 341 bt = a; 342 } else { 343 at = a; 344 bt = b; 345 } 346 347 if (bn_wexpand(r, at->top) == NULL) 348 return 0; 349 350 for (i = 0; i < bt->top; i++) { 351 r->d[i] = at->d[i] ^ bt->d[i]; 352 } 353 for (; i < at->top; i++) { 354 r->d[i] = at->d[i]; 355 } 356 357 r->top = at->top; 358 bn_correct_top(r); 359 360 return 1; 361} 362 363/*- 364 * Some functions allow for representation of the irreducible polynomials 365 * as an int[], say p. The irreducible f(t) is then of the form: 366 * t^p[0] + t^p[1] + ... + t^p[k] 367 * where m = p[0] > p[1] > ... > p[k] = 0. 368 */ 369 370/* Performs modular reduction of a and store result in r. r could be a. */ 371int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]) 372{ 373 int j, k; 374 int n, dN, d0, d1; 375 BN_ULONG zz, *z; 376 377 bn_check_top(a); 378 379 if (!p[0]) { 380 /* reduction mod 1 => return 0 */ 381 BN_zero(r); 382 return 1; 383 } 384 385 /* 386 * Since the algorithm does reduction in the r value, if a != r, copy the 387 * contents of a into r so we can do reduction in r. 388 */ 389 if (a != r) { 390 if (!bn_wexpand(r, a->top)) 391 return 0; 392 for (j = 0; j < a->top; j++) { 393 r->d[j] = a->d[j]; 394 } 395 r->top = a->top; 396 } 397 z = r->d; 398 399 /* start reduction */ 400 dN = p[0] / BN_BITS2; 401 for (j = r->top - 1; j > dN;) { 402 zz = z[j]; 403 if (z[j] == 0) { 404 j--; 405 continue; 406 } 407 z[j] = 0; 408 409 for (k = 1; p[k] != 0; k++) { 410 /* reducing component t^p[k] */ 411 n = p[0] - p[k]; 412 d0 = n % BN_BITS2; 413 d1 = BN_BITS2 - d0; 414 n /= BN_BITS2; 415 z[j - n] ^= (zz >> d0); 416 if (d0) 417 z[j - n - 1] ^= (zz << d1); 418 } 419 420 /* reducing component t^0 */ 421 n = dN; 422 d0 = p[0] % BN_BITS2; 423 d1 = BN_BITS2 - d0; 424 z[j - n] ^= (zz >> d0); 425 if (d0) 426 z[j - n - 1] ^= (zz << d1); 427 } 428 429 /* final round of reduction */ 430 while (j == dN) { 431 432 d0 = p[0] % BN_BITS2; 433 zz = z[dN] >> d0; 434 if (zz == 0) 435 break; 436 d1 = BN_BITS2 - d0; 437 438 /* clear up the top d1 bits */ 439 if (d0) 440 z[dN] = (z[dN] << d1) >> d1; 441 else 442 z[dN] = 0; 443 z[0] ^= zz; /* reduction t^0 component */ 444 445 for (k = 1; p[k] != 0; k++) { 446 BN_ULONG tmp_ulong; 447 448 /* reducing component t^p[k] */ 449 n = p[k] / BN_BITS2; 450 d0 = p[k] % BN_BITS2; 451 d1 = BN_BITS2 - d0; 452 z[n] ^= (zz << d0); 453 if (d0 && (tmp_ulong = zz >> d1)) 454 z[n + 1] ^= tmp_ulong; 455 } 456 457 } 458 459 bn_correct_top(r); 460 return 1; 461} 462 463/* 464 * Performs modular reduction of a by p and store result in r. r could be a. 465 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 466 * function is only provided for convenience; for best performance, use the 467 * BN_GF2m_mod_arr function. 468 */ 469int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 470{ 471 int ret = 0; 472 int arr[6]; 473 bn_check_top(a); 474 bn_check_top(p); 475 ret = BN_GF2m_poly2arr(p, arr, sizeof(arr) / sizeof(arr[0])); 476 if (!ret || ret > (int)(sizeof(arr) / sizeof(arr[0]))) { 477 BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH); 478 return 0; 479 } 480 ret = BN_GF2m_mod_arr(r, a, arr); 481 bn_check_top(r); 482 return ret; 483} 484 485/* 486 * Compute the product of two polynomials a and b, reduce modulo p, and store 487 * the result in r. r could be a or b; a could be b. 488 */ 489int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 490 const int p[], BN_CTX *ctx) 491{ 492 int zlen, i, j, k, ret = 0; 493 BIGNUM *s; 494 BN_ULONG x1, x0, y1, y0, zz[4]; 495 496 bn_check_top(a); 497 bn_check_top(b); 498 499 if (a == b) { 500 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 501 } 502 503 BN_CTX_start(ctx); 504 if ((s = BN_CTX_get(ctx)) == NULL) 505 goto err; 506 507 zlen = a->top + b->top + 4; 508 if (!bn_wexpand(s, zlen)) 509 goto err; 510 s->top = zlen; 511 512 for (i = 0; i < zlen; i++) 513 s->d[i] = 0; 514 515 for (j = 0; j < b->top; j += 2) { 516 y0 = b->d[j]; 517 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1]; 518 for (i = 0; i < a->top; i += 2) { 519 x0 = a->d[i]; 520 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1]; 521 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 522 for (k = 0; k < 4; k++) 523 s->d[i + j + k] ^= zz[k]; 524 } 525 } 526 527 bn_correct_top(s); 528 if (BN_GF2m_mod_arr(r, s, p)) 529 ret = 1; 530 bn_check_top(r); 531 532 err: 533 BN_CTX_end(ctx); 534 return ret; 535} 536 537/* 538 * Compute the product of two polynomials a and b, reduce modulo p, and store 539 * the result in r. r could be a or b; a could equal b. This function calls 540 * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is 541 * only provided for convenience; for best performance, use the 542 * BN_GF2m_mod_mul_arr function. 543 */ 544int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 545 const BIGNUM *p, BN_CTX *ctx) 546{ 547 int ret = 0; 548 const int max = BN_num_bits(p) + 1; 549 int *arr = NULL; 550 bn_check_top(a); 551 bn_check_top(b); 552 bn_check_top(p); 553 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 554 goto err; 555 ret = BN_GF2m_poly2arr(p, arr, max); 556 if (!ret || ret > max) { 557 BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH); 558 goto err; 559 } 560 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 561 bn_check_top(r); 562 err: 563 if (arr) 564 OPENSSL_free(arr); 565 return ret; 566} 567 568/* Square a, reduce the result mod p, and store it in a. r could be a. */ 569int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], 570 BN_CTX *ctx) 571{ 572 int i, ret = 0; 573 BIGNUM *s; 574 575 bn_check_top(a); 576 BN_CTX_start(ctx); 577 if ((s = BN_CTX_get(ctx)) == NULL) 578 return 0; 579 if (!bn_wexpand(s, 2 * a->top)) 580 goto err; 581 582 for (i = a->top - 1; i >= 0; i--) { 583 s->d[2 * i + 1] = SQR1(a->d[i]); 584 s->d[2 * i] = SQR0(a->d[i]); 585 } 586 587 s->top = 2 * a->top; 588 bn_correct_top(s); 589 if (!BN_GF2m_mod_arr(r, s, p)) 590 goto err; 591 bn_check_top(r); 592 ret = 1; 593 err: 594 BN_CTX_end(ctx); 595 return ret; 596} 597 598/* 599 * Square a, reduce the result mod p, and store it in a. r could be a. This 600 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this 601 * wrapper function is only provided for convenience; for best performance, 602 * use the BN_GF2m_mod_sqr_arr function. 603 */ 604int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 605{ 606 int ret = 0; 607 const int max = BN_num_bits(p) + 1; 608 int *arr = NULL; 609 610 bn_check_top(a); 611 bn_check_top(p); 612 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 613 goto err; 614 ret = BN_GF2m_poly2arr(p, arr, max); 615 if (!ret || ret > max) { 616 BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH); 617 goto err; 618 } 619 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 620 bn_check_top(r); 621 err: 622 if (arr) 623 OPENSSL_free(arr); 624 return ret; 625} 626 627/* 628 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses 629 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D., 630 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic 631 * Curve Cryptography Over Binary Fields". 632 */ 633int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 634{ 635 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; 636 int ret = 0; 637 638 bn_check_top(a); 639 bn_check_top(p); 640 641 BN_CTX_start(ctx); 642 643 if ((b = BN_CTX_get(ctx)) == NULL) 644 goto err; 645 if ((c = BN_CTX_get(ctx)) == NULL) 646 goto err; 647 if ((u = BN_CTX_get(ctx)) == NULL) 648 goto err; 649 if ((v = BN_CTX_get(ctx)) == NULL) 650 goto err; 651 652 if (!BN_GF2m_mod(u, a, p)) 653 goto err; 654 if (BN_is_zero(u)) 655 goto err; 656 657 if (!BN_copy(v, p)) 658 goto err; 659# if 0 660 if (!BN_one(b)) 661 goto err; 662 663 while (1) { 664 while (!BN_is_odd(u)) { 665 if (BN_is_zero(u)) 666 goto err; 667 if (!BN_rshift1(u, u)) 668 goto err; 669 if (BN_is_odd(b)) { 670 if (!BN_GF2m_add(b, b, p)) 671 goto err; 672 } 673 if (!BN_rshift1(b, b)) 674 goto err; 675 } 676 677 if (BN_abs_is_word(u, 1)) 678 break; 679 680 if (BN_num_bits(u) < BN_num_bits(v)) { 681 tmp = u; 682 u = v; 683 v = tmp; 684 tmp = b; 685 b = c; 686 c = tmp; 687 } 688 689 if (!BN_GF2m_add(u, u, v)) 690 goto err; 691 if (!BN_GF2m_add(b, b, c)) 692 goto err; 693 } 694# else 695 { 696 int i; 697 int ubits = BN_num_bits(u); 698 int vbits = BN_num_bits(v); /* v is copy of p */ 699 int top = p->top; 700 BN_ULONG *udp, *bdp, *vdp, *cdp; 701 702 bn_wexpand(u, top); 703 udp = u->d; 704 for (i = u->top; i < top; i++) 705 udp[i] = 0; 706 u->top = top; 707 bn_wexpand(b, top); 708 bdp = b->d; 709 bdp[0] = 1; 710 for (i = 1; i < top; i++) 711 bdp[i] = 0; 712 b->top = top; 713 bn_wexpand(c, top); 714 cdp = c->d; 715 for (i = 0; i < top; i++) 716 cdp[i] = 0; 717 c->top = top; 718 vdp = v->d; /* It pays off to "cache" *->d pointers, 719 * because it allows optimizer to be more 720 * aggressive. But we don't have to "cache" 721 * p->d, because *p is declared 'const'... */ 722 while (1) { 723 while (ubits && !(udp[0] & 1)) { 724 BN_ULONG u0, u1, b0, b1, mask; 725 726 u0 = udp[0]; 727 b0 = bdp[0]; 728 mask = (BN_ULONG)0 - (b0 & 1); 729 b0 ^= p->d[0] & mask; 730 for (i = 0; i < top - 1; i++) { 731 u1 = udp[i + 1]; 732 udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2; 733 u0 = u1; 734 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask); 735 bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2; 736 b0 = b1; 737 } 738 udp[i] = u0 >> 1; 739 bdp[i] = b0 >> 1; 740 ubits--; 741 } 742 743 if (ubits <= BN_BITS2) { 744 if (udp[0] == 0) /* poly was reducible */ 745 goto err; 746 if (udp[0] == 1) 747 break; 748 } 749 750 if (ubits < vbits) { 751 i = ubits; 752 ubits = vbits; 753 vbits = i; 754 tmp = u; 755 u = v; 756 v = tmp; 757 tmp = b; 758 b = c; 759 c = tmp; 760 udp = vdp; 761 vdp = v->d; 762 bdp = cdp; 763 cdp = c->d; 764 } 765 for (i = 0; i < top; i++) { 766 udp[i] ^= vdp[i]; 767 bdp[i] ^= cdp[i]; 768 } 769 if (ubits == vbits) { 770 BN_ULONG ul; 771 int utop = (ubits - 1) / BN_BITS2; 772 773 while ((ul = udp[utop]) == 0 && utop) 774 utop--; 775 ubits = utop * BN_BITS2 + BN_num_bits_word(ul); 776 } 777 } 778 bn_correct_top(b); 779 } 780# endif 781 782 if (!BN_copy(r, b)) 783 goto err; 784 bn_check_top(r); 785 ret = 1; 786 787 err: 788# ifdef BN_DEBUG /* BN_CTX_end would complain about the 789 * expanded form */ 790 bn_correct_top(c); 791 bn_correct_top(u); 792 bn_correct_top(v); 793# endif 794 BN_CTX_end(ctx); 795 return ret; 796} 797 798/* 799 * Invert xx, reduce modulo p, and store the result in r. r could be xx. 800 * This function calls down to the BN_GF2m_mod_inv implementation; this 801 * wrapper function is only provided for convenience; for best performance, 802 * use the BN_GF2m_mod_inv function. 803 */ 804int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], 805 BN_CTX *ctx) 806{ 807 BIGNUM *field; 808 int ret = 0; 809 810 bn_check_top(xx); 811 BN_CTX_start(ctx); 812 if ((field = BN_CTX_get(ctx)) == NULL) 813 goto err; 814 if (!BN_GF2m_arr2poly(p, field)) 815 goto err; 816 817 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 818 bn_check_top(r); 819 820 err: 821 BN_CTX_end(ctx); 822 return ret; 823} 824 825# ifndef OPENSSL_SUN_GF2M_DIV 826/* 827 * Divide y by x, reduce modulo p, and store the result in r. r could be x 828 * or y, x could equal y. 829 */ 830int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 831 const BIGNUM *p, BN_CTX *ctx) 832{ 833 BIGNUM *xinv = NULL; 834 int ret = 0; 835 836 bn_check_top(y); 837 bn_check_top(x); 838 bn_check_top(p); 839 840 BN_CTX_start(ctx); 841 xinv = BN_CTX_get(ctx); 842 if (xinv == NULL) 843 goto err; 844 845 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) 846 goto err; 847 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) 848 goto err; 849 bn_check_top(r); 850 ret = 1; 851 852 err: 853 BN_CTX_end(ctx); 854 return ret; 855} 856# else 857/* 858 * Divide y by x, reduce modulo p, and store the result in r. r could be x 859 * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from 860 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the 861 * Great Divide". 862 */ 863int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 864 const BIGNUM *p, BN_CTX *ctx) 865{ 866 BIGNUM *a, *b, *u, *v; 867 int ret = 0; 868 869 bn_check_top(y); 870 bn_check_top(x); 871 bn_check_top(p); 872 873 BN_CTX_start(ctx); 874 875 a = BN_CTX_get(ctx); 876 b = BN_CTX_get(ctx); 877 u = BN_CTX_get(ctx); 878 v = BN_CTX_get(ctx); 879 if (v == NULL) 880 goto err; 881 882 /* reduce x and y mod p */ 883 if (!BN_GF2m_mod(u, y, p)) 884 goto err; 885 if (!BN_GF2m_mod(a, x, p)) 886 goto err; 887 if (!BN_copy(b, p)) 888 goto err; 889 890 while (!BN_is_odd(a)) { 891 if (!BN_rshift1(a, a)) 892 goto err; 893 if (BN_is_odd(u)) 894 if (!BN_GF2m_add(u, u, p)) 895 goto err; 896 if (!BN_rshift1(u, u)) 897 goto err; 898 } 899 900 do { 901 if (BN_GF2m_cmp(b, a) > 0) { 902 if (!BN_GF2m_add(b, b, a)) 903 goto err; 904 if (!BN_GF2m_add(v, v, u)) 905 goto err; 906 do { 907 if (!BN_rshift1(b, b)) 908 goto err; 909 if (BN_is_odd(v)) 910 if (!BN_GF2m_add(v, v, p)) 911 goto err; 912 if (!BN_rshift1(v, v)) 913 goto err; 914 } while (!BN_is_odd(b)); 915 } else if (BN_abs_is_word(a, 1)) 916 break; 917 else { 918 if (!BN_GF2m_add(a, a, b)) 919 goto err; 920 if (!BN_GF2m_add(u, u, v)) 921 goto err; 922 do { 923 if (!BN_rshift1(a, a)) 924 goto err; 925 if (BN_is_odd(u)) 926 if (!BN_GF2m_add(u, u, p)) 927 goto err; 928 if (!BN_rshift1(u, u)) 929 goto err; 930 } while (!BN_is_odd(a)); 931 } 932 } while (1); 933 934 if (!BN_copy(r, u)) 935 goto err; 936 bn_check_top(r); 937 ret = 1; 938 939 err: 940 BN_CTX_end(ctx); 941 return ret; 942} 943# endif 944 945/* 946 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 947 * * or yy, xx could equal yy. This function calls down to the 948 * BN_GF2m_mod_div implementation; this wrapper function is only provided for 949 * convenience; for best performance, use the BN_GF2m_mod_div function. 950 */ 951int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, 952 const int p[], BN_CTX *ctx) 953{ 954 BIGNUM *field; 955 int ret = 0; 956 957 bn_check_top(yy); 958 bn_check_top(xx); 959 960 BN_CTX_start(ctx); 961 if ((field = BN_CTX_get(ctx)) == NULL) 962 goto err; 963 if (!BN_GF2m_arr2poly(p, field)) 964 goto err; 965 966 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 967 bn_check_top(r); 968 969 err: 970 BN_CTX_end(ctx); 971 return ret; 972} 973 974/* 975 * Compute the bth power of a, reduce modulo p, and store the result in r. r 976 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE 977 * P1363. 978 */ 979int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 980 const int p[], BN_CTX *ctx) 981{ 982 int ret = 0, i, n; 983 BIGNUM *u; 984 985 bn_check_top(a); 986 bn_check_top(b); 987 988 if (BN_is_zero(b)) 989 return (BN_one(r)); 990 991 if (BN_abs_is_word(b, 1)) 992 return (BN_copy(r, a) != NULL); 993 994 BN_CTX_start(ctx); 995 if ((u = BN_CTX_get(ctx)) == NULL) 996 goto err; 997 998 if (!BN_GF2m_mod_arr(u, a, p)) 999 goto err; 1000 1001 n = BN_num_bits(b) - 1; 1002 for (i = n - 1; i >= 0; i--) { 1003 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) 1004 goto err; 1005 if (BN_is_bit_set(b, i)) { 1006 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) 1007 goto err; 1008 } 1009 } 1010 if (!BN_copy(r, u)) 1011 goto err; 1012 bn_check_top(r); 1013 ret = 1; 1014 err: 1015 BN_CTX_end(ctx); 1016 return ret; 1017} 1018 1019/* 1020 * Compute the bth power of a, reduce modulo p, and store the result in r. r 1021 * could be a. This function calls down to the BN_GF2m_mod_exp_arr 1022 * implementation; this wrapper function is only provided for convenience; 1023 * for best performance, use the BN_GF2m_mod_exp_arr function. 1024 */ 1025int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 1026 const BIGNUM *p, BN_CTX *ctx) 1027{ 1028 int ret = 0; 1029 const int max = BN_num_bits(p) + 1; 1030 int *arr = NULL; 1031 bn_check_top(a); 1032 bn_check_top(b); 1033 bn_check_top(p); 1034 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1035 goto err; 1036 ret = BN_GF2m_poly2arr(p, arr, max); 1037 if (!ret || ret > max) { 1038 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH); 1039 goto err; 1040 } 1041 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 1042 bn_check_top(r); 1043 err: 1044 if (arr) 1045 OPENSSL_free(arr); 1046 return ret; 1047} 1048 1049/* 1050 * Compute the square root of a, reduce modulo p, and store the result in r. 1051 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 1052 */ 1053int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], 1054 BN_CTX *ctx) 1055{ 1056 int ret = 0; 1057 BIGNUM *u; 1058 1059 bn_check_top(a); 1060 1061 if (!p[0]) { 1062 /* reduction mod 1 => return 0 */ 1063 BN_zero(r); 1064 return 1; 1065 } 1066 1067 BN_CTX_start(ctx); 1068 if ((u = BN_CTX_get(ctx)) == NULL) 1069 goto err; 1070 1071 if (!BN_set_bit(u, p[0] - 1)) 1072 goto err; 1073 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 1074 bn_check_top(r); 1075 1076 err: 1077 BN_CTX_end(ctx); 1078 return ret; 1079} 1080 1081/* 1082 * Compute the square root of a, reduce modulo p, and store the result in r. 1083 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr 1084 * implementation; this wrapper function is only provided for convenience; 1085 * for best performance, use the BN_GF2m_mod_sqrt_arr function. 1086 */ 1087int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1088{ 1089 int ret = 0; 1090 const int max = BN_num_bits(p) + 1; 1091 int *arr = NULL; 1092 bn_check_top(a); 1093 bn_check_top(p); 1094 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1095 goto err; 1096 ret = BN_GF2m_poly2arr(p, arr, max); 1097 if (!ret || ret > max) { 1098 BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH); 1099 goto err; 1100 } 1101 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 1102 bn_check_top(r); 1103 err: 1104 if (arr) 1105 OPENSSL_free(arr); 1106 return ret; 1107} 1108 1109/* 1110 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1111 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 1112 */ 1113int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], 1114 BN_CTX *ctx) 1115{ 1116 int ret = 0, count = 0, j; 1117 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 1118 1119 bn_check_top(a_); 1120 1121 if (!p[0]) { 1122 /* reduction mod 1 => return 0 */ 1123 BN_zero(r); 1124 return 1; 1125 } 1126 1127 BN_CTX_start(ctx); 1128 a = BN_CTX_get(ctx); 1129 z = BN_CTX_get(ctx); 1130 w = BN_CTX_get(ctx); 1131 if (w == NULL) 1132 goto err; 1133 1134 if (!BN_GF2m_mod_arr(a, a_, p)) 1135 goto err; 1136 1137 if (BN_is_zero(a)) { 1138 BN_zero(r); 1139 ret = 1; 1140 goto err; 1141 } 1142 1143 if (p[0] & 0x1) { /* m is odd */ 1144 /* compute half-trace of a */ 1145 if (!BN_copy(z, a)) 1146 goto err; 1147 for (j = 1; j <= (p[0] - 1) / 2; j++) { 1148 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1149 goto err; 1150 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1151 goto err; 1152 if (!BN_GF2m_add(z, z, a)) 1153 goto err; 1154 } 1155 1156 } else { /* m is even */ 1157 1158 rho = BN_CTX_get(ctx); 1159 w2 = BN_CTX_get(ctx); 1160 tmp = BN_CTX_get(ctx); 1161 if (tmp == NULL) 1162 goto err; 1163 do { 1164 if (!BN_rand(rho, p[0], 0, 0)) 1165 goto err; 1166 if (!BN_GF2m_mod_arr(rho, rho, p)) 1167 goto err; 1168 BN_zero(z); 1169 if (!BN_copy(w, rho)) 1170 goto err; 1171 for (j = 1; j <= p[0] - 1; j++) { 1172 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1173 goto err; 1174 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) 1175 goto err; 1176 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) 1177 goto err; 1178 if (!BN_GF2m_add(z, z, tmp)) 1179 goto err; 1180 if (!BN_GF2m_add(w, w2, rho)) 1181 goto err; 1182 } 1183 count++; 1184 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1185 if (BN_is_zero(w)) { 1186 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS); 1187 goto err; 1188 } 1189 } 1190 1191 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) 1192 goto err; 1193 if (!BN_GF2m_add(w, z, w)) 1194 goto err; 1195 if (BN_GF2m_cmp(w, a)) { 1196 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1197 goto err; 1198 } 1199 1200 if (!BN_copy(r, z)) 1201 goto err; 1202 bn_check_top(r); 1203 1204 ret = 1; 1205 1206 err: 1207 BN_CTX_end(ctx); 1208 return ret; 1209} 1210 1211/* 1212 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1213 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr 1214 * implementation; this wrapper function is only provided for convenience; 1215 * for best performance, use the BN_GF2m_mod_solve_quad_arr function. 1216 */ 1217int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, 1218 BN_CTX *ctx) 1219{ 1220 int ret = 0; 1221 const int max = BN_num_bits(p) + 1; 1222 int *arr = NULL; 1223 bn_check_top(a); 1224 bn_check_top(p); 1225 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1226 goto err; 1227 ret = BN_GF2m_poly2arr(p, arr, max); 1228 if (!ret || ret > max) { 1229 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH); 1230 goto err; 1231 } 1232 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1233 bn_check_top(r); 1234 err: 1235 if (arr) 1236 OPENSSL_free(arr); 1237 return ret; 1238} 1239 1240/* 1241 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * 1242 * x^i) into an array of integers corresponding to the bits with non-zero 1243 * coefficient. Array is terminated with -1. Up to max elements of the array 1244 * will be filled. Return value is total number of array elements that would 1245 * be filled if array was large enough. 1246 */ 1247int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1248{ 1249 int i, j, k = 0; 1250 BN_ULONG mask; 1251 1252 if (BN_is_zero(a)) 1253 return 0; 1254 1255 for (i = a->top - 1; i >= 0; i--) { 1256 if (!a->d[i]) 1257 /* skip word if a->d[i] == 0 */ 1258 continue; 1259 mask = BN_TBIT; 1260 for (j = BN_BITS2 - 1; j >= 0; j--) { 1261 if (a->d[i] & mask) { 1262 if (k < max) 1263 p[k] = BN_BITS2 * i + j; 1264 k++; 1265 } 1266 mask >>= 1; 1267 } 1268 } 1269 1270 if (k < max) { 1271 p[k] = -1; 1272 k++; 1273 } 1274 1275 return k; 1276} 1277 1278/* 1279 * Convert the coefficient array representation of a polynomial to a 1280 * bit-string. The array must be terminated by -1. 1281 */ 1282int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1283{ 1284 int i; 1285 1286 bn_check_top(a); 1287 BN_zero(a); 1288 for (i = 0; p[i] != -1; i++) { 1289 if (BN_set_bit(a, p[i]) == 0) 1290 return 0; 1291 } 1292 bn_check_top(a); 1293 1294 return 1; 1295} 1296 1297#endif 1298