bn_gf2m.c revision 280304
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 tmp_ulong = zz >> d1; 454 if (d0 && tmp_ulong) 455 z[n + 1] ^= tmp_ulong; 456 } 457 458 } 459 460 bn_correct_top(r); 461 return 1; 462} 463 464/* 465 * Performs modular reduction of a by p and store result in r. r could be a. 466 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 467 * function is only provided for convenience; for best performance, use the 468 * BN_GF2m_mod_arr function. 469 */ 470int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 471{ 472 int ret = 0; 473 int arr[6]; 474 bn_check_top(a); 475 bn_check_top(p); 476 ret = BN_GF2m_poly2arr(p, arr, sizeof(arr) / sizeof(arr[0])); 477 if (!ret || ret > (int)(sizeof(arr) / sizeof(arr[0]))) { 478 BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH); 479 return 0; 480 } 481 ret = BN_GF2m_mod_arr(r, a, arr); 482 bn_check_top(r); 483 return ret; 484} 485 486/* 487 * Compute the product of two polynomials a and b, reduce modulo p, and store 488 * the result in r. r could be a or b; a could be b. 489 */ 490int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 491 const int p[], BN_CTX *ctx) 492{ 493 int zlen, i, j, k, ret = 0; 494 BIGNUM *s; 495 BN_ULONG x1, x0, y1, y0, zz[4]; 496 497 bn_check_top(a); 498 bn_check_top(b); 499 500 if (a == b) { 501 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 502 } 503 504 BN_CTX_start(ctx); 505 if ((s = BN_CTX_get(ctx)) == NULL) 506 goto err; 507 508 zlen = a->top + b->top + 4; 509 if (!bn_wexpand(s, zlen)) 510 goto err; 511 s->top = zlen; 512 513 for (i = 0; i < zlen; i++) 514 s->d[i] = 0; 515 516 for (j = 0; j < b->top; j += 2) { 517 y0 = b->d[j]; 518 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1]; 519 for (i = 0; i < a->top; i += 2) { 520 x0 = a->d[i]; 521 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1]; 522 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 523 for (k = 0; k < 4; k++) 524 s->d[i + j + k] ^= zz[k]; 525 } 526 } 527 528 bn_correct_top(s); 529 if (BN_GF2m_mod_arr(r, s, p)) 530 ret = 1; 531 bn_check_top(r); 532 533 err: 534 BN_CTX_end(ctx); 535 return ret; 536} 537 538/* 539 * Compute the product of two polynomials a and b, reduce modulo p, and store 540 * the result in r. r could be a or b; a could equal b. This function calls 541 * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is 542 * only provided for convenience; for best performance, use the 543 * BN_GF2m_mod_mul_arr function. 544 */ 545int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 546 const BIGNUM *p, BN_CTX *ctx) 547{ 548 int ret = 0; 549 const int max = BN_num_bits(p) + 1; 550 int *arr = NULL; 551 bn_check_top(a); 552 bn_check_top(b); 553 bn_check_top(p); 554 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 555 goto err; 556 ret = BN_GF2m_poly2arr(p, arr, max); 557 if (!ret || ret > max) { 558 BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH); 559 goto err; 560 } 561 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 562 bn_check_top(r); 563 err: 564 if (arr) 565 OPENSSL_free(arr); 566 return ret; 567} 568 569/* Square a, reduce the result mod p, and store it in a. r could be a. */ 570int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], 571 BN_CTX *ctx) 572{ 573 int i, ret = 0; 574 BIGNUM *s; 575 576 bn_check_top(a); 577 BN_CTX_start(ctx); 578 if ((s = BN_CTX_get(ctx)) == NULL) 579 return 0; 580 if (!bn_wexpand(s, 2 * a->top)) 581 goto err; 582 583 for (i = a->top - 1; i >= 0; i--) { 584 s->d[2 * i + 1] = SQR1(a->d[i]); 585 s->d[2 * i] = SQR0(a->d[i]); 586 } 587 588 s->top = 2 * a->top; 589 bn_correct_top(s); 590 if (!BN_GF2m_mod_arr(r, s, p)) 591 goto err; 592 bn_check_top(r); 593 ret = 1; 594 err: 595 BN_CTX_end(ctx); 596 return ret; 597} 598 599/* 600 * Square a, reduce the result mod p, and store it in a. r could be a. This 601 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this 602 * wrapper function is only provided for convenience; for best performance, 603 * use the BN_GF2m_mod_sqr_arr function. 604 */ 605int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 606{ 607 int ret = 0; 608 const int max = BN_num_bits(p) + 1; 609 int *arr = NULL; 610 611 bn_check_top(a); 612 bn_check_top(p); 613 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 614 goto err; 615 ret = BN_GF2m_poly2arr(p, arr, max); 616 if (!ret || ret > max) { 617 BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH); 618 goto err; 619 } 620 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 621 bn_check_top(r); 622 err: 623 if (arr) 624 OPENSSL_free(arr); 625 return ret; 626} 627 628/* 629 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses 630 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D., 631 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic 632 * Curve Cryptography Over Binary Fields". 633 */ 634int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 635{ 636 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; 637 int ret = 0; 638 639 bn_check_top(a); 640 bn_check_top(p); 641 642 BN_CTX_start(ctx); 643 644 if ((b = BN_CTX_get(ctx)) == NULL) 645 goto err; 646 if ((c = BN_CTX_get(ctx)) == NULL) 647 goto err; 648 if ((u = BN_CTX_get(ctx)) == NULL) 649 goto err; 650 if ((v = BN_CTX_get(ctx)) == NULL) 651 goto err; 652 653 if (!BN_GF2m_mod(u, a, p)) 654 goto err; 655 if (BN_is_zero(u)) 656 goto err; 657 658 if (!BN_copy(v, p)) 659 goto err; 660# if 0 661 if (!BN_one(b)) 662 goto err; 663 664 while (1) { 665 while (!BN_is_odd(u)) { 666 if (BN_is_zero(u)) 667 goto err; 668 if (!BN_rshift1(u, u)) 669 goto err; 670 if (BN_is_odd(b)) { 671 if (!BN_GF2m_add(b, b, p)) 672 goto err; 673 } 674 if (!BN_rshift1(b, b)) 675 goto err; 676 } 677 678 if (BN_abs_is_word(u, 1)) 679 break; 680 681 if (BN_num_bits(u) < BN_num_bits(v)) { 682 tmp = u; 683 u = v; 684 v = tmp; 685 tmp = b; 686 b = c; 687 c = tmp; 688 } 689 690 if (!BN_GF2m_add(u, u, v)) 691 goto err; 692 if (!BN_GF2m_add(b, b, c)) 693 goto err; 694 } 695# else 696 { 697 int i, ubits = BN_num_bits(u), vbits = BN_num_bits(v), /* v is copy 698 * of p */ 699 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 && udp[0] == 1) 744 break; 745 746 if (ubits < vbits) { 747 i = ubits; 748 ubits = vbits; 749 vbits = i; 750 tmp = u; 751 u = v; 752 v = tmp; 753 tmp = b; 754 b = c; 755 c = tmp; 756 udp = vdp; 757 vdp = v->d; 758 bdp = cdp; 759 cdp = c->d; 760 } 761 for (i = 0; i < top; i++) { 762 udp[i] ^= vdp[i]; 763 bdp[i] ^= cdp[i]; 764 } 765 if (ubits == vbits) { 766 BN_ULONG ul; 767 int utop = (ubits - 1) / BN_BITS2; 768 769 while ((ul = udp[utop]) == 0 && utop) 770 utop--; 771 ubits = utop * BN_BITS2 + BN_num_bits_word(ul); 772 } 773 } 774 bn_correct_top(b); 775 } 776# endif 777 778 if (!BN_copy(r, b)) 779 goto err; 780 bn_check_top(r); 781 ret = 1; 782 783 err: 784# ifdef BN_DEBUG /* BN_CTX_end would complain about the 785 * expanded form */ 786 bn_correct_top(c); 787 bn_correct_top(u); 788 bn_correct_top(v); 789# endif 790 BN_CTX_end(ctx); 791 return ret; 792} 793 794/* 795 * Invert xx, reduce modulo p, and store the result in r. r could be xx. 796 * This function calls down to the BN_GF2m_mod_inv implementation; this 797 * wrapper function is only provided for convenience; for best performance, 798 * use the BN_GF2m_mod_inv function. 799 */ 800int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], 801 BN_CTX *ctx) 802{ 803 BIGNUM *field; 804 int ret = 0; 805 806 bn_check_top(xx); 807 BN_CTX_start(ctx); 808 if ((field = BN_CTX_get(ctx)) == NULL) 809 goto err; 810 if (!BN_GF2m_arr2poly(p, field)) 811 goto err; 812 813 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 814 bn_check_top(r); 815 816 err: 817 BN_CTX_end(ctx); 818 return ret; 819} 820 821# ifndef OPENSSL_SUN_GF2M_DIV 822/* 823 * Divide y by x, reduce modulo p, and store the result in r. r could be x 824 * or y, x could equal y. 825 */ 826int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 827 const BIGNUM *p, BN_CTX *ctx) 828{ 829 BIGNUM *xinv = NULL; 830 int ret = 0; 831 832 bn_check_top(y); 833 bn_check_top(x); 834 bn_check_top(p); 835 836 BN_CTX_start(ctx); 837 xinv = BN_CTX_get(ctx); 838 if (xinv == NULL) 839 goto err; 840 841 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) 842 goto err; 843 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) 844 goto err; 845 bn_check_top(r); 846 ret = 1; 847 848 err: 849 BN_CTX_end(ctx); 850 return ret; 851} 852# else 853/* 854 * Divide y by x, reduce modulo p, and store the result in r. r could be x 855 * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from 856 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the 857 * Great Divide". 858 */ 859int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 860 const BIGNUM *p, BN_CTX *ctx) 861{ 862 BIGNUM *a, *b, *u, *v; 863 int ret = 0; 864 865 bn_check_top(y); 866 bn_check_top(x); 867 bn_check_top(p); 868 869 BN_CTX_start(ctx); 870 871 a = BN_CTX_get(ctx); 872 b = BN_CTX_get(ctx); 873 u = BN_CTX_get(ctx); 874 v = BN_CTX_get(ctx); 875 if (v == NULL) 876 goto err; 877 878 /* reduce x and y mod p */ 879 if (!BN_GF2m_mod(u, y, p)) 880 goto err; 881 if (!BN_GF2m_mod(a, x, p)) 882 goto err; 883 if (!BN_copy(b, p)) 884 goto err; 885 886 while (!BN_is_odd(a)) { 887 if (!BN_rshift1(a, a)) 888 goto err; 889 if (BN_is_odd(u)) 890 if (!BN_GF2m_add(u, u, p)) 891 goto err; 892 if (!BN_rshift1(u, u)) 893 goto err; 894 } 895 896 do { 897 if (BN_GF2m_cmp(b, a) > 0) { 898 if (!BN_GF2m_add(b, b, a)) 899 goto err; 900 if (!BN_GF2m_add(v, v, u)) 901 goto err; 902 do { 903 if (!BN_rshift1(b, b)) 904 goto err; 905 if (BN_is_odd(v)) 906 if (!BN_GF2m_add(v, v, p)) 907 goto err; 908 if (!BN_rshift1(v, v)) 909 goto err; 910 } while (!BN_is_odd(b)); 911 } else if (BN_abs_is_word(a, 1)) 912 break; 913 else { 914 if (!BN_GF2m_add(a, a, b)) 915 goto err; 916 if (!BN_GF2m_add(u, u, v)) 917 goto err; 918 do { 919 if (!BN_rshift1(a, a)) 920 goto err; 921 if (BN_is_odd(u)) 922 if (!BN_GF2m_add(u, u, p)) 923 goto err; 924 if (!BN_rshift1(u, u)) 925 goto err; 926 } while (!BN_is_odd(a)); 927 } 928 } while (1); 929 930 if (!BN_copy(r, u)) 931 goto err; 932 bn_check_top(r); 933 ret = 1; 934 935 err: 936 BN_CTX_end(ctx); 937 return ret; 938} 939# endif 940 941/* 942 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 943 * * or yy, xx could equal yy. This function calls down to the 944 * BN_GF2m_mod_div implementation; this wrapper function is only provided for 945 * convenience; for best performance, use the BN_GF2m_mod_div function. 946 */ 947int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, 948 const int p[], BN_CTX *ctx) 949{ 950 BIGNUM *field; 951 int ret = 0; 952 953 bn_check_top(yy); 954 bn_check_top(xx); 955 956 BN_CTX_start(ctx); 957 if ((field = BN_CTX_get(ctx)) == NULL) 958 goto err; 959 if (!BN_GF2m_arr2poly(p, field)) 960 goto err; 961 962 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 963 bn_check_top(r); 964 965 err: 966 BN_CTX_end(ctx); 967 return ret; 968} 969 970/* 971 * Compute the bth power of a, reduce modulo p, and store the result in r. r 972 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE 973 * P1363. 974 */ 975int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 976 const int p[], BN_CTX *ctx) 977{ 978 int ret = 0, i, n; 979 BIGNUM *u; 980 981 bn_check_top(a); 982 bn_check_top(b); 983 984 if (BN_is_zero(b)) 985 return (BN_one(r)); 986 987 if (BN_abs_is_word(b, 1)) 988 return (BN_copy(r, a) != NULL); 989 990 BN_CTX_start(ctx); 991 if ((u = BN_CTX_get(ctx)) == NULL) 992 goto err; 993 994 if (!BN_GF2m_mod_arr(u, a, p)) 995 goto err; 996 997 n = BN_num_bits(b) - 1; 998 for (i = n - 1; i >= 0; i--) { 999 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) 1000 goto err; 1001 if (BN_is_bit_set(b, i)) { 1002 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) 1003 goto err; 1004 } 1005 } 1006 if (!BN_copy(r, u)) 1007 goto err; 1008 bn_check_top(r); 1009 ret = 1; 1010 err: 1011 BN_CTX_end(ctx); 1012 return ret; 1013} 1014 1015/* 1016 * Compute the bth power of a, reduce modulo p, and store the result in r. r 1017 * could be a. This function calls down to the BN_GF2m_mod_exp_arr 1018 * implementation; this wrapper function is only provided for convenience; 1019 * for best performance, use the BN_GF2m_mod_exp_arr function. 1020 */ 1021int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 1022 const BIGNUM *p, BN_CTX *ctx) 1023{ 1024 int ret = 0; 1025 const int max = BN_num_bits(p) + 1; 1026 int *arr = NULL; 1027 bn_check_top(a); 1028 bn_check_top(b); 1029 bn_check_top(p); 1030 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1031 goto err; 1032 ret = BN_GF2m_poly2arr(p, arr, max); 1033 if (!ret || ret > max) { 1034 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH); 1035 goto err; 1036 } 1037 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 1038 bn_check_top(r); 1039 err: 1040 if (arr) 1041 OPENSSL_free(arr); 1042 return ret; 1043} 1044 1045/* 1046 * Compute the square root of a, reduce modulo p, and store the result in r. 1047 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 1048 */ 1049int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], 1050 BN_CTX *ctx) 1051{ 1052 int ret = 0; 1053 BIGNUM *u; 1054 1055 bn_check_top(a); 1056 1057 if (!p[0]) { 1058 /* reduction mod 1 => return 0 */ 1059 BN_zero(r); 1060 return 1; 1061 } 1062 1063 BN_CTX_start(ctx); 1064 if ((u = BN_CTX_get(ctx)) == NULL) 1065 goto err; 1066 1067 if (!BN_set_bit(u, p[0] - 1)) 1068 goto err; 1069 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 1070 bn_check_top(r); 1071 1072 err: 1073 BN_CTX_end(ctx); 1074 return ret; 1075} 1076 1077/* 1078 * Compute the square root of a, reduce modulo p, and store the result in r. 1079 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr 1080 * implementation; this wrapper function is only provided for convenience; 1081 * for best performance, use the BN_GF2m_mod_sqrt_arr function. 1082 */ 1083int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1084{ 1085 int ret = 0; 1086 const int max = BN_num_bits(p) + 1; 1087 int *arr = NULL; 1088 bn_check_top(a); 1089 bn_check_top(p); 1090 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1091 goto err; 1092 ret = BN_GF2m_poly2arr(p, arr, max); 1093 if (!ret || ret > max) { 1094 BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH); 1095 goto err; 1096 } 1097 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 1098 bn_check_top(r); 1099 err: 1100 if (arr) 1101 OPENSSL_free(arr); 1102 return ret; 1103} 1104 1105/* 1106 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1107 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 1108 */ 1109int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], 1110 BN_CTX *ctx) 1111{ 1112 int ret = 0, count = 0, j; 1113 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 1114 1115 bn_check_top(a_); 1116 1117 if (!p[0]) { 1118 /* reduction mod 1 => return 0 */ 1119 BN_zero(r); 1120 return 1; 1121 } 1122 1123 BN_CTX_start(ctx); 1124 a = BN_CTX_get(ctx); 1125 z = BN_CTX_get(ctx); 1126 w = BN_CTX_get(ctx); 1127 if (w == NULL) 1128 goto err; 1129 1130 if (!BN_GF2m_mod_arr(a, a_, p)) 1131 goto err; 1132 1133 if (BN_is_zero(a)) { 1134 BN_zero(r); 1135 ret = 1; 1136 goto err; 1137 } 1138 1139 if (p[0] & 0x1) { /* m is odd */ 1140 /* compute half-trace of a */ 1141 if (!BN_copy(z, a)) 1142 goto err; 1143 for (j = 1; j <= (p[0] - 1) / 2; j++) { 1144 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1145 goto err; 1146 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1147 goto err; 1148 if (!BN_GF2m_add(z, z, a)) 1149 goto err; 1150 } 1151 1152 } else { /* m is even */ 1153 1154 rho = BN_CTX_get(ctx); 1155 w2 = BN_CTX_get(ctx); 1156 tmp = BN_CTX_get(ctx); 1157 if (tmp == NULL) 1158 goto err; 1159 do { 1160 if (!BN_rand(rho, p[0], 0, 0)) 1161 goto err; 1162 if (!BN_GF2m_mod_arr(rho, rho, p)) 1163 goto err; 1164 BN_zero(z); 1165 if (!BN_copy(w, rho)) 1166 goto err; 1167 for (j = 1; j <= p[0] - 1; j++) { 1168 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1169 goto err; 1170 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) 1171 goto err; 1172 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) 1173 goto err; 1174 if (!BN_GF2m_add(z, z, tmp)) 1175 goto err; 1176 if (!BN_GF2m_add(w, w2, rho)) 1177 goto err; 1178 } 1179 count++; 1180 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1181 if (BN_is_zero(w)) { 1182 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS); 1183 goto err; 1184 } 1185 } 1186 1187 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) 1188 goto err; 1189 if (!BN_GF2m_add(w, z, w)) 1190 goto err; 1191 if (BN_GF2m_cmp(w, a)) { 1192 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1193 goto err; 1194 } 1195 1196 if (!BN_copy(r, z)) 1197 goto err; 1198 bn_check_top(r); 1199 1200 ret = 1; 1201 1202 err: 1203 BN_CTX_end(ctx); 1204 return ret; 1205} 1206 1207/* 1208 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1209 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr 1210 * implementation; this wrapper function is only provided for convenience; 1211 * for best performance, use the BN_GF2m_mod_solve_quad_arr function. 1212 */ 1213int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, 1214 BN_CTX *ctx) 1215{ 1216 int ret = 0; 1217 const int max = BN_num_bits(p) + 1; 1218 int *arr = NULL; 1219 bn_check_top(a); 1220 bn_check_top(p); 1221 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1222 goto err; 1223 ret = BN_GF2m_poly2arr(p, arr, max); 1224 if (!ret || ret > max) { 1225 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH); 1226 goto err; 1227 } 1228 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1229 bn_check_top(r); 1230 err: 1231 if (arr) 1232 OPENSSL_free(arr); 1233 return ret; 1234} 1235 1236/* 1237 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * 1238 * x^i) into an array of integers corresponding to the bits with non-zero 1239 * coefficient. Array is terminated with -1. Up to max elements of the array 1240 * will be filled. Return value is total number of array elements that would 1241 * be filled if array was large enough. 1242 */ 1243int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1244{ 1245 int i, j, k = 0; 1246 BN_ULONG mask; 1247 1248 if (BN_is_zero(a)) 1249 return 0; 1250 1251 for (i = a->top - 1; i >= 0; i--) { 1252 if (!a->d[i]) 1253 /* skip word if a->d[i] == 0 */ 1254 continue; 1255 mask = BN_TBIT; 1256 for (j = BN_BITS2 - 1; j >= 0; j--) { 1257 if (a->d[i] & mask) { 1258 if (k < max) 1259 p[k] = BN_BITS2 * i + j; 1260 k++; 1261 } 1262 mask >>= 1; 1263 } 1264 } 1265 1266 if (k < max) { 1267 p[k] = -1; 1268 k++; 1269 } 1270 1271 return k; 1272} 1273 1274/* 1275 * Convert the coefficient array representation of a polynomial to a 1276 * bit-string. The array must be terminated by -1. 1277 */ 1278int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1279{ 1280 int i; 1281 1282 bn_check_top(a); 1283 BN_zero(a); 1284 for (i = 0; p[i] != -1; i++) { 1285 if (BN_set_bit(a, p[i]) == 0) 1286 return 0; 1287 } 1288 bn_check_top(a); 1289 1290 return 1; 1291} 1292 1293#endif 1294