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; 698 int ubits = BN_num_bits(u); 699 int vbits = BN_num_bits(v); /* v is copy of p */ 700 int top = p->top; 701 BN_ULONG *udp, *bdp, *vdp, *cdp; 702 703 bn_wexpand(u, top); 704 udp = u->d; 705 for (i = u->top; i < top; i++) 706 udp[i] = 0; 707 u->top = top; 708 bn_wexpand(b, top); 709 bdp = b->d; 710 bdp[0] = 1; 711 for (i = 1; i < top; i++) 712 bdp[i] = 0; 713 b->top = top; 714 bn_wexpand(c, top); 715 cdp = c->d; 716 for (i = 0; i < top; i++) 717 cdp[i] = 0; 718 c->top = top; 719 vdp = v->d; /* It pays off to "cache" *->d pointers, 720 * because it allows optimizer to be more 721 * aggressive. But we don't have to "cache" 722 * p->d, because *p is declared 'const'... */ 723 while (1) { 724 while (ubits && !(udp[0] & 1)) { 725 BN_ULONG u0, u1, b0, b1, mask; 726 727 u0 = udp[0]; 728 b0 = bdp[0]; 729 mask = (BN_ULONG)0 - (b0 & 1); 730 b0 ^= p->d[0] & mask; 731 for (i = 0; i < top - 1; i++) { 732 u1 = udp[i + 1]; 733 udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2; 734 u0 = u1; 735 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask); 736 bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2; 737 b0 = b1; 738 } 739 udp[i] = u0 >> 1; 740 bdp[i] = b0 >> 1; 741 ubits--; 742 } 743 744 if (ubits <= BN_BITS2) { 745 if (udp[0] == 0) /* poly was reducible */ 746 goto err; 747 if (udp[0] == 1) 748 break; 749 } 750 751 if (ubits < vbits) { 752 i = ubits; 753 ubits = vbits; 754 vbits = i; 755 tmp = u; 756 u = v; 757 v = tmp; 758 tmp = b; 759 b = c; 760 c = tmp; 761 udp = vdp; 762 vdp = v->d; 763 bdp = cdp; 764 cdp = c->d; 765 } 766 for (i = 0; i < top; i++) { 767 udp[i] ^= vdp[i]; 768 bdp[i] ^= cdp[i]; 769 } 770 if (ubits == vbits) { 771 BN_ULONG ul; 772 int utop = (ubits - 1) / BN_BITS2; 773 774 while ((ul = udp[utop]) == 0 && utop) 775 utop--; 776 ubits = utop * BN_BITS2 + BN_num_bits_word(ul); 777 } 778 } 779 bn_correct_top(b); 780 } 781# endif 782 783 if (!BN_copy(r, b)) 784 goto err; 785 bn_check_top(r); 786 ret = 1; 787 788 err: 789# ifdef BN_DEBUG /* BN_CTX_end would complain about the 790 * expanded form */ 791 bn_correct_top(c); 792 bn_correct_top(u); 793 bn_correct_top(v); 794# endif 795 BN_CTX_end(ctx); 796 return ret; 797} 798 799/* 800 * Invert xx, reduce modulo p, and store the result in r. r could be xx. 801 * This function calls down to the BN_GF2m_mod_inv implementation; this 802 * wrapper function is only provided for convenience; for best performance, 803 * use the BN_GF2m_mod_inv function. 804 */ 805int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], 806 BN_CTX *ctx) 807{ 808 BIGNUM *field; 809 int ret = 0; 810 811 bn_check_top(xx); 812 BN_CTX_start(ctx); 813 if ((field = BN_CTX_get(ctx)) == NULL) 814 goto err; 815 if (!BN_GF2m_arr2poly(p, field)) 816 goto err; 817 818 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 819 bn_check_top(r); 820 821 err: 822 BN_CTX_end(ctx); 823 return ret; 824} 825 826# ifndef OPENSSL_SUN_GF2M_DIV 827/* 828 * Divide y by x, reduce modulo p, and store the result in r. r could be x 829 * or y, x could equal y. 830 */ 831int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 832 const BIGNUM *p, BN_CTX *ctx) 833{ 834 BIGNUM *xinv = NULL; 835 int ret = 0; 836 837 bn_check_top(y); 838 bn_check_top(x); 839 bn_check_top(p); 840 841 BN_CTX_start(ctx); 842 xinv = BN_CTX_get(ctx); 843 if (xinv == NULL) 844 goto err; 845 846 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) 847 goto err; 848 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) 849 goto err; 850 bn_check_top(r); 851 ret = 1; 852 853 err: 854 BN_CTX_end(ctx); 855 return ret; 856} 857# else 858/* 859 * Divide y by x, reduce modulo p, and store the result in r. r could be x 860 * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from 861 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the 862 * Great Divide". 863 */ 864int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, 865 const BIGNUM *p, BN_CTX *ctx) 866{ 867 BIGNUM *a, *b, *u, *v; 868 int ret = 0; 869 870 bn_check_top(y); 871 bn_check_top(x); 872 bn_check_top(p); 873 874 BN_CTX_start(ctx); 875 876 a = BN_CTX_get(ctx); 877 b = BN_CTX_get(ctx); 878 u = BN_CTX_get(ctx); 879 v = BN_CTX_get(ctx); 880 if (v == NULL) 881 goto err; 882 883 /* reduce x and y mod p */ 884 if (!BN_GF2m_mod(u, y, p)) 885 goto err; 886 if (!BN_GF2m_mod(a, x, p)) 887 goto err; 888 if (!BN_copy(b, p)) 889 goto err; 890 891 while (!BN_is_odd(a)) { 892 if (!BN_rshift1(a, a)) 893 goto err; 894 if (BN_is_odd(u)) 895 if (!BN_GF2m_add(u, u, p)) 896 goto err; 897 if (!BN_rshift1(u, u)) 898 goto err; 899 } 900 901 do { 902 if (BN_GF2m_cmp(b, a) > 0) { 903 if (!BN_GF2m_add(b, b, a)) 904 goto err; 905 if (!BN_GF2m_add(v, v, u)) 906 goto err; 907 do { 908 if (!BN_rshift1(b, b)) 909 goto err; 910 if (BN_is_odd(v)) 911 if (!BN_GF2m_add(v, v, p)) 912 goto err; 913 if (!BN_rshift1(v, v)) 914 goto err; 915 } while (!BN_is_odd(b)); 916 } else if (BN_abs_is_word(a, 1)) 917 break; 918 else { 919 if (!BN_GF2m_add(a, a, b)) 920 goto err; 921 if (!BN_GF2m_add(u, u, v)) 922 goto err; 923 do { 924 if (!BN_rshift1(a, a)) 925 goto err; 926 if (BN_is_odd(u)) 927 if (!BN_GF2m_add(u, u, p)) 928 goto err; 929 if (!BN_rshift1(u, u)) 930 goto err; 931 } while (!BN_is_odd(a)); 932 } 933 } while (1); 934 935 if (!BN_copy(r, u)) 936 goto err; 937 bn_check_top(r); 938 ret = 1; 939 940 err: 941 BN_CTX_end(ctx); 942 return ret; 943} 944# endif 945 946/* 947 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 948 * * or yy, xx could equal yy. This function calls down to the 949 * BN_GF2m_mod_div implementation; this wrapper function is only provided for 950 * convenience; for best performance, use the BN_GF2m_mod_div function. 951 */ 952int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, 953 const int p[], BN_CTX *ctx) 954{ 955 BIGNUM *field; 956 int ret = 0; 957 958 bn_check_top(yy); 959 bn_check_top(xx); 960 961 BN_CTX_start(ctx); 962 if ((field = BN_CTX_get(ctx)) == NULL) 963 goto err; 964 if (!BN_GF2m_arr2poly(p, field)) 965 goto err; 966 967 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 968 bn_check_top(r); 969 970 err: 971 BN_CTX_end(ctx); 972 return ret; 973} 974 975/* 976 * Compute the bth power of a, reduce modulo p, and store the result in r. r 977 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE 978 * P1363. 979 */ 980int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 981 const int p[], BN_CTX *ctx) 982{ 983 int ret = 0, i, n; 984 BIGNUM *u; 985 986 bn_check_top(a); 987 bn_check_top(b); 988 989 if (BN_is_zero(b)) 990 return (BN_one(r)); 991 992 if (BN_abs_is_word(b, 1)) 993 return (BN_copy(r, a) != NULL); 994 995 BN_CTX_start(ctx); 996 if ((u = BN_CTX_get(ctx)) == NULL) 997 goto err; 998 999 if (!BN_GF2m_mod_arr(u, a, p)) 1000 goto err; 1001 1002 n = BN_num_bits(b) - 1; 1003 for (i = n - 1; i >= 0; i--) { 1004 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) 1005 goto err; 1006 if (BN_is_bit_set(b, i)) { 1007 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) 1008 goto err; 1009 } 1010 } 1011 if (!BN_copy(r, u)) 1012 goto err; 1013 bn_check_top(r); 1014 ret = 1; 1015 err: 1016 BN_CTX_end(ctx); 1017 return ret; 1018} 1019 1020/* 1021 * Compute the bth power of a, reduce modulo p, and store the result in r. r 1022 * could be a. This function calls down to the BN_GF2m_mod_exp_arr 1023 * implementation; this wrapper function is only provided for convenience; 1024 * for best performance, use the BN_GF2m_mod_exp_arr function. 1025 */ 1026int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, 1027 const BIGNUM *p, BN_CTX *ctx) 1028{ 1029 int ret = 0; 1030 const int max = BN_num_bits(p) + 1; 1031 int *arr = NULL; 1032 bn_check_top(a); 1033 bn_check_top(b); 1034 bn_check_top(p); 1035 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1036 goto err; 1037 ret = BN_GF2m_poly2arr(p, arr, max); 1038 if (!ret || ret > max) { 1039 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH); 1040 goto err; 1041 } 1042 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 1043 bn_check_top(r); 1044 err: 1045 if (arr) 1046 OPENSSL_free(arr); 1047 return ret; 1048} 1049 1050/* 1051 * Compute the square root of a, reduce modulo p, and store the result in r. 1052 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 1053 */ 1054int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], 1055 BN_CTX *ctx) 1056{ 1057 int ret = 0; 1058 BIGNUM *u; 1059 1060 bn_check_top(a); 1061 1062 if (!p[0]) { 1063 /* reduction mod 1 => return 0 */ 1064 BN_zero(r); 1065 return 1; 1066 } 1067 1068 BN_CTX_start(ctx); 1069 if ((u = BN_CTX_get(ctx)) == NULL) 1070 goto err; 1071 1072 if (!BN_set_bit(u, p[0] - 1)) 1073 goto err; 1074 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 1075 bn_check_top(r); 1076 1077 err: 1078 BN_CTX_end(ctx); 1079 return ret; 1080} 1081 1082/* 1083 * Compute the square root of a, reduce modulo p, and store the result in r. 1084 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr 1085 * implementation; this wrapper function is only provided for convenience; 1086 * for best performance, use the BN_GF2m_mod_sqrt_arr function. 1087 */ 1088int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, 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 = (int *)OPENSSL_malloc(sizeof(int) * 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_SQRT, BN_R_INVALID_LENGTH); 1100 goto err; 1101 } 1102 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 1103 bn_check_top(r); 1104 err: 1105 if (arr) 1106 OPENSSL_free(arr); 1107 return ret; 1108} 1109 1110/* 1111 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1112 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 1113 */ 1114int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], 1115 BN_CTX *ctx) 1116{ 1117 int ret = 0, count = 0, j; 1118 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 1119 1120 bn_check_top(a_); 1121 1122 if (!p[0]) { 1123 /* reduction mod 1 => return 0 */ 1124 BN_zero(r); 1125 return 1; 1126 } 1127 1128 BN_CTX_start(ctx); 1129 a = BN_CTX_get(ctx); 1130 z = BN_CTX_get(ctx); 1131 w = BN_CTX_get(ctx); 1132 if (w == NULL) 1133 goto err; 1134 1135 if (!BN_GF2m_mod_arr(a, a_, p)) 1136 goto err; 1137 1138 if (BN_is_zero(a)) { 1139 BN_zero(r); 1140 ret = 1; 1141 goto err; 1142 } 1143 1144 if (p[0] & 0x1) { /* m is odd */ 1145 /* compute half-trace of a */ 1146 if (!BN_copy(z, a)) 1147 goto err; 1148 for (j = 1; j <= (p[0] - 1) / 2; j++) { 1149 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1150 goto err; 1151 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1152 goto err; 1153 if (!BN_GF2m_add(z, z, a)) 1154 goto err; 1155 } 1156 1157 } else { /* m is even */ 1158 1159 rho = BN_CTX_get(ctx); 1160 w2 = BN_CTX_get(ctx); 1161 tmp = BN_CTX_get(ctx); 1162 if (tmp == NULL) 1163 goto err; 1164 do { 1165 if (!BN_rand(rho, p[0], 0, 0)) 1166 goto err; 1167 if (!BN_GF2m_mod_arr(rho, rho, p)) 1168 goto err; 1169 BN_zero(z); 1170 if (!BN_copy(w, rho)) 1171 goto err; 1172 for (j = 1; j <= p[0] - 1; j++) { 1173 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) 1174 goto err; 1175 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) 1176 goto err; 1177 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) 1178 goto err; 1179 if (!BN_GF2m_add(z, z, tmp)) 1180 goto err; 1181 if (!BN_GF2m_add(w, w2, rho)) 1182 goto err; 1183 } 1184 count++; 1185 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1186 if (BN_is_zero(w)) { 1187 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS); 1188 goto err; 1189 } 1190 } 1191 1192 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) 1193 goto err; 1194 if (!BN_GF2m_add(w, z, w)) 1195 goto err; 1196 if (BN_GF2m_cmp(w, a)) { 1197 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1198 goto err; 1199 } 1200 1201 if (!BN_copy(r, z)) 1202 goto err; 1203 bn_check_top(r); 1204 1205 ret = 1; 1206 1207 err: 1208 BN_CTX_end(ctx); 1209 return ret; 1210} 1211 1212/* 1213 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 1214 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr 1215 * implementation; this wrapper function is only provided for convenience; 1216 * for best performance, use the BN_GF2m_mod_solve_quad_arr function. 1217 */ 1218int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, 1219 BN_CTX *ctx) 1220{ 1221 int ret = 0; 1222 const int max = BN_num_bits(p) + 1; 1223 int *arr = NULL; 1224 bn_check_top(a); 1225 bn_check_top(p); 1226 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) 1227 goto err; 1228 ret = BN_GF2m_poly2arr(p, arr, max); 1229 if (!ret || ret > max) { 1230 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH); 1231 goto err; 1232 } 1233 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1234 bn_check_top(r); 1235 err: 1236 if (arr) 1237 OPENSSL_free(arr); 1238 return ret; 1239} 1240 1241/* 1242 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * 1243 * x^i) into an array of integers corresponding to the bits with non-zero 1244 * coefficient. Array is terminated with -1. Up to max elements of the array 1245 * will be filled. Return value is total number of array elements that would 1246 * be filled if array was large enough. 1247 */ 1248int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1249{ 1250 int i, j, k = 0; 1251 BN_ULONG mask; 1252 1253 if (BN_is_zero(a)) 1254 return 0; 1255 1256 for (i = a->top - 1; i >= 0; i--) { 1257 if (!a->d[i]) 1258 /* skip word if a->d[i] == 0 */ 1259 continue; 1260 mask = BN_TBIT; 1261 for (j = BN_BITS2 - 1; j >= 0; j--) { 1262 if (a->d[i] & mask) { 1263 if (k < max) 1264 p[k] = BN_BITS2 * i + j; 1265 k++; 1266 } 1267 mask >>= 1; 1268 } 1269 } 1270 1271 if (k < max) { 1272 p[k] = -1; 1273 k++; 1274 } 1275 1276 return k; 1277} 1278 1279/* 1280 * Convert the coefficient array representation of a polynomial to a 1281 * bit-string. The array must be terminated by -1. 1282 */ 1283int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1284{ 1285 int i; 1286 1287 bn_check_top(a); 1288 BN_zero(a); 1289 for (i = 0; p[i] != -1; i++) { 1290 if (BN_set_bit(a, p[i]) == 0) 1291 return 0; 1292 } 1293 bn_check_top(a); 1294 1295 return 1; 1296} 1297 1298#endif 1299