bn_gf2m.c revision 284295
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/* NOTE: This file is licensed pursuant to the OpenSSL license below 31 * and may be modified; but after modifications, the above covenant 32 * may no longer apply! In such cases, the corresponding paragraph 33 * ["In addition, Sun covenants ... causes the infringement."] and 34 * this note can be edited out; but please keep the Sun copyright 35 * notice and attribution. */ 36 37/* ==================================================================== 38 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved. 39 * 40 * Redistribution and use in source and binary forms, with or without 41 * modification, are permitted provided that the following conditions 42 * are met: 43 * 44 * 1. Redistributions of source code must retain the above copyright 45 * notice, this list of conditions and the following disclaimer. 46 * 47 * 2. Redistributions in binary form must reproduce the above copyright 48 * notice, this list of conditions and the following disclaimer in 49 * the documentation and/or other materials provided with the 50 * distribution. 51 * 52 * 3. All advertising materials mentioning features or use of this 53 * software must display the following acknowledgment: 54 * "This product includes software developed by the OpenSSL Project 55 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 56 * 57 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 58 * endorse or promote products derived from this software without 59 * prior written permission. For written permission, please contact 60 * openssl-core@openssl.org. 61 * 62 * 5. Products derived from this software may not be called "OpenSSL" 63 * nor may "OpenSSL" appear in their names without prior written 64 * permission of the OpenSSL Project. 65 * 66 * 6. Redistributions of any form whatsoever must retain the following 67 * acknowledgment: 68 * "This product includes software developed by the OpenSSL Project 69 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 70 * 71 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 72 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 73 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 74 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 75 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 76 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 77 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 78 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 79 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 80 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 81 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 82 * OF THE POSSIBILITY OF SUCH DAMAGE. 83 * ==================================================================== 84 * 85 * This product includes cryptographic software written by Eric Young 86 * (eay@cryptsoft.com). This product includes software written by Tim 87 * Hudson (tjh@cryptsoft.com). 88 * 89 */ 90 91#include <assert.h> 92#include <limits.h> 93#include <stdio.h> 94#include "cryptlib.h" 95#include "bn_lcl.h" 96 97#ifndef OPENSSL_NO_EC2M 98 99/* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */ 100#define MAX_ITERATIONS 50 101 102static const BN_ULONG SQR_tb[16] = 103 { 0, 1, 4, 5, 16, 17, 20, 21, 104 64, 65, 68, 69, 80, 81, 84, 85 }; 105/* Platform-specific macros to accelerate squaring. */ 106#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 107#define SQR1(w) \ 108 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \ 109 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \ 110 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \ 111 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF] 112#define SQR0(w) \ 113 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \ 114 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \ 115 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ 116 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 117#endif 118#ifdef THIRTY_TWO_BIT 119#define SQR1(w) \ 120 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \ 121 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF] 122#define SQR0(w) \ 123 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ 124 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 125#endif 126 127#if !defined(OPENSSL_BN_ASM_GF2m) 128/* Product of two polynomials a, b each with degree < BN_BITS2 - 1, 129 * result is a polynomial r with degree < 2 * BN_BITS - 1 130 * The caller MUST ensure that the variables have the right amount 131 * of space allocated. 132 */ 133#ifdef THIRTY_TWO_BIT 134static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 135 { 136 register BN_ULONG h, l, s; 137 BN_ULONG tab[8], top2b = a >> 30; 138 register BN_ULONG a1, a2, a4; 139 140 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1; 141 142 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2; 143 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4; 144 145 s = tab[b & 0x7]; l = s; 146 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29; 147 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26; 148 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23; 149 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20; 150 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17; 151 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14; 152 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11; 153 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8; 154 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5; 155 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2; 156 157 /* compensate for the top two bits of a */ 158 159 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; } 160 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; } 161 162 *r1 = h; *r0 = l; 163 } 164#endif 165#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 166static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 167 { 168 register BN_ULONG h, l, s; 169 BN_ULONG tab[16], top3b = a >> 61; 170 register BN_ULONG a1, a2, a4, a8; 171 172 a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1; 173 174 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2; 175 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4; 176 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8; 177 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8; 178 179 s = tab[b & 0xF]; l = s; 180 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60; 181 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56; 182 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52; 183 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48; 184 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44; 185 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40; 186 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36; 187 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32; 188 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28; 189 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24; 190 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20; 191 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16; 192 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12; 193 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8; 194 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4; 195 196 /* compensate for the top three bits of a */ 197 198 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; } 199 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; } 200 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; } 201 202 *r1 = h; *r0 = l; 203 } 204#endif 205 206/* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, 207 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 208 * The caller MUST ensure that the variables have the right amount 209 * of space allocated. 210 */ 211static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0) 212 { 213 BN_ULONG m1, m0; 214 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ 215 bn_GF2m_mul_1x1(r+3, r+2, a1, b1); 216 bn_GF2m_mul_1x1(r+1, r, a0, b0); 217 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); 218 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ 219 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ 220 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ 221 } 222#else 223void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1, BN_ULONG b0); 224#endif 225 226/* Add polynomials a and b and store result in r; r could be a or b, a and b 227 * could be equal; r is the bitwise XOR of a and b. 228 */ 229int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) 230 { 231 int i; 232 const BIGNUM *at, *bt; 233 234 bn_check_top(a); 235 bn_check_top(b); 236 237 if (a->top < b->top) { at = b; bt = a; } 238 else { at = a; bt = b; } 239 240 if(bn_wexpand(r, at->top) == NULL) 241 return 0; 242 243 for (i = 0; i < bt->top; i++) 244 { 245 r->d[i] = at->d[i] ^ bt->d[i]; 246 } 247 for (; i < at->top; i++) 248 { 249 r->d[i] = at->d[i]; 250 } 251 252 r->top = at->top; 253 bn_correct_top(r); 254 255 return 1; 256 } 257 258 259/* Some functions allow for representation of the irreducible polynomials 260 * as an int[], say p. The irreducible f(t) is then of the form: 261 * t^p[0] + t^p[1] + ... + t^p[k] 262 * where m = p[0] > p[1] > ... > p[k] = 0. 263 */ 264 265 266/* Performs modular reduction of a and store result in r. r could be a. */ 267int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]) 268 { 269 int j, k; 270 int n, dN, d0, d1; 271 BN_ULONG zz, *z; 272 273 bn_check_top(a); 274 275 if (!p[0]) 276 { 277 /* reduction mod 1 => return 0 */ 278 BN_zero(r); 279 return 1; 280 } 281 282 /* Since the algorithm does reduction in the r value, if a != r, copy 283 * the contents of a into r so we can do reduction in r. 284 */ 285 if (a != r) 286 { 287 if (!bn_wexpand(r, a->top)) return 0; 288 for (j = 0; j < a->top; j++) 289 { 290 r->d[j] = a->d[j]; 291 } 292 r->top = a->top; 293 } 294 z = r->d; 295 296 /* start reduction */ 297 dN = p[0] / BN_BITS2; 298 for (j = r->top - 1; j > dN;) 299 { 300 zz = z[j]; 301 if (z[j] == 0) { j--; continue; } 302 z[j] = 0; 303 304 for (k = 1; p[k] != 0; k++) 305 { 306 /* reducing component t^p[k] */ 307 n = p[0] - p[k]; 308 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0; 309 n /= BN_BITS2; 310 z[j-n] ^= (zz>>d0); 311 if (d0) z[j-n-1] ^= (zz<<d1); 312 } 313 314 /* reducing component t^0 */ 315 n = dN; 316 d0 = p[0] % BN_BITS2; 317 d1 = BN_BITS2 - d0; 318 z[j-n] ^= (zz >> d0); 319 if (d0) z[j-n-1] ^= (zz << d1); 320 } 321 322 /* final round of reduction */ 323 while (j == dN) 324 { 325 326 d0 = p[0] % BN_BITS2; 327 zz = z[dN] >> d0; 328 if (zz == 0) break; 329 d1 = BN_BITS2 - d0; 330 331 /* clear up the top d1 bits */ 332 if (d0) 333 z[dN] = (z[dN] << d1) >> d1; 334 else 335 z[dN] = 0; 336 z[0] ^= zz; /* reduction t^0 component */ 337 338 for (k = 1; p[k] != 0; k++) 339 { 340 BN_ULONG tmp_ulong; 341 342 /* reducing component t^p[k]*/ 343 n = p[k] / BN_BITS2; 344 d0 = p[k] % BN_BITS2; 345 d1 = BN_BITS2 - d0; 346 z[n] ^= (zz << d0); 347 tmp_ulong = zz >> d1; 348 if (d0 && tmp_ulong) 349 z[n+1] ^= tmp_ulong; 350 } 351 352 353 } 354 355 bn_correct_top(r); 356 return 1; 357 } 358 359/* Performs modular reduction of a by p and store result in r. r could be a. 360 * 361 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 362 * function is only provided for convenience; for best performance, use the 363 * BN_GF2m_mod_arr function. 364 */ 365int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 366 { 367 int ret = 0; 368 int arr[6]; 369 bn_check_top(a); 370 bn_check_top(p); 371 ret = BN_GF2m_poly2arr(p, arr, sizeof(arr)/sizeof(arr[0])); 372 if (!ret || ret > (int)(sizeof(arr)/sizeof(arr[0]))) 373 { 374 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH); 375 return 0; 376 } 377 ret = BN_GF2m_mod_arr(r, a, arr); 378 bn_check_top(r); 379 return ret; 380 } 381 382 383/* Compute the product of two polynomials a and b, reduce modulo p, and store 384 * the result in r. r could be a or b; a could be b. 385 */ 386int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) 387 { 388 int zlen, i, j, k, ret = 0; 389 BIGNUM *s; 390 BN_ULONG x1, x0, y1, y0, zz[4]; 391 392 bn_check_top(a); 393 bn_check_top(b); 394 395 if (a == b) 396 { 397 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 398 } 399 400 BN_CTX_start(ctx); 401 if ((s = BN_CTX_get(ctx)) == NULL) goto err; 402 403 zlen = a->top + b->top + 4; 404 if (!bn_wexpand(s, zlen)) goto err; 405 s->top = zlen; 406 407 for (i = 0; i < zlen; i++) s->d[i] = 0; 408 409 for (j = 0; j < b->top; j += 2) 410 { 411 y0 = b->d[j]; 412 y1 = ((j+1) == b->top) ? 0 : b->d[j+1]; 413 for (i = 0; i < a->top; i += 2) 414 { 415 x0 = a->d[i]; 416 x1 = ((i+1) == a->top) ? 0 : a->d[i+1]; 417 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 418 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k]; 419 } 420 } 421 422 bn_correct_top(s); 423 if (BN_GF2m_mod_arr(r, s, p)) 424 ret = 1; 425 bn_check_top(r); 426 427err: 428 BN_CTX_end(ctx); 429 return ret; 430 } 431 432/* Compute the product of two polynomials a and b, reduce modulo p, and store 433 * the result in r. r could be a or b; a could equal b. 434 * 435 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper 436 * function is only provided for convenience; for best performance, use the 437 * BN_GF2m_mod_mul_arr function. 438 */ 439int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 440 { 441 int ret = 0; 442 const int max = BN_num_bits(p) + 1; 443 int *arr=NULL; 444 bn_check_top(a); 445 bn_check_top(b); 446 bn_check_top(p); 447 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 448 ret = BN_GF2m_poly2arr(p, arr, max); 449 if (!ret || ret > max) 450 { 451 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH); 452 goto err; 453 } 454 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 455 bn_check_top(r); 456err: 457 if (arr) OPENSSL_free(arr); 458 return ret; 459 } 460 461 462/* Square a, reduce the result mod p, and store it in a. r could be a. */ 463int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) 464 { 465 int i, ret = 0; 466 BIGNUM *s; 467 468 bn_check_top(a); 469 BN_CTX_start(ctx); 470 if ((s = BN_CTX_get(ctx)) == NULL) return 0; 471 if (!bn_wexpand(s, 2 * a->top)) goto err; 472 473 for (i = a->top - 1; i >= 0; i--) 474 { 475 s->d[2*i+1] = SQR1(a->d[i]); 476 s->d[2*i ] = SQR0(a->d[i]); 477 } 478 479 s->top = 2 * a->top; 480 bn_correct_top(s); 481 if (!BN_GF2m_mod_arr(r, s, p)) goto err; 482 bn_check_top(r); 483 ret = 1; 484err: 485 BN_CTX_end(ctx); 486 return ret; 487 } 488 489/* Square a, reduce the result mod p, and store it in a. r could be a. 490 * 491 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper 492 * function is only provided for convenience; for best performance, use the 493 * BN_GF2m_mod_sqr_arr function. 494 */ 495int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 496 { 497 int ret = 0; 498 const int max = BN_num_bits(p) + 1; 499 int *arr=NULL; 500 501 bn_check_top(a); 502 bn_check_top(p); 503 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 504 ret = BN_GF2m_poly2arr(p, arr, max); 505 if (!ret || ret > max) 506 { 507 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH); 508 goto err; 509 } 510 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 511 bn_check_top(r); 512err: 513 if (arr) OPENSSL_free(arr); 514 return ret; 515 } 516 517 518/* Invert a, reduce modulo p, and store the result in r. r could be a. 519 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from 520 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation 521 * of Elliptic Curve Cryptography Over Binary Fields". 522 */ 523int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 524 { 525 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; 526 int ret = 0; 527 528 bn_check_top(a); 529 bn_check_top(p); 530 531 BN_CTX_start(ctx); 532 533 if ((b = BN_CTX_get(ctx))==NULL) goto err; 534 if ((c = BN_CTX_get(ctx))==NULL) goto err; 535 if ((u = BN_CTX_get(ctx))==NULL) goto err; 536 if ((v = BN_CTX_get(ctx))==NULL) goto err; 537 538 if (!BN_GF2m_mod(u, a, p)) goto err; 539 if (BN_is_zero(u)) goto err; 540 541 if (!BN_copy(v, p)) goto err; 542#if 0 543 if (!BN_one(b)) goto err; 544 545 while (1) 546 { 547 while (!BN_is_odd(u)) 548 { 549 if (BN_is_zero(u)) goto err; 550 if (!BN_rshift1(u, u)) goto err; 551 if (BN_is_odd(b)) 552 { 553 if (!BN_GF2m_add(b, b, p)) goto err; 554 } 555 if (!BN_rshift1(b, b)) goto err; 556 } 557 558 if (BN_abs_is_word(u, 1)) break; 559 560 if (BN_num_bits(u) < BN_num_bits(v)) 561 { 562 tmp = u; u = v; v = tmp; 563 tmp = b; b = c; c = tmp; 564 } 565 566 if (!BN_GF2m_add(u, u, v)) goto err; 567 if (!BN_GF2m_add(b, b, c)) goto err; 568 } 569#else 570 { 571 int i; 572 int ubits = BN_num_bits(u); 573 int vbits = BN_num_bits(v); /* v is copy of p */ 574 int top = p->top; 575 BN_ULONG *udp, *bdp, *vdp, *cdp; 576 577 bn_wexpand(u,top); udp = u->d; 578 for (i=u->top;i<top;i++) udp[i] = 0; 579 u->top = top; 580 bn_wexpand(b,top); bdp = b->d; 581 bdp[0] = 1; 582 for (i=1;i<top;i++) bdp[i] = 0; 583 b->top = top; 584 bn_wexpand(c,top); cdp = c->d; 585 for (i=0;i<top;i++) cdp[i] = 0; 586 c->top = top; 587 vdp = v->d; /* It pays off to "cache" *->d pointers, because 588 * it allows optimizer to be more aggressive. 589 * But we don't have to "cache" p->d, because *p 590 * is declared 'const'... */ 591 while (1) 592 { 593 while (ubits && !(udp[0]&1)) 594 { 595 BN_ULONG u0,u1,b0,b1,mask; 596 597 u0 = udp[0]; 598 b0 = bdp[0]; 599 mask = (BN_ULONG)0-(b0&1); 600 b0 ^= p->d[0]&mask; 601 for (i=0;i<top-1;i++) 602 { 603 u1 = udp[i+1]; 604 udp[i] = ((u0>>1)|(u1<<(BN_BITS2-1)))&BN_MASK2; 605 u0 = u1; 606 b1 = bdp[i+1]^(p->d[i+1]&mask); 607 bdp[i] = ((b0>>1)|(b1<<(BN_BITS2-1)))&BN_MASK2; 608 b0 = b1; 609 } 610 udp[i] = u0>>1; 611 bdp[i] = b0>>1; 612 ubits--; 613 } 614 615 if (ubits <= BN_BITS2) { 616 if (udp[0] == 0) /* poly was reducible */ 617 goto err; 618 if (udp[0] == 1) 619 break; 620 } 621 622 if (ubits<vbits) 623 { 624 i = ubits; ubits = vbits; vbits = i; 625 tmp = u; u = v; v = tmp; 626 tmp = b; b = c; c = tmp; 627 udp = vdp; vdp = v->d; 628 bdp = cdp; cdp = c->d; 629 } 630 for(i=0;i<top;i++) 631 { 632 udp[i] ^= vdp[i]; 633 bdp[i] ^= cdp[i]; 634 } 635 if (ubits==vbits) 636 { 637 BN_ULONG ul; 638 int utop = (ubits-1)/BN_BITS2; 639 640 while ((ul=udp[utop])==0 && utop) utop--; 641 ubits = utop*BN_BITS2 + BN_num_bits_word(ul); 642 } 643 } 644 bn_correct_top(b); 645 } 646#endif 647 648 if (!BN_copy(r, b)) goto err; 649 bn_check_top(r); 650 ret = 1; 651 652err: 653#ifdef BN_DEBUG /* BN_CTX_end would complain about the expanded form */ 654 bn_correct_top(c); 655 bn_correct_top(u); 656 bn_correct_top(v); 657#endif 658 BN_CTX_end(ctx); 659 return ret; 660 } 661 662/* Invert xx, reduce modulo p, and store the result in r. r could be xx. 663 * 664 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 665 * function is only provided for convenience; for best performance, use the 666 * BN_GF2m_mod_inv function. 667 */ 668int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx) 669 { 670 BIGNUM *field; 671 int ret = 0; 672 673 bn_check_top(xx); 674 BN_CTX_start(ctx); 675 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 676 if (!BN_GF2m_arr2poly(p, field)) goto err; 677 678 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 679 bn_check_top(r); 680 681err: 682 BN_CTX_end(ctx); 683 return ret; 684 } 685 686 687#ifndef OPENSSL_SUN_GF2M_DIV 688/* Divide y by x, reduce modulo p, and store the result in r. r could be x 689 * or y, x could equal y. 690 */ 691int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 692 { 693 BIGNUM *xinv = NULL; 694 int ret = 0; 695 696 bn_check_top(y); 697 bn_check_top(x); 698 bn_check_top(p); 699 700 BN_CTX_start(ctx); 701 xinv = BN_CTX_get(ctx); 702 if (xinv == NULL) goto err; 703 704 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 705 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 706 bn_check_top(r); 707 ret = 1; 708 709err: 710 BN_CTX_end(ctx); 711 return ret; 712 } 713#else 714/* Divide y by x, reduce modulo p, and store the result in r. r could be x 715 * or y, x could equal y. 716 * Uses algorithm Modular_Division_GF(2^m) from 717 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 718 * the Great Divide". 719 */ 720int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 721 { 722 BIGNUM *a, *b, *u, *v; 723 int ret = 0; 724 725 bn_check_top(y); 726 bn_check_top(x); 727 bn_check_top(p); 728 729 BN_CTX_start(ctx); 730 731 a = BN_CTX_get(ctx); 732 b = BN_CTX_get(ctx); 733 u = BN_CTX_get(ctx); 734 v = BN_CTX_get(ctx); 735 if (v == NULL) goto err; 736 737 /* reduce x and y mod p */ 738 if (!BN_GF2m_mod(u, y, p)) goto err; 739 if (!BN_GF2m_mod(a, x, p)) goto err; 740 if (!BN_copy(b, p)) goto err; 741 742 while (!BN_is_odd(a)) 743 { 744 if (!BN_rshift1(a, a)) goto err; 745 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 746 if (!BN_rshift1(u, u)) goto err; 747 } 748 749 do 750 { 751 if (BN_GF2m_cmp(b, a) > 0) 752 { 753 if (!BN_GF2m_add(b, b, a)) goto err; 754 if (!BN_GF2m_add(v, v, u)) goto err; 755 do 756 { 757 if (!BN_rshift1(b, b)) goto err; 758 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 759 if (!BN_rshift1(v, v)) goto err; 760 } while (!BN_is_odd(b)); 761 } 762 else if (BN_abs_is_word(a, 1)) 763 break; 764 else 765 { 766 if (!BN_GF2m_add(a, a, b)) goto err; 767 if (!BN_GF2m_add(u, u, v)) goto err; 768 do 769 { 770 if (!BN_rshift1(a, a)) goto err; 771 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 772 if (!BN_rshift1(u, u)) goto err; 773 } while (!BN_is_odd(a)); 774 } 775 } while (1); 776 777 if (!BN_copy(r, u)) goto err; 778 bn_check_top(r); 779 ret = 1; 780 781err: 782 BN_CTX_end(ctx); 783 return ret; 784 } 785#endif 786 787/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 788 * or yy, xx could equal yy. 789 * 790 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 791 * function is only provided for convenience; for best performance, use the 792 * BN_GF2m_mod_div function. 793 */ 794int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx) 795 { 796 BIGNUM *field; 797 int ret = 0; 798 799 bn_check_top(yy); 800 bn_check_top(xx); 801 802 BN_CTX_start(ctx); 803 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 804 if (!BN_GF2m_arr2poly(p, field)) goto err; 805 806 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 807 bn_check_top(r); 808 809err: 810 BN_CTX_end(ctx); 811 return ret; 812 } 813 814 815/* Compute the bth power of a, reduce modulo p, and store 816 * the result in r. r could be a. 817 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 818 */ 819int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) 820 { 821 int ret = 0, i, n; 822 BIGNUM *u; 823 824 bn_check_top(a); 825 bn_check_top(b); 826 827 if (BN_is_zero(b)) 828 return(BN_one(r)); 829 830 if (BN_abs_is_word(b, 1)) 831 return (BN_copy(r, a) != NULL); 832 833 BN_CTX_start(ctx); 834 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 835 836 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 837 838 n = BN_num_bits(b) - 1; 839 for (i = n - 1; i >= 0; i--) 840 { 841 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 842 if (BN_is_bit_set(b, i)) 843 { 844 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 845 } 846 } 847 if (!BN_copy(r, u)) goto err; 848 bn_check_top(r); 849 ret = 1; 850err: 851 BN_CTX_end(ctx); 852 return ret; 853 } 854 855/* Compute the bth power of a, reduce modulo p, and store 856 * the result in r. r could be a. 857 * 858 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 859 * function is only provided for convenience; for best performance, use the 860 * BN_GF2m_mod_exp_arr function. 861 */ 862int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 863 { 864 int ret = 0; 865 const int max = BN_num_bits(p) + 1; 866 int *arr=NULL; 867 bn_check_top(a); 868 bn_check_top(b); 869 bn_check_top(p); 870 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 871 ret = BN_GF2m_poly2arr(p, arr, max); 872 if (!ret || ret > max) 873 { 874 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 875 goto err; 876 } 877 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 878 bn_check_top(r); 879err: 880 if (arr) OPENSSL_free(arr); 881 return ret; 882 } 883 884/* Compute the square root of a, reduce modulo p, and store 885 * the result in r. r could be a. 886 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 887 */ 888int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) 889 { 890 int ret = 0; 891 BIGNUM *u; 892 893 bn_check_top(a); 894 895 if (!p[0]) 896 { 897 /* reduction mod 1 => return 0 */ 898 BN_zero(r); 899 return 1; 900 } 901 902 BN_CTX_start(ctx); 903 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 904 905 if (!BN_set_bit(u, p[0] - 1)) goto err; 906 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 907 bn_check_top(r); 908 909err: 910 BN_CTX_end(ctx); 911 return ret; 912 } 913 914/* Compute the square root of a, reduce modulo p, and store 915 * the result in r. r could be a. 916 * 917 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 918 * function is only provided for convenience; for best performance, use the 919 * BN_GF2m_mod_sqrt_arr function. 920 */ 921int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 922 { 923 int ret = 0; 924 const int max = BN_num_bits(p) + 1; 925 int *arr=NULL; 926 bn_check_top(a); 927 bn_check_top(p); 928 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 929 ret = BN_GF2m_poly2arr(p, arr, max); 930 if (!ret || ret > max) 931 { 932 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 933 goto err; 934 } 935 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 936 bn_check_top(r); 937err: 938 if (arr) OPENSSL_free(arr); 939 return ret; 940 } 941 942/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 943 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 944 */ 945int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx) 946 { 947 int ret = 0, count = 0, j; 948 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 949 950 bn_check_top(a_); 951 952 if (!p[0]) 953 { 954 /* reduction mod 1 => return 0 */ 955 BN_zero(r); 956 return 1; 957 } 958 959 BN_CTX_start(ctx); 960 a = BN_CTX_get(ctx); 961 z = BN_CTX_get(ctx); 962 w = BN_CTX_get(ctx); 963 if (w == NULL) goto err; 964 965 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 966 967 if (BN_is_zero(a)) 968 { 969 BN_zero(r); 970 ret = 1; 971 goto err; 972 } 973 974 if (p[0] & 0x1) /* m is odd */ 975 { 976 /* compute half-trace of a */ 977 if (!BN_copy(z, a)) goto err; 978 for (j = 1; j <= (p[0] - 1) / 2; j++) 979 { 980 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 981 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 982 if (!BN_GF2m_add(z, z, a)) goto err; 983 } 984 985 } 986 else /* m is even */ 987 { 988 rho = BN_CTX_get(ctx); 989 w2 = BN_CTX_get(ctx); 990 tmp = BN_CTX_get(ctx); 991 if (tmp == NULL) goto err; 992 do 993 { 994 if (!BN_rand(rho, p[0], 0, 0)) goto err; 995 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 996 BN_zero(z); 997 if (!BN_copy(w, rho)) goto err; 998 for (j = 1; j <= p[0] - 1; j++) 999 { 1000 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 1001 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 1002 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 1003 if (!BN_GF2m_add(z, z, tmp)) goto err; 1004 if (!BN_GF2m_add(w, w2, rho)) goto err; 1005 } 1006 count++; 1007 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1008 if (BN_is_zero(w)) 1009 { 1010 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 1011 goto err; 1012 } 1013 } 1014 1015 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 1016 if (!BN_GF2m_add(w, z, w)) goto err; 1017 if (BN_GF2m_cmp(w, a)) 1018 { 1019 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1020 goto err; 1021 } 1022 1023 if (!BN_copy(r, z)) goto err; 1024 bn_check_top(r); 1025 1026 ret = 1; 1027 1028err: 1029 BN_CTX_end(ctx); 1030 return ret; 1031 } 1032 1033/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 1034 * 1035 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 1036 * function is only provided for convenience; for best performance, use the 1037 * BN_GF2m_mod_solve_quad_arr function. 1038 */ 1039int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1040 { 1041 int ret = 0; 1042 const int max = BN_num_bits(p) + 1; 1043 int *arr=NULL; 1044 bn_check_top(a); 1045 bn_check_top(p); 1046 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * 1047 max)) == NULL) goto err; 1048 ret = BN_GF2m_poly2arr(p, arr, max); 1049 if (!ret || ret > max) 1050 { 1051 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 1052 goto err; 1053 } 1054 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1055 bn_check_top(r); 1056err: 1057 if (arr) OPENSSL_free(arr); 1058 return ret; 1059 } 1060 1061/* Convert the bit-string representation of a polynomial 1062 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding 1063 * to the bits with non-zero coefficient. Array is terminated with -1. 1064 * Up to max elements of the array will be filled. Return value is total 1065 * number of array elements that would be filled if array was large enough. 1066 */ 1067int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1068 { 1069 int i, j, k = 0; 1070 BN_ULONG mask; 1071 1072 if (BN_is_zero(a)) 1073 return 0; 1074 1075 for (i = a->top - 1; i >= 0; i--) 1076 { 1077 if (!a->d[i]) 1078 /* skip word if a->d[i] == 0 */ 1079 continue; 1080 mask = BN_TBIT; 1081 for (j = BN_BITS2 - 1; j >= 0; j--) 1082 { 1083 if (a->d[i] & mask) 1084 { 1085 if (k < max) p[k] = BN_BITS2 * i + j; 1086 k++; 1087 } 1088 mask >>= 1; 1089 } 1090 } 1091 1092 if (k < max) { 1093 p[k] = -1; 1094 k++; 1095 } 1096 1097 return k; 1098 } 1099 1100/* Convert the coefficient array representation of a polynomial to a 1101 * bit-string. The array must be terminated by -1. 1102 */ 1103int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1104 { 1105 int i; 1106 1107 bn_check_top(a); 1108 BN_zero(a); 1109 for (i = 0; p[i] != -1; i++) 1110 { 1111 if (BN_set_bit(a, p[i]) == 0) 1112 return 0; 1113 } 1114 bn_check_top(a); 1115 1116 return 1; 1117 } 1118 1119#endif 1120