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, ubits = BN_num_bits(u), 572 vbits = BN_num_bits(v), /* v is copy of p */ 573 top = p->top; 574 BN_ULONG *udp,*bdp,*vdp,*cdp; 575 576 bn_wexpand(u,top); udp = u->d; 577 for (i=u->top;i<top;i++) udp[i] = 0; 578 u->top = top; 579 bn_wexpand(b,top); bdp = b->d; 580 bdp[0] = 1; 581 for (i=1;i<top;i++) bdp[i] = 0; 582 b->top = top; 583 bn_wexpand(c,top); cdp = c->d; 584 for (i=0;i<top;i++) cdp[i] = 0; 585 c->top = top; 586 vdp = v->d; /* It pays off to "cache" *->d pointers, because 587 * it allows optimizer to be more aggressive. 588 * But we don't have to "cache" p->d, because *p 589 * is declared 'const'... */ 590 while (1) 591 { 592 while (ubits && !(udp[0]&1)) 593 { 594 BN_ULONG u0,u1,b0,b1,mask; 595 596 u0 = udp[0]; 597 b0 = bdp[0]; 598 mask = (BN_ULONG)0-(b0&1); 599 b0 ^= p->d[0]&mask; 600 for (i=0;i<top-1;i++) 601 { 602 u1 = udp[i+1]; 603 udp[i] = ((u0>>1)|(u1<<(BN_BITS2-1)))&BN_MASK2; 604 u0 = u1; 605 b1 = bdp[i+1]^(p->d[i+1]&mask); 606 bdp[i] = ((b0>>1)|(b1<<(BN_BITS2-1)))&BN_MASK2; 607 b0 = b1; 608 } 609 udp[i] = u0>>1; 610 bdp[i] = b0>>1; 611 ubits--; 612 } 613 614 if (ubits<=BN_BITS2 && udp[0]==1) break; 615 616 if (ubits<vbits) 617 { 618 i = ubits; ubits = vbits; vbits = i; 619 tmp = u; u = v; v = tmp; 620 tmp = b; b = c; c = tmp; 621 udp = vdp; vdp = v->d; 622 bdp = cdp; cdp = c->d; 623 } 624 for(i=0;i<top;i++) 625 { 626 udp[i] ^= vdp[i]; 627 bdp[i] ^= cdp[i]; 628 } 629 if (ubits==vbits) 630 { 631 BN_ULONG ul; 632 int utop = (ubits-1)/BN_BITS2; 633 634 while ((ul=udp[utop])==0 && utop) utop--; 635 ubits = utop*BN_BITS2 + BN_num_bits_word(ul); 636 } 637 } 638 bn_correct_top(b); 639 } 640#endif 641 642 if (!BN_copy(r, b)) goto err; 643 bn_check_top(r); 644 ret = 1; 645 646err: 647#ifdef BN_DEBUG /* BN_CTX_end would complain about the expanded form */ 648 bn_correct_top(c); 649 bn_correct_top(u); 650 bn_correct_top(v); 651#endif 652 BN_CTX_end(ctx); 653 return ret; 654 } 655 656/* Invert xx, reduce modulo p, and store the result in r. r could be xx. 657 * 658 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 659 * function is only provided for convenience; for best performance, use the 660 * BN_GF2m_mod_inv function. 661 */ 662int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx) 663 { 664 BIGNUM *field; 665 int ret = 0; 666 667 bn_check_top(xx); 668 BN_CTX_start(ctx); 669 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 670 if (!BN_GF2m_arr2poly(p, field)) goto err; 671 672 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 673 bn_check_top(r); 674 675err: 676 BN_CTX_end(ctx); 677 return ret; 678 } 679 680 681#ifndef OPENSSL_SUN_GF2M_DIV 682/* Divide y by x, reduce modulo p, and store the result in r. r could be x 683 * or y, x could equal y. 684 */ 685int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 686 { 687 BIGNUM *xinv = NULL; 688 int ret = 0; 689 690 bn_check_top(y); 691 bn_check_top(x); 692 bn_check_top(p); 693 694 BN_CTX_start(ctx); 695 xinv = BN_CTX_get(ctx); 696 if (xinv == NULL) goto err; 697 698 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 699 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 700 bn_check_top(r); 701 ret = 1; 702 703err: 704 BN_CTX_end(ctx); 705 return ret; 706 } 707#else 708/* Divide y by x, reduce modulo p, and store the result in r. r could be x 709 * or y, x could equal y. 710 * Uses algorithm Modular_Division_GF(2^m) from 711 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 712 * the Great Divide". 713 */ 714int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 715 { 716 BIGNUM *a, *b, *u, *v; 717 int ret = 0; 718 719 bn_check_top(y); 720 bn_check_top(x); 721 bn_check_top(p); 722 723 BN_CTX_start(ctx); 724 725 a = BN_CTX_get(ctx); 726 b = BN_CTX_get(ctx); 727 u = BN_CTX_get(ctx); 728 v = BN_CTX_get(ctx); 729 if (v == NULL) goto err; 730 731 /* reduce x and y mod p */ 732 if (!BN_GF2m_mod(u, y, p)) goto err; 733 if (!BN_GF2m_mod(a, x, p)) goto err; 734 if (!BN_copy(b, p)) goto err; 735 736 while (!BN_is_odd(a)) 737 { 738 if (!BN_rshift1(a, a)) goto err; 739 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 740 if (!BN_rshift1(u, u)) goto err; 741 } 742 743 do 744 { 745 if (BN_GF2m_cmp(b, a) > 0) 746 { 747 if (!BN_GF2m_add(b, b, a)) goto err; 748 if (!BN_GF2m_add(v, v, u)) goto err; 749 do 750 { 751 if (!BN_rshift1(b, b)) goto err; 752 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 753 if (!BN_rshift1(v, v)) goto err; 754 } while (!BN_is_odd(b)); 755 } 756 else if (BN_abs_is_word(a, 1)) 757 break; 758 else 759 { 760 if (!BN_GF2m_add(a, a, b)) goto err; 761 if (!BN_GF2m_add(u, u, v)) goto err; 762 do 763 { 764 if (!BN_rshift1(a, a)) goto err; 765 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 766 if (!BN_rshift1(u, u)) goto err; 767 } while (!BN_is_odd(a)); 768 } 769 } while (1); 770 771 if (!BN_copy(r, u)) goto err; 772 bn_check_top(r); 773 ret = 1; 774 775err: 776 BN_CTX_end(ctx); 777 return ret; 778 } 779#endif 780 781/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 782 * or yy, xx could equal yy. 783 * 784 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 785 * function is only provided for convenience; for best performance, use the 786 * BN_GF2m_mod_div function. 787 */ 788int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx) 789 { 790 BIGNUM *field; 791 int ret = 0; 792 793 bn_check_top(yy); 794 bn_check_top(xx); 795 796 BN_CTX_start(ctx); 797 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 798 if (!BN_GF2m_arr2poly(p, field)) goto err; 799 800 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 801 bn_check_top(r); 802 803err: 804 BN_CTX_end(ctx); 805 return ret; 806 } 807 808 809/* Compute the bth power of a, reduce modulo p, and store 810 * the result in r. r could be a. 811 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 812 */ 813int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) 814 { 815 int ret = 0, i, n; 816 BIGNUM *u; 817 818 bn_check_top(a); 819 bn_check_top(b); 820 821 if (BN_is_zero(b)) 822 return(BN_one(r)); 823 824 if (BN_abs_is_word(b, 1)) 825 return (BN_copy(r, a) != NULL); 826 827 BN_CTX_start(ctx); 828 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 829 830 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 831 832 n = BN_num_bits(b) - 1; 833 for (i = n - 1; i >= 0; i--) 834 { 835 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 836 if (BN_is_bit_set(b, i)) 837 { 838 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 839 } 840 } 841 if (!BN_copy(r, u)) goto err; 842 bn_check_top(r); 843 ret = 1; 844err: 845 BN_CTX_end(ctx); 846 return ret; 847 } 848 849/* Compute the bth power of a, reduce modulo p, and store 850 * the result in r. r could be a. 851 * 852 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 853 * function is only provided for convenience; for best performance, use the 854 * BN_GF2m_mod_exp_arr function. 855 */ 856int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 857 { 858 int ret = 0; 859 const int max = BN_num_bits(p) + 1; 860 int *arr=NULL; 861 bn_check_top(a); 862 bn_check_top(b); 863 bn_check_top(p); 864 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 865 ret = BN_GF2m_poly2arr(p, arr, max); 866 if (!ret || ret > max) 867 { 868 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 869 goto err; 870 } 871 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 872 bn_check_top(r); 873err: 874 if (arr) OPENSSL_free(arr); 875 return ret; 876 } 877 878/* Compute the square root of a, reduce modulo p, and store 879 * the result in r. r could be a. 880 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 881 */ 882int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) 883 { 884 int ret = 0; 885 BIGNUM *u; 886 887 bn_check_top(a); 888 889 if (!p[0]) 890 { 891 /* reduction mod 1 => return 0 */ 892 BN_zero(r); 893 return 1; 894 } 895 896 BN_CTX_start(ctx); 897 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 898 899 if (!BN_set_bit(u, p[0] - 1)) goto err; 900 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 901 bn_check_top(r); 902 903err: 904 BN_CTX_end(ctx); 905 return ret; 906 } 907 908/* Compute the square root of a, reduce modulo p, and store 909 * the result in r. r could be a. 910 * 911 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 912 * function is only provided for convenience; for best performance, use the 913 * BN_GF2m_mod_sqrt_arr function. 914 */ 915int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 916 { 917 int ret = 0; 918 const int max = BN_num_bits(p) + 1; 919 int *arr=NULL; 920 bn_check_top(a); 921 bn_check_top(p); 922 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 923 ret = BN_GF2m_poly2arr(p, arr, max); 924 if (!ret || ret > max) 925 { 926 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 927 goto err; 928 } 929 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 930 bn_check_top(r); 931err: 932 if (arr) OPENSSL_free(arr); 933 return ret; 934 } 935 936/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 937 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 938 */ 939int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx) 940 { 941 int ret = 0, count = 0, j; 942 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 943 944 bn_check_top(a_); 945 946 if (!p[0]) 947 { 948 /* reduction mod 1 => return 0 */ 949 BN_zero(r); 950 return 1; 951 } 952 953 BN_CTX_start(ctx); 954 a = BN_CTX_get(ctx); 955 z = BN_CTX_get(ctx); 956 w = BN_CTX_get(ctx); 957 if (w == NULL) goto err; 958 959 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 960 961 if (BN_is_zero(a)) 962 { 963 BN_zero(r); 964 ret = 1; 965 goto err; 966 } 967 968 if (p[0] & 0x1) /* m is odd */ 969 { 970 /* compute half-trace of a */ 971 if (!BN_copy(z, a)) goto err; 972 for (j = 1; j <= (p[0] - 1) / 2; j++) 973 { 974 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 975 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 976 if (!BN_GF2m_add(z, z, a)) goto err; 977 } 978 979 } 980 else /* m is even */ 981 { 982 rho = BN_CTX_get(ctx); 983 w2 = BN_CTX_get(ctx); 984 tmp = BN_CTX_get(ctx); 985 if (tmp == NULL) goto err; 986 do 987 { 988 if (!BN_rand(rho, p[0], 0, 0)) goto err; 989 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 990 BN_zero(z); 991 if (!BN_copy(w, rho)) goto err; 992 for (j = 1; j <= p[0] - 1; j++) 993 { 994 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 995 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 996 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 997 if (!BN_GF2m_add(z, z, tmp)) goto err; 998 if (!BN_GF2m_add(w, w2, rho)) goto err; 999 } 1000 count++; 1001 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 1002 if (BN_is_zero(w)) 1003 { 1004 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 1005 goto err; 1006 } 1007 } 1008 1009 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 1010 if (!BN_GF2m_add(w, z, w)) goto err; 1011 if (BN_GF2m_cmp(w, a)) 1012 { 1013 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1014 goto err; 1015 } 1016 1017 if (!BN_copy(r, z)) goto err; 1018 bn_check_top(r); 1019 1020 ret = 1; 1021 1022err: 1023 BN_CTX_end(ctx); 1024 return ret; 1025 } 1026 1027/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 1028 * 1029 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 1030 * function is only provided for convenience; for best performance, use the 1031 * BN_GF2m_mod_solve_quad_arr function. 1032 */ 1033int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1034 { 1035 int ret = 0; 1036 const int max = BN_num_bits(p) + 1; 1037 int *arr=NULL; 1038 bn_check_top(a); 1039 bn_check_top(p); 1040 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * 1041 max)) == NULL) goto err; 1042 ret = BN_GF2m_poly2arr(p, arr, max); 1043 if (!ret || ret > max) 1044 { 1045 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 1046 goto err; 1047 } 1048 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1049 bn_check_top(r); 1050err: 1051 if (arr) OPENSSL_free(arr); 1052 return ret; 1053 } 1054 1055/* Convert the bit-string representation of a polynomial 1056 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding 1057 * to the bits with non-zero coefficient. Array is terminated with -1. 1058 * Up to max elements of the array will be filled. Return value is total 1059 * number of array elements that would be filled if array was large enough. 1060 */ 1061int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 1062 { 1063 int i, j, k = 0; 1064 BN_ULONG mask; 1065 1066 if (BN_is_zero(a)) 1067 return 0; 1068 1069 for (i = a->top - 1; i >= 0; i--) 1070 { 1071 if (!a->d[i]) 1072 /* skip word if a->d[i] == 0 */ 1073 continue; 1074 mask = BN_TBIT; 1075 for (j = BN_BITS2 - 1; j >= 0; j--) 1076 { 1077 if (a->d[i] & mask) 1078 { 1079 if (k < max) p[k] = BN_BITS2 * i + j; 1080 k++; 1081 } 1082 mask >>= 1; 1083 } 1084 } 1085 1086 if (k < max) { 1087 p[k] = -1; 1088 k++; 1089 } 1090 1091 return k; 1092 } 1093 1094/* Convert the coefficient array representation of a polynomial to a 1095 * bit-string. The array must be terminated by -1. 1096 */ 1097int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1098 { 1099 int i; 1100 1101 bn_check_top(a); 1102 BN_zero(a); 1103 for (i = 0; p[i] != -1; i++) 1104 { 1105 if (BN_set_bit(a, p[i]) == 0) 1106 return 0; 1107 } 1108 bn_check_top(a); 1109 1110 return 1; 1111 } 1112 1113#endif 1114