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/* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */ 98#define MAX_ITERATIONS 50 99 100static const BN_ULONG SQR_tb[16] = 101 { 0, 1, 4, 5, 16, 17, 20, 21, 102 64, 65, 68, 69, 80, 81, 84, 85 }; 103/* Platform-specific macros to accelerate squaring. */ 104#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 105#define SQR1(w) \ 106 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \ 107 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \ 108 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \ 109 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF] 110#define SQR0(w) \ 111 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \ 112 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \ 113 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ 114 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 115#endif 116#ifdef THIRTY_TWO_BIT 117#define SQR1(w) \ 118 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \ 119 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF] 120#define SQR0(w) \ 121 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ 122 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 123#endif 124#ifdef SIXTEEN_BIT 125#define SQR1(w) \ 126 SQR_tb[(w) >> 12 & 0xF] << 8 | SQR_tb[(w) >> 8 & 0xF] 127#define SQR0(w) \ 128 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] 129#endif 130#ifdef EIGHT_BIT 131#define SQR1(w) \ 132 SQR_tb[(w) >> 4 & 0xF] 133#define SQR0(w) \ 134 SQR_tb[(w) & 15] 135#endif 136 137/* Product of two polynomials a, b each with degree < BN_BITS2 - 1, 138 * result is a polynomial r with degree < 2 * BN_BITS - 1 139 * The caller MUST ensure that the variables have the right amount 140 * of space allocated. 141 */ 142#ifdef EIGHT_BIT 143static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 144 { 145 register BN_ULONG h, l, s; 146 BN_ULONG tab[4], top1b = a >> 7; 147 register BN_ULONG a1, a2; 148 149 a1 = a & (0x7F); a2 = a1 << 1; 150 151 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2; 152 153 s = tab[b & 0x3]; l = s; 154 s = tab[b >> 2 & 0x3]; l ^= s << 2; h = s >> 6; 155 s = tab[b >> 4 & 0x3]; l ^= s << 4; h ^= s >> 4; 156 s = tab[b >> 6 ]; l ^= s << 6; h ^= s >> 2; 157 158 /* compensate for the top bit of a */ 159 160 if (top1b & 01) { l ^= b << 7; h ^= b >> 1; } 161 162 *r1 = h; *r0 = l; 163 } 164#endif 165#ifdef SIXTEEN_BIT 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[4], top1b = a >> 15; 170 register BN_ULONG a1, a2; 171 172 a1 = a & (0x7FFF); a2 = a1 << 1; 173 174 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2; 175 176 s = tab[b & 0x3]; l = s; 177 s = tab[b >> 2 & 0x3]; l ^= s << 2; h = s >> 14; 178 s = tab[b >> 4 & 0x3]; l ^= s << 4; h ^= s >> 12; 179 s = tab[b >> 6 & 0x3]; l ^= s << 6; h ^= s >> 10; 180 s = tab[b >> 8 & 0x3]; l ^= s << 8; h ^= s >> 8; 181 s = tab[b >>10 & 0x3]; l ^= s << 10; h ^= s >> 6; 182 s = tab[b >>12 & 0x3]; l ^= s << 12; h ^= s >> 4; 183 s = tab[b >>14 ]; l ^= s << 14; h ^= s >> 2; 184 185 /* compensate for the top bit of a */ 186 187 if (top1b & 01) { l ^= b << 15; h ^= b >> 1; } 188 189 *r1 = h; *r0 = l; 190 } 191#endif 192#ifdef THIRTY_TWO_BIT 193static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 194 { 195 register BN_ULONG h, l, s; 196 BN_ULONG tab[8], top2b = a >> 30; 197 register BN_ULONG a1, a2, a4; 198 199 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1; 200 201 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2; 202 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4; 203 204 s = tab[b & 0x7]; l = s; 205 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29; 206 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26; 207 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23; 208 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20; 209 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17; 210 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14; 211 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11; 212 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8; 213 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5; 214 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2; 215 216 /* compensate for the top two bits of a */ 217 218 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; } 219 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; } 220 221 *r1 = h; *r0 = l; 222 } 223#endif 224#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 225static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 226 { 227 register BN_ULONG h, l, s; 228 BN_ULONG tab[16], top3b = a >> 61; 229 register BN_ULONG a1, a2, a4, a8; 230 231 a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1; 232 233 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2; 234 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4; 235 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8; 236 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8; 237 238 s = tab[b & 0xF]; l = s; 239 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60; 240 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56; 241 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52; 242 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48; 243 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44; 244 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40; 245 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36; 246 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32; 247 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28; 248 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24; 249 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20; 250 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16; 251 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12; 252 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8; 253 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4; 254 255 /* compensate for the top three bits of a */ 256 257 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; } 258 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; } 259 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; } 260 261 *r1 = h; *r0 = l; 262 } 263#endif 264 265/* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, 266 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 267 * The caller MUST ensure that the variables have the right amount 268 * of space allocated. 269 */ 270static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0) 271 { 272 BN_ULONG m1, m0; 273 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ 274 bn_GF2m_mul_1x1(r+3, r+2, a1, b1); 275 bn_GF2m_mul_1x1(r+1, r, a0, b0); 276 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); 277 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ 278 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ 279 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ 280 } 281 282 283/* Add polynomials a and b and store result in r; r could be a or b, a and b 284 * could be equal; r is the bitwise XOR of a and b. 285 */ 286int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) 287 { 288 int i; 289 const BIGNUM *at, *bt; 290 291 bn_check_top(a); 292 bn_check_top(b); 293 294 if (a->top < b->top) { at = b; bt = a; } 295 else { at = a; bt = b; } 296 297 if(bn_wexpand(r, at->top) == NULL) 298 return 0; 299 300 for (i = 0; i < bt->top; i++) 301 { 302 r->d[i] = at->d[i] ^ bt->d[i]; 303 } 304 for (; i < at->top; i++) 305 { 306 r->d[i] = at->d[i]; 307 } 308 309 r->top = at->top; 310 bn_correct_top(r); 311 312 return 1; 313 } 314 315 316/* Some functions allow for representation of the irreducible polynomials 317 * as an int[], say p. The irreducible f(t) is then of the form: 318 * t^p[0] + t^p[1] + ... + t^p[k] 319 * where m = p[0] > p[1] > ... > p[k] = 0. 320 */ 321 322 323/* Performs modular reduction of a and store result in r. r could be a. */ 324int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[]) 325 { 326 int j, k; 327 int n, dN, d0, d1; 328 BN_ULONG zz, *z; 329 330 bn_check_top(a); 331 332 if (!p[0]) 333 { 334 /* reduction mod 1 => return 0 */ 335 BN_zero(r); 336 return 1; 337 } 338 339 /* Since the algorithm does reduction in the r value, if a != r, copy 340 * the contents of a into r so we can do reduction in r. 341 */ 342 if (a != r) 343 { 344 if (!bn_wexpand(r, a->top)) return 0; 345 for (j = 0; j < a->top; j++) 346 { 347 r->d[j] = a->d[j]; 348 } 349 r->top = a->top; 350 } 351 z = r->d; 352 353 /* start reduction */ 354 dN = p[0] / BN_BITS2; 355 for (j = r->top - 1; j > dN;) 356 { 357 zz = z[j]; 358 if (z[j] == 0) { j--; continue; } 359 z[j] = 0; 360 361 for (k = 1; p[k] != 0; k++) 362 { 363 /* reducing component t^p[k] */ 364 n = p[0] - p[k]; 365 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0; 366 n /= BN_BITS2; 367 z[j-n] ^= (zz>>d0); 368 if (d0) z[j-n-1] ^= (zz<<d1); 369 } 370 371 /* reducing component t^0 */ 372 n = dN; 373 d0 = p[0] % BN_BITS2; 374 d1 = BN_BITS2 - d0; 375 z[j-n] ^= (zz >> d0); 376 if (d0) z[j-n-1] ^= (zz << d1); 377 } 378 379 /* final round of reduction */ 380 while (j == dN) 381 { 382 383 d0 = p[0] % BN_BITS2; 384 zz = z[dN] >> d0; 385 if (zz == 0) break; 386 d1 = BN_BITS2 - d0; 387 388 /* clear up the top d1 bits */ 389 if (d0) 390 z[dN] = (z[dN] << d1) >> d1; 391 else 392 z[dN] = 0; 393 z[0] ^= zz; /* reduction t^0 component */ 394 395 for (k = 1; p[k] != 0; k++) 396 { 397 BN_ULONG tmp_ulong; 398 399 /* reducing component t^p[k]*/ 400 n = p[k] / BN_BITS2; 401 d0 = p[k] % BN_BITS2; 402 d1 = BN_BITS2 - d0; 403 z[n] ^= (zz << d0); 404 tmp_ulong = zz >> d1; 405 if (d0 && tmp_ulong) 406 z[n+1] ^= tmp_ulong; 407 } 408 409 410 } 411 412 bn_correct_top(r); 413 return 1; 414 } 415 416/* Performs modular reduction of a by p and store result in r. r could be a. 417 * 418 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 419 * function is only provided for convenience; for best performance, use the 420 * BN_GF2m_mod_arr function. 421 */ 422int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 423 { 424 int ret = 0; 425 const int max = BN_num_bits(p); 426 unsigned int *arr=NULL; 427 bn_check_top(a); 428 bn_check_top(p); 429 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 430 ret = BN_GF2m_poly2arr(p, arr, max); 431 if (!ret || ret > max) 432 { 433 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH); 434 goto err; 435 } 436 ret = BN_GF2m_mod_arr(r, a, arr); 437 bn_check_top(r); 438err: 439 if (arr) OPENSSL_free(arr); 440 return ret; 441 } 442 443 444/* Compute the product of two polynomials a and b, reduce modulo p, and store 445 * the result in r. r could be a or b; a could be b. 446 */ 447int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx) 448 { 449 int zlen, i, j, k, ret = 0; 450 BIGNUM *s; 451 BN_ULONG x1, x0, y1, y0, zz[4]; 452 453 bn_check_top(a); 454 bn_check_top(b); 455 456 if (a == b) 457 { 458 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 459 } 460 461 BN_CTX_start(ctx); 462 if ((s = BN_CTX_get(ctx)) == NULL) goto err; 463 464 zlen = a->top + b->top + 4; 465 if (!bn_wexpand(s, zlen)) goto err; 466 s->top = zlen; 467 468 for (i = 0; i < zlen; i++) s->d[i] = 0; 469 470 for (j = 0; j < b->top; j += 2) 471 { 472 y0 = b->d[j]; 473 y1 = ((j+1) == b->top) ? 0 : b->d[j+1]; 474 for (i = 0; i < a->top; i += 2) 475 { 476 x0 = a->d[i]; 477 x1 = ((i+1) == a->top) ? 0 : a->d[i+1]; 478 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 479 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k]; 480 } 481 } 482 483 bn_correct_top(s); 484 if (BN_GF2m_mod_arr(r, s, p)) 485 ret = 1; 486 bn_check_top(r); 487 488err: 489 BN_CTX_end(ctx); 490 return ret; 491 } 492 493/* Compute the product of two polynomials a and b, reduce modulo p, and store 494 * the result in r. r could be a or b; a could equal b. 495 * 496 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper 497 * function is only provided for convenience; for best performance, use the 498 * BN_GF2m_mod_mul_arr function. 499 */ 500int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 501 { 502 int ret = 0; 503 const int max = BN_num_bits(p); 504 unsigned int *arr=NULL; 505 bn_check_top(a); 506 bn_check_top(b); 507 bn_check_top(p); 508 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 509 ret = BN_GF2m_poly2arr(p, arr, max); 510 if (!ret || ret > max) 511 { 512 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH); 513 goto err; 514 } 515 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 516 bn_check_top(r); 517err: 518 if (arr) OPENSSL_free(arr); 519 return ret; 520 } 521 522 523/* Square a, reduce the result mod p, and store it in a. r could be a. */ 524int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx) 525 { 526 int i, ret = 0; 527 BIGNUM *s; 528 529 bn_check_top(a); 530 BN_CTX_start(ctx); 531 if ((s = BN_CTX_get(ctx)) == NULL) return 0; 532 if (!bn_wexpand(s, 2 * a->top)) goto err; 533 534 for (i = a->top - 1; i >= 0; i--) 535 { 536 s->d[2*i+1] = SQR1(a->d[i]); 537 s->d[2*i ] = SQR0(a->d[i]); 538 } 539 540 s->top = 2 * a->top; 541 bn_correct_top(s); 542 if (!BN_GF2m_mod_arr(r, s, p)) goto err; 543 bn_check_top(r); 544 ret = 1; 545err: 546 BN_CTX_end(ctx); 547 return ret; 548 } 549 550/* Square a, reduce the result mod p, and store it in a. r could be a. 551 * 552 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper 553 * function is only provided for convenience; for best performance, use the 554 * BN_GF2m_mod_sqr_arr function. 555 */ 556int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 557 { 558 int ret = 0; 559 const int max = BN_num_bits(p); 560 unsigned int *arr=NULL; 561 562 bn_check_top(a); 563 bn_check_top(p); 564 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 565 ret = BN_GF2m_poly2arr(p, arr, max); 566 if (!ret || ret > max) 567 { 568 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH); 569 goto err; 570 } 571 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 572 bn_check_top(r); 573err: 574 if (arr) OPENSSL_free(arr); 575 return ret; 576 } 577 578 579/* Invert a, reduce modulo p, and store the result in r. r could be a. 580 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from 581 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation 582 * of Elliptic Curve Cryptography Over Binary Fields". 583 */ 584int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 585 { 586 BIGNUM *b, *c, *u, *v, *tmp; 587 int ret = 0; 588 589 bn_check_top(a); 590 bn_check_top(p); 591 592 BN_CTX_start(ctx); 593 594 b = BN_CTX_get(ctx); 595 c = BN_CTX_get(ctx); 596 u = BN_CTX_get(ctx); 597 v = BN_CTX_get(ctx); 598 if (v == NULL) goto err; 599 600 if (!BN_one(b)) goto err; 601 if (!BN_GF2m_mod(u, a, p)) goto err; 602 if (!BN_copy(v, p)) goto err; 603 604 if (BN_is_zero(u)) goto err; 605 606 while (1) 607 { 608 while (!BN_is_odd(u)) 609 { 610 if (BN_is_zero(u)) goto err; 611 if (!BN_rshift1(u, u)) goto err; 612 if (BN_is_odd(b)) 613 { 614 if (!BN_GF2m_add(b, b, p)) goto err; 615 } 616 if (!BN_rshift1(b, b)) goto err; 617 } 618 619 if (BN_abs_is_word(u, 1)) break; 620 621 if (BN_num_bits(u) < BN_num_bits(v)) 622 { 623 tmp = u; u = v; v = tmp; 624 tmp = b; b = c; c = tmp; 625 } 626 627 if (!BN_GF2m_add(u, u, v)) goto err; 628 if (!BN_GF2m_add(b, b, c)) goto err; 629 } 630 631 632 if (!BN_copy(r, b)) goto err; 633 bn_check_top(r); 634 ret = 1; 635 636err: 637 BN_CTX_end(ctx); 638 return ret; 639 } 640 641/* Invert xx, reduce modulo p, and store the result in r. r could be xx. 642 * 643 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 644 * function is only provided for convenience; for best performance, use the 645 * BN_GF2m_mod_inv function. 646 */ 647int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx) 648 { 649 BIGNUM *field; 650 int ret = 0; 651 652 bn_check_top(xx); 653 BN_CTX_start(ctx); 654 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 655 if (!BN_GF2m_arr2poly(p, field)) goto err; 656 657 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 658 bn_check_top(r); 659 660err: 661 BN_CTX_end(ctx); 662 return ret; 663 } 664 665 666#ifndef OPENSSL_SUN_GF2M_DIV 667/* Divide y by x, reduce modulo p, and store the result in r. r could be x 668 * or y, x could equal y. 669 */ 670int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 671 { 672 BIGNUM *xinv = NULL; 673 int ret = 0; 674 675 bn_check_top(y); 676 bn_check_top(x); 677 bn_check_top(p); 678 679 BN_CTX_start(ctx); 680 xinv = BN_CTX_get(ctx); 681 if (xinv == NULL) goto err; 682 683 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 684 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 685 bn_check_top(r); 686 ret = 1; 687 688err: 689 BN_CTX_end(ctx); 690 return ret; 691 } 692#else 693/* Divide y by x, reduce modulo p, and store the result in r. r could be x 694 * or y, x could equal y. 695 * Uses algorithm Modular_Division_GF(2^m) from 696 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 697 * the Great Divide". 698 */ 699int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 700 { 701 BIGNUM *a, *b, *u, *v; 702 int ret = 0; 703 704 bn_check_top(y); 705 bn_check_top(x); 706 bn_check_top(p); 707 708 BN_CTX_start(ctx); 709 710 a = BN_CTX_get(ctx); 711 b = BN_CTX_get(ctx); 712 u = BN_CTX_get(ctx); 713 v = BN_CTX_get(ctx); 714 if (v == NULL) goto err; 715 716 /* reduce x and y mod p */ 717 if (!BN_GF2m_mod(u, y, p)) goto err; 718 if (!BN_GF2m_mod(a, x, p)) goto err; 719 if (!BN_copy(b, p)) goto err; 720 721 while (!BN_is_odd(a)) 722 { 723 if (!BN_rshift1(a, a)) goto err; 724 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 725 if (!BN_rshift1(u, u)) goto err; 726 } 727 728 do 729 { 730 if (BN_GF2m_cmp(b, a) > 0) 731 { 732 if (!BN_GF2m_add(b, b, a)) goto err; 733 if (!BN_GF2m_add(v, v, u)) goto err; 734 do 735 { 736 if (!BN_rshift1(b, b)) goto err; 737 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 738 if (!BN_rshift1(v, v)) goto err; 739 } while (!BN_is_odd(b)); 740 } 741 else if (BN_abs_is_word(a, 1)) 742 break; 743 else 744 { 745 if (!BN_GF2m_add(a, a, b)) goto err; 746 if (!BN_GF2m_add(u, u, v)) goto err; 747 do 748 { 749 if (!BN_rshift1(a, a)) goto err; 750 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 751 if (!BN_rshift1(u, u)) goto err; 752 } while (!BN_is_odd(a)); 753 } 754 } while (1); 755 756 if (!BN_copy(r, u)) goto err; 757 bn_check_top(r); 758 ret = 1; 759 760err: 761 BN_CTX_end(ctx); 762 return ret; 763 } 764#endif 765 766/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 767 * or yy, xx could equal yy. 768 * 769 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 770 * function is only provided for convenience; for best performance, use the 771 * BN_GF2m_mod_div function. 772 */ 773int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx) 774 { 775 BIGNUM *field; 776 int ret = 0; 777 778 bn_check_top(yy); 779 bn_check_top(xx); 780 781 BN_CTX_start(ctx); 782 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 783 if (!BN_GF2m_arr2poly(p, field)) goto err; 784 785 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 786 bn_check_top(r); 787 788err: 789 BN_CTX_end(ctx); 790 return ret; 791 } 792 793 794/* Compute the bth power of a, reduce modulo p, and store 795 * the result in r. r could be a. 796 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 797 */ 798int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx) 799 { 800 int ret = 0, i, n; 801 BIGNUM *u; 802 803 bn_check_top(a); 804 bn_check_top(b); 805 806 if (BN_is_zero(b)) 807 return(BN_one(r)); 808 809 if (BN_abs_is_word(b, 1)) 810 return (BN_copy(r, a) != NULL); 811 812 BN_CTX_start(ctx); 813 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 814 815 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 816 817 n = BN_num_bits(b) - 1; 818 for (i = n - 1; i >= 0; i--) 819 { 820 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 821 if (BN_is_bit_set(b, i)) 822 { 823 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 824 } 825 } 826 if (!BN_copy(r, u)) goto err; 827 bn_check_top(r); 828 ret = 1; 829err: 830 BN_CTX_end(ctx); 831 return ret; 832 } 833 834/* Compute the bth power of a, reduce modulo p, and store 835 * the result in r. r could be a. 836 * 837 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 838 * function is only provided for convenience; for best performance, use the 839 * BN_GF2m_mod_exp_arr function. 840 */ 841int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 842 { 843 int ret = 0; 844 const int max = BN_num_bits(p); 845 unsigned int *arr=NULL; 846 bn_check_top(a); 847 bn_check_top(b); 848 bn_check_top(p); 849 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 850 ret = BN_GF2m_poly2arr(p, arr, max); 851 if (!ret || ret > max) 852 { 853 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 854 goto err; 855 } 856 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 857 bn_check_top(r); 858err: 859 if (arr) OPENSSL_free(arr); 860 return ret; 861 } 862 863/* Compute the square root of a, reduce modulo p, and store 864 * the result in r. r could be a. 865 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 866 */ 867int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx) 868 { 869 int ret = 0; 870 BIGNUM *u; 871 872 bn_check_top(a); 873 874 if (!p[0]) 875 { 876 /* reduction mod 1 => return 0 */ 877 BN_zero(r); 878 return 1; 879 } 880 881 BN_CTX_start(ctx); 882 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 883 884 if (!BN_set_bit(u, p[0] - 1)) goto err; 885 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 886 bn_check_top(r); 887 888err: 889 BN_CTX_end(ctx); 890 return ret; 891 } 892 893/* Compute the square root of a, reduce modulo p, and store 894 * the result in r. r could be a. 895 * 896 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 897 * function is only provided for convenience; for best performance, use the 898 * BN_GF2m_mod_sqrt_arr function. 899 */ 900int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 901 { 902 int ret = 0; 903 const int max = BN_num_bits(p); 904 unsigned int *arr=NULL; 905 bn_check_top(a); 906 bn_check_top(p); 907 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 908 ret = BN_GF2m_poly2arr(p, arr, max); 909 if (!ret || ret > max) 910 { 911 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 912 goto err; 913 } 914 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 915 bn_check_top(r); 916err: 917 if (arr) OPENSSL_free(arr); 918 return ret; 919 } 920 921/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 922 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 923 */ 924int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx) 925 { 926 int ret = 0, count = 0; 927 unsigned int j; 928 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 929 930 bn_check_top(a_); 931 932 if (!p[0]) 933 { 934 /* reduction mod 1 => return 0 */ 935 BN_zero(r); 936 return 1; 937 } 938 939 BN_CTX_start(ctx); 940 a = BN_CTX_get(ctx); 941 z = BN_CTX_get(ctx); 942 w = BN_CTX_get(ctx); 943 if (w == NULL) goto err; 944 945 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 946 947 if (BN_is_zero(a)) 948 { 949 BN_zero(r); 950 ret = 1; 951 goto err; 952 } 953 954 if (p[0] & 0x1) /* m is odd */ 955 { 956 /* compute half-trace of a */ 957 if (!BN_copy(z, a)) goto err; 958 for (j = 1; j <= (p[0] - 1) / 2; j++) 959 { 960 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 961 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 962 if (!BN_GF2m_add(z, z, a)) goto err; 963 } 964 965 } 966 else /* m is even */ 967 { 968 rho = BN_CTX_get(ctx); 969 w2 = BN_CTX_get(ctx); 970 tmp = BN_CTX_get(ctx); 971 if (tmp == NULL) goto err; 972 do 973 { 974 if (!BN_rand(rho, p[0], 0, 0)) goto err; 975 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 976 BN_zero(z); 977 if (!BN_copy(w, rho)) goto err; 978 for (j = 1; j <= p[0] - 1; j++) 979 { 980 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 981 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 982 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 983 if (!BN_GF2m_add(z, z, tmp)) goto err; 984 if (!BN_GF2m_add(w, w2, rho)) goto err; 985 } 986 count++; 987 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 988 if (BN_is_zero(w)) 989 { 990 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 991 goto err; 992 } 993 } 994 995 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 996 if (!BN_GF2m_add(w, z, w)) goto err; 997 if (BN_GF2m_cmp(w, a)) 998 { 999 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 1000 goto err; 1001 } 1002 1003 if (!BN_copy(r, z)) goto err; 1004 bn_check_top(r); 1005 1006 ret = 1; 1007 1008err: 1009 BN_CTX_end(ctx); 1010 return ret; 1011 } 1012 1013/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 1014 * 1015 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 1016 * function is only provided for convenience; for best performance, use the 1017 * BN_GF2m_mod_solve_quad_arr function. 1018 */ 1019int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1020 { 1021 int ret = 0; 1022 const int max = BN_num_bits(p); 1023 unsigned int *arr=NULL; 1024 bn_check_top(a); 1025 bn_check_top(p); 1026 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * 1027 max)) == NULL) goto err; 1028 ret = BN_GF2m_poly2arr(p, arr, max); 1029 if (!ret || ret > max) 1030 { 1031 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 1032 goto err; 1033 } 1034 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1035 bn_check_top(r); 1036err: 1037 if (arr) OPENSSL_free(arr); 1038 return ret; 1039 } 1040 1041/* Convert the bit-string representation of a polynomial 1042 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array 1043 * of integers corresponding to the bits with non-zero coefficient. 1044 * Up to max elements of the array will be filled. Return value is total 1045 * number of coefficients that would be extracted if array was large enough. 1046 */ 1047int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max) 1048 { 1049 int i, j, k = 0; 1050 BN_ULONG mask; 1051 1052 if (BN_is_zero(a) || !BN_is_bit_set(a, 0)) 1053 /* a_0 == 0 => return error (the unsigned int array 1054 * must be terminated by 0) 1055 */ 1056 return 0; 1057 1058 for (i = a->top - 1; i >= 0; i--) 1059 { 1060 if (!a->d[i]) 1061 /* skip word if a->d[i] == 0 */ 1062 continue; 1063 mask = BN_TBIT; 1064 for (j = BN_BITS2 - 1; j >= 0; j--) 1065 { 1066 if (a->d[i] & mask) 1067 { 1068 if (k < max) p[k] = BN_BITS2 * i + j; 1069 k++; 1070 } 1071 mask >>= 1; 1072 } 1073 } 1074 1075 return k; 1076 } 1077 1078/* Convert the coefficient array representation of a polynomial to a 1079 * bit-string. The array must be terminated by 0. 1080 */ 1081int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a) 1082 { 1083 int i; 1084 1085 bn_check_top(a); 1086 BN_zero(a); 1087 for (i = 0; p[i] != 0; i++) 1088 { 1089 if (BN_set_bit(a, p[i]) == 0) 1090 return 0; 1091 } 1092 BN_set_bit(a, 0); 1093 bn_check_top(a); 1094 1095 return 1; 1096 } 1097 1098