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 bn_wexpand(r, at->top); 298 299 for (i = 0; i < bt->top; i++) 300 { 301 r->d[i] = at->d[i] ^ bt->d[i]; 302 } 303 for (; i < at->top; i++) 304 { 305 r->d[i] = at->d[i]; 306 } 307 308 r->top = at->top; 309 bn_correct_top(r); 310 311 return 1; 312 } 313 314 315/* Some functions allow for representation of the irreducible polynomials 316 * as an int[], say p. The irreducible f(t) is then of the form: 317 * t^p[0] + t^p[1] + ... + t^p[k] 318 * where m = p[0] > p[1] > ... > p[k] = 0. 319 */ 320 321 322/* Performs modular reduction of a and store result in r. r could be a. */ 323int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[]) 324 { 325 int j, k; 326 int n, dN, d0, d1; 327 BN_ULONG zz, *z; 328 329 bn_check_top(a); 330 331 if (!p[0]) 332 { 333 /* reduction mod 1 => return 0 */ 334 BN_zero(r); 335 return 1; 336 } 337 338 /* Since the algorithm does reduction in the r value, if a != r, copy 339 * the contents of a into r so we can do reduction in r. 340 */ 341 if (a != r) 342 { 343 if (!bn_wexpand(r, a->top)) return 0; 344 for (j = 0; j < a->top; j++) 345 { 346 r->d[j] = a->d[j]; 347 } 348 r->top = a->top; 349 } 350 z = r->d; 351 352 /* start reduction */ 353 dN = p[0] / BN_BITS2; 354 for (j = r->top - 1; j > dN;) 355 { 356 zz = z[j]; 357 if (z[j] == 0) { j--; continue; } 358 z[j] = 0; 359 360 for (k = 1; p[k] != 0; k++) 361 { 362 /* reducing component t^p[k] */ 363 n = p[0] - p[k]; 364 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0; 365 n /= BN_BITS2; 366 z[j-n] ^= (zz>>d0); 367 if (d0) z[j-n-1] ^= (zz<<d1); 368 } 369 370 /* reducing component t^0 */ 371 n = dN; 372 d0 = p[0] % BN_BITS2; 373 d1 = BN_BITS2 - d0; 374 z[j-n] ^= (zz >> d0); 375 if (d0) z[j-n-1] ^= (zz << d1); 376 } 377 378 /* final round of reduction */ 379 while (j == dN) 380 { 381 382 d0 = p[0] % BN_BITS2; 383 zz = z[dN] >> d0; 384 if (zz == 0) break; 385 d1 = BN_BITS2 - d0; 386 387 if (d0) z[dN] = (z[dN] << d1) >> d1; /* clear up the top d1 bits */ 388 z[0] ^= zz; /* reduction t^0 component */ 389 390 for (k = 1; p[k] != 0; k++) 391 { 392 BN_ULONG tmp_ulong; 393 394 /* reducing component t^p[k]*/ 395 n = p[k] / BN_BITS2; 396 d0 = p[k] % BN_BITS2; 397 d1 = BN_BITS2 - d0; 398 z[n] ^= (zz << d0); 399 tmp_ulong = zz >> d1; 400 if (d0 && tmp_ulong) 401 z[n+1] ^= tmp_ulong; 402 } 403 404 405 } 406 407 bn_correct_top(r); 408 return 1; 409 } 410 411/* Performs modular reduction of a by p and store result in r. r could be a. 412 * 413 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 414 * function is only provided for convenience; for best performance, use the 415 * BN_GF2m_mod_arr function. 416 */ 417int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 418 { 419 int ret = 0; 420 const int max = BN_num_bits(p); 421 unsigned int *arr=NULL; 422 bn_check_top(a); 423 bn_check_top(p); 424 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 425 ret = BN_GF2m_poly2arr(p, arr, max); 426 if (!ret || ret > max) 427 { 428 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH); 429 goto err; 430 } 431 ret = BN_GF2m_mod_arr(r, a, arr); 432 bn_check_top(r); 433err: 434 if (arr) OPENSSL_free(arr); 435 return ret; 436 } 437 438 439/* Compute the product of two polynomials a and b, reduce modulo p, and store 440 * the result in r. r could be a or b; a could be b. 441 */ 442int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx) 443 { 444 int zlen, i, j, k, ret = 0; 445 BIGNUM *s; 446 BN_ULONG x1, x0, y1, y0, zz[4]; 447 448 bn_check_top(a); 449 bn_check_top(b); 450 451 if (a == b) 452 { 453 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 454 } 455 456 BN_CTX_start(ctx); 457 if ((s = BN_CTX_get(ctx)) == NULL) goto err; 458 459 zlen = a->top + b->top + 4; 460 if (!bn_wexpand(s, zlen)) goto err; 461 s->top = zlen; 462 463 for (i = 0; i < zlen; i++) s->d[i] = 0; 464 465 for (j = 0; j < b->top; j += 2) 466 { 467 y0 = b->d[j]; 468 y1 = ((j+1) == b->top) ? 0 : b->d[j+1]; 469 for (i = 0; i < a->top; i += 2) 470 { 471 x0 = a->d[i]; 472 x1 = ((i+1) == a->top) ? 0 : a->d[i+1]; 473 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 474 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k]; 475 } 476 } 477 478 bn_correct_top(s); 479 if (BN_GF2m_mod_arr(r, s, p)) 480 ret = 1; 481 bn_check_top(r); 482 483err: 484 BN_CTX_end(ctx); 485 return ret; 486 } 487 488/* Compute the product of two polynomials a and b, reduce modulo p, and store 489 * the result in r. r could be a or b; a could equal b. 490 * 491 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper 492 * function is only provided for convenience; for best performance, use the 493 * BN_GF2m_mod_mul_arr function. 494 */ 495int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 496 { 497 int ret = 0; 498 const int max = BN_num_bits(p); 499 unsigned int *arr=NULL; 500 bn_check_top(a); 501 bn_check_top(b); 502 bn_check_top(p); 503 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned 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_MUL,BN_R_INVALID_LENGTH); 508 goto err; 509 } 510 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 511 bn_check_top(r); 512err: 513 if (arr) OPENSSL_free(arr); 514 return ret; 515 } 516 517 518/* Square a, reduce the result mod p, and store it in a. r could be a. */ 519int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx) 520 { 521 int i, ret = 0; 522 BIGNUM *s; 523 524 bn_check_top(a); 525 BN_CTX_start(ctx); 526 if ((s = BN_CTX_get(ctx)) == NULL) return 0; 527 if (!bn_wexpand(s, 2 * a->top)) goto err; 528 529 for (i = a->top - 1; i >= 0; i--) 530 { 531 s->d[2*i+1] = SQR1(a->d[i]); 532 s->d[2*i ] = SQR0(a->d[i]); 533 } 534 535 s->top = 2 * a->top; 536 bn_correct_top(s); 537 if (!BN_GF2m_mod_arr(r, s, p)) goto err; 538 bn_check_top(r); 539 ret = 1; 540err: 541 BN_CTX_end(ctx); 542 return ret; 543 } 544 545/* Square a, reduce the result mod p, and store it in a. r could be a. 546 * 547 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper 548 * function is only provided for convenience; for best performance, use the 549 * BN_GF2m_mod_sqr_arr function. 550 */ 551int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 552 { 553 int ret = 0; 554 const int max = BN_num_bits(p); 555 unsigned int *arr=NULL; 556 557 bn_check_top(a); 558 bn_check_top(p); 559 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 560 ret = BN_GF2m_poly2arr(p, arr, max); 561 if (!ret || ret > max) 562 { 563 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH); 564 goto err; 565 } 566 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 567 bn_check_top(r); 568err: 569 if (arr) OPENSSL_free(arr); 570 return ret; 571 } 572 573 574/* Invert a, reduce modulo p, and store the result in r. r could be a. 575 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from 576 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation 577 * of Elliptic Curve Cryptography Over Binary Fields". 578 */ 579int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 580 { 581 BIGNUM *b, *c, *u, *v, *tmp; 582 int ret = 0; 583 584 bn_check_top(a); 585 bn_check_top(p); 586 587 BN_CTX_start(ctx); 588 589 b = BN_CTX_get(ctx); 590 c = BN_CTX_get(ctx); 591 u = BN_CTX_get(ctx); 592 v = BN_CTX_get(ctx); 593 if (v == NULL) goto err; 594 595 if (!BN_one(b)) goto err; 596 if (!BN_GF2m_mod(u, a, p)) goto err; 597 if (!BN_copy(v, p)) goto err; 598 599 if (BN_is_zero(u)) goto err; 600 601 while (1) 602 { 603 while (!BN_is_odd(u)) 604 { 605 if (!BN_rshift1(u, u)) goto err; 606 if (BN_is_odd(b)) 607 { 608 if (!BN_GF2m_add(b, b, p)) goto err; 609 } 610 if (!BN_rshift1(b, b)) goto err; 611 } 612 613 if (BN_abs_is_word(u, 1)) break; 614 615 if (BN_num_bits(u) < BN_num_bits(v)) 616 { 617 tmp = u; u = v; v = tmp; 618 tmp = b; b = c; c = tmp; 619 } 620 621 if (!BN_GF2m_add(u, u, v)) goto err; 622 if (!BN_GF2m_add(b, b, c)) goto err; 623 } 624 625 626 if (!BN_copy(r, b)) goto err; 627 bn_check_top(r); 628 ret = 1; 629 630err: 631 BN_CTX_end(ctx); 632 return ret; 633 } 634 635/* Invert xx, reduce modulo p, and store the result in r. r could be xx. 636 * 637 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 638 * function is only provided for convenience; for best performance, use the 639 * BN_GF2m_mod_inv function. 640 */ 641int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx) 642 { 643 BIGNUM *field; 644 int ret = 0; 645 646 bn_check_top(xx); 647 BN_CTX_start(ctx); 648 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 649 if (!BN_GF2m_arr2poly(p, field)) goto err; 650 651 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 652 bn_check_top(r); 653 654err: 655 BN_CTX_end(ctx); 656 return ret; 657 } 658 659 660#ifndef OPENSSL_SUN_GF2M_DIV 661/* Divide y by x, reduce modulo p, and store the result in r. r could be x 662 * or y, x could equal y. 663 */ 664int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 665 { 666 BIGNUM *xinv = NULL; 667 int ret = 0; 668 669 bn_check_top(y); 670 bn_check_top(x); 671 bn_check_top(p); 672 673 BN_CTX_start(ctx); 674 xinv = BN_CTX_get(ctx); 675 if (xinv == NULL) goto err; 676 677 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 678 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 679 bn_check_top(r); 680 ret = 1; 681 682err: 683 BN_CTX_end(ctx); 684 return ret; 685 } 686#else 687/* Divide y by x, reduce modulo p, and store the result in r. r could be x 688 * or y, x could equal y. 689 * Uses algorithm Modular_Division_GF(2^m) from 690 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 691 * the Great Divide". 692 */ 693int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 694 { 695 BIGNUM *a, *b, *u, *v; 696 int ret = 0; 697 698 bn_check_top(y); 699 bn_check_top(x); 700 bn_check_top(p); 701 702 BN_CTX_start(ctx); 703 704 a = BN_CTX_get(ctx); 705 b = BN_CTX_get(ctx); 706 u = BN_CTX_get(ctx); 707 v = BN_CTX_get(ctx); 708 if (v == NULL) goto err; 709 710 /* reduce x and y mod p */ 711 if (!BN_GF2m_mod(u, y, p)) goto err; 712 if (!BN_GF2m_mod(a, x, p)) goto err; 713 if (!BN_copy(b, p)) goto err; 714 715 while (!BN_is_odd(a)) 716 { 717 if (!BN_rshift1(a, a)) goto err; 718 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 719 if (!BN_rshift1(u, u)) goto err; 720 } 721 722 do 723 { 724 if (BN_GF2m_cmp(b, a) > 0) 725 { 726 if (!BN_GF2m_add(b, b, a)) goto err; 727 if (!BN_GF2m_add(v, v, u)) goto err; 728 do 729 { 730 if (!BN_rshift1(b, b)) goto err; 731 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 732 if (!BN_rshift1(v, v)) goto err; 733 } while (!BN_is_odd(b)); 734 } 735 else if (BN_abs_is_word(a, 1)) 736 break; 737 else 738 { 739 if (!BN_GF2m_add(a, a, b)) goto err; 740 if (!BN_GF2m_add(u, u, v)) goto err; 741 do 742 { 743 if (!BN_rshift1(a, a)) goto err; 744 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 745 if (!BN_rshift1(u, u)) goto err; 746 } while (!BN_is_odd(a)); 747 } 748 } while (1); 749 750 if (!BN_copy(r, u)) goto err; 751 bn_check_top(r); 752 ret = 1; 753 754err: 755 BN_CTX_end(ctx); 756 return ret; 757 } 758#endif 759 760/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 761 * or yy, xx could equal yy. 762 * 763 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 764 * function is only provided for convenience; for best performance, use the 765 * BN_GF2m_mod_div function. 766 */ 767int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx) 768 { 769 BIGNUM *field; 770 int ret = 0; 771 772 bn_check_top(yy); 773 bn_check_top(xx); 774 775 BN_CTX_start(ctx); 776 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 777 if (!BN_GF2m_arr2poly(p, field)) goto err; 778 779 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 780 bn_check_top(r); 781 782err: 783 BN_CTX_end(ctx); 784 return ret; 785 } 786 787 788/* Compute the bth power of a, reduce modulo p, and store 789 * the result in r. r could be a. 790 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 791 */ 792int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx) 793 { 794 int ret = 0, i, n; 795 BIGNUM *u; 796 797 bn_check_top(a); 798 bn_check_top(b); 799 800 if (BN_is_zero(b)) 801 return(BN_one(r)); 802 803 if (BN_abs_is_word(b, 1)) 804 return (BN_copy(r, a) != NULL); 805 806 BN_CTX_start(ctx); 807 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 808 809 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 810 811 n = BN_num_bits(b) - 1; 812 for (i = n - 1; i >= 0; i--) 813 { 814 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 815 if (BN_is_bit_set(b, i)) 816 { 817 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 818 } 819 } 820 if (!BN_copy(r, u)) goto err; 821 bn_check_top(r); 822 ret = 1; 823err: 824 BN_CTX_end(ctx); 825 return ret; 826 } 827 828/* Compute the bth power of a, reduce modulo p, and store 829 * the result in r. r could be a. 830 * 831 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 832 * function is only provided for convenience; for best performance, use the 833 * BN_GF2m_mod_exp_arr function. 834 */ 835int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 836 { 837 int ret = 0; 838 const int max = BN_num_bits(p); 839 unsigned int *arr=NULL; 840 bn_check_top(a); 841 bn_check_top(b); 842 bn_check_top(p); 843 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 844 ret = BN_GF2m_poly2arr(p, arr, max); 845 if (!ret || ret > max) 846 { 847 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 848 goto err; 849 } 850 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 851 bn_check_top(r); 852err: 853 if (arr) OPENSSL_free(arr); 854 return ret; 855 } 856 857/* Compute the square root of a, reduce modulo p, and store 858 * the result in r. r could be a. 859 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 860 */ 861int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx) 862 { 863 int ret = 0; 864 BIGNUM *u; 865 866 bn_check_top(a); 867 868 if (!p[0]) 869 { 870 /* reduction mod 1 => return 0 */ 871 BN_zero(r); 872 return 1; 873 } 874 875 BN_CTX_start(ctx); 876 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 877 878 if (!BN_set_bit(u, p[0] - 1)) goto err; 879 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 880 bn_check_top(r); 881 882err: 883 BN_CTX_end(ctx); 884 return ret; 885 } 886 887/* Compute the square root of a, reduce modulo p, and store 888 * the result in r. r could be a. 889 * 890 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 891 * function is only provided for convenience; for best performance, use the 892 * BN_GF2m_mod_sqrt_arr function. 893 */ 894int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 895 { 896 int ret = 0; 897 const int max = BN_num_bits(p); 898 unsigned int *arr=NULL; 899 bn_check_top(a); 900 bn_check_top(p); 901 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 902 ret = BN_GF2m_poly2arr(p, arr, max); 903 if (!ret || ret > max) 904 { 905 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 906 goto err; 907 } 908 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 909 bn_check_top(r); 910err: 911 if (arr) OPENSSL_free(arr); 912 return ret; 913 } 914 915/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 916 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 917 */ 918int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx) 919 { 920 int ret = 0, count = 0; 921 unsigned int j; 922 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 923 924 bn_check_top(a_); 925 926 if (!p[0]) 927 { 928 /* reduction mod 1 => return 0 */ 929 BN_zero(r); 930 return 1; 931 } 932 933 BN_CTX_start(ctx); 934 a = BN_CTX_get(ctx); 935 z = BN_CTX_get(ctx); 936 w = BN_CTX_get(ctx); 937 if (w == NULL) goto err; 938 939 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 940 941 if (BN_is_zero(a)) 942 { 943 BN_zero(r); 944 ret = 1; 945 goto err; 946 } 947 948 if (p[0] & 0x1) /* m is odd */ 949 { 950 /* compute half-trace of a */ 951 if (!BN_copy(z, a)) goto err; 952 for (j = 1; j <= (p[0] - 1) / 2; j++) 953 { 954 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 955 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 956 if (!BN_GF2m_add(z, z, a)) goto err; 957 } 958 959 } 960 else /* m is even */ 961 { 962 rho = BN_CTX_get(ctx); 963 w2 = BN_CTX_get(ctx); 964 tmp = BN_CTX_get(ctx); 965 if (tmp == NULL) goto err; 966 do 967 { 968 if (!BN_rand(rho, p[0], 0, 0)) goto err; 969 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 970 BN_zero(z); 971 if (!BN_copy(w, rho)) goto err; 972 for (j = 1; j <= p[0] - 1; j++) 973 { 974 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 975 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 976 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 977 if (!BN_GF2m_add(z, z, tmp)) goto err; 978 if (!BN_GF2m_add(w, w2, rho)) goto err; 979 } 980 count++; 981 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 982 if (BN_is_zero(w)) 983 { 984 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 985 goto err; 986 } 987 } 988 989 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 990 if (!BN_GF2m_add(w, z, w)) goto err; 991 if (BN_GF2m_cmp(w, a)) 992 { 993 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 994 goto err; 995 } 996 997 if (!BN_copy(r, z)) goto err; 998 bn_check_top(r); 999 1000 ret = 1; 1001 1002err: 1003 BN_CTX_end(ctx); 1004 return ret; 1005 } 1006 1007/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 1008 * 1009 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 1010 * function is only provided for convenience; for best performance, use the 1011 * BN_GF2m_mod_solve_quad_arr function. 1012 */ 1013int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1014 { 1015 int ret = 0; 1016 const int max = BN_num_bits(p); 1017 unsigned int *arr=NULL; 1018 bn_check_top(a); 1019 bn_check_top(p); 1020 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * 1021 max)) == NULL) goto err; 1022 ret = BN_GF2m_poly2arr(p, arr, max); 1023 if (!ret || ret > max) 1024 { 1025 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 1026 goto err; 1027 } 1028 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1029 bn_check_top(r); 1030err: 1031 if (arr) OPENSSL_free(arr); 1032 return ret; 1033 } 1034 1035/* Convert the bit-string representation of a polynomial 1036 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array 1037 * of integers corresponding to the bits with non-zero coefficient. 1038 * Up to max elements of the array will be filled. Return value is total 1039 * number of coefficients that would be extracted if array was large enough. 1040 */ 1041int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max) 1042 { 1043 int i, j, k = 0; 1044 BN_ULONG mask; 1045 1046 if (BN_is_zero(a) || !BN_is_bit_set(a, 0)) 1047 /* a_0 == 0 => return error (the unsigned int array 1048 * must be terminated by 0) 1049 */ 1050 return 0; 1051 1052 for (i = a->top - 1; i >= 0; i--) 1053 { 1054 if (!a->d[i]) 1055 /* skip word if a->d[i] == 0 */ 1056 continue; 1057 mask = BN_TBIT; 1058 for (j = BN_BITS2 - 1; j >= 0; j--) 1059 { 1060 if (a->d[i] & mask) 1061 { 1062 if (k < max) p[k] = BN_BITS2 * i + j; 1063 k++; 1064 } 1065 mask >>= 1; 1066 } 1067 } 1068 1069 return k; 1070 } 1071 1072/* Convert the coefficient array representation of a polynomial to a 1073 * bit-string. The array must be terminated by 0. 1074 */ 1075int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a) 1076 { 1077 int i; 1078 1079 bn_check_top(a); 1080 BN_zero(a); 1081 for (i = 0; p[i] != 0; i++) 1082 { 1083 if (BN_set_bit(a, p[i]) == 0) 1084 return 0; 1085 } 1086 BN_set_bit(a, 0); 1087 bn_check_top(a); 1088 1089 return 1; 1090 } 1091 1092