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 (at) 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 (at) cryptsoft.com). This product includes software written by Tim 87 * Hudson (tjh (at) 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 99 100 static 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 124 125 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1, 126 * result is a polynomial r with degree < 2 * BN_BITS - 1 127 * The caller MUST ensure that the variables have the right amount 128 * of space allocated. 129 */ 130 #ifdef THIRTY_TWO_BIT 131 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 132 { 133 register BN_ULONG h, l, s; 134 BN_ULONG tab[8], top2b = a >> 30; 135 register BN_ULONG a1, a2, a4; 136 137 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1; 138 139 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2; 140 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4; 141 142 s = tab[b & 0x7]; l = s; 143 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29; 144 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26; 145 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23; 146 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20; 147 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17; 148 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14; 149 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11; 150 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8; 151 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5; 152 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2; 153 154 /* compensate for the top two bits of a */ 155 156 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; } 157 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; } 158 159 *r1 = h; *r0 = l; 160 } 161 #endif 162 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) 163 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) 164 { 165 register BN_ULONG h, l, s; 166 BN_ULONG tab[16], top3b = a >> 61; 167 register BN_ULONG a1, a2, a4, a8; 168 169 a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1; 170 171 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2; 172 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4; 173 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8; 174 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8; 175 176 s = tab[b & 0xF]; l = s; 177 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60; 178 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56; 179 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52; 180 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48; 181 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44; 182 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40; 183 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36; 184 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32; 185 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28; 186 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24; 187 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20; 188 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16; 189 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12; 190 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8; 191 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4; 192 193 /* compensate for the top three bits of a */ 194 195 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; } 196 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; } 197 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; } 198 199 *r1 = h; *r0 = l; 200 } 201 #endif 202 203 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, 204 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 205 * The caller MUST ensure that the variables have the right amount 206 * of space allocated. 207 */ 208 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0) 209 { 210 BN_ULONG m1, m0; 211 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ 212 bn_GF2m_mul_1x1(r+3, r+2, a1, b1); 213 bn_GF2m_mul_1x1(r+1, r, a0, b0); 214 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); 215 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ 216 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ 217 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ 218 } 219 220 221 /* Add polynomials a and b and store result in r; r could be a or b, a and b 222 * could be equal; r is the bitwise XOR of a and b. 223 */ 224 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) 225 { 226 int i; 227 const BIGNUM *at, *bt; 228 229 bn_check_top(a); 230 bn_check_top(b); 231 232 if (a->top < b->top) { at = b; bt = a; } 233 else { at = a; bt = b; } 234 235 if(bn_wexpand(r, at->top) == NULL) 236 return 0; 237 238 for (i = 0; i < bt->top; i++) 239 { 240 r->d[i] = at->d[i] ^ bt->d[i]; 241 } 242 for (; i < at->top; i++) 243 { 244 r->d[i] = at->d[i]; 245 } 246 247 r->top = at->top; 248 bn_correct_top(r); 249 250 return 1; 251 } 252 253 254 /* Some functions allow for representation of the irreducible polynomials 255 * as an int[], say p. The irreducible f(t) is then of the form: 256 * t^p[0] + t^p[1] + ... + t^p[k] 257 * where m = p[0] > p[1] > ... > p[k] = 0. 258 */ 259 260 261 /* Performs modular reduction of a and store result in r. r could be a. */ 262 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]) 263 { 264 int j, k; 265 int n, dN, d0, d1; 266 BN_ULONG zz, *z; 267 268 bn_check_top(a); 269 270 if (!p[0]) 271 { 272 /* reduction mod 1 => return 0 */ 273 BN_zero(r); 274 return 1; 275 } 276 277 /* Since the algorithm does reduction in the r value, if a != r, copy 278 * the contents of a into r so we can do reduction in r. 279 */ 280 if (a != r) 281 { 282 if (!bn_wexpand(r, a->top)) return 0; 283 for (j = 0; j < a->top; j++) 284 { 285 r->d[j] = a->d[j]; 286 } 287 r->top = a->top; 288 } 289 z = r->d; 290 291 /* start reduction */ 292 dN = p[0] / BN_BITS2; 293 for (j = r->top - 1; j > dN;) 294 { 295 zz = z[j]; 296 if (z[j] == 0) { j--; continue; } 297 z[j] = 0; 298 299 for (k = 1; p[k] != 0; k++) 300 { 301 /* reducing component t^p[k] */ 302 n = p[0] - p[k]; 303 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0; 304 n /= BN_BITS2; 305 z[j-n] ^= (zz>>d0); 306 if (d0) z[j-n-1] ^= (zz<<d1); 307 } 308 309 /* reducing component t^0 */ 310 n = dN; 311 d0 = p[0] % BN_BITS2; 312 d1 = BN_BITS2 - d0; 313 z[j-n] ^= (zz >> d0); 314 if (d0) z[j-n-1] ^= (zz << d1); 315 } 316 317 /* final round of reduction */ 318 while (j == dN) 319 { 320 321 d0 = p[0] % BN_BITS2; 322 zz = z[dN] >> d0; 323 if (zz == 0) break; 324 d1 = BN_BITS2 - d0; 325 326 /* clear up the top d1 bits */ 327 if (d0) 328 z[dN] = (z[dN] << d1) >> d1; 329 else 330 z[dN] = 0; 331 z[0] ^= zz; /* reduction t^0 component */ 332 333 for (k = 1; p[k] != 0; k++) 334 { 335 BN_ULONG tmp_ulong; 336 337 /* reducing component t^p[k]*/ 338 n = p[k] / BN_BITS2; 339 d0 = p[k] % BN_BITS2; 340 d1 = BN_BITS2 - d0; 341 z[n] ^= (zz << d0); 342 tmp_ulong = zz >> d1; 343 if (d0 && tmp_ulong) 344 z[n+1] ^= tmp_ulong; 345 } 346 347 348 } 349 350 bn_correct_top(r); 351 return 1; 352 } 353 354 /* Performs modular reduction of a by p and store result in r. r could be a. 355 * 356 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 357 * function is only provided for convenience; for best performance, use the 358 * BN_GF2m_mod_arr function. 359 */ 360 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 361 { 362 int ret = 0; 363 const int max = BN_num_bits(p) + 1; 364 int *arr=NULL; 365 bn_check_top(a); 366 bn_check_top(p); 367 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 368 ret = BN_GF2m_poly2arr(p, arr, max); 369 if (!ret || ret > max) 370 { 371 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH); 372 goto err; 373 } 374 ret = BN_GF2m_mod_arr(r, a, arr); 375 bn_check_top(r); 376 err: 377 if (arr) OPENSSL_free(arr); 378 return ret; 379 } 380 381 382 /* Compute the product of two polynomials a and b, reduce modulo p, and store 383 * the result in r. r could be a or b; a could be b. 384 */ 385 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) 386 { 387 int zlen, i, j, k, ret = 0; 388 BIGNUM *s; 389 BN_ULONG x1, x0, y1, y0, zz[4]; 390 391 bn_check_top(a); 392 bn_check_top(b); 393 394 if (a == b) 395 { 396 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 397 } 398 399 BN_CTX_start(ctx); 400 if ((s = BN_CTX_get(ctx)) == NULL) goto err; 401 402 zlen = a->top + b->top + 4; 403 if (!bn_wexpand(s, zlen)) goto err; 404 s->top = zlen; 405 406 for (i = 0; i < zlen; i++) s->d[i] = 0; 407 408 for (j = 0; j < b->top; j += 2) 409 { 410 y0 = b->d[j]; 411 y1 = ((j+1) == b->top) ? 0 : b->d[j+1]; 412 for (i = 0; i < a->top; i += 2) 413 { 414 x0 = a->d[i]; 415 x1 = ((i+1) == a->top) ? 0 : a->d[i+1]; 416 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 417 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k]; 418 } 419 } 420 421 bn_correct_top(s); 422 if (BN_GF2m_mod_arr(r, s, p)) 423 ret = 1; 424 bn_check_top(r); 425 426 err: 427 BN_CTX_end(ctx); 428 return ret; 429 } 430 431 /* Compute the product of two polynomials a and b, reduce modulo p, and store 432 * the result in r. r could be a or b; a could equal b. 433 * 434 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper 435 * function is only provided for convenience; for best performance, use the 436 * BN_GF2m_mod_mul_arr function. 437 */ 438 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 439 { 440 int ret = 0; 441 const int max = BN_num_bits(p) + 1; 442 int *arr=NULL; 443 bn_check_top(a); 444 bn_check_top(b); 445 bn_check_top(p); 446 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 447 ret = BN_GF2m_poly2arr(p, arr, max); 448 if (!ret || ret > max) 449 { 450 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH); 451 goto err; 452 } 453 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 454 bn_check_top(r); 455 err: 456 if (arr) OPENSSL_free(arr); 457 return ret; 458 } 459 460 461 /* Square a, reduce the result mod p, and store it in a. r could be a. */ 462 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) 463 { 464 int i, ret = 0; 465 BIGNUM *s; 466 467 bn_check_top(a); 468 BN_CTX_start(ctx); 469 if ((s = BN_CTX_get(ctx)) == NULL) return 0; 470 if (!bn_wexpand(s, 2 * a->top)) goto err; 471 472 for (i = a->top - 1; i >= 0; i--) 473 { 474 s->d[2*i+1] = SQR1(a->d[i]); 475 s->d[2*i ] = SQR0(a->d[i]); 476 } 477 478 s->top = 2 * a->top; 479 bn_correct_top(s); 480 if (!BN_GF2m_mod_arr(r, s, p)) goto err; 481 bn_check_top(r); 482 ret = 1; 483 err: 484 BN_CTX_end(ctx); 485 return ret; 486 } 487 488 /* Square a, reduce the result mod p, and store it in a. r could be a. 489 * 490 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper 491 * function is only provided for convenience; for best performance, use the 492 * BN_GF2m_mod_sqr_arr function. 493 */ 494 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 495 { 496 int ret = 0; 497 const int max = BN_num_bits(p) + 1; 498 int *arr=NULL; 499 500 bn_check_top(a); 501 bn_check_top(p); 502 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 503 ret = BN_GF2m_poly2arr(p, arr, max); 504 if (!ret || ret > max) 505 { 506 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH); 507 goto err; 508 } 509 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 510 bn_check_top(r); 511 err: 512 if (arr) OPENSSL_free(arr); 513 return ret; 514 } 515 516 517 /* Invert a, reduce modulo p, and store the result in r. r could be a. 518 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from 519 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation 520 * of Elliptic Curve Cryptography Over Binary Fields". 521 */ 522 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 523 { 524 BIGNUM *b, *c, *u, *v, *tmp; 525 int ret = 0; 526 527 bn_check_top(a); 528 bn_check_top(p); 529 530 BN_CTX_start(ctx); 531 532 b = BN_CTX_get(ctx); 533 c = BN_CTX_get(ctx); 534 u = BN_CTX_get(ctx); 535 v = BN_CTX_get(ctx); 536 if (v == NULL) goto err; 537 538 if (!BN_one(b)) goto err; 539 if (!BN_GF2m_mod(u, a, p)) goto err; 540 if (!BN_copy(v, p)) goto err; 541 542 if (BN_is_zero(u)) goto err; 543 544 while (1) 545 { 546 while (!BN_is_odd(u)) 547 { 548 if (BN_is_zero(u)) goto err; 549 if (!BN_rshift1(u, u)) goto err; 550 if (BN_is_odd(b)) 551 { 552 if (!BN_GF2m_add(b, b, p)) goto err; 553 } 554 if (!BN_rshift1(b, b)) goto err; 555 } 556 557 if (BN_abs_is_word(u, 1)) break; 558 559 if (BN_num_bits(u) < BN_num_bits(v)) 560 { 561 tmp = u; u = v; v = tmp; 562 tmp = b; b = c; c = tmp; 563 } 564 565 if (!BN_GF2m_add(u, u, v)) goto err; 566 if (!BN_GF2m_add(b, b, c)) goto err; 567 } 568 569 570 if (!BN_copy(r, b)) goto err; 571 bn_check_top(r); 572 ret = 1; 573 574 err: 575 BN_CTX_end(ctx); 576 return ret; 577 } 578 579 /* Invert xx, reduce modulo p, and store the result in r. r could be xx. 580 * 581 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 582 * function is only provided for convenience; for best performance, use the 583 * BN_GF2m_mod_inv function. 584 */ 585 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx) 586 { 587 BIGNUM *field; 588 int ret = 0; 589 590 bn_check_top(xx); 591 BN_CTX_start(ctx); 592 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 593 if (!BN_GF2m_arr2poly(p, field)) goto err; 594 595 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 596 bn_check_top(r); 597 598 err: 599 BN_CTX_end(ctx); 600 return ret; 601 } 602 603 604 #ifndef OPENSSL_SUN_GF2M_DIV 605 /* Divide y by x, reduce modulo p, and store the result in r. r could be x 606 * or y, x could equal y. 607 */ 608 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 609 { 610 BIGNUM *xinv = NULL; 611 int ret = 0; 612 613 bn_check_top(y); 614 bn_check_top(x); 615 bn_check_top(p); 616 617 BN_CTX_start(ctx); 618 xinv = BN_CTX_get(ctx); 619 if (xinv == NULL) goto err; 620 621 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 622 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 623 bn_check_top(r); 624 ret = 1; 625 626 err: 627 BN_CTX_end(ctx); 628 return ret; 629 } 630 #else 631 /* Divide y by x, reduce modulo p, and store the result in r. r could be x 632 * or y, x could equal y. 633 * Uses algorithm Modular_Division_GF(2^m) from 634 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 635 * the Great Divide". 636 */ 637 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 638 { 639 BIGNUM *a, *b, *u, *v; 640 int ret = 0; 641 642 bn_check_top(y); 643 bn_check_top(x); 644 bn_check_top(p); 645 646 BN_CTX_start(ctx); 647 648 a = BN_CTX_get(ctx); 649 b = BN_CTX_get(ctx); 650 u = BN_CTX_get(ctx); 651 v = BN_CTX_get(ctx); 652 if (v == NULL) goto err; 653 654 /* reduce x and y mod p */ 655 if (!BN_GF2m_mod(u, y, p)) goto err; 656 if (!BN_GF2m_mod(a, x, p)) goto err; 657 if (!BN_copy(b, p)) goto err; 658 659 while (!BN_is_odd(a)) 660 { 661 if (!BN_rshift1(a, a)) goto err; 662 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 663 if (!BN_rshift1(u, u)) goto err; 664 } 665 666 do 667 { 668 if (BN_GF2m_cmp(b, a) > 0) 669 { 670 if (!BN_GF2m_add(b, b, a)) goto err; 671 if (!BN_GF2m_add(v, v, u)) goto err; 672 do 673 { 674 if (!BN_rshift1(b, b)) goto err; 675 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 676 if (!BN_rshift1(v, v)) goto err; 677 } while (!BN_is_odd(b)); 678 } 679 else if (BN_abs_is_word(a, 1)) 680 break; 681 else 682 { 683 if (!BN_GF2m_add(a, a, b)) goto err; 684 if (!BN_GF2m_add(u, u, v)) goto err; 685 do 686 { 687 if (!BN_rshift1(a, a)) goto err; 688 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 689 if (!BN_rshift1(u, u)) goto err; 690 } while (!BN_is_odd(a)); 691 } 692 } while (1); 693 694 if (!BN_copy(r, u)) goto err; 695 bn_check_top(r); 696 ret = 1; 697 698 err: 699 BN_CTX_end(ctx); 700 return ret; 701 } 702 #endif 703 704 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 705 * or yy, xx could equal yy. 706 * 707 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 708 * function is only provided for convenience; for best performance, use the 709 * BN_GF2m_mod_div function. 710 */ 711 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx) 712 { 713 BIGNUM *field; 714 int ret = 0; 715 716 bn_check_top(yy); 717 bn_check_top(xx); 718 719 BN_CTX_start(ctx); 720 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 721 if (!BN_GF2m_arr2poly(p, field)) goto err; 722 723 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 724 bn_check_top(r); 725 726 err: 727 BN_CTX_end(ctx); 728 return ret; 729 } 730 731 732 /* Compute the bth power of a, reduce modulo p, and store 733 * the result in r. r could be a. 734 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 735 */ 736 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) 737 { 738 int ret = 0, i, n; 739 BIGNUM *u; 740 741 bn_check_top(a); 742 bn_check_top(b); 743 744 if (BN_is_zero(b)) 745 return(BN_one(r)); 746 747 if (BN_abs_is_word(b, 1)) 748 return (BN_copy(r, a) != NULL); 749 750 BN_CTX_start(ctx); 751 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 752 753 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 754 755 n = BN_num_bits(b) - 1; 756 for (i = n - 1; i >= 0; i--) 757 { 758 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 759 if (BN_is_bit_set(b, i)) 760 { 761 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 762 } 763 } 764 if (!BN_copy(r, u)) goto err; 765 bn_check_top(r); 766 ret = 1; 767 err: 768 BN_CTX_end(ctx); 769 return ret; 770 } 771 772 /* Compute the bth power of a, reduce modulo p, and store 773 * the result in r. r could be a. 774 * 775 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 776 * function is only provided for convenience; for best performance, use the 777 * BN_GF2m_mod_exp_arr function. 778 */ 779 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 780 { 781 int ret = 0; 782 const int max = BN_num_bits(p) + 1; 783 int *arr=NULL; 784 bn_check_top(a); 785 bn_check_top(b); 786 bn_check_top(p); 787 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 788 ret = BN_GF2m_poly2arr(p, arr, max); 789 if (!ret || ret > max) 790 { 791 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 792 goto err; 793 } 794 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 795 bn_check_top(r); 796 err: 797 if (arr) OPENSSL_free(arr); 798 return ret; 799 } 800 801 /* Compute the square root of a, reduce modulo p, and store 802 * the result in r. r could be a. 803 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 804 */ 805 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) 806 { 807 int ret = 0; 808 BIGNUM *u; 809 810 bn_check_top(a); 811 812 if (!p[0]) 813 { 814 /* reduction mod 1 => return 0 */ 815 BN_zero(r); 816 return 1; 817 } 818 819 BN_CTX_start(ctx); 820 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 821 822 if (!BN_set_bit(u, p[0] - 1)) goto err; 823 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 824 bn_check_top(r); 825 826 err: 827 BN_CTX_end(ctx); 828 return ret; 829 } 830 831 /* Compute the square root of a, reduce modulo p, and store 832 * the result in r. r could be a. 833 * 834 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 835 * function is only provided for convenience; for best performance, use the 836 * BN_GF2m_mod_sqrt_arr function. 837 */ 838 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 839 { 840 int ret = 0; 841 const int max = BN_num_bits(p) + 1; 842 int *arr=NULL; 843 bn_check_top(a); 844 bn_check_top(p); 845 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 846 ret = BN_GF2m_poly2arr(p, arr, max); 847 if (!ret || ret > max) 848 { 849 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 850 goto err; 851 } 852 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 853 bn_check_top(r); 854 err: 855 if (arr) OPENSSL_free(arr); 856 return ret; 857 } 858 859 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 860 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 861 */ 862 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx) 863 { 864 int ret = 0, count = 0, j; 865 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 866 867 bn_check_top(a_); 868 869 if (!p[0]) 870 { 871 /* reduction mod 1 => return 0 */ 872 BN_zero(r); 873 return 1; 874 } 875 876 BN_CTX_start(ctx); 877 a = BN_CTX_get(ctx); 878 z = BN_CTX_get(ctx); 879 w = BN_CTX_get(ctx); 880 if (w == NULL) goto err; 881 882 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 883 884 if (BN_is_zero(a)) 885 { 886 BN_zero(r); 887 ret = 1; 888 goto err; 889 } 890 891 if (p[0] & 0x1) /* m is odd */ 892 { 893 /* compute half-trace of a */ 894 if (!BN_copy(z, a)) goto err; 895 for (j = 1; j <= (p[0] - 1) / 2; j++) 896 { 897 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 898 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 899 if (!BN_GF2m_add(z, z, a)) goto err; 900 } 901 902 } 903 else /* m is even */ 904 { 905 rho = BN_CTX_get(ctx); 906 w2 = BN_CTX_get(ctx); 907 tmp = BN_CTX_get(ctx); 908 if (tmp == NULL) goto err; 909 do 910 { 911 if (!BN_rand(rho, p[0], 0, 0)) goto err; 912 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 913 BN_zero(z); 914 if (!BN_copy(w, rho)) goto err; 915 for (j = 1; j <= p[0] - 1; j++) 916 { 917 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 918 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 919 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 920 if (!BN_GF2m_add(z, z, tmp)) goto err; 921 if (!BN_GF2m_add(w, w2, rho)) goto err; 922 } 923 count++; 924 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 925 if (BN_is_zero(w)) 926 { 927 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 928 goto err; 929 } 930 } 931 932 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 933 if (!BN_GF2m_add(w, z, w)) goto err; 934 if (BN_GF2m_cmp(w, a)) 935 { 936 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 937 goto err; 938 } 939 940 if (!BN_copy(r, z)) goto err; 941 bn_check_top(r); 942 943 ret = 1; 944 945 err: 946 BN_CTX_end(ctx); 947 return ret; 948 } 949 950 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 951 * 952 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 953 * function is only provided for convenience; for best performance, use the 954 * BN_GF2m_mod_solve_quad_arr function. 955 */ 956 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 957 { 958 int ret = 0; 959 const int max = BN_num_bits(p) + 1; 960 int *arr=NULL; 961 bn_check_top(a); 962 bn_check_top(p); 963 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * 964 max)) == NULL) goto err; 965 ret = BN_GF2m_poly2arr(p, arr, max); 966 if (!ret || ret > max) 967 { 968 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 969 goto err; 970 } 971 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 972 bn_check_top(r); 973 err: 974 if (arr) OPENSSL_free(arr); 975 return ret; 976 } 977 978 /* Convert the bit-string representation of a polynomial 979 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding 980 * to the bits with non-zero coefficient. Array is terminated with -1. 981 * Up to max elements of the array will be filled. Return value is total 982 * number of array elements that would be filled if array was large enough. 983 */ 984 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 985 { 986 int i, j, k = 0; 987 BN_ULONG mask; 988 989 if (BN_is_zero(a)) 990 return 0; 991 992 for (i = a->top - 1; i >= 0; i--) 993 { 994 if (!a->d[i]) 995 /* skip word if a->d[i] == 0 */ 996 continue; 997 mask = BN_TBIT; 998 for (j = BN_BITS2 - 1; j >= 0; j--) 999 { 1000 if (a->d[i] & mask) 1001 { 1002 if (k < max) p[k] = BN_BITS2 * i + j; 1003 k++; 1004 } 1005 mask >>= 1; 1006 } 1007 } 1008 1009 if (k < max) { 1010 p[k] = -1; 1011 k++; 1012 } 1013 1014 return k; 1015 } 1016 1017 /* Convert the coefficient array representation of a polynomial to a 1018 * bit-string. The array must be terminated by -1. 1019 */ 1020 int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1021 { 1022 int i; 1023 1024 bn_check_top(a); 1025 BN_zero(a); 1026 for (i = 0; p[i] != -1; i++) 1027 { 1028 if (BN_set_bit(a, p[i]) == 0) 1029 return 0; 1030 } 1031 bn_check_top(a); 1032 1033 return 1; 1034 } 1035 1036
