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_rshift1(u, u)) goto err; 549 if (BN_is_odd(b)) 550 { 551 if (!BN_GF2m_add(b, b, p)) goto err; 552 } 553 if (!BN_rshift1(b, b)) goto err; 554 } 555 556 if (BN_abs_is_word(u, 1)) break; 557 558 if (BN_num_bits(u) < BN_num_bits(v)) 559 { 560 tmp = u; u = v; v = tmp; 561 tmp = b; b = c; c = tmp; 562 } 563 564 if (!BN_GF2m_add(u, u, v)) goto err; 565 if (!BN_GF2m_add(b, b, c)) goto err; 566 } 567 568 569 if (!BN_copy(r, b)) goto err; 570 bn_check_top(r); 571 ret = 1; 572 573 err: 574 BN_CTX_end(ctx); 575 return ret; 576 } 577 578 /* Invert xx, reduce modulo p, and store the result in r. r could be xx. 579 * 580 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 581 * function is only provided for convenience; for best performance, use the 582 * BN_GF2m_mod_inv function. 583 */ 584 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx) 585 { 586 BIGNUM *field; 587 int ret = 0; 588 589 bn_check_top(xx); 590 BN_CTX_start(ctx); 591 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 592 if (!BN_GF2m_arr2poly(p, field)) goto err; 593 594 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 595 bn_check_top(r); 596 597 err: 598 BN_CTX_end(ctx); 599 return ret; 600 } 601 602 603 #ifndef OPENSSL_SUN_GF2M_DIV 604 /* Divide y by x, reduce modulo p, and store the result in r. r could be x 605 * or y, x could equal y. 606 */ 607 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 608 { 609 BIGNUM *xinv = NULL; 610 int ret = 0; 611 612 bn_check_top(y); 613 bn_check_top(x); 614 bn_check_top(p); 615 616 BN_CTX_start(ctx); 617 xinv = BN_CTX_get(ctx); 618 if (xinv == NULL) goto err; 619 620 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 621 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 622 bn_check_top(r); 623 ret = 1; 624 625 err: 626 BN_CTX_end(ctx); 627 return ret; 628 } 629 #else 630 /* Divide y by x, reduce modulo p, and store the result in r. r could be x 631 * or y, x could equal y. 632 * Uses algorithm Modular_Division_GF(2^m) from 633 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 634 * the Great Divide". 635 */ 636 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 637 { 638 BIGNUM *a, *b, *u, *v; 639 int ret = 0; 640 641 bn_check_top(y); 642 bn_check_top(x); 643 bn_check_top(p); 644 645 BN_CTX_start(ctx); 646 647 a = BN_CTX_get(ctx); 648 b = BN_CTX_get(ctx); 649 u = BN_CTX_get(ctx); 650 v = BN_CTX_get(ctx); 651 if (v == NULL) goto err; 652 653 /* reduce x and y mod p */ 654 if (!BN_GF2m_mod(u, y, p)) goto err; 655 if (!BN_GF2m_mod(a, x, p)) goto err; 656 if (!BN_copy(b, p)) goto err; 657 658 while (!BN_is_odd(a)) 659 { 660 if (!BN_rshift1(a, a)) goto err; 661 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 662 if (!BN_rshift1(u, u)) goto err; 663 } 664 665 do 666 { 667 if (BN_GF2m_cmp(b, a) > 0) 668 { 669 if (!BN_GF2m_add(b, b, a)) goto err; 670 if (!BN_GF2m_add(v, v, u)) goto err; 671 do 672 { 673 if (!BN_rshift1(b, b)) goto err; 674 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 675 if (!BN_rshift1(v, v)) goto err; 676 } while (!BN_is_odd(b)); 677 } 678 else if (BN_abs_is_word(a, 1)) 679 break; 680 else 681 { 682 if (!BN_GF2m_add(a, a, b)) goto err; 683 if (!BN_GF2m_add(u, u, v)) goto err; 684 do 685 { 686 if (!BN_rshift1(a, a)) goto err; 687 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 688 if (!BN_rshift1(u, u)) goto err; 689 } while (!BN_is_odd(a)); 690 } 691 } while (1); 692 693 if (!BN_copy(r, u)) goto err; 694 bn_check_top(r); 695 ret = 1; 696 697 err: 698 BN_CTX_end(ctx); 699 return ret; 700 } 701 #endif 702 703 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 704 * or yy, xx could equal yy. 705 * 706 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 707 * function is only provided for convenience; for best performance, use the 708 * BN_GF2m_mod_div function. 709 */ 710 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx) 711 { 712 BIGNUM *field; 713 int ret = 0; 714 715 bn_check_top(yy); 716 bn_check_top(xx); 717 718 BN_CTX_start(ctx); 719 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 720 if (!BN_GF2m_arr2poly(p, field)) goto err; 721 722 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 723 bn_check_top(r); 724 725 err: 726 BN_CTX_end(ctx); 727 return ret; 728 } 729 730 731 /* Compute the bth power of a, reduce modulo p, and store 732 * the result in r. r could be a. 733 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 734 */ 735 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) 736 { 737 int ret = 0, i, n; 738 BIGNUM *u; 739 740 bn_check_top(a); 741 bn_check_top(b); 742 743 if (BN_is_zero(b)) 744 return(BN_one(r)); 745 746 if (BN_abs_is_word(b, 1)) 747 return (BN_copy(r, a) != NULL); 748 749 BN_CTX_start(ctx); 750 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 751 752 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 753 754 n = BN_num_bits(b) - 1; 755 for (i = n - 1; i >= 0; i--) 756 { 757 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 758 if (BN_is_bit_set(b, i)) 759 { 760 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 761 } 762 } 763 if (!BN_copy(r, u)) goto err; 764 bn_check_top(r); 765 ret = 1; 766 err: 767 BN_CTX_end(ctx); 768 return ret; 769 } 770 771 /* Compute the bth power of a, reduce modulo p, and store 772 * the result in r. r could be a. 773 * 774 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 775 * function is only provided for convenience; for best performance, use the 776 * BN_GF2m_mod_exp_arr function. 777 */ 778 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 779 { 780 int ret = 0; 781 const int max = BN_num_bits(p) + 1; 782 int *arr=NULL; 783 bn_check_top(a); 784 bn_check_top(b); 785 bn_check_top(p); 786 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 787 ret = BN_GF2m_poly2arr(p, arr, max); 788 if (!ret || ret > max) 789 { 790 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 791 goto err; 792 } 793 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 794 bn_check_top(r); 795 err: 796 if (arr) OPENSSL_free(arr); 797 return ret; 798 } 799 800 /* Compute the square root of a, reduce modulo p, and store 801 * the result in r. r could be a. 802 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 803 */ 804 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) 805 { 806 int ret = 0; 807 BIGNUM *u; 808 809 bn_check_top(a); 810 811 if (!p[0]) 812 { 813 /* reduction mod 1 => return 0 */ 814 BN_zero(r); 815 return 1; 816 } 817 818 BN_CTX_start(ctx); 819 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 820 821 if (!BN_set_bit(u, p[0] - 1)) goto err; 822 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 823 bn_check_top(r); 824 825 err: 826 BN_CTX_end(ctx); 827 return ret; 828 } 829 830 /* Compute the square root of a, reduce modulo p, and store 831 * the result in r. r could be a. 832 * 833 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 834 * function is only provided for convenience; for best performance, use the 835 * BN_GF2m_mod_sqrt_arr function. 836 */ 837 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 838 { 839 int ret = 0; 840 const int max = BN_num_bits(p) + 1; 841 int *arr=NULL; 842 bn_check_top(a); 843 bn_check_top(p); 844 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err; 845 ret = BN_GF2m_poly2arr(p, arr, max); 846 if (!ret || ret > max) 847 { 848 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 849 goto err; 850 } 851 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 852 bn_check_top(r); 853 err: 854 if (arr) OPENSSL_free(arr); 855 return ret; 856 } 857 858 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 859 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 860 */ 861 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx) 862 { 863 int ret = 0, count = 0, j; 864 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 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 a = BN_CTX_get(ctx); 877 z = BN_CTX_get(ctx); 878 w = BN_CTX_get(ctx); 879 if (w == NULL) goto err; 880 881 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 882 883 if (BN_is_zero(a)) 884 { 885 BN_zero(r); 886 ret = 1; 887 goto err; 888 } 889 890 if (p[0] & 0x1) /* m is odd */ 891 { 892 /* compute half-trace of a */ 893 if (!BN_copy(z, a)) goto err; 894 for (j = 1; j <= (p[0] - 1) / 2; j++) 895 { 896 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 897 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 898 if (!BN_GF2m_add(z, z, a)) goto err; 899 } 900 901 } 902 else /* m is even */ 903 { 904 rho = BN_CTX_get(ctx); 905 w2 = BN_CTX_get(ctx); 906 tmp = BN_CTX_get(ctx); 907 if (tmp == NULL) goto err; 908 do 909 { 910 if (!BN_rand(rho, p[0], 0, 0)) goto err; 911 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 912 BN_zero(z); 913 if (!BN_copy(w, rho)) goto err; 914 for (j = 1; j <= p[0] - 1; j++) 915 { 916 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 917 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 918 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 919 if (!BN_GF2m_add(z, z, tmp)) goto err; 920 if (!BN_GF2m_add(w, w2, rho)) goto err; 921 } 922 count++; 923 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 924 if (BN_is_zero(w)) 925 { 926 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 927 goto err; 928 } 929 } 930 931 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 932 if (!BN_GF2m_add(w, z, w)) goto err; 933 if (BN_GF2m_cmp(w, a)) 934 { 935 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 936 goto err; 937 } 938 939 if (!BN_copy(r, z)) goto err; 940 bn_check_top(r); 941 942 ret = 1; 943 944 err: 945 BN_CTX_end(ctx); 946 return ret; 947 } 948 949 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 950 * 951 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 952 * function is only provided for convenience; for best performance, use the 953 * BN_GF2m_mod_solve_quad_arr function. 954 */ 955 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 956 { 957 int ret = 0; 958 const int max = BN_num_bits(p) + 1; 959 int *arr=NULL; 960 bn_check_top(a); 961 bn_check_top(p); 962 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * 963 max)) == NULL) goto err; 964 ret = BN_GF2m_poly2arr(p, arr, max); 965 if (!ret || ret > max) 966 { 967 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 968 goto err; 969 } 970 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 971 bn_check_top(r); 972 err: 973 if (arr) OPENSSL_free(arr); 974 return ret; 975 } 976 977 /* Convert the bit-string representation of a polynomial 978 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding 979 * to the bits with non-zero coefficient. Array is terminated with -1. 980 * Up to max elements of the array will be filled. Return value is total 981 * number of array elements that would be filled if array was large enough. 982 */ 983 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) 984 { 985 int i, j, k = 0; 986 BN_ULONG mask; 987 988 if (BN_is_zero(a)) 989 return 0; 990 991 for (i = a->top - 1; i >= 0; i--) 992 { 993 if (!a->d[i]) 994 /* skip word if a->d[i] == 0 */ 995 continue; 996 mask = BN_TBIT; 997 for (j = BN_BITS2 - 1; j >= 0; j--) 998 { 999 if (a->d[i] & mask) 1000 { 1001 if (k < max) p[k] = BN_BITS2 * i + j; 1002 k++; 1003 } 1004 mask >>= 1; 1005 } 1006 } 1007 1008 if (k < max) { 1009 p[k] = -1; 1010 k++; 1011 } 1012 1013 return k; 1014 } 1015 1016 /* Convert the coefficient array representation of a polynomial to a 1017 * bit-string. The array must be terminated by -1. 1018 */ 1019 int BN_GF2m_arr2poly(const int p[], BIGNUM *a) 1020 { 1021 int i; 1022 1023 bn_check_top(a); 1024 BN_zero(a); 1025 for (i = 0; p[i] != -1; i++) 1026 { 1027 if (BN_set_bit(a, p[i]) == 0) 1028 return 0; 1029 } 1030 bn_check_top(a); 1031 1032 return 1; 1033 } 1034 1035
