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 99 100 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 #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 143 static 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 166 static 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 193 static 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) 225 static 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 */ 270 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) 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 */ 286 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) 287 { 288 int i; 289 const BIGNUM *at, *bt; 290 291 bn_check_top(a); 292 bn_check_top(b); 293 294 if (a->top < b->top) { at = b; bt = a; } 295 else { at = a; bt = b; } 296 297 if(bn_wexpand(r, at->top) == NULL) 298 return 0; 299 300 for (i = 0; i < bt->top; i++) 301 { 302 r->d[i] = at->d[i] ^ bt->d[i]; 303 } 304 for (; i < at->top; i++) 305 { 306 r->d[i] = at->d[i]; 307 } 308 309 r->top = at->top; 310 bn_correct_top(r); 311 312 return 1; 313 } 314 315 316 /* Some functions allow for representation of the irreducible polynomials 317 * as an int[], say p. The irreducible f(t) is then of the form: 318 * t^p[0] + t^p[1] + ... + t^p[k] 319 * where m = p[0] > p[1] > ... > p[k] = 0. 320 */ 321 322 323 /* Performs modular reduction of a and store result in r. r could be a. */ 324 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[]) 325 { 326 int j, k; 327 int n, dN, d0, d1; 328 BN_ULONG zz, *z; 329 330 bn_check_top(a); 331 332 if (!p[0]) 333 { 334 /* reduction mod 1 => return 0 */ 335 BN_zero(r); 336 return 1; 337 } 338 339 /* Since the algorithm does reduction in the r value, if a != r, copy 340 * the contents of a into r so we can do reduction in r. 341 */ 342 if (a != r) 343 { 344 if (!bn_wexpand(r, a->top)) return 0; 345 for (j = 0; j < a->top; j++) 346 { 347 r->d[j] = a->d[j]; 348 } 349 r->top = a->top; 350 } 351 z = r->d; 352 353 /* start reduction */ 354 dN = p[0] / BN_BITS2; 355 for (j = r->top - 1; j > dN;) 356 { 357 zz = z[j]; 358 if (z[j] == 0) { j--; continue; } 359 z[j] = 0; 360 361 for (k = 1; p[k] != 0; k++) 362 { 363 /* reducing component t^p[k] */ 364 n = p[0] - p[k]; 365 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0; 366 n /= BN_BITS2; 367 z[j-n] ^= (zz>>d0); 368 if (d0) z[j-n-1] ^= (zz<<d1); 369 } 370 371 /* reducing component t^0 */ 372 n = dN; 373 d0 = p[0] % BN_BITS2; 374 d1 = BN_BITS2 - d0; 375 z[j-n] ^= (zz >> d0); 376 if (d0) z[j-n-1] ^= (zz << d1); 377 } 378 379 /* final round of reduction */ 380 while (j == dN) 381 { 382 383 d0 = p[0] % BN_BITS2; 384 zz = z[dN] >> d0; 385 if (zz == 0) break; 386 d1 = BN_BITS2 - d0; 387 388 /* clear up the top d1 bits */ 389 if (d0) 390 z[dN] = (z[dN] << d1) >> d1; 391 else 392 z[dN] = 0; 393 z[0] ^= zz; /* reduction t^0 component */ 394 395 for (k = 1; p[k] != 0; k++) 396 { 397 BN_ULONG tmp_ulong; 398 399 /* reducing component t^p[k]*/ 400 n = p[k] / BN_BITS2; 401 d0 = p[k] % BN_BITS2; 402 d1 = BN_BITS2 - d0; 403 z[n] ^= (zz << d0); 404 tmp_ulong = zz >> d1; 405 if (d0 && tmp_ulong) 406 z[n+1] ^= tmp_ulong; 407 } 408 409 410 } 411 412 bn_correct_top(r); 413 return 1; 414 } 415 416 /* Performs modular reduction of a by p and store result in r. r could be a. 417 * 418 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper 419 * function is only provided for convenience; for best performance, use the 420 * BN_GF2m_mod_arr function. 421 */ 422 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) 423 { 424 int ret = 0; 425 const int max = BN_num_bits(p); 426 unsigned int *arr=NULL; 427 bn_check_top(a); 428 bn_check_top(p); 429 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 430 ret = BN_GF2m_poly2arr(p, arr, max); 431 if (!ret || ret > max) 432 { 433 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH); 434 goto err; 435 } 436 ret = BN_GF2m_mod_arr(r, a, arr); 437 bn_check_top(r); 438 err: 439 if (arr) OPENSSL_free(arr); 440 return ret; 441 } 442 443 444 /* Compute the product of two polynomials a and b, reduce modulo p, and store 445 * the result in r. r could be a or b; a could be b. 446 */ 447 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx) 448 { 449 int zlen, i, j, k, ret = 0; 450 BIGNUM *s; 451 BN_ULONG x1, x0, y1, y0, zz[4]; 452 453 bn_check_top(a); 454 bn_check_top(b); 455 456 if (a == b) 457 { 458 return BN_GF2m_mod_sqr_arr(r, a, p, ctx); 459 } 460 461 BN_CTX_start(ctx); 462 if ((s = BN_CTX_get(ctx)) == NULL) goto err; 463 464 zlen = a->top + b->top + 4; 465 if (!bn_wexpand(s, zlen)) goto err; 466 s->top = zlen; 467 468 for (i = 0; i < zlen; i++) s->d[i] = 0; 469 470 for (j = 0; j < b->top; j += 2) 471 { 472 y0 = b->d[j]; 473 y1 = ((j+1) == b->top) ? 0 : b->d[j+1]; 474 for (i = 0; i < a->top; i += 2) 475 { 476 x0 = a->d[i]; 477 x1 = ((i+1) == a->top) ? 0 : a->d[i+1]; 478 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); 479 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k]; 480 } 481 } 482 483 bn_correct_top(s); 484 if (BN_GF2m_mod_arr(r, s, p)) 485 ret = 1; 486 bn_check_top(r); 487 488 err: 489 BN_CTX_end(ctx); 490 return ret; 491 } 492 493 /* Compute the product of two polynomials a and b, reduce modulo p, and store 494 * the result in r. r could be a or b; a could equal b. 495 * 496 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper 497 * function is only provided for convenience; for best performance, use the 498 * BN_GF2m_mod_mul_arr function. 499 */ 500 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 501 { 502 int ret = 0; 503 const int max = BN_num_bits(p); 504 unsigned int *arr=NULL; 505 bn_check_top(a); 506 bn_check_top(b); 507 bn_check_top(p); 508 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 509 ret = BN_GF2m_poly2arr(p, arr, max); 510 if (!ret || ret > max) 511 { 512 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH); 513 goto err; 514 } 515 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); 516 bn_check_top(r); 517 err: 518 if (arr) OPENSSL_free(arr); 519 return ret; 520 } 521 522 523 /* Square a, reduce the result mod p, and store it in a. r could be a. */ 524 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx) 525 { 526 int i, ret = 0; 527 BIGNUM *s; 528 529 bn_check_top(a); 530 BN_CTX_start(ctx); 531 if ((s = BN_CTX_get(ctx)) == NULL) return 0; 532 if (!bn_wexpand(s, 2 * a->top)) goto err; 533 534 for (i = a->top - 1; i >= 0; i--) 535 { 536 s->d[2*i+1] = SQR1(a->d[i]); 537 s->d[2*i ] = SQR0(a->d[i]); 538 } 539 540 s->top = 2 * a->top; 541 bn_correct_top(s); 542 if (!BN_GF2m_mod_arr(r, s, p)) goto err; 543 bn_check_top(r); 544 ret = 1; 545 err: 546 BN_CTX_end(ctx); 547 return ret; 548 } 549 550 /* Square a, reduce the result mod p, and store it in a. r could be a. 551 * 552 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper 553 * function is only provided for convenience; for best performance, use the 554 * BN_GF2m_mod_sqr_arr function. 555 */ 556 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 557 { 558 int ret = 0; 559 const int max = BN_num_bits(p); 560 unsigned int *arr=NULL; 561 562 bn_check_top(a); 563 bn_check_top(p); 564 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 565 ret = BN_GF2m_poly2arr(p, arr, max); 566 if (!ret || ret > max) 567 { 568 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH); 569 goto err; 570 } 571 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); 572 bn_check_top(r); 573 err: 574 if (arr) OPENSSL_free(arr); 575 return ret; 576 } 577 578 579 /* Invert a, reduce modulo p, and store the result in r. r could be a. 580 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from 581 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation 582 * of Elliptic Curve Cryptography Over Binary Fields". 583 */ 584 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 585 { 586 BIGNUM *b, *c, *u, *v, *tmp; 587 int ret = 0; 588 589 bn_check_top(a); 590 bn_check_top(p); 591 592 BN_CTX_start(ctx); 593 594 b = BN_CTX_get(ctx); 595 c = BN_CTX_get(ctx); 596 u = BN_CTX_get(ctx); 597 v = BN_CTX_get(ctx); 598 if (v == NULL) goto err; 599 600 if (!BN_one(b)) goto err; 601 if (!BN_GF2m_mod(u, a, p)) goto err; 602 if (!BN_copy(v, p)) goto err; 603 604 if (BN_is_zero(u)) goto err; 605 606 while (1) 607 { 608 while (!BN_is_odd(u)) 609 { 610 if (!BN_rshift1(u, u)) goto err; 611 if (BN_is_odd(b)) 612 { 613 if (!BN_GF2m_add(b, b, p)) goto err; 614 } 615 if (!BN_rshift1(b, b)) goto err; 616 } 617 618 if (BN_abs_is_word(u, 1)) break; 619 620 if (BN_num_bits(u) < BN_num_bits(v)) 621 { 622 tmp = u; u = v; v = tmp; 623 tmp = b; b = c; c = tmp; 624 } 625 626 if (!BN_GF2m_add(u, u, v)) goto err; 627 if (!BN_GF2m_add(b, b, c)) goto err; 628 } 629 630 631 if (!BN_copy(r, b)) goto err; 632 bn_check_top(r); 633 ret = 1; 634 635 err: 636 BN_CTX_end(ctx); 637 return ret; 638 } 639 640 /* Invert xx, reduce modulo p, and store the result in r. r could be xx. 641 * 642 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper 643 * function is only provided for convenience; for best performance, use the 644 * BN_GF2m_mod_inv function. 645 */ 646 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx) 647 { 648 BIGNUM *field; 649 int ret = 0; 650 651 bn_check_top(xx); 652 BN_CTX_start(ctx); 653 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 654 if (!BN_GF2m_arr2poly(p, field)) goto err; 655 656 ret = BN_GF2m_mod_inv(r, xx, field, ctx); 657 bn_check_top(r); 658 659 err: 660 BN_CTX_end(ctx); 661 return ret; 662 } 663 664 665 #ifndef OPENSSL_SUN_GF2M_DIV 666 /* Divide y by x, reduce modulo p, and store the result in r. r could be x 667 * or y, x could equal y. 668 */ 669 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 670 { 671 BIGNUM *xinv = NULL; 672 int ret = 0; 673 674 bn_check_top(y); 675 bn_check_top(x); 676 bn_check_top(p); 677 678 BN_CTX_start(ctx); 679 xinv = BN_CTX_get(ctx); 680 if (xinv == NULL) goto err; 681 682 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; 683 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; 684 bn_check_top(r); 685 ret = 1; 686 687 err: 688 BN_CTX_end(ctx); 689 return ret; 690 } 691 #else 692 /* Divide y by x, reduce modulo p, and store the result in r. r could be x 693 * or y, x could equal y. 694 * Uses algorithm Modular_Division_GF(2^m) from 695 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to 696 * the Great Divide". 697 */ 698 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) 699 { 700 BIGNUM *a, *b, *u, *v; 701 int ret = 0; 702 703 bn_check_top(y); 704 bn_check_top(x); 705 bn_check_top(p); 706 707 BN_CTX_start(ctx); 708 709 a = BN_CTX_get(ctx); 710 b = BN_CTX_get(ctx); 711 u = BN_CTX_get(ctx); 712 v = BN_CTX_get(ctx); 713 if (v == NULL) goto err; 714 715 /* reduce x and y mod p */ 716 if (!BN_GF2m_mod(u, y, p)) goto err; 717 if (!BN_GF2m_mod(a, x, p)) goto err; 718 if (!BN_copy(b, p)) goto err; 719 720 while (!BN_is_odd(a)) 721 { 722 if (!BN_rshift1(a, a)) goto err; 723 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 724 if (!BN_rshift1(u, u)) goto err; 725 } 726 727 do 728 { 729 if (BN_GF2m_cmp(b, a) > 0) 730 { 731 if (!BN_GF2m_add(b, b, a)) goto err; 732 if (!BN_GF2m_add(v, v, u)) goto err; 733 do 734 { 735 if (!BN_rshift1(b, b)) goto err; 736 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; 737 if (!BN_rshift1(v, v)) goto err; 738 } while (!BN_is_odd(b)); 739 } 740 else if (BN_abs_is_word(a, 1)) 741 break; 742 else 743 { 744 if (!BN_GF2m_add(a, a, b)) goto err; 745 if (!BN_GF2m_add(u, u, v)) goto err; 746 do 747 { 748 if (!BN_rshift1(a, a)) goto err; 749 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; 750 if (!BN_rshift1(u, u)) goto err; 751 } while (!BN_is_odd(a)); 752 } 753 } while (1); 754 755 if (!BN_copy(r, u)) goto err; 756 bn_check_top(r); 757 ret = 1; 758 759 err: 760 BN_CTX_end(ctx); 761 return ret; 762 } 763 #endif 764 765 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx 766 * or yy, xx could equal yy. 767 * 768 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper 769 * function is only provided for convenience; for best performance, use the 770 * BN_GF2m_mod_div function. 771 */ 772 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx) 773 { 774 BIGNUM *field; 775 int ret = 0; 776 777 bn_check_top(yy); 778 bn_check_top(xx); 779 780 BN_CTX_start(ctx); 781 if ((field = BN_CTX_get(ctx)) == NULL) goto err; 782 if (!BN_GF2m_arr2poly(p, field)) goto err; 783 784 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); 785 bn_check_top(r); 786 787 err: 788 BN_CTX_end(ctx); 789 return ret; 790 } 791 792 793 /* Compute the bth power of a, reduce modulo p, and store 794 * the result in r. r could be a. 795 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363. 796 */ 797 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx) 798 { 799 int ret = 0, i, n; 800 BIGNUM *u; 801 802 bn_check_top(a); 803 bn_check_top(b); 804 805 if (BN_is_zero(b)) 806 return(BN_one(r)); 807 808 if (BN_abs_is_word(b, 1)) 809 return (BN_copy(r, a) != NULL); 810 811 BN_CTX_start(ctx); 812 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 813 814 if (!BN_GF2m_mod_arr(u, a, p)) goto err; 815 816 n = BN_num_bits(b) - 1; 817 for (i = n - 1; i >= 0; i--) 818 { 819 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; 820 if (BN_is_bit_set(b, i)) 821 { 822 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; 823 } 824 } 825 if (!BN_copy(r, u)) goto err; 826 bn_check_top(r); 827 ret = 1; 828 err: 829 BN_CTX_end(ctx); 830 return ret; 831 } 832 833 /* Compute the bth power of a, reduce modulo p, and store 834 * the result in r. r could be a. 835 * 836 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper 837 * function is only provided for convenience; for best performance, use the 838 * BN_GF2m_mod_exp_arr function. 839 */ 840 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) 841 { 842 int ret = 0; 843 const int max = BN_num_bits(p); 844 unsigned int *arr=NULL; 845 bn_check_top(a); 846 bn_check_top(b); 847 bn_check_top(p); 848 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 849 ret = BN_GF2m_poly2arr(p, arr, max); 850 if (!ret || ret > max) 851 { 852 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH); 853 goto err; 854 } 855 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); 856 bn_check_top(r); 857 err: 858 if (arr) OPENSSL_free(arr); 859 return ret; 860 } 861 862 /* Compute the square root of a, reduce modulo p, and store 863 * the result in r. r could be a. 864 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363. 865 */ 866 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx) 867 { 868 int ret = 0; 869 BIGNUM *u; 870 871 bn_check_top(a); 872 873 if (!p[0]) 874 { 875 /* reduction mod 1 => return 0 */ 876 BN_zero(r); 877 return 1; 878 } 879 880 BN_CTX_start(ctx); 881 if ((u = BN_CTX_get(ctx)) == NULL) goto err; 882 883 if (!BN_set_bit(u, p[0] - 1)) goto err; 884 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); 885 bn_check_top(r); 886 887 err: 888 BN_CTX_end(ctx); 889 return ret; 890 } 891 892 /* Compute the square root of a, reduce modulo p, and store 893 * the result in r. r could be a. 894 * 895 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper 896 * function is only provided for convenience; for best performance, use the 897 * BN_GF2m_mod_sqrt_arr function. 898 */ 899 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 900 { 901 int ret = 0; 902 const int max = BN_num_bits(p); 903 unsigned int *arr=NULL; 904 bn_check_top(a); 905 bn_check_top(p); 906 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err; 907 ret = BN_GF2m_poly2arr(p, arr, max); 908 if (!ret || ret > max) 909 { 910 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH); 911 goto err; 912 } 913 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); 914 bn_check_top(r); 915 err: 916 if (arr) OPENSSL_free(arr); 917 return ret; 918 } 919 920 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 921 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363. 922 */ 923 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx) 924 { 925 int ret = 0, count = 0; 926 unsigned int j; 927 BIGNUM *a, *z, *rho, *w, *w2, *tmp; 928 929 bn_check_top(a_); 930 931 if (!p[0]) 932 { 933 /* reduction mod 1 => return 0 */ 934 BN_zero(r); 935 return 1; 936 } 937 938 BN_CTX_start(ctx); 939 a = BN_CTX_get(ctx); 940 z = BN_CTX_get(ctx); 941 w = BN_CTX_get(ctx); 942 if (w == NULL) goto err; 943 944 if (!BN_GF2m_mod_arr(a, a_, p)) goto err; 945 946 if (BN_is_zero(a)) 947 { 948 BN_zero(r); 949 ret = 1; 950 goto err; 951 } 952 953 if (p[0] & 0x1) /* m is odd */ 954 { 955 /* compute half-trace of a */ 956 if (!BN_copy(z, a)) goto err; 957 for (j = 1; j <= (p[0] - 1) / 2; j++) 958 { 959 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 960 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 961 if (!BN_GF2m_add(z, z, a)) goto err; 962 } 963 964 } 965 else /* m is even */ 966 { 967 rho = BN_CTX_get(ctx); 968 w2 = BN_CTX_get(ctx); 969 tmp = BN_CTX_get(ctx); 970 if (tmp == NULL) goto err; 971 do 972 { 973 if (!BN_rand(rho, p[0], 0, 0)) goto err; 974 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; 975 BN_zero(z); 976 if (!BN_copy(w, rho)) goto err; 977 for (j = 1; j <= p[0] - 1; j++) 978 { 979 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; 980 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; 981 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; 982 if (!BN_GF2m_add(z, z, tmp)) goto err; 983 if (!BN_GF2m_add(w, w2, rho)) goto err; 984 } 985 count++; 986 } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); 987 if (BN_is_zero(w)) 988 { 989 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS); 990 goto err; 991 } 992 } 993 994 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; 995 if (!BN_GF2m_add(w, z, w)) goto err; 996 if (BN_GF2m_cmp(w, a)) 997 { 998 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); 999 goto err; 1000 } 1001 1002 if (!BN_copy(r, z)) goto err; 1003 bn_check_top(r); 1004 1005 ret = 1; 1006 1007 err: 1008 BN_CTX_end(ctx); 1009 return ret; 1010 } 1011 1012 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0. 1013 * 1014 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper 1015 * function is only provided for convenience; for best performance, use the 1016 * BN_GF2m_mod_solve_quad_arr function. 1017 */ 1018 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 1019 { 1020 int ret = 0; 1021 const int max = BN_num_bits(p); 1022 unsigned int *arr=NULL; 1023 bn_check_top(a); 1024 bn_check_top(p); 1025 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * 1026 max)) == NULL) goto err; 1027 ret = BN_GF2m_poly2arr(p, arr, max); 1028 if (!ret || ret > max) 1029 { 1030 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH); 1031 goto err; 1032 } 1033 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); 1034 bn_check_top(r); 1035 err: 1036 if (arr) OPENSSL_free(arr); 1037 return ret; 1038 } 1039 1040 /* Convert the bit-string representation of a polynomial 1041 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array 1042 * of integers corresponding to the bits with non-zero coefficient. 1043 * Up to max elements of the array will be filled. Return value is total 1044 * number of coefficients that would be extracted if array was large enough. 1045 */ 1046 int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max) 1047 { 1048 int i, j, k = 0; 1049 BN_ULONG mask; 1050 1051 if (BN_is_zero(a) || !BN_is_bit_set(a, 0)) 1052 /* a_0 == 0 => return error (the unsigned int array 1053 * must be terminated by 0) 1054 */ 1055 return 0; 1056 1057 for (i = a->top - 1; i >= 0; i--) 1058 { 1059 if (!a->d[i]) 1060 /* skip word if a->d[i] == 0 */ 1061 continue; 1062 mask = BN_TBIT; 1063 for (j = BN_BITS2 - 1; j >= 0; j--) 1064 { 1065 if (a->d[i] & mask) 1066 { 1067 if (k < max) p[k] = BN_BITS2 * i + j; 1068 k++; 1069 } 1070 mask >>= 1; 1071 } 1072 } 1073 1074 return k; 1075 } 1076 1077 /* Convert the coefficient array representation of a polynomial to a 1078 * bit-string. The array must be terminated by 0. 1079 */ 1080 int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a) 1081 { 1082 int i; 1083 1084 bn_check_top(a); 1085 BN_zero(a); 1086 for (i = 0; p[i] != 0; i++) 1087 { 1088 if (BN_set_bit(a, p[i]) == 0) 1089 return 0; 1090 } 1091 BN_set_bit(a, 0); 1092 bn_check_top(a); 1093 1094 return 1; 1095 } 1096 1097