1 /* crypto/bn/bn_gcd.c */ 2 /* Copyright (C) 1995-1998 Eric Young (eay (at) cryptsoft.com) 3 * All rights reserved. 4 * 5 * This package is an SSL implementation written 6 * by Eric Young (eay (at) cryptsoft.com). 7 * The implementation was written so as to conform with Netscapes SSL. 8 * 9 * This library is free for commercial and non-commercial use as long as 10 * the following conditions are aheared to. The following conditions 11 * apply to all code found in this distribution, be it the RC4, RSA, 12 * lhash, DES, etc., code; not just the SSL code. The SSL documentation 13 * included with this distribution is covered by the same copyright terms 14 * except that the holder is Tim Hudson (tjh (at) cryptsoft.com). 15 * 16 * Copyright remains Eric Young's, and as such any Copyright notices in 17 * the code are not to be removed. 18 * If this package is used in a product, Eric Young should be given attribution 19 * as the author of the parts of the library used. 20 * This can be in the form of a textual message at program startup or 21 * in documentation (online or textual) provided with the package. 22 * 23 * Redistribution and use in source and binary forms, with or without 24 * modification, are permitted provided that the following conditions 25 * are met: 26 * 1. Redistributions of source code must retain the copyright 27 * notice, this list of conditions and the following disclaimer. 28 * 2. Redistributions in binary form must reproduce the above copyright 29 * notice, this list of conditions and the following disclaimer in the 30 * documentation and/or other materials provided with the distribution. 31 * 3. All advertising materials mentioning features or use of this software 32 * must display the following acknowledgement: 33 * "This product includes cryptographic software written by 34 * Eric Young (eay (at) cryptsoft.com)" 35 * The word 'cryptographic' can be left out if the rouines from the library 36 * being used are not cryptographic related :-). 37 * 4. If you include any Windows specific code (or a derivative thereof) from 38 * the apps directory (application code) you must include an acknowledgement: 39 * "This product includes software written by Tim Hudson (tjh (at) cryptsoft.com)" 40 * 41 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND 42 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 43 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 44 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 45 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 46 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 47 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 48 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 49 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 50 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 51 * SUCH DAMAGE. 52 * 53 * The licence and distribution terms for any publically available version or 54 * derivative of this code cannot be changed. i.e. this code cannot simply be 55 * copied and put under another distribution licence 56 * [including the GNU Public Licence.] 57 */ 58 /* ==================================================================== 59 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved. 60 * 61 * Redistribution and use in source and binary forms, with or without 62 * modification, are permitted provided that the following conditions 63 * are met: 64 * 65 * 1. Redistributions of source code must retain the above copyright 66 * notice, this list of conditions and the following disclaimer. 67 * 68 * 2. Redistributions in binary form must reproduce the above copyright 69 * notice, this list of conditions and the following disclaimer in 70 * the documentation and/or other materials provided with the 71 * distribution. 72 * 73 * 3. All advertising materials mentioning features or use of this 74 * software must display the following acknowledgment: 75 * "This product includes software developed by the OpenSSL Project 76 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 77 * 78 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 79 * endorse or promote products derived from this software without 80 * prior written permission. For written permission, please contact 81 * openssl-core (at) openssl.org. 82 * 83 * 5. Products derived from this software may not be called "OpenSSL" 84 * nor may "OpenSSL" appear in their names without prior written 85 * permission of the OpenSSL Project. 86 * 87 * 6. Redistributions of any form whatsoever must retain the following 88 * acknowledgment: 89 * "This product includes software developed by the OpenSSL Project 90 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 91 * 92 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 93 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 94 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 95 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 96 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 97 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 98 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 99 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 100 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 101 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 102 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 103 * OF THE POSSIBILITY OF SUCH DAMAGE. 104 * ==================================================================== 105 * 106 * This product includes cryptographic software written by Eric Young 107 * (eay (at) cryptsoft.com). This product includes software written by Tim 108 * Hudson (tjh (at) cryptsoft.com). 109 * 110 */ 111 112 #include "cryptlib.h" 113 #include "bn_lcl.h" 114 115 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b); 116 117 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) 118 { 119 BIGNUM *a,*b,*t; 120 int ret=0; 121 122 bn_check_top(in_a); 123 bn_check_top(in_b); 124 125 BN_CTX_start(ctx); 126 a = BN_CTX_get(ctx); 127 b = BN_CTX_get(ctx); 128 if (a == NULL || b == NULL) goto err; 129 130 if (BN_copy(a,in_a) == NULL) goto err; 131 if (BN_copy(b,in_b) == NULL) goto err; 132 a->neg = 0; 133 b->neg = 0; 134 135 if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; } 136 t=euclid(a,b); 137 if (t == NULL) goto err; 138 139 if (BN_copy(r,t) == NULL) goto err; 140 ret=1; 141 err: 142 BN_CTX_end(ctx); 143 bn_check_top(r); 144 return(ret); 145 } 146 147 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) 148 { 149 BIGNUM *t; 150 int shifts=0; 151 152 bn_check_top(a); 153 bn_check_top(b); 154 155 /* 0 <= b <= a */ 156 while (!BN_is_zero(b)) 157 { 158 /* 0 < b <= a */ 159 160 if (BN_is_odd(a)) 161 { 162 if (BN_is_odd(b)) 163 { 164 if (!BN_sub(a,a,b)) goto err; 165 if (!BN_rshift1(a,a)) goto err; 166 if (BN_cmp(a,b) < 0) 167 { t=a; a=b; b=t; } 168 } 169 else /* a odd - b even */ 170 { 171 if (!BN_rshift1(b,b)) goto err; 172 if (BN_cmp(a,b) < 0) 173 { t=a; a=b; b=t; } 174 } 175 } 176 else /* a is even */ 177 { 178 if (BN_is_odd(b)) 179 { 180 if (!BN_rshift1(a,a)) goto err; 181 if (BN_cmp(a,b) < 0) 182 { t=a; a=b; b=t; } 183 } 184 else /* a even - b even */ 185 { 186 if (!BN_rshift1(a,a)) goto err; 187 if (!BN_rshift1(b,b)) goto err; 188 shifts++; 189 } 190 } 191 /* 0 <= b <= a */ 192 } 193 194 if (shifts) 195 { 196 if (!BN_lshift(a,a,shifts)) goto err; 197 } 198 bn_check_top(a); 199 return(a); 200 err: 201 return(NULL); 202 } 203 204 205 /* solves ax == 1 (mod n) */ 206 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in, 207 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx); 208 BIGNUM *BN_mod_inverse(BIGNUM *in, 209 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 210 { 211 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; 212 BIGNUM *ret=NULL; 213 int sign; 214 215 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) 216 { 217 return BN_mod_inverse_no_branch(in, a, n, ctx); 218 } 219 220 bn_check_top(a); 221 bn_check_top(n); 222 223 BN_CTX_start(ctx); 224 A = BN_CTX_get(ctx); 225 B = BN_CTX_get(ctx); 226 X = BN_CTX_get(ctx); 227 D = BN_CTX_get(ctx); 228 M = BN_CTX_get(ctx); 229 Y = BN_CTX_get(ctx); 230 T = BN_CTX_get(ctx); 231 if (T == NULL) goto err; 232 233 if (in == NULL) 234 R=BN_new(); 235 else 236 R=in; 237 if (R == NULL) goto err; 238 239 BN_one(X); 240 BN_zero(Y); 241 if (BN_copy(B,a) == NULL) goto err; 242 if (BN_copy(A,n) == NULL) goto err; 243 A->neg = 0; 244 if (B->neg || (BN_ucmp(B, A) >= 0)) 245 { 246 if (!BN_nnmod(B, B, A, ctx)) goto err; 247 } 248 sign = -1; 249 /* From B = a mod |n|, A = |n| it follows that 250 * 251 * 0 <= B < A, 252 * -sign*X*a == B (mod |n|), 253 * sign*Y*a == A (mod |n|). 254 */ 255 256 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) 257 { 258 /* Binary inversion algorithm; requires odd modulus. 259 * This is faster than the general algorithm if the modulus 260 * is sufficiently small (about 400 .. 500 bits on 32-bit 261 * sytems, but much more on 64-bit systems) */ 262 int shift; 263 264 while (!BN_is_zero(B)) 265 { 266 /* 267 * 0 < B < |n|, 268 * 0 < A <= |n|, 269 * (1) -sign*X*a == B (mod |n|), 270 * (2) sign*Y*a == A (mod |n|) 271 */ 272 273 /* Now divide B by the maximum possible power of two in the integers, 274 * and divide X by the same value mod |n|. 275 * When we're done, (1) still holds. */ 276 shift = 0; 277 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */ 278 { 279 shift++; 280 281 if (BN_is_odd(X)) 282 { 283 if (!BN_uadd(X, X, n)) goto err; 284 } 285 /* now X is even, so we can easily divide it by two */ 286 if (!BN_rshift1(X, X)) goto err; 287 } 288 if (shift > 0) 289 { 290 if (!BN_rshift(B, B, shift)) goto err; 291 } 292 293 294 /* Same for A and Y. Afterwards, (2) still holds. */ 295 shift = 0; 296 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */ 297 { 298 shift++; 299 300 if (BN_is_odd(Y)) 301 { 302 if (!BN_uadd(Y, Y, n)) goto err; 303 } 304 /* now Y is even */ 305 if (!BN_rshift1(Y, Y)) goto err; 306 } 307 if (shift > 0) 308 { 309 if (!BN_rshift(A, A, shift)) goto err; 310 } 311 312 313 /* We still have (1) and (2). 314 * Both A and B are odd. 315 * The following computations ensure that 316 * 317 * 0 <= B < |n|, 318 * 0 < A < |n|, 319 * (1) -sign*X*a == B (mod |n|), 320 * (2) sign*Y*a == A (mod |n|), 321 * 322 * and that either A or B is even in the next iteration. 323 */ 324 if (BN_ucmp(B, A) >= 0) 325 { 326 /* -sign*(X + Y)*a == B - A (mod |n|) */ 327 if (!BN_uadd(X, X, Y)) goto err; 328 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that 329 * actually makes the algorithm slower */ 330 if (!BN_usub(B, B, A)) goto err; 331 } 332 else 333 { 334 /* sign*(X + Y)*a == A - B (mod |n|) */ 335 if (!BN_uadd(Y, Y, X)) goto err; 336 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */ 337 if (!BN_usub(A, A, B)) goto err; 338 } 339 } 340 } 341 else 342 { 343 /* general inversion algorithm */ 344 345 while (!BN_is_zero(B)) 346 { 347 BIGNUM *tmp; 348 349 /* 350 * 0 < B < A, 351 * (*) -sign*X*a == B (mod |n|), 352 * sign*Y*a == A (mod |n|) 353 */ 354 355 /* (D, M) := (A/B, A%B) ... */ 356 if (BN_num_bits(A) == BN_num_bits(B)) 357 { 358 if (!BN_one(D)) goto err; 359 if (!BN_sub(M,A,B)) goto err; 360 } 361 else if (BN_num_bits(A) == BN_num_bits(B) + 1) 362 { 363 /* A/B is 1, 2, or 3 */ 364 if (!BN_lshift1(T,B)) goto err; 365 if (BN_ucmp(A,T) < 0) 366 { 367 /* A < 2*B, so D=1 */ 368 if (!BN_one(D)) goto err; 369 if (!BN_sub(M,A,B)) goto err; 370 } 371 else 372 { 373 /* A >= 2*B, so D=2 or D=3 */ 374 if (!BN_sub(M,A,T)) goto err; 375 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */ 376 if (BN_ucmp(A,D) < 0) 377 { 378 /* A < 3*B, so D=2 */ 379 if (!BN_set_word(D,2)) goto err; 380 /* M (= A - 2*B) already has the correct value */ 381 } 382 else 383 { 384 /* only D=3 remains */ 385 if (!BN_set_word(D,3)) goto err; 386 /* currently M = A - 2*B, but we need M = A - 3*B */ 387 if (!BN_sub(M,M,B)) goto err; 388 } 389 } 390 } 391 else 392 { 393 if (!BN_div(D,M,A,B,ctx)) goto err; 394 } 395 396 /* Now 397 * A = D*B + M; 398 * thus we have 399 * (**) sign*Y*a == D*B + M (mod |n|). 400 */ 401 402 tmp=A; /* keep the BIGNUM object, the value does not matter */ 403 404 /* (A, B) := (B, A mod B) ... */ 405 A=B; 406 B=M; 407 /* ... so we have 0 <= B < A again */ 408 409 /* Since the former M is now B and the former B is now A, 410 * (**) translates into 411 * sign*Y*a == D*A + B (mod |n|), 412 * i.e. 413 * sign*Y*a - D*A == B (mod |n|). 414 * Similarly, (*) translates into 415 * -sign*X*a == A (mod |n|). 416 * 417 * Thus, 418 * sign*Y*a + D*sign*X*a == B (mod |n|), 419 * i.e. 420 * sign*(Y + D*X)*a == B (mod |n|). 421 * 422 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 423 * -sign*X*a == B (mod |n|), 424 * sign*Y*a == A (mod |n|). 425 * Note that X and Y stay non-negative all the time. 426 */ 427 428 /* most of the time D is very small, so we can optimize tmp := D*X+Y */ 429 if (BN_is_one(D)) 430 { 431 if (!BN_add(tmp,X,Y)) goto err; 432 } 433 else 434 { 435 if (BN_is_word(D,2)) 436 { 437 if (!BN_lshift1(tmp,X)) goto err; 438 } 439 else if (BN_is_word(D,4)) 440 { 441 if (!BN_lshift(tmp,X,2)) goto err; 442 } 443 else if (D->top == 1) 444 { 445 if (!BN_copy(tmp,X)) goto err; 446 if (!BN_mul_word(tmp,D->d[0])) goto err; 447 } 448 else 449 { 450 if (!BN_mul(tmp,D,X,ctx)) goto err; 451 } 452 if (!BN_add(tmp,tmp,Y)) goto err; 453 } 454 455 M=Y; /* keep the BIGNUM object, the value does not matter */ 456 Y=X; 457 X=tmp; 458 sign = -sign; 459 } 460 } 461 462 /* 463 * The while loop (Euclid's algorithm) ends when 464 * A == gcd(a,n); 465 * we have 466 * sign*Y*a == A (mod |n|), 467 * where Y is non-negative. 468 */ 469 470 if (sign < 0) 471 { 472 if (!BN_sub(Y,n,Y)) goto err; 473 } 474 /* Now Y*a == A (mod |n|). */ 475 476 477 if (BN_is_one(A)) 478 { 479 /* Y*a == 1 (mod |n|) */ 480 if (!Y->neg && BN_ucmp(Y,n) < 0) 481 { 482 if (!BN_copy(R,Y)) goto err; 483 } 484 else 485 { 486 if (!BN_nnmod(R,Y,n,ctx)) goto err; 487 } 488 } 489 else 490 { 491 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE); 492 goto err; 493 } 494 ret=R; 495 err: 496 if ((ret == NULL) && (in == NULL)) BN_free(R); 497 BN_CTX_end(ctx); 498 bn_check_top(ret); 499 return(ret); 500 } 501 502 503 /* BN_mod_inverse_no_branch is a special version of BN_mod_inverse. 504 * It does not contain branches that may leak sensitive information. 505 */ 506 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in, 507 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 508 { 509 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; 510 BIGNUM local_A, local_B; 511 BIGNUM *pA, *pB; 512 BIGNUM *ret=NULL; 513 int sign; 514 515 bn_check_top(a); 516 bn_check_top(n); 517 518 BN_CTX_start(ctx); 519 A = BN_CTX_get(ctx); 520 B = BN_CTX_get(ctx); 521 X = BN_CTX_get(ctx); 522 D = BN_CTX_get(ctx); 523 M = BN_CTX_get(ctx); 524 Y = BN_CTX_get(ctx); 525 T = BN_CTX_get(ctx); 526 if (T == NULL) goto err; 527 528 if (in == NULL) 529 R=BN_new(); 530 else 531 R=in; 532 if (R == NULL) goto err; 533 534 BN_one(X); 535 BN_zero(Y); 536 if (BN_copy(B,a) == NULL) goto err; 537 if (BN_copy(A,n) == NULL) goto err; 538 A->neg = 0; 539 540 if (B->neg || (BN_ucmp(B, A) >= 0)) 541 { 542 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked, 543 * BN_div_no_branch will be called eventually. 544 */ 545 pB = &local_B; 546 BN_with_flags(pB, B, BN_FLG_CONSTTIME); 547 if (!BN_nnmod(B, pB, A, ctx)) goto err; 548 } 549 sign = -1; 550 /* From B = a mod |n|, A = |n| it follows that 551 * 552 * 0 <= B < A, 553 * -sign*X*a == B (mod |n|), 554 * sign*Y*a == A (mod |n|). 555 */ 556 557 while (!BN_is_zero(B)) 558 { 559 BIGNUM *tmp; 560 561 /* 562 * 0 < B < A, 563 * (*) -sign*X*a == B (mod |n|), 564 * sign*Y*a == A (mod |n|) 565 */ 566 567 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked, 568 * BN_div_no_branch will be called eventually. 569 */ 570 pA = &local_A; 571 BN_with_flags(pA, A, BN_FLG_CONSTTIME); 572 573 /* (D, M) := (A/B, A%B) ... */ 574 if (!BN_div(D,M,pA,B,ctx)) goto err; 575 576 /* Now 577 * A = D*B + M; 578 * thus we have 579 * (**) sign*Y*a == D*B + M (mod |n|). 580 */ 581 582 tmp=A; /* keep the BIGNUM object, the value does not matter */ 583 584 /* (A, B) := (B, A mod B) ... */ 585 A=B; 586 B=M; 587 /* ... so we have 0 <= B < A again */ 588 589 /* Since the former M is now B and the former B is now A, 590 * (**) translates into 591 * sign*Y*a == D*A + B (mod |n|), 592 * i.e. 593 * sign*Y*a - D*A == B (mod |n|). 594 * Similarly, (*) translates into 595 * -sign*X*a == A (mod |n|). 596 * 597 * Thus, 598 * sign*Y*a + D*sign*X*a == B (mod |n|), 599 * i.e. 600 * sign*(Y + D*X)*a == B (mod |n|). 601 * 602 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 603 * -sign*X*a == B (mod |n|), 604 * sign*Y*a == A (mod |n|). 605 * Note that X and Y stay non-negative all the time. 606 */ 607 608 if (!BN_mul(tmp,D,X,ctx)) goto err; 609 if (!BN_add(tmp,tmp,Y)) goto err; 610 611 M=Y; /* keep the BIGNUM object, the value does not matter */ 612 Y=X; 613 X=tmp; 614 sign = -sign; 615 } 616 617 /* 618 * The while loop (Euclid's algorithm) ends when 619 * A == gcd(a,n); 620 * we have 621 * sign*Y*a == A (mod |n|), 622 * where Y is non-negative. 623 */ 624 625 if (sign < 0) 626 { 627 if (!BN_sub(Y,n,Y)) goto err; 628 } 629 /* Now Y*a == A (mod |n|). */ 630 631 if (BN_is_one(A)) 632 { 633 /* Y*a == 1 (mod |n|) */ 634 if (!Y->neg && BN_ucmp(Y,n) < 0) 635 { 636 if (!BN_copy(R,Y)) goto err; 637 } 638 else 639 { 640 if (!BN_nnmod(R,Y,n,ctx)) goto err; 641 } 642 } 643 else 644 { 645 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE); 646 goto err; 647 } 648 ret=R; 649 err: 650 if ((ret == NULL) && (in == NULL)) BN_free(R); 651 BN_CTX_end(ctx); 652 bn_check_top(ret); 653 return(ret); 654 } 655
