1/* crypto/bn/bn_gcd.c */ 2/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com) 3 * All rights reserved. 4 * 5 * This package is an SSL implementation written 6 * by Eric Young (eay@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@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@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@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@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@cryptsoft.com). This product includes software written by Tim 108 * Hudson (tjh@cryptsoft.com). 109 * 110 */ 111 112#include "cryptlib.h" 113#include "bn_lcl.h" 114 115static BIGNUM *euclid(BIGNUM *a, BIGNUM *b); 116 117int 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; 141err: 142 BN_CTX_end(ctx); 143 bn_check_top(r); 144 return(ret); 145 } 146 147static 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); 200err: 201 return(NULL); 202 } 203 204 205/* solves ax == 1 (mod n) */ 206static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in, 207 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx); 208 209BIGNUM *BN_mod_inverse(BIGNUM *in, 210 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 211 { 212 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; 213 BIGNUM *ret=NULL; 214 int sign; 215 216 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) 217 { 218 return BN_mod_inverse_no_branch(in, a, n, ctx); 219 } 220 221 bn_check_top(a); 222 bn_check_top(n); 223 224 BN_CTX_start(ctx); 225 A = BN_CTX_get(ctx); 226 B = BN_CTX_get(ctx); 227 X = BN_CTX_get(ctx); 228 D = BN_CTX_get(ctx); 229 M = BN_CTX_get(ctx); 230 Y = BN_CTX_get(ctx); 231 T = BN_CTX_get(ctx); 232 if (T == NULL) goto err; 233 234 if (in == NULL) 235 R=BN_new(); 236 else 237 R=in; 238 if (R == NULL) goto err; 239 240 BN_one(X); 241 BN_zero(Y); 242 if (BN_copy(B,a) == NULL) goto err; 243 if (BN_copy(A,n) == NULL) goto err; 244 A->neg = 0; 245 if (B->neg || (BN_ucmp(B, A) >= 0)) 246 { 247 if (!BN_nnmod(B, B, A, ctx)) goto err; 248 } 249 sign = -1; 250 /* From B = a mod |n|, A = |n| it follows that 251 * 252 * 0 <= B < A, 253 * -sign*X*a == B (mod |n|), 254 * sign*Y*a == A (mod |n|). 255 */ 256 257 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) 258 { 259 /* Binary inversion algorithm; requires odd modulus. 260 * This is faster than the general algorithm if the modulus 261 * is sufficiently small (about 400 .. 500 bits on 32-bit 262 * sytems, but much more on 64-bit systems) */ 263 int shift; 264 265 while (!BN_is_zero(B)) 266 { 267 /* 268 * 0 < B < |n|, 269 * 0 < A <= |n|, 270 * (1) -sign*X*a == B (mod |n|), 271 * (2) sign*Y*a == A (mod |n|) 272 */ 273 274 /* Now divide B by the maximum possible power of two in the integers, 275 * and divide X by the same value mod |n|. 276 * When we're done, (1) still holds. */ 277 shift = 0; 278 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */ 279 { 280 shift++; 281 282 if (BN_is_odd(X)) 283 { 284 if (!BN_uadd(X, X, n)) goto err; 285 } 286 /* now X is even, so we can easily divide it by two */ 287 if (!BN_rshift1(X, X)) goto err; 288 } 289 if (shift > 0) 290 { 291 if (!BN_rshift(B, B, shift)) goto err; 292 } 293 294 295 /* Same for A and Y. Afterwards, (2) still holds. */ 296 shift = 0; 297 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */ 298 { 299 shift++; 300 301 if (BN_is_odd(Y)) 302 { 303 if (!BN_uadd(Y, Y, n)) goto err; 304 } 305 /* now Y is even */ 306 if (!BN_rshift1(Y, Y)) goto err; 307 } 308 if (shift > 0) 309 { 310 if (!BN_rshift(A, A, shift)) goto err; 311 } 312 313 314 /* We still have (1) and (2). 315 * Both A and B are odd. 316 * The following computations ensure that 317 * 318 * 0 <= B < |n|, 319 * 0 < A < |n|, 320 * (1) -sign*X*a == B (mod |n|), 321 * (2) sign*Y*a == A (mod |n|), 322 * 323 * and that either A or B is even in the next iteration. 324 */ 325 if (BN_ucmp(B, A) >= 0) 326 { 327 /* -sign*(X + Y)*a == B - A (mod |n|) */ 328 if (!BN_uadd(X, X, Y)) goto err; 329 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that 330 * actually makes the algorithm slower */ 331 if (!BN_usub(B, B, A)) goto err; 332 } 333 else 334 { 335 /* sign*(X + Y)*a == A - B (mod |n|) */ 336 if (!BN_uadd(Y, Y, X)) goto err; 337 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */ 338 if (!BN_usub(A, A, B)) goto err; 339 } 340 } 341 } 342 else 343 { 344 /* general inversion algorithm */ 345 346 while (!BN_is_zero(B)) 347 { 348 BIGNUM *tmp; 349 350 /* 351 * 0 < B < A, 352 * (*) -sign*X*a == B (mod |n|), 353 * sign*Y*a == A (mod |n|) 354 */ 355 356 /* (D, M) := (A/B, A%B) ... */ 357 if (BN_num_bits(A) == BN_num_bits(B)) 358 { 359 if (!BN_one(D)) goto err; 360 if (!BN_sub(M,A,B)) goto err; 361 } 362 else if (BN_num_bits(A) == BN_num_bits(B) + 1) 363 { 364 /* A/B is 1, 2, or 3 */ 365 if (!BN_lshift1(T,B)) goto err; 366 if (BN_ucmp(A,T) < 0) 367 { 368 /* A < 2*B, so D=1 */ 369 if (!BN_one(D)) goto err; 370 if (!BN_sub(M,A,B)) goto err; 371 } 372 else 373 { 374 /* A >= 2*B, so D=2 or D=3 */ 375 if (!BN_sub(M,A,T)) goto err; 376 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */ 377 if (BN_ucmp(A,D) < 0) 378 { 379 /* A < 3*B, so D=2 */ 380 if (!BN_set_word(D,2)) goto err; 381 /* M (= A - 2*B) already has the correct value */ 382 } 383 else 384 { 385 /* only D=3 remains */ 386 if (!BN_set_word(D,3)) goto err; 387 /* currently M = A - 2*B, but we need M = A - 3*B */ 388 if (!BN_sub(M,M,B)) goto err; 389 } 390 } 391 } 392 else 393 { 394 if (!BN_div(D,M,A,B,ctx)) goto err; 395 } 396 397 /* Now 398 * A = D*B + M; 399 * thus we have 400 * (**) sign*Y*a == D*B + M (mod |n|). 401 */ 402 403 tmp=A; /* keep the BIGNUM object, the value does not matter */ 404 405 /* (A, B) := (B, A mod B) ... */ 406 A=B; 407 B=M; 408 /* ... so we have 0 <= B < A again */ 409 410 /* Since the former M is now B and the former B is now A, 411 * (**) translates into 412 * sign*Y*a == D*A + B (mod |n|), 413 * i.e. 414 * sign*Y*a - D*A == B (mod |n|). 415 * Similarly, (*) translates into 416 * -sign*X*a == A (mod |n|). 417 * 418 * Thus, 419 * sign*Y*a + D*sign*X*a == B (mod |n|), 420 * i.e. 421 * sign*(Y + D*X)*a == B (mod |n|). 422 * 423 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 424 * -sign*X*a == B (mod |n|), 425 * sign*Y*a == A (mod |n|). 426 * Note that X and Y stay non-negative all the time. 427 */ 428 429 /* most of the time D is very small, so we can optimize tmp := D*X+Y */ 430 if (BN_is_one(D)) 431 { 432 if (!BN_add(tmp,X,Y)) goto err; 433 } 434 else 435 { 436 if (BN_is_word(D,2)) 437 { 438 if (!BN_lshift1(tmp,X)) goto err; 439 } 440 else if (BN_is_word(D,4)) 441 { 442 if (!BN_lshift(tmp,X,2)) goto err; 443 } 444 else if (D->top == 1) 445 { 446 if (!BN_copy(tmp,X)) goto err; 447 if (!BN_mul_word(tmp,D->d[0])) goto err; 448 } 449 else 450 { 451 if (!BN_mul(tmp,D,X,ctx)) goto err; 452 } 453 if (!BN_add(tmp,tmp,Y)) goto err; 454 } 455 456 M=Y; /* keep the BIGNUM object, the value does not matter */ 457 Y=X; 458 X=tmp; 459 sign = -sign; 460 } 461 } 462 463 /* 464 * The while loop (Euclid's algorithm) ends when 465 * A == gcd(a,n); 466 * we have 467 * sign*Y*a == A (mod |n|), 468 * where Y is non-negative. 469 */ 470 471 if (sign < 0) 472 { 473 if (!BN_sub(Y,n,Y)) goto err; 474 } 475 /* Now Y*a == A (mod |n|). */ 476 477 478 if (BN_is_one(A)) 479 { 480 /* Y*a == 1 (mod |n|) */ 481 if (!Y->neg && BN_ucmp(Y,n) < 0) 482 { 483 if (!BN_copy(R,Y)) goto err; 484 } 485 else 486 { 487 if (!BN_nnmod(R,Y,n,ctx)) goto err; 488 } 489 } 490 else 491 { 492 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE); 493 goto err; 494 } 495 ret=R; 496err: 497 if ((ret == NULL) && (in == NULL)) BN_free(R); 498 BN_CTX_end(ctx); 499 bn_check_top(ret); 500 return(ret); 501 } 502 503 504/* BN_mod_inverse_no_branch is a special version of BN_mod_inverse. 505 * It does not contain branches that may leak sensitive information. 506 */ 507static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in, 508 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 509 { 510 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; 511 BIGNUM local_A, local_B; 512 BIGNUM *pA, *pB; 513 BIGNUM *ret=NULL; 514 int sign; 515 516 bn_check_top(a); 517 bn_check_top(n); 518 519 BN_CTX_start(ctx); 520 A = BN_CTX_get(ctx); 521 B = BN_CTX_get(ctx); 522 X = BN_CTX_get(ctx); 523 D = BN_CTX_get(ctx); 524 M = BN_CTX_get(ctx); 525 Y = BN_CTX_get(ctx); 526 T = BN_CTX_get(ctx); 527 if (T == NULL) goto err; 528 529 if (in == NULL) 530 R=BN_new(); 531 else 532 R=in; 533 if (R == NULL) goto err; 534 535 BN_one(X); 536 BN_zero(Y); 537 if (BN_copy(B,a) == NULL) goto err; 538 if (BN_copy(A,n) == NULL) goto err; 539 A->neg = 0; 540 541 if (B->neg || (BN_ucmp(B, A) >= 0)) 542 { 543 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked, 544 * BN_div_no_branch will be called eventually. 545 */ 546 pB = &local_B; 547 BN_with_flags(pB, B, BN_FLG_CONSTTIME); 548 if (!BN_nnmod(B, pB, A, ctx)) goto err; 549 } 550 sign = -1; 551 /* From B = a mod |n|, A = |n| it follows that 552 * 553 * 0 <= B < A, 554 * -sign*X*a == B (mod |n|), 555 * sign*Y*a == A (mod |n|). 556 */ 557 558 while (!BN_is_zero(B)) 559 { 560 BIGNUM *tmp; 561 562 /* 563 * 0 < B < A, 564 * (*) -sign*X*a == B (mod |n|), 565 * sign*Y*a == A (mod |n|) 566 */ 567 568 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked, 569 * BN_div_no_branch will be called eventually. 570 */ 571 pA = &local_A; 572 BN_with_flags(pA, A, BN_FLG_CONSTTIME); 573 574 /* (D, M) := (A/B, A%B) ... */ 575 if (!BN_div(D,M,pA,B,ctx)) goto err; 576 577 /* Now 578 * A = D*B + M; 579 * thus we have 580 * (**) sign*Y*a == D*B + M (mod |n|). 581 */ 582 583 tmp=A; /* keep the BIGNUM object, the value does not matter */ 584 585 /* (A, B) := (B, A mod B) ... */ 586 A=B; 587 B=M; 588 /* ... so we have 0 <= B < A again */ 589 590 /* Since the former M is now B and the former B is now A, 591 * (**) translates into 592 * sign*Y*a == D*A + B (mod |n|), 593 * i.e. 594 * sign*Y*a - D*A == B (mod |n|). 595 * Similarly, (*) translates into 596 * -sign*X*a == A (mod |n|). 597 * 598 * Thus, 599 * sign*Y*a + D*sign*X*a == B (mod |n|), 600 * i.e. 601 * sign*(Y + D*X)*a == B (mod |n|). 602 * 603 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 604 * -sign*X*a == B (mod |n|), 605 * sign*Y*a == A (mod |n|). 606 * Note that X and Y stay non-negative all the time. 607 */ 608 609 if (!BN_mul(tmp,D,X,ctx)) goto err; 610 if (!BN_add(tmp,tmp,Y)) goto err; 611 612 M=Y; /* keep the BIGNUM object, the value does not matter */ 613 Y=X; 614 X=tmp; 615 sign = -sign; 616 } 617 618 /* 619 * The while loop (Euclid's algorithm) ends when 620 * A == gcd(a,n); 621 * we have 622 * sign*Y*a == A (mod |n|), 623 * where Y is non-negative. 624 */ 625 626 if (sign < 0) 627 { 628 if (!BN_sub(Y,n,Y)) goto err; 629 } 630 /* Now Y*a == A (mod |n|). */ 631 632 if (BN_is_one(A)) 633 { 634 /* Y*a == 1 (mod |n|) */ 635 if (!Y->neg && BN_ucmp(Y,n) < 0) 636 { 637 if (!BN_copy(R,Y)) goto err; 638 } 639 else 640 { 641 if (!BN_nnmod(R,Y,n,ctx)) goto err; 642 } 643 } 644 else 645 { 646 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE); 647 goto err; 648 } 649 ret=R; 650err: 651 if ((ret == NULL) && (in == NULL)) BN_free(R); 652 BN_CTX_end(ctx); 653 bn_check_top(ret); 654 return(ret); 655 } 656