1/* 2 * Copyright 1995-2023 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the Apache License 2.0 (the "License"). You may not use 5 * this file except in compliance with the License. You can obtain a copy 6 * in the file LICENSE in the source distribution or at 7 * https://www.openssl.org/source/license.html 8 */ 9 10#include "internal/cryptlib.h" 11#include "bn_local.h" 12 13/* 14 * bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does 15 * not contain branches that may leak sensitive information. 16 * 17 * This is a static function, we ensure all callers in this file pass valid 18 * arguments: all passed pointers here are non-NULL. 19 */ 20static ossl_inline 21BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in, 22 const BIGNUM *a, const BIGNUM *n, 23 BN_CTX *ctx, int *pnoinv) 24{ 25 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL; 26 BIGNUM *ret = NULL; 27 int sign; 28 29 bn_check_top(a); 30 bn_check_top(n); 31 32 BN_CTX_start(ctx); 33 A = BN_CTX_get(ctx); 34 B = BN_CTX_get(ctx); 35 X = BN_CTX_get(ctx); 36 D = BN_CTX_get(ctx); 37 M = BN_CTX_get(ctx); 38 Y = BN_CTX_get(ctx); 39 T = BN_CTX_get(ctx); 40 if (T == NULL) 41 goto err; 42 43 if (in == NULL) 44 R = BN_new(); 45 else 46 R = in; 47 if (R == NULL) 48 goto err; 49 50 if (!BN_one(X)) 51 goto err; 52 BN_zero(Y); 53 if (BN_copy(B, a) == NULL) 54 goto err; 55 if (BN_copy(A, n) == NULL) 56 goto err; 57 A->neg = 0; 58 59 if (B->neg || (BN_ucmp(B, A) >= 0)) { 60 /* 61 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked, 62 * BN_div_no_branch will be called eventually. 63 */ 64 { 65 BIGNUM local_B; 66 bn_init(&local_B); 67 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME); 68 if (!BN_nnmod(B, &local_B, A, ctx)) 69 goto err; 70 /* Ensure local_B goes out of scope before any further use of B */ 71 } 72 } 73 sign = -1; 74 /*- 75 * From B = a mod |n|, A = |n| it follows that 76 * 77 * 0 <= B < A, 78 * -sign*X*a == B (mod |n|), 79 * sign*Y*a == A (mod |n|). 80 */ 81 82 while (!BN_is_zero(B)) { 83 BIGNUM *tmp; 84 85 /*- 86 * 0 < B < A, 87 * (*) -sign*X*a == B (mod |n|), 88 * sign*Y*a == A (mod |n|) 89 */ 90 91 /* 92 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked, 93 * BN_div_no_branch will be called eventually. 94 */ 95 { 96 BIGNUM local_A; 97 bn_init(&local_A); 98 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME); 99 100 /* (D, M) := (A/B, A%B) ... */ 101 if (!BN_div(D, M, &local_A, B, ctx)) 102 goto err; 103 /* Ensure local_A goes out of scope before any further use of A */ 104 } 105 106 /*- 107 * Now 108 * A = D*B + M; 109 * thus we have 110 * (**) sign*Y*a == D*B + M (mod |n|). 111 */ 112 113 tmp = A; /* keep the BIGNUM object, the value does not 114 * matter */ 115 116 /* (A, B) := (B, A mod B) ... */ 117 A = B; 118 B = M; 119 /* ... so we have 0 <= B < A again */ 120 121 /*- 122 * Since the former M is now B and the former B is now A, 123 * (**) translates into 124 * sign*Y*a == D*A + B (mod |n|), 125 * i.e. 126 * sign*Y*a - D*A == B (mod |n|). 127 * Similarly, (*) translates into 128 * -sign*X*a == A (mod |n|). 129 * 130 * Thus, 131 * sign*Y*a + D*sign*X*a == B (mod |n|), 132 * i.e. 133 * sign*(Y + D*X)*a == B (mod |n|). 134 * 135 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 136 * -sign*X*a == B (mod |n|), 137 * sign*Y*a == A (mod |n|). 138 * Note that X and Y stay non-negative all the time. 139 */ 140 141 if (!BN_mul(tmp, D, X, ctx)) 142 goto err; 143 if (!BN_add(tmp, tmp, Y)) 144 goto err; 145 146 M = Y; /* keep the BIGNUM object, the value does not 147 * matter */ 148 Y = X; 149 X = tmp; 150 sign = -sign; 151 } 152 153 /*- 154 * The while loop (Euclid's algorithm) ends when 155 * A == gcd(a,n); 156 * we have 157 * sign*Y*a == A (mod |n|), 158 * where Y is non-negative. 159 */ 160 161 if (sign < 0) { 162 if (!BN_sub(Y, n, Y)) 163 goto err; 164 } 165 /* Now Y*a == A (mod |n|). */ 166 167 if (BN_is_one(A)) { 168 /* Y*a == 1 (mod |n|) */ 169 if (!Y->neg && BN_ucmp(Y, n) < 0) { 170 if (!BN_copy(R, Y)) 171 goto err; 172 } else { 173 if (!BN_nnmod(R, Y, n, ctx)) 174 goto err; 175 } 176 } else { 177 *pnoinv = 1; 178 /* caller sets the BN_R_NO_INVERSE error */ 179 goto err; 180 } 181 182 ret = R; 183 *pnoinv = 0; 184 185 err: 186 if ((ret == NULL) && (in == NULL)) 187 BN_free(R); 188 BN_CTX_end(ctx); 189 bn_check_top(ret); 190 return ret; 191} 192 193/* 194 * This is an internal function, we assume all callers pass valid arguments: 195 * all pointers passed here are assumed non-NULL. 196 */ 197BIGNUM *int_bn_mod_inverse(BIGNUM *in, 198 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx, 199 int *pnoinv) 200{ 201 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL; 202 BIGNUM *ret = NULL; 203 int sign; 204 205 /* This is invalid input so we don't worry about constant time here */ 206 if (BN_abs_is_word(n, 1) || BN_is_zero(n)) { 207 *pnoinv = 1; 208 return NULL; 209 } 210 211 *pnoinv = 0; 212 213 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) 214 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) { 215 return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv); 216 } 217 218 bn_check_top(a); 219 bn_check_top(n); 220 221 BN_CTX_start(ctx); 222 A = BN_CTX_get(ctx); 223 B = BN_CTX_get(ctx); 224 X = BN_CTX_get(ctx); 225 D = BN_CTX_get(ctx); 226 M = BN_CTX_get(ctx); 227 Y = BN_CTX_get(ctx); 228 T = BN_CTX_get(ctx); 229 if (T == NULL) 230 goto err; 231 232 if (in == NULL) 233 R = BN_new(); 234 else 235 R = in; 236 if (R == NULL) 237 goto err; 238 239 if (!BN_one(X)) 240 goto err; 241 BN_zero(Y); 242 if (BN_copy(B, a) == NULL) 243 goto err; 244 if (BN_copy(A, n) == NULL) 245 goto err; 246 A->neg = 0; 247 if (B->neg || (BN_ucmp(B, A) >= 0)) { 248 if (!BN_nnmod(B, B, A, ctx)) 249 goto err; 250 } 251 sign = -1; 252 /*- 253 * From B = a mod |n|, A = |n| it follows that 254 * 255 * 0 <= B < A, 256 * -sign*X*a == B (mod |n|), 257 * sign*Y*a == A (mod |n|). 258 */ 259 260 if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) { 261 /* 262 * Binary inversion algorithm; requires odd modulus. This is faster 263 * than the general algorithm if the modulus is sufficiently small 264 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit 265 * systems) 266 */ 267 int shift; 268 269 while (!BN_is_zero(B)) { 270 /*- 271 * 0 < B < |n|, 272 * 0 < A <= |n|, 273 * (1) -sign*X*a == B (mod |n|), 274 * (2) sign*Y*a == A (mod |n|) 275 */ 276 277 /* 278 * Now divide B by the maximum possible power of two in the 279 * integers, and divide X by the same value mod |n|. When we're 280 * done, (1) still holds. 281 */ 282 shift = 0; 283 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */ 284 shift++; 285 286 if (BN_is_odd(X)) { 287 if (!BN_uadd(X, X, n)) 288 goto err; 289 } 290 /* 291 * now X is even, so we can easily divide it by two 292 */ 293 if (!BN_rshift1(X, X)) 294 goto err; 295 } 296 if (shift > 0) { 297 if (!BN_rshift(B, B, shift)) 298 goto err; 299 } 300 301 /* 302 * Same for A and Y. Afterwards, (2) still holds. 303 */ 304 shift = 0; 305 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */ 306 shift++; 307 308 if (BN_is_odd(Y)) { 309 if (!BN_uadd(Y, Y, n)) 310 goto err; 311 } 312 /* now Y is even */ 313 if (!BN_rshift1(Y, Y)) 314 goto err; 315 } 316 if (shift > 0) { 317 if (!BN_rshift(A, A, shift)) 318 goto err; 319 } 320 321 /*- 322 * We still have (1) and (2). 323 * Both A and B are odd. 324 * The following computations ensure that 325 * 326 * 0 <= B < |n|, 327 * 0 < A < |n|, 328 * (1) -sign*X*a == B (mod |n|), 329 * (2) sign*Y*a == A (mod |n|), 330 * 331 * and that either A or B is even in the next iteration. 332 */ 333 if (BN_ucmp(B, A) >= 0) { 334 /* -sign*(X + Y)*a == B - A (mod |n|) */ 335 if (!BN_uadd(X, X, Y)) 336 goto err; 337 /* 338 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that 339 * actually makes the algorithm slower 340 */ 341 if (!BN_usub(B, B, A)) 342 goto err; 343 } else { 344 /* sign*(X + Y)*a == A - B (mod |n|) */ 345 if (!BN_uadd(Y, Y, X)) 346 goto err; 347 /* 348 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down 349 */ 350 if (!BN_usub(A, A, B)) 351 goto err; 352 } 353 } 354 } else { 355 /* general inversion algorithm */ 356 357 while (!BN_is_zero(B)) { 358 BIGNUM *tmp; 359 360 /*- 361 * 0 < B < A, 362 * (*) -sign*X*a == B (mod |n|), 363 * sign*Y*a == A (mod |n|) 364 */ 365 366 /* (D, M) := (A/B, A%B) ... */ 367 if (BN_num_bits(A) == BN_num_bits(B)) { 368 if (!BN_one(D)) 369 goto err; 370 if (!BN_sub(M, A, B)) 371 goto err; 372 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) { 373 /* A/B is 1, 2, or 3 */ 374 if (!BN_lshift1(T, B)) 375 goto err; 376 if (BN_ucmp(A, T) < 0) { 377 /* A < 2*B, so D=1 */ 378 if (!BN_one(D)) 379 goto err; 380 if (!BN_sub(M, A, B)) 381 goto err; 382 } else { 383 /* A >= 2*B, so D=2 or D=3 */ 384 if (!BN_sub(M, A, T)) 385 goto err; 386 if (!BN_add(D, T, B)) 387 goto err; /* use D (:= 3*B) as temp */ 388 if (BN_ucmp(A, D) < 0) { 389 /* A < 3*B, so D=2 */ 390 if (!BN_set_word(D, 2)) 391 goto err; 392 /* 393 * M (= A - 2*B) already has the correct value 394 */ 395 } else { 396 /* only D=3 remains */ 397 if (!BN_set_word(D, 3)) 398 goto err; 399 /* 400 * currently M = A - 2*B, but we need M = A - 3*B 401 */ 402 if (!BN_sub(M, M, B)) 403 goto err; 404 } 405 } 406 } else { 407 if (!BN_div(D, M, A, B, ctx)) 408 goto err; 409 } 410 411 /*- 412 * Now 413 * A = D*B + M; 414 * thus we have 415 * (**) sign*Y*a == D*B + M (mod |n|). 416 */ 417 418 tmp = A; /* keep the BIGNUM object, the value does not matter */ 419 420 /* (A, B) := (B, A mod B) ... */ 421 A = B; 422 B = M; 423 /* ... so we have 0 <= B < A again */ 424 425 /*- 426 * Since the former M is now B and the former B is now A, 427 * (**) translates into 428 * sign*Y*a == D*A + B (mod |n|), 429 * i.e. 430 * sign*Y*a - D*A == B (mod |n|). 431 * Similarly, (*) translates into 432 * -sign*X*a == A (mod |n|). 433 * 434 * Thus, 435 * sign*Y*a + D*sign*X*a == B (mod |n|), 436 * i.e. 437 * sign*(Y + D*X)*a == B (mod |n|). 438 * 439 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 440 * -sign*X*a == B (mod |n|), 441 * sign*Y*a == A (mod |n|). 442 * Note that X and Y stay non-negative all the time. 443 */ 444 445 /* 446 * most of the time D is very small, so we can optimize tmp := D*X+Y 447 */ 448 if (BN_is_one(D)) { 449 if (!BN_add(tmp, X, Y)) 450 goto err; 451 } else { 452 if (BN_is_word(D, 2)) { 453 if (!BN_lshift1(tmp, X)) 454 goto err; 455 } else if (BN_is_word(D, 4)) { 456 if (!BN_lshift(tmp, X, 2)) 457 goto err; 458 } else if (D->top == 1) { 459 if (!BN_copy(tmp, X)) 460 goto err; 461 if (!BN_mul_word(tmp, D->d[0])) 462 goto err; 463 } else { 464 if (!BN_mul(tmp, D, X, ctx)) 465 goto err; 466 } 467 if (!BN_add(tmp, tmp, Y)) 468 goto err; 469 } 470 471 M = Y; /* keep the BIGNUM object, the value does not matter */ 472 Y = X; 473 X = tmp; 474 sign = -sign; 475 } 476 } 477 478 /*- 479 * The while loop (Euclid's algorithm) ends when 480 * A == gcd(a,n); 481 * we have 482 * sign*Y*a == A (mod |n|), 483 * where Y is non-negative. 484 */ 485 486 if (sign < 0) { 487 if (!BN_sub(Y, n, Y)) 488 goto err; 489 } 490 /* Now Y*a == A (mod |n|). */ 491 492 if (BN_is_one(A)) { 493 /* Y*a == 1 (mod |n|) */ 494 if (!Y->neg && BN_ucmp(Y, n) < 0) { 495 if (!BN_copy(R, Y)) 496 goto err; 497 } else { 498 if (!BN_nnmod(R, Y, n, ctx)) 499 goto err; 500 } 501 } else { 502 *pnoinv = 1; 503 goto err; 504 } 505 ret = R; 506 err: 507 if ((ret == NULL) && (in == NULL)) 508 BN_free(R); 509 BN_CTX_end(ctx); 510 bn_check_top(ret); 511 return ret; 512} 513 514/* solves ax == 1 (mod n) */ 515BIGNUM *BN_mod_inverse(BIGNUM *in, 516 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 517{ 518 BN_CTX *new_ctx = NULL; 519 BIGNUM *rv; 520 int noinv = 0; 521 522 if (ctx == NULL) { 523 ctx = new_ctx = BN_CTX_new_ex(NULL); 524 if (ctx == NULL) { 525 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); 526 return NULL; 527 } 528 } 529 530 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv); 531 if (noinv) 532 ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE); 533 BN_CTX_free(new_ctx); 534 return rv; 535} 536 537/*- 538 * This function is based on the constant-time GCD work by Bernstein and Yang: 539 * https://eprint.iacr.org/2019/266 540 * Generalized fast GCD function to allow even inputs. 541 * The algorithm first finds the shared powers of 2 between 542 * the inputs, and removes them, reducing at least one of the 543 * inputs to an odd value. Then it proceeds to calculate the GCD. 544 * Before returning the resulting GCD, we take care of adding 545 * back the powers of two removed at the beginning. 546 * Note 1: we assume the bit length of both inputs is public information, 547 * since access to top potentially leaks this information. 548 */ 549int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) 550{ 551 BIGNUM *g, *temp = NULL; 552 BN_ULONG mask = 0; 553 int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0; 554 555 /* Note 2: zero input corner cases are not constant-time since they are 556 * handled immediately. An attacker can run an attack under this 557 * assumption without the need of side-channel information. */ 558 if (BN_is_zero(in_b)) { 559 ret = BN_copy(r, in_a) != NULL; 560 r->neg = 0; 561 return ret; 562 } 563 if (BN_is_zero(in_a)) { 564 ret = BN_copy(r, in_b) != NULL; 565 r->neg = 0; 566 return ret; 567 } 568 569 bn_check_top(in_a); 570 bn_check_top(in_b); 571 572 BN_CTX_start(ctx); 573 temp = BN_CTX_get(ctx); 574 g = BN_CTX_get(ctx); 575 576 /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */ 577 if (g == NULL 578 || !BN_lshift1(g, in_b) 579 || !BN_lshift1(r, in_a)) 580 goto err; 581 582 /* find shared powers of two, i.e. "shifts" >= 1 */ 583 for (i = 0; i < r->dmax && i < g->dmax; i++) { 584 mask = ~(r->d[i] | g->d[i]); 585 for (j = 0; j < BN_BITS2; j++) { 586 bit &= mask; 587 shifts += bit; 588 mask >>= 1; 589 } 590 } 591 592 /* subtract shared powers of two; shifts >= 1 */ 593 if (!BN_rshift(r, r, shifts) 594 || !BN_rshift(g, g, shifts)) 595 goto err; 596 597 /* expand to biggest nword, with room for a possible extra word */ 598 top = 1 + ((r->top >= g->top) ? r->top : g->top); 599 if (bn_wexpand(r, top) == NULL 600 || bn_wexpand(g, top) == NULL 601 || bn_wexpand(temp, top) == NULL) 602 goto err; 603 604 /* re arrange inputs s.t. r is odd */ 605 BN_consttime_swap((~r->d[0]) & 1, r, g, top); 606 607 /* compute the number of iterations */ 608 rlen = BN_num_bits(r); 609 glen = BN_num_bits(g); 610 m = 4 + 3 * ((rlen >= glen) ? rlen : glen); 611 612 for (i = 0; i < m; i++) { 613 /* conditionally flip signs if delta is positive and g is odd */ 614 cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1 615 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */ 616 & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))); 617 delta = (-cond & -delta) | ((cond - 1) & delta); 618 r->neg ^= cond; 619 /* swap */ 620 BN_consttime_swap(cond, r, g, top); 621 622 /* elimination step */ 623 delta++; 624 if (!BN_add(temp, g, r)) 625 goto err; 626 BN_consttime_swap(g->d[0] & 1 /* g is odd */ 627 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */ 628 & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))), 629 g, temp, top); 630 if (!BN_rshift1(g, g)) 631 goto err; 632 } 633 634 /* remove possible negative sign */ 635 r->neg = 0; 636 /* add powers of 2 removed, then correct the artificial shift */ 637 if (!BN_lshift(r, r, shifts) 638 || !BN_rshift1(r, r)) 639 goto err; 640 641 ret = 1; 642 643 err: 644 BN_CTX_end(ctx); 645 bn_check_top(r); 646 return ret; 647} 648