2 * Copyright 1995-2020 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (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
10 #include "internal/cryptlib.h"
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.
17 * This is a static function, we ensure all callers in this file pass valid
18 * arguments: all passed pointers here are non-NULL.
21 BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in,
22 const BIGNUM *a, const BIGNUM *n,
23 BN_CTX *ctx, int *pnoinv)
25 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
52 if (BN_copy(B, a) == NULL)
54 if (BN_copy(A, n) == NULL)
58 if (B->neg || (BN_ucmp(B, A) >= 0)) {
60 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
61 * BN_div_no_branch will be called eventually.
66 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
67 if (!BN_nnmod(B, &local_B, A, ctx))
69 /* Ensure local_B goes out of scope before any further use of B */
74 * From B = a mod |n|, A = |n| it follows that
77 * -sign*X*a == B (mod |n|),
78 * sign*Y*a == A (mod |n|).
81 while (!BN_is_zero(B)) {
86 * (*) -sign*X*a == B (mod |n|),
87 * sign*Y*a == A (mod |n|)
91 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
92 * BN_div_no_branch will be called eventually.
97 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
99 /* (D, M) := (A/B, A%B) ... */
100 if (!BN_div(D, M, &local_A, B, ctx))
102 /* Ensure local_A goes out of scope before any further use of A */
109 * (**) sign*Y*a == D*B + M (mod |n|).
112 tmp = A; /* keep the BIGNUM object, the value does not
115 /* (A, B) := (B, A mod B) ... */
118 /* ... so we have 0 <= B < A again */
121 * Since the former M is now B and the former B is now A,
122 * (**) translates into
123 * sign*Y*a == D*A + B (mod |n|),
125 * sign*Y*a - D*A == B (mod |n|).
126 * Similarly, (*) translates into
127 * -sign*X*a == A (mod |n|).
130 * sign*Y*a + D*sign*X*a == B (mod |n|),
132 * sign*(Y + D*X)*a == B (mod |n|).
134 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
135 * -sign*X*a == B (mod |n|),
136 * sign*Y*a == A (mod |n|).
137 * Note that X and Y stay non-negative all the time.
140 if (!BN_mul(tmp, D, X, ctx))
142 if (!BN_add(tmp, tmp, Y))
145 M = Y; /* keep the BIGNUM object, the value does not
153 * The while loop (Euclid's algorithm) ends when
156 * sign*Y*a == A (mod |n|),
157 * where Y is non-negative.
161 if (!BN_sub(Y, n, Y))
164 /* Now Y*a == A (mod |n|). */
167 /* Y*a == 1 (mod |n|) */
168 if (!Y->neg && BN_ucmp(Y, n) < 0) {
172 if (!BN_nnmod(R, Y, n, ctx))
177 /* caller sets the BN_R_NO_INVERSE error */
185 if ((ret == NULL) && (in == NULL))
193 * This is an internal function, we assume all callers pass valid arguments:
194 * all pointers passed here are assumed non-NULL.
196 BIGNUM *int_bn_mod_inverse(BIGNUM *in,
197 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
200 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
204 /* This is invalid input so we don't worry about constant time here */
205 if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
212 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
213 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
214 return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
240 if (BN_copy(B, a) == NULL)
242 if (BN_copy(A, n) == NULL)
245 if (B->neg || (BN_ucmp(B, A) >= 0)) {
246 if (!BN_nnmod(B, B, A, ctx))
251 * From B = a mod |n|, A = |n| it follows that
254 * -sign*X*a == B (mod |n|),
255 * sign*Y*a == A (mod |n|).
258 if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
260 * Binary inversion algorithm; requires odd modulus. This is faster
261 * than the general algorithm if the modulus is sufficiently small
262 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
267 while (!BN_is_zero(B)) {
271 * (1) -sign*X*a == B (mod |n|),
272 * (2) sign*Y*a == A (mod |n|)
276 * Now divide B by the maximum possible power of two in the
277 * integers, and divide X by the same value mod |n|. When we're
278 * done, (1) still holds.
281 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
285 if (!BN_uadd(X, X, n))
289 * now X is even, so we can easily divide it by two
291 if (!BN_rshift1(X, X))
295 if (!BN_rshift(B, B, shift))
300 * Same for A and Y. Afterwards, (2) still holds.
303 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
307 if (!BN_uadd(Y, Y, n))
311 if (!BN_rshift1(Y, Y))
315 if (!BN_rshift(A, A, shift))
320 * We still have (1) and (2).
321 * Both A and B are odd.
322 * The following computations ensure that
326 * (1) -sign*X*a == B (mod |n|),
327 * (2) sign*Y*a == A (mod |n|),
329 * and that either A or B is even in the next iteration.
331 if (BN_ucmp(B, A) >= 0) {
332 /* -sign*(X + Y)*a == B - A (mod |n|) */
333 if (!BN_uadd(X, X, Y))
336 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
337 * actually makes the algorithm slower
339 if (!BN_usub(B, B, A))
342 /* sign*(X + Y)*a == A - B (mod |n|) */
343 if (!BN_uadd(Y, Y, X))
346 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
348 if (!BN_usub(A, A, B))
353 /* general inversion algorithm */
355 while (!BN_is_zero(B)) {
360 * (*) -sign*X*a == B (mod |n|),
361 * sign*Y*a == A (mod |n|)
364 /* (D, M) := (A/B, A%B) ... */
365 if (BN_num_bits(A) == BN_num_bits(B)) {
368 if (!BN_sub(M, A, B))
370 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
371 /* A/B is 1, 2, or 3 */
372 if (!BN_lshift1(T, B))
374 if (BN_ucmp(A, T) < 0) {
375 /* A < 2*B, so D=1 */
378 if (!BN_sub(M, A, B))
381 /* A >= 2*B, so D=2 or D=3 */
382 if (!BN_sub(M, A, T))
384 if (!BN_add(D, T, B))
385 goto err; /* use D (:= 3*B) as temp */
386 if (BN_ucmp(A, D) < 0) {
387 /* A < 3*B, so D=2 */
388 if (!BN_set_word(D, 2))
391 * M (= A - 2*B) already has the correct value
394 /* only D=3 remains */
395 if (!BN_set_word(D, 3))
398 * currently M = A - 2*B, but we need M = A - 3*B
400 if (!BN_sub(M, M, B))
405 if (!BN_div(D, M, A, B, ctx))
413 * (**) sign*Y*a == D*B + M (mod |n|).
416 tmp = A; /* keep the BIGNUM object, the value does not matter */
418 /* (A, B) := (B, A mod B) ... */
421 /* ... so we have 0 <= B < A again */
424 * Since the former M is now B and the former B is now A,
425 * (**) translates into
426 * sign*Y*a == D*A + B (mod |n|),
428 * sign*Y*a - D*A == B (mod |n|).
429 * Similarly, (*) translates into
430 * -sign*X*a == A (mod |n|).
433 * sign*Y*a + D*sign*X*a == B (mod |n|),
435 * sign*(Y + D*X)*a == B (mod |n|).
437 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
438 * -sign*X*a == B (mod |n|),
439 * sign*Y*a == A (mod |n|).
440 * Note that X and Y stay non-negative all the time.
444 * most of the time D is very small, so we can optimize tmp := D*X+Y
447 if (!BN_add(tmp, X, Y))
450 if (BN_is_word(D, 2)) {
451 if (!BN_lshift1(tmp, X))
453 } else if (BN_is_word(D, 4)) {
454 if (!BN_lshift(tmp, X, 2))
456 } else if (D->top == 1) {
457 if (!BN_copy(tmp, X))
459 if (!BN_mul_word(tmp, D->d[0]))
462 if (!BN_mul(tmp, D, X, ctx))
465 if (!BN_add(tmp, tmp, Y))
469 M = Y; /* keep the BIGNUM object, the value does not matter */
477 * The while loop (Euclid's algorithm) ends when
480 * sign*Y*a == A (mod |n|),
481 * where Y is non-negative.
485 if (!BN_sub(Y, n, Y))
488 /* Now Y*a == A (mod |n|). */
491 /* Y*a == 1 (mod |n|) */
492 if (!Y->neg && BN_ucmp(Y, n) < 0) {
496 if (!BN_nnmod(R, Y, n, ctx))
505 if ((ret == NULL) && (in == NULL))
512 /* solves ax == 1 (mod n) */
513 BIGNUM *BN_mod_inverse(BIGNUM *in,
514 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
516 BN_CTX *new_ctx = NULL;
521 ctx = new_ctx = BN_CTX_new();
523 BNerr(BN_F_BN_MOD_INVERSE, ERR_R_MALLOC_FAILURE);
528 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
530 BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
531 BN_CTX_free(new_ctx);
536 * This function is based on the constant-time GCD work by Bernstein and Yang:
537 * https://eprint.iacr.org/2019/266
538 * Generalized fast GCD function to allow even inputs.
539 * The algorithm first finds the shared powers of 2 between
540 * the inputs, and removes them, reducing at least one of the
541 * inputs to an odd value. Then it proceeds to calculate the GCD.
542 * Before returning the resulting GCD, we take care of adding
543 * back the powers of two removed at the beginning.
544 * Note 1: we assume the bit length of both inputs is public information,
545 * since access to top potentially leaks this information.
547 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
549 BIGNUM *g, *temp = NULL;
551 int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
553 /* Note 2: zero input corner cases are not constant-time since they are
554 * handled immediately. An attacker can run an attack under this
555 * assumption without the need of side-channel information. */
556 if (BN_is_zero(in_b)) {
557 ret = BN_copy(r, in_a) != NULL;
561 if (BN_is_zero(in_a)) {
562 ret = BN_copy(r, in_b) != NULL;
571 temp = BN_CTX_get(ctx);
574 /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
576 || !BN_lshift1(g, in_b)
577 || !BN_lshift1(r, in_a))
580 /* find shared powers of two, i.e. "shifts" >= 1 */
581 for (i = 0; i < r->dmax && i < g->dmax; i++) {
582 mask = ~(r->d[i] | g->d[i]);
583 for (j = 0; j < BN_BITS2; j++) {
590 /* subtract shared powers of two; shifts >= 1 */
591 if (!BN_rshift(r, r, shifts)
592 || !BN_rshift(g, g, shifts))
595 /* expand to biggest nword, with room for a possible extra word */
596 top = 1 + ((r->top >= g->top) ? r->top : g->top);
597 if (bn_wexpand(r, top) == NULL
598 || bn_wexpand(g, top) == NULL
599 || bn_wexpand(temp, top) == NULL)
602 /* re arrange inputs s.t. r is odd */
603 BN_consttime_swap((~r->d[0]) & 1, r, g, top);
605 /* compute the number of iterations */
606 rlen = BN_num_bits(r);
607 glen = BN_num_bits(g);
608 m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
610 for (i = 0; i < m; i++) {
611 /* conditionally flip signs if delta is positive and g is odd */
612 cond = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
613 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
614 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1)));
615 delta = (-cond & -delta) | ((cond - 1) & delta);
618 BN_consttime_swap(cond, r, g, top);
620 /* elimination step */
622 if (!BN_add(temp, g, r))
624 BN_consttime_swap(g->d[0] & 1 /* g is odd */
625 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
626 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1))),
628 if (!BN_rshift1(g, g))
632 /* remove possible negative sign */
634 /* add powers of 2 removed, then correct the artificial shift */
635 if (!BN_lshift(r, r, shifts)
636 || !BN_rshift1(r, r))