2 * Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
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
10 #include "internal/cryptlib.h"
13 /* solves ax == 1 (mod n) */
14 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
15 const BIGNUM *a, const BIGNUM *n,
18 BIGNUM *BN_mod_inverse(BIGNUM *in,
19 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
23 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
25 BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
29 BIGNUM *int_bn_mod_inverse(BIGNUM *in,
30 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
33 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
37 /* This is invalid input so we don't worry about constant time here */
38 if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
47 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
48 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
49 return BN_mod_inverse_no_branch(in, a, n, ctx);
75 if (BN_copy(B, a) == NULL)
77 if (BN_copy(A, n) == NULL)
80 if (B->neg || (BN_ucmp(B, A) >= 0)) {
81 if (!BN_nnmod(B, B, A, ctx))
86 * From B = a mod |n|, A = |n| it follows that
89 * -sign*X*a == B (mod |n|),
90 * sign*Y*a == A (mod |n|).
93 if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
95 * Binary inversion algorithm; requires odd modulus. This is faster
96 * than the general algorithm if the modulus is sufficiently small
97 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
102 while (!BN_is_zero(B)) {
106 * (1) -sign*X*a == B (mod |n|),
107 * (2) sign*Y*a == A (mod |n|)
111 * Now divide B by the maximum possible power of two in the
112 * integers, and divide X by the same value mod |n|. When we're
113 * done, (1) still holds.
116 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
120 if (!BN_uadd(X, X, n))
124 * now X is even, so we can easily divide it by two
126 if (!BN_rshift1(X, X))
130 if (!BN_rshift(B, B, shift))
135 * Same for A and Y. Afterwards, (2) still holds.
138 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
142 if (!BN_uadd(Y, Y, n))
146 if (!BN_rshift1(Y, Y))
150 if (!BN_rshift(A, A, shift))
155 * We still have (1) and (2).
156 * Both A and B are odd.
157 * The following computations ensure that
161 * (1) -sign*X*a == B (mod |n|),
162 * (2) sign*Y*a == A (mod |n|),
164 * and that either A or B is even in the next iteration.
166 if (BN_ucmp(B, A) >= 0) {
167 /* -sign*(X + Y)*a == B - A (mod |n|) */
168 if (!BN_uadd(X, X, Y))
171 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
172 * actually makes the algorithm slower
174 if (!BN_usub(B, B, A))
177 /* sign*(X + Y)*a == A - B (mod |n|) */
178 if (!BN_uadd(Y, Y, X))
181 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
183 if (!BN_usub(A, A, B))
188 /* general inversion algorithm */
190 while (!BN_is_zero(B)) {
195 * (*) -sign*X*a == B (mod |n|),
196 * sign*Y*a == A (mod |n|)
199 /* (D, M) := (A/B, A%B) ... */
200 if (BN_num_bits(A) == BN_num_bits(B)) {
203 if (!BN_sub(M, A, B))
205 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
206 /* A/B is 1, 2, or 3 */
207 if (!BN_lshift1(T, B))
209 if (BN_ucmp(A, T) < 0) {
210 /* A < 2*B, so D=1 */
213 if (!BN_sub(M, A, B))
216 /* A >= 2*B, so D=2 or D=3 */
217 if (!BN_sub(M, A, T))
219 if (!BN_add(D, T, B))
220 goto err; /* use D (:= 3*B) as temp */
221 if (BN_ucmp(A, D) < 0) {
222 /* A < 3*B, so D=2 */
223 if (!BN_set_word(D, 2))
226 * M (= A - 2*B) already has the correct value
229 /* only D=3 remains */
230 if (!BN_set_word(D, 3))
233 * currently M = A - 2*B, but we need M = A - 3*B
235 if (!BN_sub(M, M, B))
240 if (!BN_div(D, M, A, B, ctx))
248 * (**) sign*Y*a == D*B + M (mod |n|).
251 tmp = A; /* keep the BIGNUM object, the value does not matter */
253 /* (A, B) := (B, A mod B) ... */
256 /* ... so we have 0 <= B < A again */
259 * Since the former M is now B and the former B is now A,
260 * (**) translates into
261 * sign*Y*a == D*A + B (mod |n|),
263 * sign*Y*a - D*A == B (mod |n|).
264 * Similarly, (*) translates into
265 * -sign*X*a == A (mod |n|).
268 * sign*Y*a + D*sign*X*a == B (mod |n|),
270 * sign*(Y + D*X)*a == B (mod |n|).
272 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
273 * -sign*X*a == B (mod |n|),
274 * sign*Y*a == A (mod |n|).
275 * Note that X and Y stay non-negative all the time.
279 * most of the time D is very small, so we can optimize tmp := D*X+Y
282 if (!BN_add(tmp, X, Y))
285 if (BN_is_word(D, 2)) {
286 if (!BN_lshift1(tmp, X))
288 } else if (BN_is_word(D, 4)) {
289 if (!BN_lshift(tmp, X, 2))
291 } else if (D->top == 1) {
292 if (!BN_copy(tmp, X))
294 if (!BN_mul_word(tmp, D->d[0]))
297 if (!BN_mul(tmp, D, X, ctx))
300 if (!BN_add(tmp, tmp, Y))
304 M = Y; /* keep the BIGNUM object, the value does not matter */
312 * The while loop (Euclid's algorithm) ends when
315 * sign*Y*a == A (mod |n|),
316 * where Y is non-negative.
320 if (!BN_sub(Y, n, Y))
323 /* Now Y*a == A (mod |n|). */
326 /* Y*a == 1 (mod |n|) */
327 if (!Y->neg && BN_ucmp(Y, n) < 0) {
331 if (!BN_nnmod(R, Y, n, ctx))
341 if ((ret == NULL) && (in == NULL))
349 * BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
350 * not contain branches that may leak sensitive information.
352 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
353 const BIGNUM *a, const BIGNUM *n,
356 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
383 if (BN_copy(B, a) == NULL)
385 if (BN_copy(A, n) == NULL)
389 if (B->neg || (BN_ucmp(B, A) >= 0)) {
391 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
392 * BN_div_no_branch will be called eventually.
397 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
398 if (!BN_nnmod(B, &local_B, A, ctx))
400 /* Ensure local_B goes out of scope before any further use of B */
405 * From B = a mod |n|, A = |n| it follows that
408 * -sign*X*a == B (mod |n|),
409 * sign*Y*a == A (mod |n|).
412 while (!BN_is_zero(B)) {
417 * (*) -sign*X*a == B (mod |n|),
418 * sign*Y*a == A (mod |n|)
422 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
423 * BN_div_no_branch will be called eventually.
428 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
430 /* (D, M) := (A/B, A%B) ... */
431 if (!BN_div(D, M, &local_A, B, ctx))
433 /* Ensure local_A goes out of scope before any further use of A */
440 * (**) sign*Y*a == D*B + M (mod |n|).
443 tmp = A; /* keep the BIGNUM object, the value does not
446 /* (A, B) := (B, A mod B) ... */
449 /* ... so we have 0 <= B < A again */
452 * Since the former M is now B and the former B is now A,
453 * (**) translates into
454 * sign*Y*a == D*A + B (mod |n|),
456 * sign*Y*a - D*A == B (mod |n|).
457 * Similarly, (*) translates into
458 * -sign*X*a == A (mod |n|).
461 * sign*Y*a + D*sign*X*a == B (mod |n|),
463 * sign*(Y + D*X)*a == B (mod |n|).
465 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
466 * -sign*X*a == B (mod |n|),
467 * sign*Y*a == A (mod |n|).
468 * Note that X and Y stay non-negative all the time.
471 if (!BN_mul(tmp, D, X, ctx))
473 if (!BN_add(tmp, tmp, Y))
476 M = Y; /* keep the BIGNUM object, the value does not
484 * The while loop (Euclid's algorithm) ends when
487 * sign*Y*a == A (mod |n|),
488 * where Y is non-negative.
492 if (!BN_sub(Y, n, Y))
495 /* Now Y*a == A (mod |n|). */
498 /* Y*a == 1 (mod |n|) */
499 if (!Y->neg && BN_ucmp(Y, n) < 0) {
503 if (!BN_nnmod(R, Y, n, ctx))
507 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH, BN_R_NO_INVERSE);
512 if ((ret == NULL) && (in == NULL))
520 * This function is based on the constant-time GCD work by Bernstein and Yang:
521 * https://eprint.iacr.org/2019/266
522 * Generalized fast GCD function to allow even inputs.
523 * The algorithm first finds the shared powers of 2 between
524 * the inputs, and removes them, reducing at least one of the
525 * inputs to an odd value. Then it proceeds to calculate the GCD.
526 * Before returning the resulting GCD, we take care of adding
527 * back the powers of two removed at the beginning.
528 * Note 1: we assume the bit length of both inputs is public information,
529 * since access to top potentially leaks this information.
531 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
533 BIGNUM *g, *temp = NULL;
535 int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
537 /* Note 2: zero input corner cases are not constant-time since they are
538 * handled immediately. An attacker can run an attack under this
539 * assumption without the need of side-channel information. */
540 if (BN_is_zero(in_b)) {
541 ret = BN_copy(r, in_a) != NULL;
545 if (BN_is_zero(in_a)) {
546 ret = BN_copy(r, in_b) != NULL;
555 temp = BN_CTX_get(ctx);
558 /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
560 || !BN_lshift1(g, in_b)
561 || !BN_lshift1(r, in_a))
564 /* find shared powers of two, i.e. "shifts" >= 1 */
565 for (i = 0; i < r->dmax && i < g->dmax; i++) {
566 mask = ~(r->d[i] | g->d[i]);
567 for (j = 0; j < BN_BITS2; j++) {
574 /* subtract shared powers of two; shifts >= 1 */
575 if (!BN_rshift(r, r, shifts)
576 || !BN_rshift(g, g, shifts))
579 /* expand to biggest nword, with room for a possible extra word */
580 top = 1 + ((r->top >= g->top) ? r->top : g->top);
581 if (bn_wexpand(r, top) == NULL
582 || bn_wexpand(g, top) == NULL
583 || bn_wexpand(temp, top) == NULL)
586 /* re arrange inputs s.t. r is odd */
587 BN_consttime_swap((~r->d[0]) & 1, r, g, top);
589 /* compute the number of iterations */
590 rlen = BN_num_bits(r);
591 glen = BN_num_bits(g);
592 m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
594 for (i = 0; i < m; i++) {
595 /* conditionally flip signs if delta is positive and g is odd */
596 cond = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
597 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
598 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1)));
599 delta = (-cond & -delta) | ((cond - 1) & delta);
602 BN_consttime_swap(cond, r, g, top);
604 /* elimination step */
606 if (!BN_add(temp, g, r))
608 BN_consttime_swap(g->d[0] & 1 /* g is odd */
609 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
610 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1))),
612 if (!BN_rshift1(g, g))
616 /* remove possible negative sign */
618 /* add powers of 2 removed, then correct the artificial shift */
619 if (!BN_lshift(r, r, shifts)
620 || !BN_rshift1(r, r))