1 /* crypto/bn/bn_gcd.c */
2 /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
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.
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).
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.
23 * Redistribution and use in source and binary forms, with or without
24 * modification, are permitted provided that the following conditions
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)"
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
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.]
58 /* ====================================================================
59 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
61 * Redistribution and use in source and binary forms, with or without
62 * modification, are permitted provided that the following conditions
65 * 1. Redistributions of source code must retain the above copyright
66 * notice, this list of conditions and the following disclaimer.
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
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/)"
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.
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.
87 * 6. Redistributions of any form whatsoever must retain the following
89 * "This product includes software developed by the OpenSSL Project
90 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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 * ====================================================================
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).
112 #include "internal/cryptlib.h"
115 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
117 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
128 if (a == NULL || b == NULL)
131 if (BN_copy(a, in_a) == NULL)
133 if (BN_copy(b, in_b) == NULL)
138 if (BN_cmp(a, b) < 0) {
147 if (BN_copy(r, t) == NULL)
156 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
165 while (!BN_is_zero(b)) {
170 if (!BN_sub(a, a, b))
172 if (!BN_rshift1(a, a))
174 if (BN_cmp(a, b) < 0) {
179 } else { /* a odd - b even */
181 if (!BN_rshift1(b, b))
183 if (BN_cmp(a, b) < 0) {
189 } else { /* a is even */
192 if (!BN_rshift1(a, a))
194 if (BN_cmp(a, b) < 0) {
199 } else { /* a even - b even */
201 if (!BN_rshift1(a, a))
203 if (!BN_rshift1(b, b))
212 if (!BN_lshift(a, a, shifts))
221 /* solves ax == 1 (mod n) */
222 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
223 const BIGNUM *a, const BIGNUM *n,
226 BIGNUM *BN_mod_inverse(BIGNUM *in,
227 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
231 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
233 BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
237 BIGNUM *int_bn_mod_inverse(BIGNUM *in,
238 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
241 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
248 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
249 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
250 return BN_mod_inverse_no_branch(in, a, n, ctx);
276 if (BN_copy(B, a) == NULL)
278 if (BN_copy(A, n) == NULL)
281 if (B->neg || (BN_ucmp(B, A) >= 0)) {
282 if (!BN_nnmod(B, B, A, ctx))
287 * From B = a mod |n|, A = |n| it follows that
290 * -sign*X*a == B (mod |n|),
291 * sign*Y*a == A (mod |n|).
294 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) {
296 * Binary inversion algorithm; requires odd modulus. This is faster
297 * than the general algorithm if the modulus is sufficiently small
298 * (about 400 .. 500 bits on 32-bit sytems, but much more on 64-bit
303 while (!BN_is_zero(B)) {
307 * (1) -sign*X*a == B (mod |n|),
308 * (2) sign*Y*a == A (mod |n|)
312 * Now divide B by the maximum possible power of two in the
313 * integers, and divide X by the same value mod |n|. When we're
314 * done, (1) still holds.
317 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
321 if (!BN_uadd(X, X, n))
325 * now X is even, so we can easily divide it by two
327 if (!BN_rshift1(X, X))
331 if (!BN_rshift(B, B, shift))
336 * Same for A and Y. Afterwards, (2) still holds.
339 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
343 if (!BN_uadd(Y, Y, n))
347 if (!BN_rshift1(Y, Y))
351 if (!BN_rshift(A, A, shift))
356 * We still have (1) and (2).
357 * Both A and B are odd.
358 * The following computations ensure that
362 * (1) -sign*X*a == B (mod |n|),
363 * (2) sign*Y*a == A (mod |n|),
365 * and that either A or B is even in the next iteration.
367 if (BN_ucmp(B, A) >= 0) {
368 /* -sign*(X + Y)*a == B - A (mod |n|) */
369 if (!BN_uadd(X, X, Y))
372 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
373 * actually makes the algorithm slower
375 if (!BN_usub(B, B, A))
378 /* sign*(X + Y)*a == A - B (mod |n|) */
379 if (!BN_uadd(Y, Y, X))
382 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things
385 if (!BN_usub(A, A, B))
390 /* general inversion algorithm */
392 while (!BN_is_zero(B)) {
397 * (*) -sign*X*a == B (mod |n|),
398 * sign*Y*a == A (mod |n|)
401 /* (D, M) := (A/B, A%B) ... */
402 if (BN_num_bits(A) == BN_num_bits(B)) {
405 if (!BN_sub(M, A, B))
407 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
408 /* A/B is 1, 2, or 3 */
409 if (!BN_lshift1(T, B))
411 if (BN_ucmp(A, T) < 0) {
412 /* A < 2*B, so D=1 */
415 if (!BN_sub(M, A, B))
418 /* A >= 2*B, so D=2 or D=3 */
419 if (!BN_sub(M, A, T))
421 if (!BN_add(D, T, B))
422 goto err; /* use D (:= 3*B) as temp */
423 if (BN_ucmp(A, D) < 0) {
424 /* A < 3*B, so D=2 */
425 if (!BN_set_word(D, 2))
428 * M (= A - 2*B) already has the correct value
431 /* only D=3 remains */
432 if (!BN_set_word(D, 3))
435 * currently M = A - 2*B, but we need M = A - 3*B
437 if (!BN_sub(M, M, B))
442 if (!BN_div(D, M, A, B, ctx))
450 * (**) sign*Y*a == D*B + M (mod |n|).
453 tmp = A; /* keep the BIGNUM object, the value does not
456 /* (A, B) := (B, A mod B) ... */
459 /* ... so we have 0 <= B < A again */
462 * Since the former M is now B and the former B is now A,
463 * (**) translates into
464 * sign*Y*a == D*A + B (mod |n|),
466 * sign*Y*a - D*A == B (mod |n|).
467 * Similarly, (*) translates into
468 * -sign*X*a == A (mod |n|).
471 * sign*Y*a + D*sign*X*a == B (mod |n|),
473 * sign*(Y + D*X)*a == B (mod |n|).
475 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
476 * -sign*X*a == B (mod |n|),
477 * sign*Y*a == A (mod |n|).
478 * Note that X and Y stay non-negative all the time.
482 * most of the time D is very small, so we can optimize tmp :=
486 if (!BN_add(tmp, X, Y))
489 if (BN_is_word(D, 2)) {
490 if (!BN_lshift1(tmp, X))
492 } else if (BN_is_word(D, 4)) {
493 if (!BN_lshift(tmp, X, 2))
495 } else if (D->top == 1) {
496 if (!BN_copy(tmp, X))
498 if (!BN_mul_word(tmp, D->d[0]))
501 if (!BN_mul(tmp, D, X, ctx))
504 if (!BN_add(tmp, tmp, Y))
508 M = Y; /* keep the BIGNUM object, the value does not
517 * The while loop (Euclid's algorithm) ends when
520 * sign*Y*a == A (mod |n|),
521 * where Y is non-negative.
525 if (!BN_sub(Y, n, Y))
528 /* Now Y*a == A (mod |n|). */
531 /* Y*a == 1 (mod |n|) */
532 if (!Y->neg && BN_ucmp(Y, n) < 0) {
536 if (!BN_nnmod(R, Y, n, ctx))
546 if ((ret == NULL) && (in == NULL))
554 * BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
555 * not contain branches that may leak sensitive information.
557 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
558 const BIGNUM *a, const BIGNUM *n,
561 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
562 BIGNUM local_A, local_B;
590 if (BN_copy(B, a) == NULL)
592 if (BN_copy(A, n) == NULL)
596 if (B->neg || (BN_ucmp(B, A) >= 0)) {
598 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
599 * BN_div_no_branch will be called eventually.
603 BN_with_flags(pB, B, BN_FLG_CONSTTIME);
604 if (!BN_nnmod(B, pB, A, ctx))
609 * From B = a mod |n|, A = |n| it follows that
612 * -sign*X*a == B (mod |n|),
613 * sign*Y*a == A (mod |n|).
616 while (!BN_is_zero(B)) {
621 * (*) -sign*X*a == B (mod |n|),
622 * sign*Y*a == A (mod |n|)
626 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
627 * BN_div_no_branch will be called eventually.
631 BN_with_flags(pA, A, BN_FLG_CONSTTIME);
633 /* (D, M) := (A/B, A%B) ... */
634 if (!BN_div(D, M, pA, B, ctx))
641 * (**) sign*Y*a == D*B + M (mod |n|).
644 tmp = A; /* keep the BIGNUM object, the value does not
647 /* (A, B) := (B, A mod B) ... */
650 /* ... so we have 0 <= B < A again */
653 * Since the former M is now B and the former B is now A,
654 * (**) translates into
655 * sign*Y*a == D*A + B (mod |n|),
657 * sign*Y*a - D*A == B (mod |n|).
658 * Similarly, (*) translates into
659 * -sign*X*a == A (mod |n|).
662 * sign*Y*a + D*sign*X*a == B (mod |n|),
664 * sign*(Y + D*X)*a == B (mod |n|).
666 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
667 * -sign*X*a == B (mod |n|),
668 * sign*Y*a == A (mod |n|).
669 * Note that X and Y stay non-negative all the time.
672 if (!BN_mul(tmp, D, X, ctx))
674 if (!BN_add(tmp, tmp, Y))
677 M = Y; /* keep the BIGNUM object, the value does not
685 * The while loop (Euclid's algorithm) ends when
688 * sign*Y*a == A (mod |n|),
689 * where Y is non-negative.
693 if (!BN_sub(Y, n, Y))
696 /* Now Y*a == A (mod |n|). */
699 /* Y*a == 1 (mod |n|) */
700 if (!Y->neg && BN_ucmp(Y, n) < 0) {
704 if (!BN_nnmod(R, Y, n, ctx))
708 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH, BN_R_NO_INVERSE);
713 if ((ret == NULL) && (in == NULL))