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 "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)
229 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
233 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
234 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
235 return BN_mod_inverse_no_branch(in, a, n, ctx);
261 if (BN_copy(B, a) == NULL)
263 if (BN_copy(A, n) == NULL)
266 if (B->neg || (BN_ucmp(B, A) >= 0)) {
267 if (!BN_nnmod(B, B, A, ctx))
272 * From B = a mod |n|, A = |n| it follows that
275 * -sign*X*a == B (mod |n|),
276 * sign*Y*a == A (mod |n|).
279 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) {
281 * Binary inversion algorithm; requires odd modulus. This is faster
282 * than the general algorithm if the modulus is sufficiently small
283 * (about 400 .. 500 bits on 32-bit sytems, but much more on 64-bit
288 while (!BN_is_zero(B)) {
292 * (1) -sign*X*a == B (mod |n|),
293 * (2) sign*Y*a == A (mod |n|)
297 * Now divide B by the maximum possible power of two in the
298 * integers, and divide X by the same value mod |n|. When we're
299 * done, (1) still holds.
302 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
306 if (!BN_uadd(X, X, n))
310 * now X is even, so we can easily divide it by two
312 if (!BN_rshift1(X, X))
316 if (!BN_rshift(B, B, shift))
321 * Same for A and Y. Afterwards, (2) still holds.
324 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
328 if (!BN_uadd(Y, Y, n))
332 if (!BN_rshift1(Y, Y))
336 if (!BN_rshift(A, A, shift))
341 * We still have (1) and (2).
342 * Both A and B are odd.
343 * The following computations ensure that
347 * (1) -sign*X*a == B (mod |n|),
348 * (2) sign*Y*a == A (mod |n|),
350 * and that either A or B is even in the next iteration.
352 if (BN_ucmp(B, A) >= 0) {
353 /* -sign*(X + Y)*a == B - A (mod |n|) */
354 if (!BN_uadd(X, X, Y))
357 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
358 * actually makes the algorithm slower
360 if (!BN_usub(B, B, A))
363 /* sign*(X + Y)*a == A - B (mod |n|) */
364 if (!BN_uadd(Y, Y, X))
367 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things
370 if (!BN_usub(A, A, B))
375 /* general inversion algorithm */
377 while (!BN_is_zero(B)) {
382 * (*) -sign*X*a == B (mod |n|),
383 * sign*Y*a == A (mod |n|)
386 /* (D, M) := (A/B, A%B) ... */
387 if (BN_num_bits(A) == BN_num_bits(B)) {
390 if (!BN_sub(M, A, B))
392 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
393 /* A/B is 1, 2, or 3 */
394 if (!BN_lshift1(T, B))
396 if (BN_ucmp(A, T) < 0) {
397 /* A < 2*B, so D=1 */
400 if (!BN_sub(M, A, B))
403 /* A >= 2*B, so D=2 or D=3 */
404 if (!BN_sub(M, A, T))
406 if (!BN_add(D, T, B))
407 goto err; /* use D (:= 3*B) as temp */
408 if (BN_ucmp(A, D) < 0) {
409 /* A < 3*B, so D=2 */
410 if (!BN_set_word(D, 2))
413 * M (= A - 2*B) already has the correct value
416 /* only D=3 remains */
417 if (!BN_set_word(D, 3))
420 * currently M = A - 2*B, but we need M = A - 3*B
422 if (!BN_sub(M, M, B))
427 if (!BN_div(D, M, A, B, ctx))
435 * (**) sign*Y*a == D*B + M (mod |n|).
438 tmp = A; /* keep the BIGNUM object, the value does not
441 /* (A, B) := (B, A mod B) ... */
444 /* ... so we have 0 <= B < A again */
447 * Since the former M is now B and the former B is now A,
448 * (**) translates into
449 * sign*Y*a == D*A + B (mod |n|),
451 * sign*Y*a - D*A == B (mod |n|).
452 * Similarly, (*) translates into
453 * -sign*X*a == A (mod |n|).
456 * sign*Y*a + D*sign*X*a == B (mod |n|),
458 * sign*(Y + D*X)*a == B (mod |n|).
460 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
461 * -sign*X*a == B (mod |n|),
462 * sign*Y*a == A (mod |n|).
463 * Note that X and Y stay non-negative all the time.
467 * most of the time D is very small, so we can optimize tmp :=
471 if (!BN_add(tmp, X, Y))
474 if (BN_is_word(D, 2)) {
475 if (!BN_lshift1(tmp, X))
477 } else if (BN_is_word(D, 4)) {
478 if (!BN_lshift(tmp, X, 2))
480 } else if (D->top == 1) {
481 if (!BN_copy(tmp, X))
483 if (!BN_mul_word(tmp, D->d[0]))
486 if (!BN_mul(tmp, D, X, ctx))
489 if (!BN_add(tmp, tmp, Y))
493 M = Y; /* keep the BIGNUM object, the value does not
502 * The while loop (Euclid's algorithm) ends when
505 * sign*Y*a == A (mod |n|),
506 * where Y is non-negative.
510 if (!BN_sub(Y, n, Y))
513 /* Now Y*a == A (mod |n|). */
516 /* Y*a == 1 (mod |n|) */
517 if (!Y->neg && BN_ucmp(Y, n) < 0) {
521 if (!BN_nnmod(R, Y, n, ctx))
525 BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
530 if ((ret == NULL) && (in == NULL))
538 * BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
539 * not contain branches that may leak sensitive information.
541 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
542 const BIGNUM *a, const BIGNUM *n,
545 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
546 BIGNUM local_A, local_B;
574 if (BN_copy(B, a) == NULL)
576 if (BN_copy(A, n) == NULL)
580 if (B->neg || (BN_ucmp(B, A) >= 0)) {
582 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
583 * BN_div_no_branch will be called eventually.
587 BN_with_flags(pB, B, BN_FLG_CONSTTIME);
588 if (!BN_nnmod(B, pB, A, ctx))
593 * From B = a mod |n|, A = |n| it follows that
596 * -sign*X*a == B (mod |n|),
597 * sign*Y*a == A (mod |n|).
600 while (!BN_is_zero(B)) {
605 * (*) -sign*X*a == B (mod |n|),
606 * sign*Y*a == A (mod |n|)
610 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
611 * BN_div_no_branch will be called eventually.
615 BN_with_flags(pA, A, BN_FLG_CONSTTIME);
617 /* (D, M) := (A/B, A%B) ... */
618 if (!BN_div(D, M, pA, B, ctx))
625 * (**) sign*Y*a == D*B + M (mod |n|).
628 tmp = A; /* keep the BIGNUM object, the value does not
631 /* (A, B) := (B, A mod B) ... */
634 /* ... so we have 0 <= B < A again */
637 * Since the former M is now B and the former B is now A,
638 * (**) translates into
639 * sign*Y*a == D*A + B (mod |n|),
641 * sign*Y*a - D*A == B (mod |n|).
642 * Similarly, (*) translates into
643 * -sign*X*a == A (mod |n|).
646 * sign*Y*a + D*sign*X*a == B (mod |n|),
648 * sign*(Y + D*X)*a == B (mod |n|).
650 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
651 * -sign*X*a == B (mod |n|),
652 * sign*Y*a == A (mod |n|).
653 * Note that X and Y stay non-negative all the time.
656 if (!BN_mul(tmp, D, X, ctx))
658 if (!BN_add(tmp, tmp, Y))
661 M = Y; /* keep the BIGNUM object, the value does not
669 * The while loop (Euclid's algorithm) ends when
672 * sign*Y*a == A (mod |n|),
673 * where Y is non-negative.
677 if (!BN_sub(Y, n, Y))
680 /* Now Y*a == A (mod |n|). */
683 /* Y*a == 1 (mod |n|) */
684 if (!Y->neg && BN_ucmp(Y, n) < 0) {
688 if (!BN_nnmod(R, Y, n, ctx))
692 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH, BN_R_NO_INVERSE);
697 if ((ret == NULL) && (in == NULL))