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.
586 BN_with_flags(pB
, B
, BN_FLG_CONSTTIME
);
587 if (!BN_nnmod(B
, pB
, A
, ctx
))
592 * From B = a mod |n|, A = |n| it follows that
595 * -sign*X*a == B (mod |n|),
596 * sign*Y*a == A (mod |n|).
599 while (!BN_is_zero(B
)) {
604 * (*) -sign*X*a == B (mod |n|),
605 * sign*Y*a == A (mod |n|)
609 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
610 * BN_div_no_branch will be called eventually.
613 BN_with_flags(pA
, A
, BN_FLG_CONSTTIME
);
615 /* (D, M) := (A/B, A%B) ... */
616 if (!BN_div(D
, M
, pA
, B
, ctx
))
623 * (**) sign*Y*a == D*B + M (mod |n|).
626 tmp
= A
; /* keep the BIGNUM object, the value does not
629 /* (A, B) := (B, A mod B) ... */
632 /* ... so we have 0 <= B < A again */
635 * Since the former M is now B and the former B is now A,
636 * (**) translates into
637 * sign*Y*a == D*A + B (mod |n|),
639 * sign*Y*a - D*A == B (mod |n|).
640 * Similarly, (*) translates into
641 * -sign*X*a == A (mod |n|).
644 * sign*Y*a + D*sign*X*a == B (mod |n|),
646 * sign*(Y + D*X)*a == B (mod |n|).
648 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
649 * -sign*X*a == B (mod |n|),
650 * sign*Y*a == A (mod |n|).
651 * Note that X and Y stay non-negative all the time.
654 if (!BN_mul(tmp
, D
, X
, ctx
))
656 if (!BN_add(tmp
, tmp
, Y
))
659 M
= Y
; /* keep the BIGNUM object, the value does not
667 * The while loop (Euclid's algorithm) ends when
670 * sign*Y*a == A (mod |n|),
671 * where Y is non-negative.
675 if (!BN_sub(Y
, n
, Y
))
678 /* Now Y*a == A (mod |n|). */
681 /* Y*a == 1 (mod |n|) */
682 if (!Y
->neg
&& BN_ucmp(Y
, n
) < 0) {
686 if (!BN_nnmod(R
, Y
, n
, ctx
))
690 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH
, BN_R_NO_INVERSE
);
695 if ((ret
== NULL
) && (in
== NULL
))