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) goto err;
130 if (BN_copy(a,in_a) == NULL) goto err;
131 if (BN_copy(b,in_b) == NULL) goto err;
135 if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; }
137 if (t == NULL) goto err;
139 if (BN_copy(r,t) == NULL) goto err;
147 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
156 while (!BN_is_zero(b))
164 if (!BN_sub(a,a,b)) goto err;
165 if (!BN_rshift1(a,a)) goto err;
169 else /* a odd - b even */
171 if (!BN_rshift1(b,b)) goto err;
180 if (!BN_rshift1(a,a)) goto err;
184 else /* a even - b even */
186 if (!BN_rshift1(a,a)) goto err;
187 if (!BN_rshift1(b,b)) goto err;
196 if (!BN_lshift(a,a,shifts)) goto err;
205 /* solves ax == 1 (mod n) */
206 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
207 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx);
209 BIGNUM *BN_mod_inverse(BIGNUM *in,
210 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
212 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
216 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0))
218 return BN_mod_inverse_no_branch(in, a, n, ctx);
232 if (T == NULL) goto err;
238 if (R == NULL) goto err;
242 if (BN_copy(B,a) == NULL) goto err;
243 if (BN_copy(A,n) == NULL) goto err;
245 if (B->neg || (BN_ucmp(B, A) >= 0))
247 if (!BN_nnmod(B, B, A, ctx)) goto err;
250 /* From B = a mod |n|, A = |n| it follows that
253 * -sign*X*a == B (mod |n|),
254 * sign*Y*a == A (mod |n|).
257 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048)))
259 /* Binary inversion algorithm; requires odd modulus.
260 * This is faster than the general algorithm if the modulus
261 * is sufficiently small (about 400 .. 500 bits on 32-bit
262 * sytems, but much more on 64-bit systems) */
265 while (!BN_is_zero(B))
270 * (1) -sign*X*a == B (mod |n|),
271 * (2) sign*Y*a == A (mod |n|)
274 /* Now divide B by the maximum possible power of two in the integers,
275 * and divide X by the same value mod |n|.
276 * When we're done, (1) still holds. */
278 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */
284 if (!BN_uadd(X, X, n)) goto err;
286 /* now X is even, so we can easily divide it by two */
287 if (!BN_rshift1(X, X)) goto err;
291 if (!BN_rshift(B, B, shift)) goto err;
295 /* Same for A and Y. Afterwards, (2) still holds. */
297 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */
303 if (!BN_uadd(Y, Y, n)) goto err;
306 if (!BN_rshift1(Y, Y)) goto err;
310 if (!BN_rshift(A, A, shift)) goto err;
314 /* We still have (1) and (2).
315 * Both A and B are odd.
316 * The following computations ensure that
320 * (1) -sign*X*a == B (mod |n|),
321 * (2) sign*Y*a == A (mod |n|),
323 * and that either A or B is even in the next iteration.
325 if (BN_ucmp(B, A) >= 0)
327 /* -sign*(X + Y)*a == B - A (mod |n|) */
328 if (!BN_uadd(X, X, Y)) goto err;
329 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
330 * actually makes the algorithm slower */
331 if (!BN_usub(B, B, A)) goto err;
335 /* sign*(X + Y)*a == A - B (mod |n|) */
336 if (!BN_uadd(Y, Y, X)) goto err;
337 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
338 if (!BN_usub(A, A, B)) goto err;
344 /* general inversion algorithm */
346 while (!BN_is_zero(B))
352 * (*) -sign*X*a == B (mod |n|),
353 * sign*Y*a == A (mod |n|)
356 /* (D, M) := (A/B, A%B) ... */
357 if (BN_num_bits(A) == BN_num_bits(B))
359 if (!BN_one(D)) goto err;
360 if (!BN_sub(M,A,B)) goto err;
362 else if (BN_num_bits(A) == BN_num_bits(B) + 1)
364 /* A/B is 1, 2, or 3 */
365 if (!BN_lshift1(T,B)) goto err;
366 if (BN_ucmp(A,T) < 0)
368 /* A < 2*B, so D=1 */
369 if (!BN_one(D)) goto err;
370 if (!BN_sub(M,A,B)) goto err;
374 /* A >= 2*B, so D=2 or D=3 */
375 if (!BN_sub(M,A,T)) goto err;
376 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */
377 if (BN_ucmp(A,D) < 0)
379 /* A < 3*B, so D=2 */
380 if (!BN_set_word(D,2)) goto err;
381 /* M (= A - 2*B) already has the correct value */
385 /* only D=3 remains */
386 if (!BN_set_word(D,3)) goto err;
387 /* currently M = A - 2*B, but we need M = A - 3*B */
388 if (!BN_sub(M,M,B)) goto err;
394 if (!BN_div(D,M,A,B,ctx)) goto err;
400 * (**) sign*Y*a == D*B + M (mod |n|).
403 tmp=A; /* keep the BIGNUM object, the value does not matter */
405 /* (A, B) := (B, A mod B) ... */
408 /* ... so we have 0 <= B < A again */
410 /* Since the former M is now B and the former B is now A,
411 * (**) translates into
412 * sign*Y*a == D*A + B (mod |n|),
414 * sign*Y*a - D*A == B (mod |n|).
415 * Similarly, (*) translates into
416 * -sign*X*a == A (mod |n|).
419 * sign*Y*a + D*sign*X*a == B (mod |n|),
421 * sign*(Y + D*X)*a == B (mod |n|).
423 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
424 * -sign*X*a == B (mod |n|),
425 * sign*Y*a == A (mod |n|).
426 * Note that X and Y stay non-negative all the time.
429 /* most of the time D is very small, so we can optimize tmp := D*X+Y */
432 if (!BN_add(tmp,X,Y)) goto err;
438 if (!BN_lshift1(tmp,X)) goto err;
440 else if (BN_is_word(D,4))
442 if (!BN_lshift(tmp,X,2)) goto err;
444 else if (D->top == 1)
446 if (!BN_copy(tmp,X)) goto err;
447 if (!BN_mul_word(tmp,D->d[0])) goto err;
451 if (!BN_mul(tmp,D,X,ctx)) goto err;
453 if (!BN_add(tmp,tmp,Y)) goto err;
456 M=Y; /* keep the BIGNUM object, the value does not matter */
464 * The while loop (Euclid's algorithm) ends when
467 * sign*Y*a == A (mod |n|),
468 * where Y is non-negative.
473 if (!BN_sub(Y,n,Y)) goto err;
475 /* Now Y*a == A (mod |n|). */
480 /* Y*a == 1 (mod |n|) */
481 if (!Y->neg && BN_ucmp(Y,n) < 0)
483 if (!BN_copy(R,Y)) goto err;
487 if (!BN_nnmod(R,Y,n,ctx)) goto err;
492 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE);
497 if ((ret == NULL) && (in == NULL)) BN_free(R);
504 /* BN_mod_inverse_no_branch is a special version of BN_mod_inverse.
505 * It does not contain branches that may leak sensitive information.
507 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
508 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
510 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
511 BIGNUM local_A, local_B;
527 if (T == NULL) goto err;
533 if (R == NULL) goto err;
537 if (BN_copy(B,a) == NULL) goto err;
538 if (BN_copy(A,n) == NULL) goto err;
541 if (B->neg || (BN_ucmp(B, A) >= 0))
543 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
544 * BN_div_no_branch will be called eventually.
547 BN_with_flags(pB, B, BN_FLG_CONSTTIME);
548 if (!BN_nnmod(B, pB, A, ctx)) goto err;
551 /* From B = a mod |n|, A = |n| it follows that
554 * -sign*X*a == B (mod |n|),
555 * sign*Y*a == A (mod |n|).
558 while (!BN_is_zero(B))
564 * (*) -sign*X*a == B (mod |n|),
565 * sign*Y*a == A (mod |n|)
568 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
569 * BN_div_no_branch will be called eventually.
572 BN_with_flags(pA, A, BN_FLG_CONSTTIME);
574 /* (D, M) := (A/B, A%B) ... */
575 if (!BN_div(D,M,pA,B,ctx)) goto err;
580 * (**) sign*Y*a == D*B + M (mod |n|).
583 tmp=A; /* keep the BIGNUM object, the value does not matter */
585 /* (A, B) := (B, A mod B) ... */
588 /* ... so we have 0 <= B < A again */
590 /* Since the former M is now B and the former B is now A,
591 * (**) translates into
592 * sign*Y*a == D*A + B (mod |n|),
594 * sign*Y*a - D*A == B (mod |n|).
595 * Similarly, (*) translates into
596 * -sign*X*a == A (mod |n|).
599 * sign*Y*a + D*sign*X*a == B (mod |n|),
601 * sign*(Y + D*X)*a == B (mod |n|).
603 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
604 * -sign*X*a == B (mod |n|),
605 * sign*Y*a == A (mod |n|).
606 * Note that X and Y stay non-negative all the time.
609 if (!BN_mul(tmp,D,X,ctx)) goto err;
610 if (!BN_add(tmp,tmp,Y)) goto err;
612 M=Y; /* keep the BIGNUM object, the value does not matter */
619 * The while loop (Euclid's algorithm) ends when
622 * sign*Y*a == A (mod |n|),
623 * where Y is non-negative.
628 if (!BN_sub(Y,n,Y)) goto err;
630 /* Now Y*a == A (mod |n|). */
634 /* Y*a == 1 (mod |n|) */
635 if (!Y->neg && BN_ucmp(Y,n) < 0)
637 if (!BN_copy(R,Y)) goto err;
641 if (!BN_nnmod(R,Y,n,ctx)) goto err;
646 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE);
651 if ((ret == NULL) && (in == NULL)) BN_free(R);