722 lines
22 KiB
C
722 lines
22 KiB
C
/* crypto/bn/bn_gcd.c */
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/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
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* All rights reserved.
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*
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* This package is an SSL implementation written
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* by Eric Young (eay@cryptsoft.com).
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* The implementation was written so as to conform with Netscapes SSL.
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*
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* This library is free for commercial and non-commercial use as long as
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* the following conditions are aheared to. The following conditions
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* apply to all code found in this distribution, be it the RC4, RSA,
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* lhash, DES, etc., code; not just the SSL code. The SSL documentation
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* included with this distribution is covered by the same copyright terms
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* except that the holder is Tim Hudson (tjh@cryptsoft.com).
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*
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* Copyright remains Eric Young's, and as such any Copyright notices in
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* the code are not to be removed.
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* If this package is used in a product, Eric Young should be given attribution
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* as the author of the parts of the library used.
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* This can be in the form of a textual message at program startup or
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* in documentation (online or textual) provided with the package.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* 3. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* "This product includes cryptographic software written by
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* Eric Young (eay@cryptsoft.com)"
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* The word 'cryptographic' can be left out if the rouines from the library
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* being used are not cryptographic related :-).
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* 4. If you include any Windows specific code (or a derivative thereof) from
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* the apps directory (application code) you must include an acknowledgement:
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* "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
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*
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* THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*
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* The licence and distribution terms for any publically available version or
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* derivative of this code cannot be changed. i.e. this code cannot simply be
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* copied and put under another distribution licence
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* [including the GNU Public Licence.]
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*/
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/* ====================================================================
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* Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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*
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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*
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in
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* the documentation and/or other materials provided with the
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* distribution.
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*
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* 3. All advertising materials mentioning features or use of this
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* software must display the following acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
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*
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* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
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* endorse or promote products derived from this software without
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* prior written permission. For written permission, please contact
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* openssl-core@openssl.org.
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*
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* 5. Products derived from this software may not be called "OpenSSL"
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* nor may "OpenSSL" appear in their names without prior written
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* permission of the OpenSSL Project.
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*
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* 6. Redistributions of any form whatsoever must retain the following
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* acknowledgment:
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* "This product includes software developed by the OpenSSL Project
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* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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*
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* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
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* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
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* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
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* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
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* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
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* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
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* OF THE POSSIBILITY OF SUCH DAMAGE.
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* ====================================================================
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*
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* This product includes cryptographic software written by Eric Young
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* (eay@cryptsoft.com). This product includes software written by Tim
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* Hudson (tjh@cryptsoft.com).
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*
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*/
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#include "internal/cryptlib.h"
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#include "bn_lcl.h"
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static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
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int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
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{
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BIGNUM *a, *b, *t;
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int ret = 0;
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bn_check_top(in_a);
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bn_check_top(in_b);
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BN_CTX_start(ctx);
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a = BN_CTX_get(ctx);
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b = BN_CTX_get(ctx);
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if (a == NULL || b == NULL)
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goto err;
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if (BN_copy(a, in_a) == NULL)
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goto err;
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if (BN_copy(b, in_b) == NULL)
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goto err;
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a->neg = 0;
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b->neg = 0;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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t = euclid(a, b);
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if (t == NULL)
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goto err;
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if (BN_copy(r, t) == NULL)
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goto err;
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ret = 1;
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err:
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BN_CTX_end(ctx);
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bn_check_top(r);
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return (ret);
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}
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static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
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{
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BIGNUM *t;
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int shifts = 0;
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bn_check_top(a);
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bn_check_top(b);
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/* 0 <= b <= a */
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while (!BN_is_zero(b)) {
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/* 0 < b <= a */
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if (BN_is_odd(a)) {
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if (BN_is_odd(b)) {
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if (!BN_sub(a, a, b))
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goto err;
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if (!BN_rshift1(a, a))
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goto err;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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} else { /* a odd - b even */
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if (!BN_rshift1(b, b))
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goto err;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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}
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} else { /* a is even */
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if (BN_is_odd(b)) {
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if (!BN_rshift1(a, a))
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goto err;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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} else { /* a even - b even */
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if (!BN_rshift1(a, a))
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goto err;
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if (!BN_rshift1(b, b))
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goto err;
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shifts++;
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}
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}
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/* 0 <= b <= a */
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}
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if (shifts) {
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if (!BN_lshift(a, a, shifts))
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goto err;
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}
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bn_check_top(a);
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return (a);
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err:
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return (NULL);
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}
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/* solves ax == 1 (mod n) */
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static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n,
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BN_CTX *ctx);
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BIGNUM *BN_mod_inverse(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
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{
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BIGNUM *rv;
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int noinv;
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rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
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if (noinv)
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BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
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return rv;
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}
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BIGNUM *int_bn_mod_inverse(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
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int *pnoinv)
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{
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BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
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BIGNUM *ret = NULL;
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int sign;
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if (pnoinv)
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*pnoinv = 0;
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if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
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|| (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
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return BN_mod_inverse_no_branch(in, a, n, ctx);
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}
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bn_check_top(a);
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bn_check_top(n);
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BN_CTX_start(ctx);
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A = BN_CTX_get(ctx);
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B = BN_CTX_get(ctx);
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X = BN_CTX_get(ctx);
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D = BN_CTX_get(ctx);
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M = BN_CTX_get(ctx);
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Y = BN_CTX_get(ctx);
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T = BN_CTX_get(ctx);
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if (T == NULL)
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goto err;
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if (in == NULL)
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R = BN_new();
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else
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R = in;
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if (R == NULL)
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goto err;
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BN_one(X);
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BN_zero(Y);
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if (BN_copy(B, a) == NULL)
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goto err;
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if (BN_copy(A, n) == NULL)
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goto err;
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A->neg = 0;
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if (B->neg || (BN_ucmp(B, A) >= 0)) {
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if (!BN_nnmod(B, B, A, ctx))
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goto err;
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}
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sign = -1;
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/*-
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* From B = a mod |n|, A = |n| it follows that
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*
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* 0 <= B < A,
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* -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|).
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*/
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if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) {
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/*
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* Binary inversion algorithm; requires odd modulus. This is faster
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* than the general algorithm if the modulus is sufficiently small
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* (about 400 .. 500 bits on 32-bit sytems, but much more on 64-bit
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* systems)
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*/
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int shift;
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while (!BN_is_zero(B)) {
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/*-
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* 0 < B < |n|,
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* 0 < A <= |n|,
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* (1) -sign*X*a == B (mod |n|),
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* (2) sign*Y*a == A (mod |n|)
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*/
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/*
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* Now divide B by the maximum possible power of two in the
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* integers, and divide X by the same value mod |n|. When we're
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* done, (1) still holds.
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*/
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shift = 0;
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while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
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shift++;
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if (BN_is_odd(X)) {
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if (!BN_uadd(X, X, n))
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goto err;
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}
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/*
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* now X is even, so we can easily divide it by two
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*/
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if (!BN_rshift1(X, X))
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goto err;
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}
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if (shift > 0) {
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if (!BN_rshift(B, B, shift))
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goto err;
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}
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/*
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* Same for A and Y. Afterwards, (2) still holds.
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*/
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shift = 0;
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while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
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shift++;
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if (BN_is_odd(Y)) {
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if (!BN_uadd(Y, Y, n))
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goto err;
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}
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/* now Y is even */
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if (!BN_rshift1(Y, Y))
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goto err;
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}
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if (shift > 0) {
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if (!BN_rshift(A, A, shift))
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goto err;
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}
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/*-
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* We still have (1) and (2).
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* Both A and B are odd.
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* The following computations ensure that
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*
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* 0 <= B < |n|,
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* 0 < A < |n|,
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* (1) -sign*X*a == B (mod |n|),
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* (2) sign*Y*a == A (mod |n|),
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*
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* and that either A or B is even in the next iteration.
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*/
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if (BN_ucmp(B, A) >= 0) {
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/* -sign*(X + Y)*a == B - A (mod |n|) */
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if (!BN_uadd(X, X, Y))
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goto err;
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/*
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* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
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* actually makes the algorithm slower
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*/
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if (!BN_usub(B, B, A))
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goto err;
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} else {
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/* sign*(X + Y)*a == A - B (mod |n|) */
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if (!BN_uadd(Y, Y, X))
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goto err;
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/*
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* as above, BN_mod_add_quick(Y, Y, X, n) would slow things
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* down
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*/
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if (!BN_usub(A, A, B))
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goto err;
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}
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}
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} else {
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/* general inversion algorithm */
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while (!BN_is_zero(B)) {
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BIGNUM *tmp;
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/*-
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* 0 < B < A,
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* (*) -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|)
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*/
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/* (D, M) := (A/B, A%B) ... */
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if (BN_num_bits(A) == BN_num_bits(B)) {
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if (!BN_one(D))
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goto err;
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if (!BN_sub(M, A, B))
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goto err;
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} else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
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/* A/B is 1, 2, or 3 */
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if (!BN_lshift1(T, B))
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goto err;
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if (BN_ucmp(A, T) < 0) {
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/* A < 2*B, so D=1 */
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if (!BN_one(D))
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goto err;
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if (!BN_sub(M, A, B))
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goto err;
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} else {
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/* A >= 2*B, so D=2 or D=3 */
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if (!BN_sub(M, A, T))
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goto err;
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if (!BN_add(D, T, B))
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goto err; /* use D (:= 3*B) as temp */
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if (BN_ucmp(A, D) < 0) {
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/* A < 3*B, so D=2 */
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if (!BN_set_word(D, 2))
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goto err;
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/*
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* M (= A - 2*B) already has the correct value
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*/
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} else {
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/* only D=3 remains */
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if (!BN_set_word(D, 3))
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goto err;
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/*
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* currently M = A - 2*B, but we need M = A - 3*B
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*/
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if (!BN_sub(M, M, B))
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goto err;
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}
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}
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} else {
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if (!BN_div(D, M, A, B, ctx))
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goto err;
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}
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/*-
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* Now
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* A = D*B + M;
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* thus we have
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* (**) sign*Y*a == D*B + M (mod |n|).
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*/
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tmp = A; /* keep the BIGNUM object, the value does not
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* matter */
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/* (A, B) := (B, A mod B) ... */
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A = B;
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B = M;
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/* ... so we have 0 <= B < A again */
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/*-
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* Since the former M is now B and the former B is now A,
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* (**) translates into
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* sign*Y*a == D*A + B (mod |n|),
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* i.e.
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* sign*Y*a - D*A == B (mod |n|).
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* Similarly, (*) translates into
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* -sign*X*a == A (mod |n|).
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*
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* Thus,
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* sign*Y*a + D*sign*X*a == B (mod |n|),
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* i.e.
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* sign*(Y + D*X)*a == B (mod |n|).
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*
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* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
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* -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|).
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* Note that X and Y stay non-negative all the time.
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*/
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/*
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* most of the time D is very small, so we can optimize tmp :=
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* D*X+Y
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*/
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if (BN_is_one(D)) {
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if (!BN_add(tmp, X, Y))
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goto err;
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} else {
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if (BN_is_word(D, 2)) {
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if (!BN_lshift1(tmp, X))
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goto err;
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} else if (BN_is_word(D, 4)) {
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if (!BN_lshift(tmp, X, 2))
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goto err;
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} else if (D->top == 1) {
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if (!BN_copy(tmp, X))
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goto err;
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if (!BN_mul_word(tmp, D->d[0]))
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goto err;
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} else {
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if (!BN_mul(tmp, D, X, ctx))
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goto err;
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}
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if (!BN_add(tmp, tmp, Y))
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goto err;
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}
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M = Y; /* keep the BIGNUM object, the value does not
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* matter */
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Y = X;
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X = tmp;
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sign = -sign;
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}
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}
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/*-
|
|
* The while loop (Euclid's algorithm) ends when
|
|
* A == gcd(a,n);
|
|
* we have
|
|
* sign*Y*a == A (mod |n|),
|
|
* where Y is non-negative.
|
|
*/
|
|
|
|
if (sign < 0) {
|
|
if (!BN_sub(Y, n, Y))
|
|
goto err;
|
|
}
|
|
/* Now Y*a == A (mod |n|). */
|
|
|
|
if (BN_is_one(A)) {
|
|
/* Y*a == 1 (mod |n|) */
|
|
if (!Y->neg && BN_ucmp(Y, n) < 0) {
|
|
if (!BN_copy(R, Y))
|
|
goto err;
|
|
} else {
|
|
if (!BN_nnmod(R, Y, n, ctx))
|
|
goto err;
|
|
}
|
|
} else {
|
|
if (pnoinv)
|
|
*pnoinv = 1;
|
|
goto err;
|
|
}
|
|
ret = R;
|
|
err:
|
|
if ((ret == NULL) && (in == NULL))
|
|
BN_free(R);
|
|
BN_CTX_end(ctx);
|
|
bn_check_top(ret);
|
|
return (ret);
|
|
}
|
|
|
|
/*
|
|
* BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
|
|
* not contain branches that may leak sensitive information.
|
|
*/
|
|
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
|
|
const BIGNUM *a, const BIGNUM *n,
|
|
BN_CTX *ctx)
|
|
{
|
|
BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
|
|
BIGNUM *ret = NULL;
|
|
int sign;
|
|
|
|
bn_check_top(a);
|
|
bn_check_top(n);
|
|
|
|
BN_CTX_start(ctx);
|
|
A = BN_CTX_get(ctx);
|
|
B = BN_CTX_get(ctx);
|
|
X = BN_CTX_get(ctx);
|
|
D = BN_CTX_get(ctx);
|
|
M = BN_CTX_get(ctx);
|
|
Y = BN_CTX_get(ctx);
|
|
T = BN_CTX_get(ctx);
|
|
if (T == NULL)
|
|
goto err;
|
|
|
|
if (in == NULL)
|
|
R = BN_new();
|
|
else
|
|
R = in;
|
|
if (R == NULL)
|
|
goto err;
|
|
|
|
BN_one(X);
|
|
BN_zero(Y);
|
|
if (BN_copy(B, a) == NULL)
|
|
goto err;
|
|
if (BN_copy(A, n) == NULL)
|
|
goto err;
|
|
A->neg = 0;
|
|
|
|
if (B->neg || (BN_ucmp(B, A) >= 0)) {
|
|
/*
|
|
* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
|
|
* BN_div_no_branch will be called eventually.
|
|
*/
|
|
{
|
|
BIGNUM local_B;
|
|
bn_init(&local_B);
|
|
BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
|
|
if (!BN_nnmod(B, &local_B, A, ctx))
|
|
goto err;
|
|
/* Ensure local_B goes out of scope before any further use of B */
|
|
}
|
|
}
|
|
sign = -1;
|
|
/*-
|
|
* From B = a mod |n|, A = |n| it follows that
|
|
*
|
|
* 0 <= B < A,
|
|
* -sign*X*a == B (mod |n|),
|
|
* sign*Y*a == A (mod |n|).
|
|
*/
|
|
|
|
while (!BN_is_zero(B)) {
|
|
BIGNUM *tmp;
|
|
|
|
/*-
|
|
* 0 < B < A,
|
|
* (*) -sign*X*a == B (mod |n|),
|
|
* sign*Y*a == A (mod |n|)
|
|
*/
|
|
|
|
/*
|
|
* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
|
|
* BN_div_no_branch will be called eventually.
|
|
*/
|
|
{
|
|
BIGNUM local_A;
|
|
bn_init(&local_A);
|
|
BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
|
|
|
|
/* (D, M) := (A/B, A%B) ... */
|
|
if (!BN_div(D, M, &local_A, B, ctx))
|
|
goto err;
|
|
/* Ensure local_A goes out of scope before any further use of A */
|
|
}
|
|
|
|
/*-
|
|
* Now
|
|
* A = D*B + M;
|
|
* thus we have
|
|
* (**) sign*Y*a == D*B + M (mod |n|).
|
|
*/
|
|
|
|
tmp = A; /* keep the BIGNUM object, the value does not
|
|
* matter */
|
|
|
|
/* (A, B) := (B, A mod B) ... */
|
|
A = B;
|
|
B = M;
|
|
/* ... so we have 0 <= B < A again */
|
|
|
|
/*-
|
|
* Since the former M is now B and the former B is now A,
|
|
* (**) translates into
|
|
* sign*Y*a == D*A + B (mod |n|),
|
|
* i.e.
|
|
* sign*Y*a - D*A == B (mod |n|).
|
|
* Similarly, (*) translates into
|
|
* -sign*X*a == A (mod |n|).
|
|
*
|
|
* Thus,
|
|
* sign*Y*a + D*sign*X*a == B (mod |n|),
|
|
* i.e.
|
|
* sign*(Y + D*X)*a == B (mod |n|).
|
|
*
|
|
* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
|
|
* -sign*X*a == B (mod |n|),
|
|
* sign*Y*a == A (mod |n|).
|
|
* Note that X and Y stay non-negative all the time.
|
|
*/
|
|
|
|
if (!BN_mul(tmp, D, X, ctx))
|
|
goto err;
|
|
if (!BN_add(tmp, tmp, Y))
|
|
goto err;
|
|
|
|
M = Y; /* keep the BIGNUM object, the value does not
|
|
* matter */
|
|
Y = X;
|
|
X = tmp;
|
|
sign = -sign;
|
|
}
|
|
|
|
/*-
|
|
* The while loop (Euclid's algorithm) ends when
|
|
* A == gcd(a,n);
|
|
* we have
|
|
* sign*Y*a == A (mod |n|),
|
|
* where Y is non-negative.
|
|
*/
|
|
|
|
if (sign < 0) {
|
|
if (!BN_sub(Y, n, Y))
|
|
goto err;
|
|
}
|
|
/* Now Y*a == A (mod |n|). */
|
|
|
|
if (BN_is_one(A)) {
|
|
/* Y*a == 1 (mod |n|) */
|
|
if (!Y->neg && BN_ucmp(Y, n) < 0) {
|
|
if (!BN_copy(R, Y))
|
|
goto err;
|
|
} else {
|
|
if (!BN_nnmod(R, Y, n, ctx))
|
|
goto err;
|
|
}
|
|
} else {
|
|
BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH, BN_R_NO_INVERSE);
|
|
goto err;
|
|
}
|
|
ret = R;
|
|
err:
|
|
if ((ret == NULL) && (in == NULL))
|
|
BN_free(R);
|
|
BN_CTX_end(ctx);
|
|
bn_check_top(ret);
|
|
return (ret);
|
|
}
|