/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
* All rights reserved.
*
* This package is an SSL implementation written
* by Eric Young (eay@cryptsoft.com).
* The implementation was written so as to conform with Netscapes SSL.
*
* This library is free for commercial and non-commercial use as long as
* the following conditions are aheared to. The following conditions
* apply to all code found in this distribution, be it the RC4, RSA,
* lhash, DES, etc., code; not just the SSL code. The SSL documentation
* included with this distribution is covered by the same copyright terms
* except that the holder is Tim Hudson (tjh@cryptsoft.com).
*
* Copyright remains Eric Young's, and as such any Copyright notices in
* the code are not to be removed.
* If this package is used in a product, Eric Young should be given attribution
* as the author of the parts of the library used.
* This can be in the form of a textual message at program startup or
* in documentation (online or textual) provided with the package.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* "This product includes cryptographic software written by
* Eric Young (eay@cryptsoft.com)"
* The word 'cryptographic' can be left out if the rouines from the library
* being used are not cryptographic related :-).
* 4. If you include any Windows specific code (or a derivative thereof) from
* the apps directory (application code) you must include an acknowledgement:
* "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
*
* THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*
* The licence and distribution terms for any publically available version or
* derivative of this code cannot be changed. i.e. this code cannot simply be
* copied and put under another distribution licence
* [including the GNU Public Licence.]
*/
/* ====================================================================
* Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
*
* 3. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* openssl-core@openssl.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
* OF THE POSSIBILITY OF SUCH DAMAGE.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com). */
#include <openssl/bn.h>
#include <openssl/err.h>
#include "internal.h"
static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) {
BIGNUM *t;
int shifts = 0;
/* 0 <= b <= a */
while (!BN_is_zero(b)) {
/* 0 < b <= a */
if (BN_is_odd(a)) {
if (BN_is_odd(b)) {
if (!BN_sub(a, a, b)) {
goto err;
}
if (!BN_rshift1(a, a)) {
goto err;
}
if (BN_cmp(a, b) < 0) {
t = a;
a = b;
b = t;
}
} else {
/* a odd - b even */
if (!BN_rshift1(b, b)) {
goto err;
}
if (BN_cmp(a, b) < 0) {
t = a;
a = b;
b = t;
}
}
} else {
/* a is even */
if (BN_is_odd(b)) {
if (!BN_rshift1(a, a)) {
goto err;
}
if (BN_cmp(a, b) < 0) {
t = a;
a = b;
b = t;
}
} else {
/* a even - b even */
if (!BN_rshift1(a, a)) {
goto err;
}
if (!BN_rshift1(b, b)) {
goto err;
}
shifts++;
}
}
/* 0 <= b <= a */
}
if (shifts) {
if (!BN_lshift(a, a, shifts)) {
goto err;
}
}
return a;
err:
return NULL;
}
int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) {
BIGNUM *a, *b, *t;
int ret = 0;
BN_CTX_start(ctx);
a = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
if (a == NULL || b == NULL) {
goto err;
}
if (BN_copy(a, in_a) == NULL) {
goto err;
}
if (BN_copy(b, in_b) == NULL) {
goto err;
}
a->neg = 0;
b->neg = 0;
if (BN_cmp(a, b) < 0) {
t = a;
a = b;
b = t;
}
t = euclid(a, b);
if (t == NULL) {
goto err;
}
if (BN_copy(r, t) == NULL) {
goto err;
}
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
/* solves ax == 1 (mod n) */
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *out, int *out_no_inverse,
const BIGNUM *a, const BIGNUM *n,
BN_CTX *ctx);
BIGNUM *BN_mod_inverse_ex(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
const BIGNUM *n, BN_CTX *ctx) {
BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
BIGNUM *ret = NULL;
int sign;
if ((a->flags & BN_FLG_CONSTTIME) != 0 ||
(n->flags & BN_FLG_CONSTTIME) != 0) {
return BN_mod_inverse_no_branch(out, out_no_inverse, a, n, ctx);
}
*out_no_inverse = 0;
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 (out == NULL) {
R = BN_new();
} else {
R = out;
}
if (R == NULL) {
goto err;
}
BN_zero(Y);
if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
goto err;
}
A->neg = 0;
if (B->neg || (BN_ucmp(B, A) >= 0)) {
if (!BN_nnmod(B, B, A, ctx)) {
goto err;
}
}
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|).
*/
if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS2 <= 32 ? 450 : 2048))) {
/* Binary inversion algorithm; requires odd modulus.
* This is faster than the general algorithm if the modulus
* is sufficiently small (about 400 .. 500 bits on 32-bit
* sytems, but much more on 64-bit systems) */
int shift;
while (!BN_is_zero(B)) {
/* 0 < B < |n|,
* 0 < A <= |n|,
* (1) -sign*X*a == B (mod |n|),
* (2) sign*Y*a == A (mod |n|) */
/* Now divide B by the maximum possible power of two in the integers,
* and divide X by the same value mod |n|.
* When we're done, (1) still holds. */
shift = 0;
while (!BN_is_bit_set(B, shift)) {
/* note that 0 < B */
shift++;
if (BN_is_odd(X)) {
if (!BN_uadd(X, X, n)) {
goto err;
}
}
/* now X is even, so we can easily divide it by two */
if (!BN_rshift1(X, X)) {
goto err;
}
}
if (shift > 0) {
if (!BN_rshift(B, B, shift)) {
goto err;
}
}
/* Same for A and Y. Afterwards, (2) still holds. */
shift = 0;
while (!BN_is_bit_set(A, shift)) {
/* note that 0 < A */
shift++;
if (BN_is_odd(Y)) {
if (!BN_uadd(Y, Y, n)) {
goto err;
}
}
/* now Y is even */
if (!BN_rshift1(Y, Y)) {
goto err;
}
}
if (shift > 0) {
if (!BN_rshift(A, A, shift)) {
goto err;
}
}
/* We still have (1) and (2).
* Both A and B are odd.
* The following computations ensure that
*
* 0 <= B < |n|,
* 0 < A < |n|,
* (1) -sign*X*a == B (mod |n|),
* (2) sign*Y*a == A (mod |n|),
*
* and that either A or B is even in the next iteration. */
if (BN_ucmp(B, A) >= 0) {
/* -sign*(X + Y)*a == B - A (mod |n|) */
if (!BN_uadd(X, X, Y)) {
goto err;
}
/* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
* actually makes the algorithm slower */
if (!BN_usub(B, B, A)) {
goto err;
}
} else {
/* sign*(X + Y)*a == A - B (mod |n|) */
if (!BN_uadd(Y, Y, X)) {
goto err;
}
/* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
if (!BN_usub(A, A, B)) {
goto err;
}
}
}
} else {
/* general inversion algorithm */
while (!BN_is_zero(B)) {
BIGNUM *tmp;
/*
* 0 < B < A,
* (*) -sign*X*a == B (mod |n|),
* sign*Y*a == A (mod |n|) */
/* (D, M) := (A/B, A%B) ... */
if (BN_num_bits(A) == BN_num_bits(B)) {
if (!BN_one(D)) {
goto err;
}
if (!BN_sub(M, A, B)) {
goto err;
}
} else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
/* A/B is 1, 2, or 3 */
if (!BN_lshift1(T, B)) {
goto err;
}
if (BN_ucmp(A, T) < 0) {
/* A < 2*B, so D=1 */
if (!BN_one(D)) {
goto err;
}
if (!BN_sub(M, A, B)) {
goto err;
}
} else {
/* A >= 2*B, so D=2 or D=3 */
if (!BN_sub(M, A, T)) {
goto err;
}
if (!BN_add(D, T, B)) {
goto err; /* use D (:= 3*B) as temp */
}
if (BN_ucmp(A, D) < 0) {
/* A < 3*B, so D=2 */
if (!BN_set_word(D, 2)) {
goto err;
}
/* M (= A - 2*B) already has the correct value */
} else {
/* only D=3 remains */
if (!BN_set_word(D, 3)) {
goto err;
}
/* currently M = A - 2*B, but we need M = A - 3*B */
if (!BN_sub(M, M, B)) {
goto err;
}
}
}
} else {
if (!BN_div(D, M, A, B, ctx)) {
goto err;
}
}
/* 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. */
/* most of the time D is very small, so we can optimize tmp := D*X+Y */
if (BN_is_one(D)) {
if (!BN_add(tmp, X, Y)) {
goto err;
}
} else {
if (BN_is_word(D, 2)) {
if (!BN_lshift1(tmp, X)) {
goto err;
}
} else if (BN_is_word(D, 4)) {
if (!BN_lshift(tmp, X, 2)) {
goto err;
}
} else if (D->top == 1) {
if (!BN_copy(tmp, X)) {
goto err;
}
if (!BN_mul_word(tmp, D->d[0])) {
goto err;
}
} else {
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 {
*out_no_inverse = 1;
OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
goto err;
}
ret = R;
err:
if (ret == NULL && out == NULL) {
BN_free(R);
}
BN_CTX_end(ctx);
return ret;
}
BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n,
BN_CTX *ctx) {
int no_inverse;
return BN_mod_inverse_ex(out, &no_inverse, a, n, ctx);
}
/* 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 *out, int *out_no_inverse,
const BIGNUM *a, const BIGNUM *n,
BN_CTX *ctx) {
BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
BIGNUM local_A, local_B;
BIGNUM *pA, *pB;
BIGNUM *ret = NULL;
int sign;
*out_no_inverse = 0;
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 (out == NULL) {
R = BN_new();
} else {
R = out;
}
if (R == NULL) {
goto err;
}
BN_zero(Y);
if (!BN_one(X) || BN_copy(B, a) == NULL || 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.
*/
pB = &local_B;
BN_with_flags(pB, B, BN_FLG_CONSTTIME);
if (!BN_nnmod(B, pB, A, ctx)) {
goto err;
}
}
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.
*/
pA = &local_A;
BN_with_flags(pA, A, BN_FLG_CONSTTIME);
/* (D, M) := (A/B, A%B) ... */
if (!BN_div(D, M, pA, B, ctx)) {
goto err;
}
/* 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;
}
if (!BN_is_one(A)) {
*out_no_inverse = 1;
OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
goto err;
}
/*
* 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|). */
/* 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;
}
}
ret = R;
err:
if (ret == NULL && out == NULL) {
BN_free(R);
}
BN_CTX_end(ctx);
return ret;
}