| /* crypto/bn/bn_gcd.c */ |
| /* 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 "cryptlib.h" |
| #include "bn_lcl.h" |
| |
| static BIGNUM *euclid(BIGNUM *a, BIGNUM *b); |
| |
| int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) |
| { |
| BIGNUM *a, *b, *t; |
| int ret = 0; |
| |
| bn_check_top(in_a); |
| bn_check_top(in_b); |
| |
| 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); |
| bn_check_top(r); |
| return (ret); |
| } |
| |
| static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) |
| { |
| BIGNUM *t; |
| int shifts = 0; |
| |
| bn_check_top(a); |
| bn_check_top(b); |
| |
| /* 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; |
| } |
| bn_check_top(a); |
| return (a); |
| err: |
| return (NULL); |
| } |
| |
| /* solves ax == 1 (mod n) */ |
| static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in, |
| const BIGNUM *a, const BIGNUM *n, |
| BN_CTX *ctx); |
| |
| BIGNUM *BN_mod_inverse(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; |
| |
| if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) |
| || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) { |
| return BN_mod_inverse_no_branch(in, a, n, ctx); |
| } |
| |
| 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)) { |
| 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_BITS <= 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 { |
| BNerr(BN_F_BN_MOD_INVERSE, 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); |
| } |
| |
| /* |
| * 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 local_A, local_B; |
| BIGNUM *pA, *pB; |
| 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. |
| */ |
| 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; |
| } |
| |
| /*- |
| * 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); |
| } |
