/* $OpenBSD: bn_gcd.c,v 1.31 2025/06/02 12:40:10 tb Exp $ */ /* 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 "bn_local.h" #include "err_local.h" static BIGNUM * euclid(BIGNUM *a, BIGNUM *b) { BIGNUM *t; int shifts = 0; /* Loop invariant: 0 <= b <= a. */ while (!BN_is_zero(b)) { if (BN_is_odd(a) && BN_is_odd(b)) { if (!BN_sub(a, a, b)) goto err; if (!BN_rshift1(a, a)) goto err; } else if (BN_is_odd(a) && !BN_is_odd(b)) { if (!BN_rshift1(b, b)) goto err; } else if (!BN_is_odd(a) && BN_is_odd(b)) { if (!BN_rshift1(a, a)) goto err; } else { if (!BN_rshift1(a, a)) goto err; if (!BN_rshift1(b, b)) goto err; shifts++; continue; } if (BN_cmp(a, b) < 0) { t = a; a = b; b = t; } } 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); if ((a = BN_CTX_get(ctx)) == NULL) goto err; if ((b = BN_CTX_get(ctx)) == NULL) goto err; if (!bn_copy(a, in_a)) goto err; if (!bn_copy(b, in_b)) 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)) goto err; ret = 1; err: BN_CTX_end(ctx); return (ret); } LCRYPTO_ALIAS(BN_gcd); /* * BN_gcd_no_branch is a special version of BN_mod_inverse_no_branch. * that returns the GCD. */ static BIGNUM * BN_gcd_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; if (in == NULL) goto err; R = in; BN_init(&local_A); BN_init(&local_B); BN_CTX_start(ctx); if ((A = BN_CTX_get(ctx)) == NULL) goto err; if ((B = BN_CTX_get(ctx)) == NULL) goto err; if ((X = BN_CTX_get(ctx)) == NULL) goto err; if ((D = BN_CTX_get(ctx)) == NULL) goto err; if ((M = BN_CTX_get(ctx)) == NULL) goto err; if ((Y = BN_CTX_get(ctx)) == NULL) goto err; if ((T = BN_CTX_get(ctx)) == NULL) goto err; if (!BN_one(X)) goto err; BN_zero(Y); if (!bn_copy(B, a)) goto err; if (!bn_copy(A, n)) 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_init() done at the top of the function. */ 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_init() done at the top of the function. */ BN_with_flags(pA, A, BN_FLG_CONSTTIME); /* (D, M) := (A/B, A%B) ... */ if (!BN_div_ct(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); */ if (!bn_copy(R, A)) goto err; ret = R; err: if ((ret == NULL) && (in == NULL)) BN_free(R); BN_CTX_end(ctx); return (ret); } int BN_gcd_ct(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) { if (BN_gcd_no_branch(r, in_a, in_b, ctx) == NULL) return 0; return 1; } /* 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_init(&local_A); BN_init(&local_B); BN_CTX_start(ctx); if ((A = BN_CTX_get(ctx)) == NULL) goto err; if ((B = BN_CTX_get(ctx)) == NULL) goto err; if ((X = BN_CTX_get(ctx)) == NULL) goto err; if ((D = BN_CTX_get(ctx)) == NULL) goto err; if ((M = BN_CTX_get(ctx)) == NULL) goto err; if ((Y = BN_CTX_get(ctx)) == NULL) goto err; if ((T = BN_CTX_get(ctx)) == NULL) goto err; if (in == NULL) R = BN_new(); else R = in; if (R == NULL) goto err; if (!BN_one(X)) goto err; BN_zero(Y); if (!bn_copy(B, a)) goto err; if (!bn_copy(A, n)) 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_init() done at the top of the function. */ 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_init() done at the top of the function. */ BN_with_flags(pA, A, BN_FLG_CONSTTIME); /* (D, M) := (A/B, A%B) ... */ if (!BN_div_ct(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)) { BNerror(BN_R_NO_INVERSE); goto err; } if (!BN_nnmod(Y, Y, n, ctx)) goto err; if (!bn_copy(R, Y)) goto err; ret = R; err: if ((ret == NULL) && (in == NULL)) BN_free(R); BN_CTX_end(ctx); return (ret); } /* solves ax == 1 (mod n) */ static BIGNUM * BN_mod_inverse_internal(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx, int ct) { BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL; BIGNUM *ret = NULL; int sign; if (ct) return BN_mod_inverse_no_branch(in, a, n, ctx); BN_CTX_start(ctx); if ((A = BN_CTX_get(ctx)) == NULL) goto err; if ((B = BN_CTX_get(ctx)) == NULL) goto err; if ((X = BN_CTX_get(ctx)) == NULL) goto err; if ((D = BN_CTX_get(ctx)) == NULL) goto err; if ((M = BN_CTX_get(ctx)) == NULL) goto err; if ((Y = BN_CTX_get(ctx)) == NULL) goto err; if ((T = BN_CTX_get(ctx)) == NULL) goto err; if (in == NULL) R = BN_new(); else R = in; if (R == NULL) goto err; if (!BN_one(X)) goto err; BN_zero(Y); if (!bn_copy(B, a)) goto err; if (!bn_copy(A, n)) 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 * systems, 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; /* use D (:= 3*B) as temp */ if (!BN_add(D, T, B)) goto err; 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_nonct(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)) { BNerror(BN_R_NO_INVERSE); goto err; } if (!BN_nnmod(Y, Y, n, ctx)) goto err; if (!bn_copy(R, Y)) goto err; ret = R; err: if ((ret == NULL) && (in == NULL)) BN_free(R); BN_CTX_end(ctx); return (ret); } BIGNUM * BN_mod_inverse(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { int ct = ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)); return BN_mod_inverse_internal(in, a, n, ctx, ct); } LCRYPTO_ALIAS(BN_mod_inverse); BIGNUM * BN_mod_inverse_nonct(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { return BN_mod_inverse_internal(in, a, n, ctx, 0); } BIGNUM * BN_mod_inverse_ct(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { return BN_mod_inverse_internal(in, a, n, ctx, 1); }
