/* 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 #include #include #include "internal.h" static BN_ULONG word_is_odd_mask(BN_ULONG a) { return (BN_ULONG)0 - (a & 1); } static void maybe_rshift1_words(BN_ULONG *a, BN_ULONG mask, BN_ULONG *tmp, size_t num) { bn_rshift1_words(tmp, a, num); bn_select_words(a, mask, tmp, a, num); } static void maybe_rshift1_words_carry(BN_ULONG *a, BN_ULONG carry, BN_ULONG mask, BN_ULONG *tmp, size_t num) { maybe_rshift1_words(a, mask, tmp, num); if (num != 0) { carry &= mask; a[num - 1] |= carry << (BN_BITS2-1); } } static BN_ULONG maybe_add_words(BN_ULONG *a, BN_ULONG mask, const BN_ULONG *b, BN_ULONG *tmp, size_t num) { BN_ULONG carry = bn_add_words(tmp, a, b, num); bn_select_words(a, mask, tmp, a, num); return carry & mask; } static int bn_gcd_consttime(BIGNUM *r, unsigned *out_shift, const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx) { size_t width = x->width > y->width ? x->width : y->width; if (width == 0) { *out_shift = 0; BN_zero(r); return 1; } // This is a constant-time implementation of Stein's algorithm (binary GCD). int ret = 0; BN_CTX_start(ctx); BIGNUM *u = BN_CTX_get(ctx); BIGNUM *v = BN_CTX_get(ctx); BIGNUM *tmp = BN_CTX_get(ctx); if (u == NULL || v == NULL || tmp == NULL || !BN_copy(u, x) || !BN_copy(v, y) || !bn_resize_words(u, width) || !bn_resize_words(v, width) || !bn_resize_words(tmp, width)) { goto err; } // Each loop iteration halves at least one of |u| and |v|. Thus we need at // most the combined bit width of inputs for at least one value to be zero. unsigned x_bits = x->width * BN_BITS2, y_bits = y->width * BN_BITS2; unsigned num_iters = x_bits + y_bits; if (num_iters < x_bits) { OPENSSL_PUT_ERROR(BN, BN_R_BIGNUM_TOO_LONG); goto err; } unsigned shift = 0; for (unsigned i = 0; i < num_iters; i++) { BN_ULONG both_odd = word_is_odd_mask(u->d[0]) & word_is_odd_mask(v->d[0]); // If both |u| and |v| are odd, subtract the smaller from the larger. BN_ULONG u_less_than_v = (BN_ULONG)0 - bn_sub_words(tmp->d, u->d, v->d, width); bn_select_words(u->d, both_odd & ~u_less_than_v, tmp->d, u->d, width); bn_sub_words(tmp->d, v->d, u->d, width); bn_select_words(v->d, both_odd & u_less_than_v, tmp->d, v->d, width); // At least one of |u| and |v| is now even. BN_ULONG u_is_odd = word_is_odd_mask(u->d[0]); BN_ULONG v_is_odd = word_is_odd_mask(v->d[0]); assert(!(u_is_odd & v_is_odd)); // If both are even, the final GCD gains a factor of two. shift += 1 & (~u_is_odd & ~v_is_odd); // Halve any which are even. maybe_rshift1_words(u->d, ~u_is_odd, tmp->d, width); maybe_rshift1_words(v->d, ~v_is_odd, tmp->d, width); } // One of |u| or |v| is zero at this point. The algorithm usually makes |u| // zero, unless |y| was already zero on input. Fix this by combining the // values. assert(BN_is_zero(u) || BN_is_zero(v)); for (size_t i = 0; i < width; i++) { v->d[i] |= u->d[i]; } *out_shift = shift; ret = bn_set_words(r, v->d, width); err: BN_CTX_end(ctx); return ret; } int BN_gcd(BIGNUM *r, const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx) { unsigned shift; return bn_gcd_consttime(r, &shift, x, y, ctx) && BN_lshift(r, r, shift); } int bn_is_relatively_prime(int *out_relatively_prime, const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx) { int ret = 0; BN_CTX_start(ctx); unsigned shift; BIGNUM *gcd = BN_CTX_get(ctx); if (gcd == NULL || !bn_gcd_consttime(gcd, &shift, x, y, ctx)) { goto err; } // Check that 2^|shift| * |gcd| is one. if (gcd->width == 0) { *out_relatively_prime = 0; } else { BN_ULONG mask = shift | (gcd->d[0] ^ 1); for (int i = 1; i < gcd->width; i++) { mask |= gcd->d[i]; } *out_relatively_prime = mask == 0; } ret = 1; err: BN_CTX_end(ctx); return ret; } int bn_lcm_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { BN_CTX_start(ctx); unsigned shift; BIGNUM *gcd = BN_CTX_get(ctx); int ret = gcd != NULL && bn_mul_consttime(r, a, b, ctx) && bn_gcd_consttime(gcd, &shift, a, b, ctx) && bn_div_consttime(r, NULL, r, gcd, ctx) && bn_rshift_secret_shift(r, r, shift, ctx); BN_CTX_end(ctx); return ret; } int bn_mod_inverse_consttime(BIGNUM *r, int *out_no_inverse, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { *out_no_inverse = 0; if (BN_is_negative(a) || BN_ucmp(a, n) >= 0) { OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED); return 0; } if (BN_is_zero(a)) { if (BN_is_one(n)) { BN_zero(r); return 1; } *out_no_inverse = 1; OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); return 0; } // This is a constant-time implementation of the extended binary GCD // algorithm. It is adapted from the Handbook of Applied Cryptography, section // 14.4.3, algorithm 14.51, and modified to bound coefficients and avoid // negative numbers. // // For more details and proof of correctness, see // https://github.com/mit-plv/fiat-crypto/pull/333. In particular, see |step| // and |mod_inverse_consttime| for the algorithm in Gallina and see // |mod_inverse_consttime_spec| for the correctness result. if (!BN_is_odd(a) && !BN_is_odd(n)) { *out_no_inverse = 1; OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); return 0; } // This function exists to compute the RSA private exponent, where |a| is one // word. We'll thus use |a_width| when available. size_t n_width = n->width, a_width = a->width; if (a_width > n_width) { a_width = n_width; } int ret = 0; BN_CTX_start(ctx); BIGNUM *u = BN_CTX_get(ctx); BIGNUM *v = BN_CTX_get(ctx); BIGNUM *A = BN_CTX_get(ctx); BIGNUM *B = BN_CTX_get(ctx); BIGNUM *C = BN_CTX_get(ctx); BIGNUM *D = BN_CTX_get(ctx); BIGNUM *tmp = BN_CTX_get(ctx); BIGNUM *tmp2 = BN_CTX_get(ctx); if (u == NULL || v == NULL || A == NULL || B == NULL || C == NULL || D == NULL || tmp == NULL || tmp2 == NULL || !BN_copy(u, a) || !BN_copy(v, n) || !BN_one(A) || !BN_one(D) || // For convenience, size |u| and |v| equivalently. !bn_resize_words(u, n_width) || !bn_resize_words(v, n_width) || // |A| and |C| are bounded by |m|. !bn_resize_words(A, n_width) || !bn_resize_words(C, n_width) || // |B| and |D| are bounded by |a|. !bn_resize_words(B, a_width) || !bn_resize_words(D, a_width) || // |tmp| and |tmp2| may be used at either size. !bn_resize_words(tmp, n_width) || !bn_resize_words(tmp2, n_width)) { goto err; } // Each loop iteration halves at least one of |u| and |v|. Thus we need at // most the combined bit width of inputs for at least one value to be zero. unsigned a_bits = a_width * BN_BITS2, n_bits = n_width * BN_BITS2; unsigned num_iters = a_bits + n_bits; if (num_iters < a_bits) { OPENSSL_PUT_ERROR(BN, BN_R_BIGNUM_TOO_LONG); goto err; } // Before and after each loop iteration, the following hold: // // u = A*a - B*n // v = D*n - C*a // 0 < u <= a // 0 <= v <= n // 0 <= A < n // 0 <= B <= a // 0 <= C < n // 0 <= D <= a // // After each loop iteration, u and v only get smaller, and at least one of // them shrinks by at least a factor of two. for (unsigned i = 0; i < num_iters; i++) { BN_ULONG both_odd = word_is_odd_mask(u->d[0]) & word_is_odd_mask(v->d[0]); // If both |u| and |v| are odd, subtract the smaller from the larger. BN_ULONG v_less_than_u = (BN_ULONG)0 - bn_sub_words(tmp->d, v->d, u->d, n_width); bn_select_words(v->d, both_odd & ~v_less_than_u, tmp->d, v->d, n_width); bn_sub_words(tmp->d, u->d, v->d, n_width); bn_select_words(u->d, both_odd & v_less_than_u, tmp->d, u->d, n_width); // If we updated one of the values, update the corresponding coefficient. BN_ULONG carry = bn_add_words(tmp->d, A->d, C->d, n_width); carry -= bn_sub_words(tmp2->d, tmp->d, n->d, n_width); bn_select_words(tmp->d, carry, tmp->d, tmp2->d, n_width); bn_select_words(A->d, both_odd & v_less_than_u, tmp->d, A->d, n_width); bn_select_words(C->d, both_odd & ~v_less_than_u, tmp->d, C->d, n_width); bn_add_words(tmp->d, B->d, D->d, a_width); bn_sub_words(tmp2->d, tmp->d, a->d, a_width); bn_select_words(tmp->d, carry, tmp->d, tmp2->d, a_width); bn_select_words(B->d, both_odd & v_less_than_u, tmp->d, B->d, a_width); bn_select_words(D->d, both_odd & ~v_less_than_u, tmp->d, D->d, a_width); // Our loop invariants hold at this point. Additionally, exactly one of |u| // and |v| is now even. BN_ULONG u_is_even = ~word_is_odd_mask(u->d[0]); BN_ULONG v_is_even = ~word_is_odd_mask(v->d[0]); assert(u_is_even != v_is_even); // Halve the even one and adjust the corresponding coefficient. maybe_rshift1_words(u->d, u_is_even, tmp->d, n_width); BN_ULONG A_or_B_is_odd = word_is_odd_mask(A->d[0]) | word_is_odd_mask(B->d[0]); BN_ULONG A_carry = maybe_add_words(A->d, A_or_B_is_odd & u_is_even, n->d, tmp->d, n_width); BN_ULONG B_carry = maybe_add_words(B->d, A_or_B_is_odd & u_is_even, a->d, tmp->d, a_width); maybe_rshift1_words_carry(A->d, A_carry, u_is_even, tmp->d, n_width); maybe_rshift1_words_carry(B->d, B_carry, u_is_even, tmp->d, a_width); maybe_rshift1_words(v->d, v_is_even, tmp->d, n_width); BN_ULONG C_or_D_is_odd = word_is_odd_mask(C->d[0]) | word_is_odd_mask(D->d[0]); BN_ULONG C_carry = maybe_add_words(C->d, C_or_D_is_odd & v_is_even, n->d, tmp->d, n_width); BN_ULONG D_carry = maybe_add_words(D->d, C_or_D_is_odd & v_is_even, a->d, tmp->d, a_width); maybe_rshift1_words_carry(C->d, C_carry, v_is_even, tmp->d, n_width); maybe_rshift1_words_carry(D->d, D_carry, v_is_even, tmp->d, a_width); } assert(BN_is_zero(v)); if (!BN_is_one(u)) { *out_no_inverse = 1; OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); goto err; } ret = BN_copy(r, A) != NULL; err: BN_CTX_end(ctx); return ret; } int BN_mod_inverse_odd(BIGNUM *out, int *out_no_inverse, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { *out_no_inverse = 0; if (!BN_is_odd(n)) { OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS); return 0; } if (BN_is_negative(a) || BN_cmp(a, n) >= 0) { OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED); return 0; } BIGNUM *A, *B, *X, *Y; int ret = 0; int sign; BN_CTX_start(ctx); A = BN_CTX_get(ctx); B = BN_CTX_get(ctx); X = BN_CTX_get(ctx); Y = BN_CTX_get(ctx); if (Y == NULL) { goto err; } BIGNUM *R = out; BN_zero(Y); if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) { goto err; } A->neg = 0; 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|). // 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; } } } 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 = 1; err: BN_CTX_end(ctx); return ret; } BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { BIGNUM *new_out = NULL; if (out == NULL) { new_out = BN_new(); if (new_out == NULL) { OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE); return NULL; } out = new_out; } int ok = 0; BIGNUM *a_reduced = NULL; if (a->neg || BN_ucmp(a, n) >= 0) { a_reduced = BN_dup(a); if (a_reduced == NULL) { goto err; } if (!BN_nnmod(a_reduced, a_reduced, n, ctx)) { goto err; } a = a_reduced; } int no_inverse; if (!BN_is_odd(n)) { if (!bn_mod_inverse_consttime(out, &no_inverse, a, n, ctx)) { goto err; } } else if (!BN_mod_inverse_odd(out, &no_inverse, a, n, ctx)) { goto err; } ok = 1; err: if (!ok) { BN_free(new_out); out = NULL; } BN_free(a_reduced); return out; } int BN_mod_inverse_blinded(BIGNUM *out, int *out_no_inverse, const BIGNUM *a, const BN_MONT_CTX *mont, BN_CTX *ctx) { *out_no_inverse = 0; if (BN_is_negative(a) || BN_cmp(a, &mont->N) >= 0) { OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED); return 0; } int ret = 0; BIGNUM blinding_factor; BN_init(&blinding_factor); if (!BN_rand_range_ex(&blinding_factor, 1, &mont->N) || !BN_mod_mul_montgomery(out, &blinding_factor, a, mont, ctx) || !BN_mod_inverse_odd(out, out_no_inverse, out, &mont->N, ctx) || !BN_mod_mul_montgomery(out, &blinding_factor, out, mont, ctx)) { OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB); goto err; } ret = 1; err: BN_free(&blinding_factor); return ret; } int bn_mod_inverse_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx, const BN_MONT_CTX *mont_p) { BN_CTX_start(ctx); BIGNUM *p_minus_2 = BN_CTX_get(ctx); int ok = p_minus_2 != NULL && BN_copy(p_minus_2, p) && BN_sub_word(p_minus_2, 2) && BN_mod_exp_mont(out, a, p_minus_2, p, ctx, mont_p); BN_CTX_end(ctx); return ok; } int bn_mod_inverse_secret_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx, const BN_MONT_CTX *mont_p) { BN_CTX_start(ctx); BIGNUM *p_minus_2 = BN_CTX_get(ctx); int ok = p_minus_2 != NULL && BN_copy(p_minus_2, p) && BN_sub_word(p_minus_2, 2) && BN_mod_exp_mont_consttime(out, a, p_minus_2, p, ctx, mont_p); BN_CTX_end(ctx); return ok; }