/* * Copyright 1995-2023 The OpenSSL Project Authors. All Rights Reserved. * * Licensed under the Apache License 2.0 (the "License"). You may not use * this file except in compliance with the License. You can obtain a copy * in the file LICENSE in the source distribution or at * https://www.openssl.org/source/license.html */ #include "internal/cryptlib.h" #include "bn_local.h" /* * bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does * not contain branches that may leak sensitive information. * * This is a static function, we ensure all callers in this file pass valid * arguments: all passed pointers here are non-NULL. */ static ossl_inline BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx, int *pnoinv) { 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; if (!BN_one(X)) goto err; 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 { *pnoinv = 1; /* caller sets the BN_R_NO_INVERSE error */ goto err; } ret = R; *pnoinv = 0; err: if ((ret == NULL) && (in == NULL)) BN_free(R); BN_CTX_end(ctx); bn_check_top(ret); return ret; } /* * This is an internal function, we assume all callers pass valid arguments: * all pointers passed here are assumed non-NULL. */ BIGNUM *int_bn_mod_inverse(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx, int *pnoinv) { BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL; BIGNUM *ret = NULL; int sign; /* This is invalid input so we don't worry about constant time here */ if (BN_abs_is_word(n, 1) || BN_is_zero(n)) { *pnoinv = 1; return NULL; } *pnoinv = 0; 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, pnoinv); } 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; if (!BN_one(X)) goto err; 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) <= 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; 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 { *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; } /* solves ax == 1 (mod n) */ BIGNUM *BN_mod_inverse(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { BN_CTX *new_ctx = NULL; BIGNUM *rv; int noinv = 0; if (ctx == NULL) { ctx = new_ctx = BN_CTX_new_ex(NULL); if (ctx == NULL) { ERR_raise(ERR_LIB_BN, ERR_R_BN_LIB); return NULL; } } rv = int_bn_mod_inverse(in, a, n, ctx, &noinv); if (noinv) ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE); BN_CTX_free(new_ctx); return rv; } /* * The numbers a and b are coprime if the only positive integer that is a * divisor of both of them is 1. * i.e. gcd(a,b) = 1. * * Coprimes have the property: b has a multiplicative inverse modulo a * i.e there is some value x such that bx = 1 (mod a). * * Testing the modulo inverse is currently much faster than the constant * time version of BN_gcd(). */ int BN_are_coprime(BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { int ret = 0; BIGNUM *tmp; BN_CTX_start(ctx); tmp = BN_CTX_get(ctx); if (tmp == NULL) goto end; ERR_set_mark(); BN_set_flags(a, BN_FLG_CONSTTIME); ret = (BN_mod_inverse(tmp, a, b, ctx) != NULL); /* Clear any errors (an error is returned if there is no inverse) */ ERR_pop_to_mark(); end: BN_CTX_end(ctx); return ret; } /*- * This function is based on the constant-time GCD work by Bernstein and Yang: * https://eprint.iacr.org/2019/266 * Generalized fast GCD function to allow even inputs. * The algorithm first finds the shared powers of 2 between * the inputs, and removes them, reducing at least one of the * inputs to an odd value. Then it proceeds to calculate the GCD. * Before returning the resulting GCD, we take care of adding * back the powers of two removed at the beginning. * Note 1: we assume the bit length of both inputs is public information, * since access to top potentially leaks this information. */ int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) { BIGNUM *g, *temp = NULL; BN_ULONG mask = 0; int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0; /* Note 2: zero input corner cases are not constant-time since they are * handled immediately. An attacker can run an attack under this * assumption without the need of side-channel information. */ if (BN_is_zero(in_b)) { ret = BN_copy(r, in_a) != NULL; r->neg = 0; return ret; } if (BN_is_zero(in_a)) { ret = BN_copy(r, in_b) != NULL; r->neg = 0; return ret; } bn_check_top(in_a); bn_check_top(in_b); BN_CTX_start(ctx); temp = BN_CTX_get(ctx); g = BN_CTX_get(ctx); /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */ if (g == NULL || !BN_lshift1(g, in_b) || !BN_lshift1(r, in_a)) goto err; /* find shared powers of two, i.e. "shifts" >= 1 */ for (i = 0; i < r->dmax && i < g->dmax; i++) { mask = ~(r->d[i] | g->d[i]); for (j = 0; j < BN_BITS2; j++) { bit &= mask; shifts += bit; mask >>= 1; } } /* subtract shared powers of two; shifts >= 1 */ if (!BN_rshift(r, r, shifts) || !BN_rshift(g, g, shifts)) goto err; /* expand to biggest nword, with room for a possible extra word */ top = 1 + ((r->top >= g->top) ? r->top : g->top); if (bn_wexpand(r, top) == NULL || bn_wexpand(g, top) == NULL || bn_wexpand(temp, top) == NULL) goto err; /* re arrange inputs s.t. r is odd */ BN_consttime_swap((~r->d[0]) & 1, r, g, top); /* compute the number of iterations */ rlen = BN_num_bits(r); glen = BN_num_bits(g); m = 4 + 3 * ((rlen >= glen) ? rlen : glen); for (i = 0; i < m; i++) { /* conditionally flip signs if delta is positive and g is odd */ cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */ & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))); delta = (-cond & -delta) | ((cond - 1) & delta); r->neg ^= cond; /* swap */ BN_consttime_swap(cond, r, g, top); /* elimination step */ delta++; if (!BN_add(temp, g, r)) goto err; BN_consttime_swap(g->d[0] & 1 /* g is odd */ /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */ & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))), g, temp, top); if (!BN_rshift1(g, g)) goto err; } /* remove possible negative sign */ r->neg = 0; /* add powers of 2 removed, then correct the artificial shift */ if (!BN_lshift(r, r, shifts) || !BN_rshift1(r, r)) goto err; ret = 1; err: BN_CTX_end(ctx); bn_check_top(r); return ret; }