| /* |
| * Copyright 1995-2020 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; |
| |
| 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. |
| */ |
| { |
| 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; |
| |
| 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) <= 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_MALLOC_FAILURE); |
| 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; |
| } |
| |
| /*- |
| * 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 = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1 |
| /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */ |
| & (~((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) */ |
| & (~((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; |
| } |
