| /* |
| * Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved. |
| * |
| * Licensed under the OpenSSL license (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_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 (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 *rv; |
| int noinv; |
| rv = int_bn_mod_inverse(in, a, n, ctx, &noinv); |
| if (noinv) |
| BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE); |
| return rv; |
| } |
| |
| 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)) { |
| if (pnoinv != NULL) |
| *pnoinv = 1; |
| return NULL; |
| } |
| |
| if (pnoinv != 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); |
| } |
| |
| 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 { |
| if (pnoinv) |
| *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; |
| } |
| |
| /* |
| * 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 *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 { |
| 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; |
| } |
