624 lines
17 KiB
C
624 lines
17 KiB
C
/*
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* Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
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*
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* Licensed under the OpenSSL license (the "License"). You may not use
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* this file except in compliance with the License. You can obtain a copy
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* in the file LICENSE in the source distribution or at
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* https://www.openssl.org/source/license.html
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*/
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#include "internal/cryptlib.h"
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#include "bn_lcl.h"
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static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
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int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
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{
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BIGNUM *a, *b, *t;
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int ret = 0;
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bn_check_top(in_a);
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bn_check_top(in_b);
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BN_CTX_start(ctx);
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a = BN_CTX_get(ctx);
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b = BN_CTX_get(ctx);
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if (b == NULL)
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goto err;
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if (BN_copy(a, in_a) == NULL)
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goto err;
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if (BN_copy(b, in_b) == NULL)
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goto err;
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a->neg = 0;
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b->neg = 0;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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t = euclid(a, b);
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if (t == NULL)
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goto err;
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if (BN_copy(r, t) == NULL)
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goto err;
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ret = 1;
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err:
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BN_CTX_end(ctx);
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bn_check_top(r);
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return ret;
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}
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static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
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{
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BIGNUM *t;
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int shifts = 0;
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bn_check_top(a);
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bn_check_top(b);
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/* 0 <= b <= a */
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while (!BN_is_zero(b)) {
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/* 0 < b <= a */
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if (BN_is_odd(a)) {
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if (BN_is_odd(b)) {
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if (!BN_sub(a, a, b))
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goto err;
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if (!BN_rshift1(a, a))
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goto err;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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} else { /* a odd - b even */
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if (!BN_rshift1(b, b))
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goto err;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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}
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} else { /* a is even */
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if (BN_is_odd(b)) {
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if (!BN_rshift1(a, a))
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goto err;
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if (BN_cmp(a, b) < 0) {
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t = a;
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a = b;
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b = t;
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}
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} else { /* a even - b even */
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if (!BN_rshift1(a, a))
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goto err;
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if (!BN_rshift1(b, b))
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goto err;
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shifts++;
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}
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}
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/* 0 <= b <= a */
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}
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if (shifts) {
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if (!BN_lshift(a, a, shifts))
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goto err;
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}
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bn_check_top(a);
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return a;
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err:
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return NULL;
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}
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/* solves ax == 1 (mod n) */
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static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n,
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BN_CTX *ctx);
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BIGNUM *BN_mod_inverse(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
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{
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BIGNUM *rv;
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int noinv;
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rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
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if (noinv)
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BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
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return rv;
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}
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BIGNUM *int_bn_mod_inverse(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
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int *pnoinv)
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{
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BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
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BIGNUM *ret = NULL;
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int sign;
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/* This is invalid input so we don't worry about constant time here */
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if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
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if (pnoinv != NULL)
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*pnoinv = 1;
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return NULL;
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}
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if (pnoinv != NULL)
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*pnoinv = 0;
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if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
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|| (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
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return BN_mod_inverse_no_branch(in, a, n, ctx);
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}
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bn_check_top(a);
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bn_check_top(n);
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BN_CTX_start(ctx);
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A = BN_CTX_get(ctx);
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B = BN_CTX_get(ctx);
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X = BN_CTX_get(ctx);
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D = BN_CTX_get(ctx);
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M = BN_CTX_get(ctx);
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Y = BN_CTX_get(ctx);
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T = BN_CTX_get(ctx);
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if (T == NULL)
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goto err;
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if (in == NULL)
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R = BN_new();
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else
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R = in;
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if (R == NULL)
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goto err;
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BN_one(X);
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BN_zero(Y);
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if (BN_copy(B, a) == NULL)
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goto err;
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if (BN_copy(A, n) == NULL)
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goto err;
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A->neg = 0;
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if (B->neg || (BN_ucmp(B, A) >= 0)) {
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if (!BN_nnmod(B, B, A, ctx))
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goto err;
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}
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sign = -1;
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/*-
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* From B = a mod |n|, A = |n| it follows that
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*
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* 0 <= B < A,
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* -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|).
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*/
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if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
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/*
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* Binary inversion algorithm; requires odd modulus. This is faster
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* than the general algorithm if the modulus is sufficiently small
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* (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
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* systems)
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*/
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int shift;
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while (!BN_is_zero(B)) {
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/*-
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* 0 < B < |n|,
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* 0 < A <= |n|,
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* (1) -sign*X*a == B (mod |n|),
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* (2) sign*Y*a == A (mod |n|)
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*/
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/*
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* Now divide B by the maximum possible power of two in the
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* integers, and divide X by the same value mod |n|. When we're
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* done, (1) still holds.
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*/
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shift = 0;
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while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
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shift++;
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if (BN_is_odd(X)) {
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if (!BN_uadd(X, X, n))
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goto err;
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}
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/*
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* now X is even, so we can easily divide it by two
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*/
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if (!BN_rshift1(X, X))
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goto err;
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}
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if (shift > 0) {
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if (!BN_rshift(B, B, shift))
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goto err;
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}
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/*
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* Same for A and Y. Afterwards, (2) still holds.
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*/
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shift = 0;
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while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
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shift++;
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if (BN_is_odd(Y)) {
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if (!BN_uadd(Y, Y, n))
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goto err;
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}
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/* now Y is even */
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if (!BN_rshift1(Y, Y))
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goto err;
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}
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if (shift > 0) {
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if (!BN_rshift(A, A, shift))
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goto err;
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}
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/*-
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* We still have (1) and (2).
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* Both A and B are odd.
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* The following computations ensure that
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*
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* 0 <= B < |n|,
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* 0 < A < |n|,
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* (1) -sign*X*a == B (mod |n|),
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* (2) sign*Y*a == A (mod |n|),
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*
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* and that either A or B is even in the next iteration.
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*/
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if (BN_ucmp(B, A) >= 0) {
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/* -sign*(X + Y)*a == B - A (mod |n|) */
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if (!BN_uadd(X, X, Y))
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goto err;
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/*
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* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
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* actually makes the algorithm slower
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*/
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if (!BN_usub(B, B, A))
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goto err;
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} else {
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/* sign*(X + Y)*a == A - B (mod |n|) */
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if (!BN_uadd(Y, Y, X))
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goto err;
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/*
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* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
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*/
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if (!BN_usub(A, A, B))
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goto err;
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}
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}
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} else {
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/* general inversion algorithm */
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while (!BN_is_zero(B)) {
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BIGNUM *tmp;
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/*-
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* 0 < B < A,
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* (*) -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|)
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*/
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/* (D, M) := (A/B, A%B) ... */
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if (BN_num_bits(A) == BN_num_bits(B)) {
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if (!BN_one(D))
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goto err;
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if (!BN_sub(M, A, B))
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goto err;
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} else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
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/* A/B is 1, 2, or 3 */
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if (!BN_lshift1(T, B))
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goto err;
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if (BN_ucmp(A, T) < 0) {
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/* A < 2*B, so D=1 */
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if (!BN_one(D))
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goto err;
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if (!BN_sub(M, A, B))
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goto err;
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} else {
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/* A >= 2*B, so D=2 or D=3 */
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if (!BN_sub(M, A, T))
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goto err;
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if (!BN_add(D, T, B))
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goto err; /* use D (:= 3*B) as temp */
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if (BN_ucmp(A, D) < 0) {
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/* A < 3*B, so D=2 */
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if (!BN_set_word(D, 2))
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goto err;
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/*
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* M (= A - 2*B) already has the correct value
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*/
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} else {
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/* only D=3 remains */
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if (!BN_set_word(D, 3))
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goto err;
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/*
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* currently M = A - 2*B, but we need M = A - 3*B
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*/
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if (!BN_sub(M, M, B))
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goto err;
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}
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}
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} else {
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if (!BN_div(D, M, A, B, ctx))
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goto err;
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}
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/*-
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* Now
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* A = D*B + M;
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* thus we have
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* (**) sign*Y*a == D*B + M (mod |n|).
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*/
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tmp = A; /* keep the BIGNUM object, the value does not matter */
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/* (A, B) := (B, A mod B) ... */
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A = B;
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B = M;
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/* ... so we have 0 <= B < A again */
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/*-
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* Since the former M is now B and the former B is now A,
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* (**) translates into
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* sign*Y*a == D*A + B (mod |n|),
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* i.e.
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* sign*Y*a - D*A == B (mod |n|).
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* Similarly, (*) translates into
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* -sign*X*a == A (mod |n|).
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*
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* Thus,
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* sign*Y*a + D*sign*X*a == B (mod |n|),
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* i.e.
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* sign*(Y + D*X)*a == B (mod |n|).
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*
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* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
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* -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|).
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* Note that X and Y stay non-negative all the time.
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*/
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/*
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* most of the time D is very small, so we can optimize tmp := D*X+Y
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*/
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if (BN_is_one(D)) {
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if (!BN_add(tmp, X, Y))
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goto err;
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} else {
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if (BN_is_word(D, 2)) {
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if (!BN_lshift1(tmp, X))
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goto err;
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} else if (BN_is_word(D, 4)) {
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if (!BN_lshift(tmp, X, 2))
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goto err;
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} else if (D->top == 1) {
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if (!BN_copy(tmp, X))
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goto err;
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if (!BN_mul_word(tmp, D->d[0]))
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goto err;
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} else {
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if (!BN_mul(tmp, D, X, ctx))
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goto err;
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}
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if (!BN_add(tmp, tmp, Y))
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goto err;
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}
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M = Y; /* keep the BIGNUM object, the value does not matter */
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Y = X;
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X = tmp;
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sign = -sign;
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}
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}
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/*-
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* The while loop (Euclid's algorithm) ends when
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* A == gcd(a,n);
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* we have
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* sign*Y*a == A (mod |n|),
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* where Y is non-negative.
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*/
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if (sign < 0) {
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if (!BN_sub(Y, n, Y))
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goto err;
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}
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/* Now Y*a == A (mod |n|). */
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if (BN_is_one(A)) {
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/* Y*a == 1 (mod |n|) */
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if (!Y->neg && BN_ucmp(Y, n) < 0) {
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if (!BN_copy(R, Y))
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goto err;
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} else {
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if (!BN_nnmod(R, Y, n, ctx))
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goto err;
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}
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} else {
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if (pnoinv)
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*pnoinv = 1;
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goto err;
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}
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ret = R;
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err:
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if ((ret == NULL) && (in == NULL))
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BN_free(R);
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BN_CTX_end(ctx);
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bn_check_top(ret);
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return ret;
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}
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/*
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* BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
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* not contain branches that may leak sensitive information.
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*/
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static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
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const BIGNUM *a, const BIGNUM *n,
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BN_CTX *ctx)
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{
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BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
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BIGNUM *ret = NULL;
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int sign;
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bn_check_top(a);
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bn_check_top(n);
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BN_CTX_start(ctx);
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A = BN_CTX_get(ctx);
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B = BN_CTX_get(ctx);
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X = BN_CTX_get(ctx);
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D = BN_CTX_get(ctx);
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M = BN_CTX_get(ctx);
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Y = BN_CTX_get(ctx);
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T = BN_CTX_get(ctx);
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if (T == NULL)
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goto err;
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if (in == NULL)
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R = BN_new();
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else
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R = in;
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if (R == NULL)
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goto err;
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BN_one(X);
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BN_zero(Y);
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if (BN_copy(B, a) == NULL)
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goto err;
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if (BN_copy(A, n) == NULL)
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goto err;
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A->neg = 0;
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if (B->neg || (BN_ucmp(B, A) >= 0)) {
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/*
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* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
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* BN_div_no_branch will be called eventually.
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*/
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{
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BIGNUM local_B;
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bn_init(&local_B);
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BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
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if (!BN_nnmod(B, &local_B, A, ctx))
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goto err;
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/* Ensure local_B goes out of scope before any further use of B */
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}
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}
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sign = -1;
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/*-
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* From B = a mod |n|, A = |n| it follows that
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*
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* 0 <= B < A,
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* -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|).
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*/
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while (!BN_is_zero(B)) {
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BIGNUM *tmp;
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/*-
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* 0 < B < A,
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* (*) -sign*X*a == B (mod |n|),
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* sign*Y*a == A (mod |n|)
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*/
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/*
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* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
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* BN_div_no_branch will be called eventually.
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*/
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{
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BIGNUM local_A;
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bn_init(&local_A);
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BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
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/* (D, M) := (A/B, A%B) ... */
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if (!BN_div(D, M, &local_A, B, ctx))
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goto err;
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/* Ensure local_A goes out of scope before any further use of A */
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}
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|
/*-
|
|
* 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;
|
|
}
|