/* 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) 1998-2005 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 <openssl/bn.h>
#include <assert.h>
#include <string.h>
#include <openssl/cpu.h>
#include <openssl/err.h>
#include <openssl/mem.h>
#include "internal.h"
#if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64)
#define OPENSSL_BN_ASM_MONT5
#define RSAZ_ENABLED
#include "rsaz_exp.h"
void bn_mul_mont_gather5(BN_ULONG *rp, const BN_ULONG *ap, const void *table,
const BN_ULONG *np, const BN_ULONG *n0, int num,
int power);
void bn_scatter5(const BN_ULONG *inp, size_t num, void *table, size_t power);
void bn_gather5(BN_ULONG *out, size_t num, void *table, size_t power);
void bn_power5(BN_ULONG *rp, const BN_ULONG *ap, const void *table,
const BN_ULONG *np, const BN_ULONG *n0, int num, int power);
int bn_from_montgomery(BN_ULONG *rp, const BN_ULONG *ap,
const BN_ULONG *not_used, const BN_ULONG *np,
const BN_ULONG *n0, int num);
#endif
int BN_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
int i, bits, ret = 0;
BIGNUM *v, *rr;
BN_CTX_start(ctx);
if (r == a || r == p) {
rr = BN_CTX_get(ctx);
} else {
rr = r;
}
v = BN_CTX_get(ctx);
if (rr == NULL || v == NULL) {
goto err;
}
if (BN_copy(v, a) == NULL) {
goto err;
}
bits = BN_num_bits(p);
if (BN_is_odd(p)) {
if (BN_copy(rr, a) == NULL) {
goto err;
}
} else {
if (!BN_one(rr)) {
goto err;
}
}
for (i = 1; i < bits; i++) {
if (!BN_sqr(v, v, ctx)) {
goto err;
}
if (BN_is_bit_set(p, i)) {
if (!BN_mul(rr, rr, v, ctx)) {
goto err;
}
}
}
if (r != rr && !BN_copy(r, rr)) {
goto err;
}
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
typedef struct bn_recp_ctx_st {
BIGNUM N; // the divisor
BIGNUM Nr; // the reciprocal
int num_bits;
int shift;
int flags;
} BN_RECP_CTX;
static void BN_RECP_CTX_init(BN_RECP_CTX *recp) {
BN_init(&recp->N);
BN_init(&recp->Nr);
recp->num_bits = 0;
recp->shift = 0;
recp->flags = 0;
}
static void BN_RECP_CTX_free(BN_RECP_CTX *recp) {
if (recp == NULL) {
return;
}
BN_free(&recp->N);
BN_free(&recp->Nr);
}
static int BN_RECP_CTX_set(BN_RECP_CTX *recp, const BIGNUM *d, BN_CTX *ctx) {
if (!BN_copy(&(recp->N), d)) {
return 0;
}
BN_zero(&recp->Nr);
recp->num_bits = BN_num_bits(d);
recp->shift = 0;
return 1;
}
// len is the expected size of the result We actually calculate with an extra
// word of precision, so we can do faster division if the remainder is not
// required.
// r := 2^len / m
static int BN_reciprocal(BIGNUM *r, const BIGNUM *m, int len, BN_CTX *ctx) {
int ret = -1;
BIGNUM *t;
BN_CTX_start(ctx);
t = BN_CTX_get(ctx);
if (t == NULL) {
goto err;
}
if (!BN_set_bit(t, len)) {
goto err;
}
if (!BN_div(r, NULL, t, m, ctx)) {
goto err;
}
ret = len;
err:
BN_CTX_end(ctx);
return ret;
}
static int BN_div_recp(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m,
BN_RECP_CTX *recp, BN_CTX *ctx) {
int i, j, ret = 0;
BIGNUM *a, *b, *d, *r;
BN_CTX_start(ctx);
a = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
if (dv != NULL) {
d = dv;
} else {
d = BN_CTX_get(ctx);
}
if (rem != NULL) {
r = rem;
} else {
r = BN_CTX_get(ctx);
}
if (a == NULL || b == NULL || d == NULL || r == NULL) {
goto err;
}
if (BN_ucmp(m, &recp->N) < 0) {
BN_zero(d);
if (!BN_copy(r, m)) {
goto err;
}
BN_CTX_end(ctx);
return 1;
}
// We want the remainder
// Given input of ABCDEF / ab
// we need multiply ABCDEF by 3 digests of the reciprocal of ab
// i := max(BN_num_bits(m), 2*BN_num_bits(N))
i = BN_num_bits(m);
j = recp->num_bits << 1;
if (j > i) {
i = j;
}
// Nr := round(2^i / N)
if (i != recp->shift) {
recp->shift =
BN_reciprocal(&(recp->Nr), &(recp->N), i,
ctx); // BN_reciprocal returns i, or -1 for an error
}
if (recp->shift == -1) {
goto err;
}
// d := |round(round(m / 2^BN_num_bits(N)) * recp->Nr / 2^(i -
// BN_num_bits(N)))|
// = |round(round(m / 2^BN_num_bits(N)) * round(2^i / N) / 2^(i -
// BN_num_bits(N)))|
// <= |(m / 2^BN_num_bits(N)) * (2^i / N) * (2^BN_num_bits(N) / 2^i)|
// = |m/N|
if (!BN_rshift(a, m, recp->num_bits)) {
goto err;
}
if (!BN_mul(b, a, &(recp->Nr), ctx)) {
goto err;
}
if (!BN_rshift(d, b, i - recp->num_bits)) {
goto err;
}
d->neg = 0;
if (!BN_mul(b, &(recp->N), d, ctx)) {
goto err;
}
if (!BN_usub(r, m, b)) {
goto err;
}
r->neg = 0;
j = 0;
while (BN_ucmp(r, &(recp->N)) >= 0) {
if (j++ > 2) {
OPENSSL_PUT_ERROR(BN, BN_R_BAD_RECIPROCAL);
goto err;
}
if (!BN_usub(r, r, &(recp->N))) {
goto err;
}
if (!BN_add_word(d, 1)) {
goto err;
}
}
r->neg = BN_is_zero(r) ? 0 : m->neg;
d->neg = m->neg ^ recp->N.neg;
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
}
static int BN_mod_mul_reciprocal(BIGNUM *r, const BIGNUM *x, const BIGNUM *y,
BN_RECP_CTX *recp, BN_CTX *ctx) {
int ret = 0;
BIGNUM *a;
const BIGNUM *ca;
BN_CTX_start(ctx);
a = BN_CTX_get(ctx);
if (a == NULL) {
goto err;
}
if (y != NULL) {
if (x == y) {
if (!BN_sqr(a, x, ctx)) {
goto err;
}
} else {
if (!BN_mul(a, x, y, ctx)) {
goto err;
}
}
ca = a;
} else {
ca = x; // Just do the mod
}
ret = BN_div_recp(NULL, r, ca, recp, ctx);
err:
BN_CTX_end(ctx);
return ret;
}
// BN_window_bits_for_exponent_size returns sliding window size for mod_exp with
// a |b| bit exponent.
//
// For window size 'w' (w >= 2) and a random 'b' bits exponent, the number of
// multiplications is a constant plus on average
//
// 2^(w-1) + (b-w)/(w+1);
//
// here 2^(w-1) is for precomputing the table (we actually need entries only
// for windows that have the lowest bit set), and (b-w)/(w+1) is an
// approximation for the expected number of w-bit windows, not counting the
// first one.
//
// Thus we should use
//
// w >= 6 if b > 671
// w = 5 if 671 > b > 239
// w = 4 if 239 > b > 79
// w = 3 if 79 > b > 23
// w <= 2 if 23 > b
//
// (with draws in between). Very small exponents are often selected
// with low Hamming weight, so we use w = 1 for b <= 23.
static int BN_window_bits_for_exponent_size(int b) {
if (b > 671) {
return 6;
}
if (b > 239) {
return 5;
}
if (b > 79) {
return 4;
}
if (b > 23) {
return 3;
}
return 1;
}
// TABLE_SIZE is the maximum precomputation table size for *variable* sliding
// windows. This must be 2^(max_window - 1), where max_window is the largest
// value returned from |BN_window_bits_for_exponent_size|.
#define TABLE_SIZE 32
// TABLE_BITS_SMALL is the smallest value returned from
// |BN_window_bits_for_exponent_size| when |b| is at most |BN_BITS2| *
// |BN_SMALL_MAX_WORDS| words.
#define TABLE_BITS_SMALL 5
// TABLE_SIZE_SMALL is the same as |TABLE_SIZE|, but when |b| is at most
// |BN_BITS2| * |BN_SMALL_MAX_WORDS|.
#define TABLE_SIZE_SMALL (1 << (TABLE_BITS_SMALL - 1))
static int mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
const BIGNUM *m, BN_CTX *ctx) {
int i, j, bits, ret = 0, wstart, window;
int start = 1;
BIGNUM *aa;
// Table of variables obtained from 'ctx'
BIGNUM *val[TABLE_SIZE];
BN_RECP_CTX recp;
bits = BN_num_bits(p);
if (bits == 0) {
// x**0 mod 1 is still zero.
if (BN_is_one(m)) {
BN_zero(r);
return 1;
}
return BN_one(r);
}
BN_CTX_start(ctx);
aa = BN_CTX_get(ctx);
val[0] = BN_CTX_get(ctx);
if (!aa || !val[0]) {
goto err;
}
BN_RECP_CTX_init(&recp);
if (m->neg) {
// ignore sign of 'm'
if (!BN_copy(aa, m)) {
goto err;
}
aa->neg = 0;
if (BN_RECP_CTX_set(&recp, aa, ctx) <= 0) {
goto err;
}
} else {
if (BN_RECP_CTX_set(&recp, m, ctx) <= 0) {
goto err;
}
}
if (!BN_nnmod(val[0], a, m, ctx)) {
goto err; // 1
}
if (BN_is_zero(val[0])) {
BN_zero(r);
ret = 1;
goto err;
}
window = BN_window_bits_for_exponent_size(bits);
if (window > 1) {
if (!BN_mod_mul_reciprocal(aa, val[0], val[0], &recp, ctx)) {
goto err; // 2
}
j = 1 << (window - 1);
for (i = 1; i < j; i++) {
if (((val[i] = BN_CTX_get(ctx)) == NULL) ||
!BN_mod_mul_reciprocal(val[i], val[i - 1], aa, &recp, ctx)) {
goto err;
}
}
}
start = 1; // This is used to avoid multiplication etc
// when there is only the value '1' in the
// buffer.
wstart = bits - 1; // The top bit of the window
if (!BN_one(r)) {
goto err;
}
for (;;) {
int wvalue; // The 'value' of the window
int wend; // The bottom bit of the window
if (!BN_is_bit_set(p, wstart)) {
if (!start) {
if (!BN_mod_mul_reciprocal(r, r, r, &recp, ctx)) {
goto err;
}
}
if (wstart == 0) {
break;
}
wstart--;
continue;
}
// We now have wstart on a 'set' bit, we now need to work out
// how bit a window to do. To do this we need to scan
// forward until the last set bit before the end of the
// window
wvalue = 1;
wend = 0;
for (i = 1; i < window; i++) {
if (wstart - i < 0) {
break;
}
if (BN_is_bit_set(p, wstart - i)) {
wvalue <<= (i - wend);
wvalue |= 1;
wend = i;
}
}
// wend is the size of the current window
j = wend + 1;
// add the 'bytes above'
if (!start) {
for (i = 0; i < j; i++) {
if (!BN_mod_mul_reciprocal(r, r, r, &recp, ctx)) {
goto err;
}
}
}
// wvalue will be an odd number < 2^window
if (!BN_mod_mul_reciprocal(r, r, val[wvalue >> 1], &recp, ctx)) {
goto err;
}
// move the 'window' down further
wstart -= wend + 1;
start = 0;
if (wstart < 0) {
break;
}
}
ret = 1;
err:
BN_CTX_end(ctx);
BN_RECP_CTX_free(&recp);
return ret;
}
int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
BN_CTX *ctx) {
if (BN_is_odd(m)) {
return BN_mod_exp_mont(r, a, p, m, ctx, NULL);
}
return mod_exp_recp(r, a, p, m, ctx);
}
int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
const BIGNUM *m, BN_CTX *ctx, const BN_MONT_CTX *mont) {
if (!BN_is_odd(m)) {
OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
return 0;
}
int bits = BN_num_bits(p);
if (bits == 0) {
// x**0 mod 1 is still zero.
if (BN_is_one(m)) {
BN_zero(rr);
return 1;
}
return BN_one(rr);
}
int ret = 0;
BIGNUM *val[TABLE_SIZE];
BN_MONT_CTX *new_mont = NULL;
BN_CTX_start(ctx);
BIGNUM *d = BN_CTX_get(ctx);
BIGNUM *r = BN_CTX_get(ctx);
val[0] = BN_CTX_get(ctx);
if (!d || !r || !val[0]) {
goto err;
}
// Allocate a montgomery context if it was not supplied by the caller.
if (mont == NULL) {
new_mont = BN_MONT_CTX_new_for_modulus(m, ctx);
if (new_mont == NULL) {
goto err;
}
mont = new_mont;
}
const BIGNUM *aa;
if (a->neg || BN_ucmp(a, m) >= 0) {
if (!BN_nnmod(val[0], a, m, ctx)) {
goto err;
}
aa = val[0];
} else {
aa = a;
}
if (BN_is_zero(aa)) {
BN_zero(rr);
ret = 1;
goto err;
}
// We exponentiate by looking at sliding windows of the exponent and
// precomputing powers of |aa|. Windows may be shifted so they always end on a
// set bit, so only precompute odd powers. We compute val[i] = aa^(2*i + 1)
// for i = 0 to 2^(window-1), all in Montgomery form.
int window = BN_window_bits_for_exponent_size(bits);
if (!BN_to_montgomery(val[0], aa, mont, ctx)) {
goto err;
}
if (window > 1) {
if (!BN_mod_mul_montgomery(d, val[0], val[0], mont, ctx)) {
goto err;
}
for (int i = 1; i < 1 << (window - 1); i++) {
val[i] = BN_CTX_get(ctx);
if (val[i] == NULL ||
!BN_mod_mul_montgomery(val[i], val[i - 1], d, mont, ctx)) {
goto err;
}
}
}
if (!bn_one_to_montgomery(r, mont, ctx)) {
goto err;
}
int r_is_one = 1;
int wstart = bits - 1; // The top bit of the window.
for (;;) {
if (!BN_is_bit_set(p, wstart)) {
if (!r_is_one && !BN_mod_mul_montgomery(r, r, r, mont, ctx)) {
goto err;
}
if (wstart == 0) {
break;
}
wstart--;
continue;
}
// We now have wstart on a set bit. Find the largest window we can use.
int wvalue = 1;
int wsize = 0;
for (int i = 1; i < window && i <= wstart; i++) {
if (BN_is_bit_set(p, wstart - i)) {
wvalue <<= (i - wsize);
wvalue |= 1;
wsize = i;
}
}
// Shift |r| to the end of the window.
if (!r_is_one) {
for (int i = 0; i < wsize + 1; i++) {
if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) {
goto err;
}
}
}
assert(wvalue & 1);
assert(wvalue < (1 << window));
if (!BN_mod_mul_montgomery(r, r, val[wvalue >> 1], mont, ctx)) {
goto err;
}
r_is_one = 0;
if (wstart == wsize) {
break;
}
wstart -= wsize + 1;
}
if (!BN_from_montgomery(rr, r, mont, ctx)) {
goto err;
}
ret = 1;
err:
BN_MONT_CTX_free(new_mont);
BN_CTX_end(ctx);
return ret;
}
int bn_mod_exp_mont_small(BN_ULONG *r, size_t num_r, const BN_ULONG *a,
size_t num_a, const BN_ULONG *p, size_t num_p,
const BN_MONT_CTX *mont) {
size_t num_n = mont->N.top;
if (num_n != num_a || num_n != num_r || num_n > BN_SMALL_MAX_WORDS) {
OPENSSL_PUT_ERROR(BN, ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
return 0;
}
if (!BN_is_odd(&mont->N)) {
OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
return 0;
}
unsigned bits = 0;
if (num_p != 0) {
bits = BN_num_bits_word(p[num_p - 1]) + (num_p - 1) * BN_BITS2;
}
if (bits == 0) {
OPENSSL_memset(r, 0, num_r * sizeof(BN_ULONG));
if (!BN_is_one(&mont->N)) {
r[0] = 1;
}
return 1;
}
// We exponentiate by looking at sliding windows of the exponent and
// precomputing powers of |a|. Windows may be shifted so they always end on a
// set bit, so only precompute odd powers. We compute val[i] = a^(2*i + 1) for
// i = 0 to 2^(window-1), all in Montgomery form.
unsigned window = BN_window_bits_for_exponent_size(bits);
if (window > TABLE_BITS_SMALL) {
window = TABLE_BITS_SMALL; // Tolerate excessively large |p|.
}
int ret = 0;
BN_ULONG val[TABLE_SIZE_SMALL][BN_SMALL_MAX_WORDS];
OPENSSL_memcpy(val[0], a, num_n * sizeof(BN_ULONG));
if (window > 1) {
BN_ULONG d[BN_SMALL_MAX_WORDS];
if (!bn_mod_mul_montgomery_small(d, num_n, val[0], num_n, val[0], num_n,
mont)) {
goto err;
}
for (unsigned i = 1; i < 1u << (window - 1); i++) {
if (!bn_mod_mul_montgomery_small(val[i], num_n, val[i - 1], num_n, d,
num_n, mont)) {
goto err;
}
}
}
if (!bn_one_to_montgomery_small(r, num_r, mont)) {
goto err;
}
int r_is_one = 1;
unsigned wstart = bits - 1; // The top bit of the window.
for (;;) {
if (!bn_is_bit_set_words(p, num_p, wstart)) {
if (!r_is_one &&
!bn_mod_mul_montgomery_small(r, num_r, r, num_r, r, num_r, mont)) {
goto err;
}
if (wstart == 0) {
break;
}
wstart--;
continue;
}
// We now have wstart on a set bit. Find the largest window we can use.
unsigned wvalue = 1;
unsigned wsize = 0;
for (unsigned i = 1; i < window && i <= wstart; i++) {
if (bn_is_bit_set_words(p, num_p, wstart - i)) {
wvalue <<= (i - wsize);
wvalue |= 1;
wsize = i;
}
}
// Shift |r| to the end of the window.
if (!r_is_one) {
for (unsigned i = 0; i < wsize + 1; i++) {
if (!bn_mod_mul_montgomery_small(r, num_r, r, num_r, r, num_r, mont)) {
goto err;
}
}
}
assert(wvalue & 1);
assert(wvalue < (1u << window));
if (!bn_mod_mul_montgomery_small(r, num_r, r, num_r, val[wvalue >> 1],
num_n, mont)) {
goto err;
}
r_is_one = 0;
if (wstart == wsize) {
break;
}
wstart -= wsize + 1;
}
ret = 1;
err:
OPENSSL_cleanse(val, sizeof(val));
return ret;
}
int bn_mod_inverse_prime_mont_small(BN_ULONG *r, size_t num_r,
const BN_ULONG *a, size_t num_a,
const BN_MONT_CTX *mont) {
const BN_ULONG *p = mont->N.d;
size_t num_p = mont->N.top;
if (num_p > BN_SMALL_MAX_WORDS || num_p == 0) {
OPENSSL_PUT_ERROR(BN, ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
return 0;
}
// Per Fermat's Little Theorem, a^-1 = a^(p-2) (mod p) for p prime.
BN_ULONG p_minus_two[BN_SMALL_MAX_WORDS];
OPENSSL_memcpy(p_minus_two, p, num_p * sizeof(BN_ULONG));
if (p_minus_two[0] >= 2) {
p_minus_two[0] -= 2;
} else {
p_minus_two[0] -= 2;
for (size_t i = 1; i < num_p; i++) {
if (p_minus_two[i]-- != 0) {
break;
}
}
}
return bn_mod_exp_mont_small(r, num_r, a, num_a, p_minus_two, num_p, mont);
}
// |BN_mod_exp_mont_consttime| stores the precomputed powers in a specific
// layout so that accessing any of these table values shows the same access
// pattern as far as cache lines are concerned. The following functions are
// used to transfer a BIGNUM from/to that table.
static void copy_to_prebuf(const BIGNUM *b, int top, unsigned char *buf,
int idx, int window) {
int i, j;
const int width = 1 << window;
BN_ULONG *table = (BN_ULONG *) buf;
if (top > b->top) {
top = b->top; // this works because 'buf' is explicitly zeroed
}
for (i = 0, j = idx; i < top; i++, j += width) {
table[j] = b->d[i];
}
}
static int copy_from_prebuf(BIGNUM *b, int top, unsigned char *buf, int idx,
int window) {
int i, j;
const int width = 1 << window;
volatile BN_ULONG *table = (volatile BN_ULONG *)buf;
if (!bn_wexpand(b, top)) {
return 0;
}
if (window <= 3) {
for (i = 0; i < top; i++, table += width) {
BN_ULONG acc = 0;
for (j = 0; j < width; j++) {
acc |= table[j] & ((BN_ULONG)0 - (constant_time_eq_int(j, idx) & 1));
}
b->d[i] = acc;
}
} else {
int xstride = 1 << (window - 2);
BN_ULONG y0, y1, y2, y3;
i = idx >> (window - 2); // equivalent of idx / xstride
idx &= xstride - 1; // equivalent of idx % xstride
y0 = (BN_ULONG)0 - (constant_time_eq_int(i, 0) & 1);
y1 = (BN_ULONG)0 - (constant_time_eq_int(i, 1) & 1);
y2 = (BN_ULONG)0 - (constant_time_eq_int(i, 2) & 1);
y3 = (BN_ULONG)0 - (constant_time_eq_int(i, 3) & 1);
for (i = 0; i < top; i++, table += width) {
BN_ULONG acc = 0;
for (j = 0; j < xstride; j++) {
acc |= ((table[j + 0 * xstride] & y0) | (table[j + 1 * xstride] & y1) |
(table[j + 2 * xstride] & y2) | (table[j + 3 * xstride] & y3)) &
((BN_ULONG)0 - (constant_time_eq_int(j, idx) & 1));
}
b->d[i] = acc;
}
}
b->top = top;
bn_correct_top(b);
return 1;
}
// BN_mod_exp_mont_conttime is based on the assumption that the L1 data cache
// line width of the target processor is at least the following value.
#define MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH (64)
#define MOD_EXP_CTIME_MIN_CACHE_LINE_MASK \
(MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH - 1)
// Window sizes optimized for fixed window size modular exponentiation
// algorithm (BN_mod_exp_mont_consttime).
//
// To achieve the security goals of BN_mode_exp_mont_consttime, the maximum
// size of the window must not exceed
// log_2(MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH).
//
// Window size thresholds are defined for cache line sizes of 32 and 64, cache
// line sizes where log_2(32)=5 and log_2(64)=6 respectively. A window size of
// 7 should only be used on processors that have a 128 byte or greater cache
// line size.
#if MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH == 64
#define BN_window_bits_for_ctime_exponent_size(b) \
((b) > 937 ? 6 : (b) > 306 ? 5 : (b) > 89 ? 4 : (b) > 22 ? 3 : 1)
#define BN_MAX_WINDOW_BITS_FOR_CTIME_EXPONENT_SIZE (6)
#elif MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH == 32
#define BN_window_bits_for_ctime_exponent_size(b) \
((b) > 306 ? 5 : (b) > 89 ? 4 : (b) > 22 ? 3 : 1)
#define BN_MAX_WINDOW_BITS_FOR_CTIME_EXPONENT_SIZE (5)
#endif
// Given a pointer value, compute the next address that is a cache line
// multiple.
#define MOD_EXP_CTIME_ALIGN(x_) \
((unsigned char *)(x_) + \
(MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH - \
(((size_t)(x_)) & (MOD_EXP_CTIME_MIN_CACHE_LINE_MASK))))
// This variant of BN_mod_exp_mont() uses fixed windows and the special
// precomputation memory layout to limit data-dependency to a minimum
// to protect secret exponents (cf. the hyper-threading timing attacks
// pointed out by Colin Percival,
// http://www.daemonology.net/hyperthreading-considered-harmful/)
int BN_mod_exp_mont_consttime(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
const BIGNUM *m, BN_CTX *ctx,
const BN_MONT_CTX *mont) {
int i, ret = 0, window, wvalue;
int top;
BN_MONT_CTX *new_mont = NULL;
int numPowers;
unsigned char *powerbufFree = NULL;
int powerbufLen = 0;
unsigned char *powerbuf = NULL;
BIGNUM tmp, am;
BIGNUM *new_a = NULL;
if (!BN_is_odd(m)) {
OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
return 0;
}
top = m->top;
// Use all bits stored in |p|, rather than |BN_num_bits|, so we do not leak
// whether the top bits are zero.
int max_bits = p->top * BN_BITS2;
int bits = max_bits;
if (bits == 0) {
// x**0 mod 1 is still zero.
if (BN_is_one(m)) {
BN_zero(rr);
return 1;
}
return BN_one(rr);
}
// Allocate a montgomery context if it was not supplied by the caller.
if (mont == NULL) {
new_mont = BN_MONT_CTX_new_for_modulus(m, ctx);
if (new_mont == NULL) {
goto err;
}
mont = new_mont;
}
if (a->neg || BN_ucmp(a, m) >= 0) {
new_a = BN_new();
if (new_a == NULL ||
!BN_nnmod(new_a, a, m, ctx)) {
goto err;
}
a = new_a;
}
#ifdef RSAZ_ENABLED
// If the size of the operands allow it, perform the optimized
// RSAZ exponentiation. For further information see
// crypto/bn/rsaz_exp.c and accompanying assembly modules.
if ((16 == a->top) && (16 == p->top) && (BN_num_bits(m) == 1024) &&
rsaz_avx2_eligible()) {
if (!bn_wexpand(rr, 16)) {
goto err;
}
RSAZ_1024_mod_exp_avx2(rr->d, a->d, p->d, m->d, mont->RR.d, mont->n0[0]);
rr->top = 16;
rr->neg = 0;
bn_correct_top(rr);
ret = 1;
goto err;
}
#endif
// Get the window size to use with size of p.
window = BN_window_bits_for_ctime_exponent_size(bits);
#if defined(OPENSSL_BN_ASM_MONT5)
if (window >= 5) {
window = 5; // ~5% improvement for RSA2048 sign, and even for RSA4096
// reserve space for mont->N.d[] copy
powerbufLen += top * sizeof(mont->N.d[0]);
}
#endif
// Allocate a buffer large enough to hold all of the pre-computed
// powers of am, am itself and tmp.
numPowers = 1 << window;
powerbufLen +=
sizeof(m->d[0]) *
(top * numPowers + ((2 * top) > numPowers ? (2 * top) : numPowers));
#ifdef alloca
if (powerbufLen < 3072) {
powerbufFree = alloca(powerbufLen + MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH);
} else
#endif
{
if ((powerbufFree = OPENSSL_malloc(
powerbufLen + MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH)) == NULL) {
goto err;
}
}
powerbuf = MOD_EXP_CTIME_ALIGN(powerbufFree);
OPENSSL_memset(powerbuf, 0, powerbufLen);
#ifdef alloca
if (powerbufLen < 3072) {
powerbufFree = NULL;
}
#endif
// lay down tmp and am right after powers table
tmp.d = (BN_ULONG *)(powerbuf + sizeof(m->d[0]) * top * numPowers);
am.d = tmp.d + top;
tmp.top = am.top = 0;
tmp.dmax = am.dmax = top;
tmp.neg = am.neg = 0;
tmp.flags = am.flags = BN_FLG_STATIC_DATA;
if (!bn_one_to_montgomery(&tmp, mont, ctx)) {
goto err;
}
// prepare a^1 in Montgomery domain
assert(!a->neg);
assert(BN_ucmp(a, m) < 0);
if (!BN_to_montgomery(&am, a, mont, ctx)) {
goto err;
}
#if defined(OPENSSL_BN_ASM_MONT5)
// This optimization uses ideas from http://eprint.iacr.org/2011/239,
// specifically optimization of cache-timing attack countermeasures
// and pre-computation optimization.
// Dedicated window==4 case improves 512-bit RSA sign by ~15%, but as
// 512-bit RSA is hardly relevant, we omit it to spare size...
if (window == 5 && top > 1) {
const BN_ULONG *n0 = mont->n0;
BN_ULONG *np;
// BN_to_montgomery can contaminate words above .top
// [in BN_DEBUG[_DEBUG] build]...
for (i = am.top; i < top; i++) {
am.d[i] = 0;
}
for (i = tmp.top; i < top; i++) {
tmp.d[i] = 0;
}
// copy mont->N.d[] to improve cache locality
for (np = am.d + top, i = 0; i < top; i++) {
np[i] = mont->N.d[i];
}
bn_scatter5(tmp.d, top, powerbuf, 0);
bn_scatter5(am.d, am.top, powerbuf, 1);
bn_mul_mont(tmp.d, am.d, am.d, np, n0, top);
bn_scatter5(tmp.d, top, powerbuf, 2);
// same as above, but uses squaring for 1/2 of operations
for (i = 4; i < 32; i *= 2) {
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_scatter5(tmp.d, top, powerbuf, i);
}
for (i = 3; i < 8; i += 2) {
int j;
bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np, n0, top, i - 1);
bn_scatter5(tmp.d, top, powerbuf, i);
for (j = 2 * i; j < 32; j *= 2) {
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_scatter5(tmp.d, top, powerbuf, j);
}
}
for (; i < 16; i += 2) {
bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np, n0, top, i - 1);
bn_scatter5(tmp.d, top, powerbuf, i);
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_scatter5(tmp.d, top, powerbuf, 2 * i);
}
for (; i < 32; i += 2) {
bn_mul_mont_gather5(tmp.d, am.d, powerbuf, np, n0, top, i - 1);
bn_scatter5(tmp.d, top, powerbuf, i);
}
bits--;
for (wvalue = 0, i = bits % 5; i >= 0; i--, bits--) {
wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
}
bn_gather5(tmp.d, top, powerbuf, wvalue);
// At this point |bits| is 4 mod 5 and at least -1. (|bits| is the first bit
// that has not been read yet.)
assert(bits >= -1 && (bits == -1 || bits % 5 == 4));
// Scan the exponent one window at a time starting from the most
// significant bits.
if (top & 7) {
while (bits >= 0) {
for (wvalue = 0, i = 0; i < 5; i++, bits--) {
wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
}
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_mul_mont(tmp.d, tmp.d, tmp.d, np, n0, top);
bn_mul_mont_gather5(tmp.d, tmp.d, powerbuf, np, n0, top, wvalue);
}
} else {
const uint8_t *p_bytes = (const uint8_t *)p->d;
assert(bits < max_bits);
// |p = 0| has been handled as a special case, so |max_bits| is at least
// one word.
assert(max_bits >= 64);
// If the first bit to be read lands in the last byte, unroll the first
// iteration to avoid reading past the bounds of |p->d|. (After the first
// iteration, we are guaranteed to be past the last byte.) Note |bits|
// here is the top bit, inclusive.
if (bits - 4 >= max_bits - 8) {
// Read five bits from |bits-4| through |bits|, inclusive.
wvalue = p_bytes[p->top * BN_BYTES - 1];
wvalue >>= (bits - 4) & 7;
wvalue &= 0x1f;
bits -= 5;
bn_power5(tmp.d, tmp.d, powerbuf, np, n0, top, wvalue);
}
while (bits >= 0) {
// Read five bits from |bits-4| through |bits|, inclusive.
int first_bit = bits - 4;
uint16_t val;
OPENSSL_memcpy(&val, p_bytes + (first_bit >> 3), sizeof(val));
val >>= first_bit & 7;
val &= 0x1f;
bits -= 5;
bn_power5(tmp.d, tmp.d, powerbuf, np, n0, top, val);
}
}
ret = bn_from_montgomery(tmp.d, tmp.d, NULL, np, n0, top);
tmp.top = top;
bn_correct_top(&tmp);
if (ret) {
if (!BN_copy(rr, &tmp)) {
ret = 0;
}
goto err; // non-zero ret means it's not error
}
} else
#endif
{
copy_to_prebuf(&tmp, top, powerbuf, 0, window);
copy_to_prebuf(&am, top, powerbuf, 1, window);
// If the window size is greater than 1, then calculate
// val[i=2..2^winsize-1]. Powers are computed as a*a^(i-1)
// (even powers could instead be computed as (a^(i/2))^2
// to use the slight performance advantage of sqr over mul).
if (window > 1) {
if (!BN_mod_mul_montgomery(&tmp, &am, &am, mont, ctx)) {
goto err;
}
copy_to_prebuf(&tmp, top, powerbuf, 2, window);
for (i = 3; i < numPowers; i++) {
// Calculate a^i = a^(i-1) * a
if (!BN_mod_mul_montgomery(&tmp, &am, &tmp, mont, ctx)) {
goto err;
}
copy_to_prebuf(&tmp, top, powerbuf, i, window);
}
}
bits--;
for (wvalue = 0, i = bits % window; i >= 0; i--, bits--) {
wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
}
if (!copy_from_prebuf(&tmp, top, powerbuf, wvalue, window)) {
goto err;
}
// Scan the exponent one window at a time starting from the most
// significant bits.
while (bits >= 0) {
wvalue = 0; // The 'value' of the window
// Scan the window, squaring the result as we go
for (i = 0; i < window; i++, bits--) {
if (!BN_mod_mul_montgomery(&tmp, &tmp, &tmp, mont, ctx)) {
goto err;
}
wvalue = (wvalue << 1) + BN_is_bit_set(p, bits);
}
// Fetch the appropriate pre-computed value from the pre-buf
if (!copy_from_prebuf(&am, top, powerbuf, wvalue, window)) {
goto err;
}
// Multiply the result into the intermediate result
if (!BN_mod_mul_montgomery(&tmp, &tmp, &am, mont, ctx)) {
goto err;
}
}
}
// Convert the final result from montgomery to standard format
if (!BN_from_montgomery(rr, &tmp, mont, ctx)) {
goto err;
}
ret = 1;
err:
BN_MONT_CTX_free(new_mont);
BN_clear_free(new_a);
OPENSSL_free(powerbufFree);
return (ret);
}
int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p,
const BIGNUM *m, BN_CTX *ctx,
const BN_MONT_CTX *mont) {
BIGNUM a_bignum;
BN_init(&a_bignum);
int ret = 0;
if (!BN_set_word(&a_bignum, a)) {
OPENSSL_PUT_ERROR(BN, ERR_R_INTERNAL_ERROR);
goto err;
}
ret = BN_mod_exp_mont(rr, &a_bignum, p, m, ctx, mont);
err:
BN_free(&a_bignum);
return ret;
}
#define TABLE_SIZE 32
int BN_mod_exp2_mont(BIGNUM *rr, const BIGNUM *a1, const BIGNUM *p1,
const BIGNUM *a2, const BIGNUM *p2, const BIGNUM *m,
BN_CTX *ctx, const BN_MONT_CTX *mont) {
BIGNUM tmp;
BN_init(&tmp);
int ret = 0;
BN_MONT_CTX *new_mont = NULL;
// Allocate a montgomery context if it was not supplied by the caller.
if (mont == NULL) {
new_mont = BN_MONT_CTX_new_for_modulus(m, ctx);
if (new_mont == NULL) {
goto err;
}
mont = new_mont;
}
// BN_mod_mul_montgomery removes one Montgomery factor, so passing one
// Montgomery-encoded and one non-Montgomery-encoded value gives a
// non-Montgomery-encoded result.
if (!BN_mod_exp_mont(rr, a1, p1, m, ctx, mont) ||
!BN_mod_exp_mont(&tmp, a2, p2, m, ctx, mont) ||
!BN_to_montgomery(rr, rr, mont, ctx) ||
!BN_mod_mul_montgomery(rr, rr, &tmp, mont, ctx)) {
goto err;
}
ret = 1;
err:
BN_MONT_CTX_free(new_mont);
BN_free(&tmp);
return ret;
}