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Calculate inverse in |BN_MONT_CTX_set| in constant time w.r.t. modulus.

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Simplify the calculation of the Montgomery constants in
|BN_MONT_CTX_set|, making the inversion constant-time. It should also
be faster by avoiding any use of the |BIGNUM| API in favor of using
only 64-bit arithmetic.

Now it's obvious how it works. /s

Change-Id: I59a1e1c3631f426fbeabd0c752e0de44bcb5fd75
Reviewed-on: https://boringssl-review.googlesource.com/9031
Reviewed-by: Adam Langley <agl@google.com>
Commit-Queue: Adam Langley <agl@google.com>
CQ-Verified: CQ bot account: commit-bot@chromium.org <commit-bot@chromium.org>
This commit is contained in:
Brian Smith 2016-07-29 16:19:46 -10:00 committed by CQ bot account: commit-bot@chromium.org
parent 0375127606
commit 7fcbfdbdf3
5 changed files with 208 additions and 112 deletions
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1
crypto/bn/CMakeLists.txt
View File

@ -57,6 +57,7 @@ add_library(
gcd.c
kronecker.c
montgomery.c
montgomery_inv.c
mul.c
prime.c
random.c

10
crypto/bn/internal.h
View File

@ -156,6 +156,7 @@ BIGNUM *bn_expand(BIGNUM *bn, size_t bits);
#define BN_MASK2l (0xffffffffUL)
#define BN_MASK2h (0xffffffff00000000UL)
#define BN_MASK2h1 (0xffffffff80000000UL)
#define BN_MONT_CTX_N0_LIMBS 1
#define BN_TBIT (0x8000000000000000UL)
#define BN_DEC_CONV (10000000000000000000UL)
#define BN_DEC_NUM 19
@ -171,6 +172,12 @@ BIGNUM *bn_expand(BIGNUM *bn, size_t bits);
#define BN_MASK2l (0xffffUL)
#define BN_MASK2h1 (0xffff8000UL)
#define BN_MASK2h (0xffff0000UL)
/* On some 32-bit platforms, Montgomery multiplication is done using 64-bit
* arithmetic with SIMD instructions. On such platforms, |BN_MONT_CTX::n0|
* needs to be two words long. Only certain 32-bit platforms actually make use
* of n0[1] and shorter R value would suffice for the others. However,
* currently only the assembly files know which is which. */
#define BN_MONT_CTX_N0_LIMBS 2
#define BN_TBIT (0x80000000UL)
#define BN_DEC_CONV (1000000000UL)
#define BN_DEC_NUM 9
@ -192,7 +199,6 @@ BIGNUM *bn_expand(BIGNUM *bn, size_t bits);
#define Hw(t) (((BN_ULONG)((t)>>BN_BITS2))&BN_MASK2)
#endif
/* bn_set_words sets |bn| to the value encoded in the |num| words in |words|,
* least significant word first. */
int bn_set_words(BIGNUM *bn, const BN_ULONG *words, size_t num);
@ -221,6 +227,8 @@ int bn_cmp_part_words(const BN_ULONG *a, const BN_ULONG *b, int cl, int dl);
int bn_mul_mont(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *bp,
const BN_ULONG *np, const BN_ULONG *n0, int num);
uint64_t bn_mont_n0(const BIGNUM *n);
#if defined(OPENSSL_X86_64) && defined(_MSC_VER)
#define BN_UMULT_LOHI(low, high, a, b) ((low) = _umul128((a), (b), &(high)))
#endif

148
crypto/bn/montgomery.c
View File

@ -162,131 +162,61 @@ BN_MONT_CTX *BN_MONT_CTX_copy(BN_MONT_CTX *to, const BN_MONT_CTX *from) {
return to;
}
int BN_MONT_CTX_set(BN_MONT_CTX *mont, const BIGNUM *mod, BN_CTX *ctx) {
int ret = 0;
BIGNUM *Ri, *R;
BIGNUM tmod;
BN_ULONG buf[2];
OPENSSL_COMPILE_ASSERT(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2,
BN_MONT_CTX_N0_LIMBS_VALUE_INVALID);
OPENSSL_COMPILE_ASSERT(sizeof(BN_ULONG) * BN_MONT_CTX_N0_LIMBS ==
sizeof(uint64_t), BN_MONT_CTX_set_64_bit_mismatch);
int BN_MONT_CTX_set(BN_MONT_CTX *mont, const BIGNUM *mod, BN_CTX *ctx) {
if (BN_is_zero(mod)) {
OPENSSL_PUT_ERROR(BN, BN_R_DIV_BY_ZERO);
return 0;
}
BN_CTX_start(ctx);
Ri = BN_CTX_get(ctx);
if (Ri == NULL) {
goto err;
if (!BN_is_odd(mod)) {
OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
return 0;
}
R = &mont->RR; /* grab RR as a temp */
if (BN_is_negative(mod)) {
OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
return 0;
}
/* Save the modulus. */
if (!BN_copy(&mont->N, mod)) {
goto err; /* Set N */
OPENSSL_PUT_ERROR(BN, ERR_R_INTERNAL_ERROR);
return 0;
}
mont->N.neg = 0;
BN_init(&tmod);
tmod.d = buf;
tmod.dmax = 2;
tmod.neg = 0;
#if defined(OPENSSL_BN_ASM_MONT) && (BN_BITS2 <= 32)
/* Only certain BN_BITS2<=32 platforms actually make use of
* n0[1], and we could use the #else case (with a shorter R
* value) for the others. However, currently only the assembler
* files do know which is which. */
BN_zero(R);
if (!BN_set_bit(R, 2 * BN_BITS2)) {
goto err;
if (BN_get_flags(mod, BN_FLG_CONSTTIME)) {
BN_set_flags(&mont->N, BN_FLG_CONSTTIME);
}
tmod.top = 0;
if ((buf[0] = mod->d[0])) {
tmod.top = 1;
}
if ((buf[1] = mod->top > 1 ? mod->d[1] : 0)) {
tmod.top = 2;
}
if (BN_mod_inverse(Ri, R, &tmod, ctx) == NULL) {
goto err;
}
if (!BN_lshift(Ri, Ri, 2 * BN_BITS2)) {
goto err; /* R*Ri */
}
if (!BN_is_zero(Ri)) {
if (!BN_sub_word(Ri, 1)) {
goto err;
}
} else {
/* if N mod word size == 1 */
if (bn_expand(Ri, (int)sizeof(BN_ULONG) * 2) == NULL) {
goto err;
}
/* Ri-- (mod double word size) */
Ri->neg = 0;
Ri->d[0] = BN_MASK2;
Ri->d[1] = BN_MASK2;
Ri->top = 2;
}
if (!BN_div(Ri, NULL, Ri, &tmod, ctx)) {
goto err;
}
/* Ni = (R*Ri-1)/N,
* keep only couple of least significant words: */
mont->n0[0] = (Ri->top > 0) ? Ri->d[0] : 0;
mont->n0[1] = (Ri->top > 1) ? Ri->d[1] : 0;
/* Find n0 such that n0 * N == -1 (mod r).
*
* Only certain BN_BITS2<=32 platforms actually make use of n0[1]. For the
* others, we could use a shorter R value and use faster |BN_ULONG|-based
* math instead of |uint64_t|-based math, which would be double-precision.
* However, currently only the assembler files know which is which. */
uint64_t n0 = bn_mont_n0(mod);
mont->n0[0] = (BN_ULONG)n0;
#if BN_MONT_CTX_N0_LIMBS == 2
mont->n0[1] = (BN_ULONG)(n0 >> BN_BITS2);
#else
BN_zero(R);
if (!BN_set_bit(R, BN_BITS2)) {
goto err; /* R */
}
buf[0] = mod->d[0]; /* tmod = N mod word size */
buf[1] = 0;
tmod.top = buf[0] != 0 ? 1 : 0;
/* Ri = R^-1 mod N*/
if (BN_mod_inverse(Ri, R, &tmod, ctx) == NULL) {
goto err;
}
if (!BN_lshift(Ri, Ri, BN_BITS2)) {
goto err; /* R*Ri */
}
if (!BN_is_zero(Ri)) {
if (!BN_sub_word(Ri, 1)) {
goto err;
}
} else {
/* if N mod word size == 1 */
if (!BN_set_word(Ri, BN_MASK2)) {
goto err; /* Ri-- (mod word size) */
}
}
if (!BN_div(Ri, NULL, Ri, &tmod, ctx)) {
goto err;
}
/* Ni = (R*Ri-1)/N,
* keep only least significant word: */
mont->n0[0] = (Ri->top > 0) ? Ri->d[0] : 0;
mont->n0[1] = 0;
#endif
/* RR = (2^ri)^2 == 2^(ri*2) == 1 << (ri*2), which has its (ri*2)th bit set. */
int ri = (BN_num_bits(mod) + (BN_BITS2 - 1)) / BN_BITS2 * BN_BITS2;
BN_zero(&(mont->RR));
if (!BN_set_bit(&(mont->RR), ri * 2)) {
goto err;
}
if (!BN_mod(&(mont->RR), &(mont->RR), &(mont->N), ctx)) {
goto err;
/* Save RR = R**2 (mod N). R is the smallest power of 2**BN_BITS such that R
* > mod. Even though the assembly on some 32-bit platforms works with 64-bit
* values, using |BN_BITS2| here, rather than |BN_MONT_CTX_N0_LIMBS *
* BN_BITS2|, is correct because because R^2 will still be a multiple of the
* latter as |BN_MONT_CTX_N0_LIMBS| is either one or two. */
unsigned lgBigR = (BN_num_bits(mod) + (BN_BITS2 - 1)) / BN_BITS2 * BN_BITS2;
BN_zero(&mont->RR);
if (!BN_set_bit(&mont->RR, lgBigR * 2) ||
!BN_mod(&mont->RR, &mont->RR, &mont->N, ctx)) {
return 0;
}
ret = 1;
err:
BN_CTX_end(ctx);
return ret;
return 1;
}
int BN_MONT_CTX_set_locked(BN_MONT_CTX **pmont, CRYPTO_MUTEX *lock,

158
crypto/bn/montgomery_inv.c Normal file
View File

@ -0,0 +1,158 @@
/* Copyright 2016 Brian Smith.
*
* Permission to use, copy, modify, and/or distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
#include <openssl/bn.h>
#include <assert.h>
#include "internal.h"
#include "../internal.h"
static uint64_t bn_neg_inv_mod_r_u64(uint64_t n);
OPENSSL_COMPILE_ASSERT(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2,
BN_MONT_CTX_N0_LIMBS_VALUE_INVALID);
OPENSSL_COMPILE_ASSERT(sizeof(uint64_t) ==
BN_MONT_CTX_N0_LIMBS * sizeof(BN_ULONG),
BN_MONT_CTX_N0_LIMBS_DOES_NOT_MATCH_UINT64_T);
/* LG_LITTLE_R is log_2(r). */
#define LG_LITTLE_R (BN_MONT_CTX_N0_LIMBS * BN_BITS2)
uint64_t bn_mont_n0(const BIGNUM *n) {
/* These conditions are checked by the caller, |BN_MONT_CTX_set|. */
assert(!BN_is_zero(n));
assert(!BN_is_negative(n));
assert(BN_is_odd(n));
/* r == 2**(BN_MONT_CTX_N0_LIMBS * BN_BITS2) and LG_LITTLE_R == lg(r). This
* ensures that we can do integer division by |r| by simply ignoring
* |BN_MONT_CTX_N0_LIMBS| limbs. Similarly, we can calculate values modulo
* |r| by just looking at the lowest |BN_MONT_CTX_N0_LIMBS| limbs. This is
* what makes Montgomery multiplication efficient.
*
* As shown in Algorithm 1 of "Fast Prime Field Elliptic Curve Cryptography
* with 256 Bit Primes" by Shay Gueron and Vlad Krasnov, in the loop of a
* multi-limb Montgomery multiplication of |a * b (mod n)|, given the
* unreduced product |t == a * b|, we repeatedly calculate:
*
* t1 := t % r |t1| is |t|'s lowest limb (see previous paragraph).
* t2 := t1*n0*n
* t3 := t + t2
* t := t3 / r copy all limbs of |t3| except the lowest to |t|.
*
* In the last step, it would only make sense to ignore the lowest limb of
* |t3| if it were zero. The middle steps ensure that this is the case:
*
* t3 == 0 (mod r)
* t + t2 == 0 (mod r)
* t + t1*n0*n == 0 (mod r)
* t1*n0*n == -t (mod r)
* t*n0*n == -t (mod r)
* n0*n == -1 (mod r)
* n0 == -1/n (mod r)
*
* Thus, in each iteration of the loop, we multiply by the constant factor
* |n0|, the negative inverse of n (mod r). */
/* n_mod_r = n % r. As explained above, this is done by taking the lowest
* |BN_MONT_CTX_N0_LIMBS| limbs of |n|. */
uint64_t n_mod_r = n->d[0];
#if BN_MONT_CTX_N0_LIMBS == 2
if (n->top > 1) {
n_mod_r |= (uint64_t)n->d[1] << BN_BITS2;
}
#endif
return bn_neg_inv_mod_r_u64(n_mod_r);
}
/* bn_neg_inv_r_mod_n_u64 calculates the -1/n mod r; i.e. it calculates |v|
* such that u*r - v*n == 1. |r| is the constant defined in |bn_mont_n0|. |n|
* must be odd.
*
* This is derived from |xbinGCD| in the "Montgomery Multiplication" chapter of
* "Hacker's Delight" by Henry S. Warren, Jr.:
* http://www.hackersdelight.org/MontgomeryMultiplication.pdf.
*
* This is inspired by Joppe W. Bos's "Constant Time Modular Inversion"
* (http://www.joppebos.com/files/CTInversion.pdf) so that the inversion is
* constant-time with respect to |n|. We assume uint64_t additions,
* subtractions, shifts, and bitwise operations are all constant time, which
* may be a large leap of faith on 32-bit targets. We avoid division and
* multiplication, which tend to be the most problematic in terms of timing
* leaks.
*
* Most GCD implementations return values such that |u*r + v*n == 1|, so the
* caller would have to negate the resultant |v| for the purpose of Montgomery
* multiplication. This implementation does the negation implicitly by doing
* the computations as a difference instead of a sum. */
static uint64_t bn_neg_inv_mod_r_u64(uint64_t n) {
assert(n % 2 == 1);
/* alpha == 2**(lg r - 1) == r / 2. */
static const uint64_t alpha = UINT64_C(1) << (LG_LITTLE_R - 1);
const uint64_t beta = n;
uint64_t u = 1;
uint64_t v = 0;
/* The invariant maintained from here on is:
* 2**(lg r - i) == u*2*alpha - v*beta. */
for (size_t i = 0; i < LG_LITTLE_R; ++i) {
#if BN_BITS2 == 64 && defined(BN_ULLONG)
assert((BN_ULLONG)(1) << (LG_LITTLE_R - i) ==
((BN_ULLONG)u * 2 * alpha) - ((BN_ULLONG)v * beta));
#endif
/* Delete a common factor of 2 in u and v if |u| is even. Otherwise, set
* |u = (u + beta) / 2| and |v = (v / 2) + alpha|. */
uint64_t u_is_odd = UINT64_C(0) - (u & 1); /* Either 0xff..ff or 0. */
/* The addition can overflow, so use Dietz's method for it.
*
* Dietz calculates (x+y)/2 by (x⊕y)>>1 + x&y. This is valid for all
* (unsigned) x and y, even when x+y overflows. Evidence for 32-bit values
* (embedded in 64 bits to so that overflow can be ignored):
*
* (declare-fun x () (_ BitVec 64))
* (declare-fun y () (_ BitVec 64))
* (assert (let (
* (one (_ bv1 64))
* (thirtyTwo (_ bv32 64)))
* (and
* (bvult x (bvshl one thirtyTwo))
* (bvult y (bvshl one thirtyTwo))
* (not (=
* (bvadd (bvlshr (bvxor x y) one) (bvand x y))
* (bvlshr (bvadd x y) one)))
* )))
* (check-sat) */
uint64_t beta_if_u_is_odd = beta & u_is_odd; /* Either |beta| or 0. */
u = ((u ^ beta_if_u_is_odd) >> 1) + (u & beta_if_u_is_odd);
uint64_t alpha_if_u_is_odd = alpha & u_is_odd; /* Either |alpha| or 0. */
v = (v >> 1) + alpha_if_u_is_odd;
}
/* The invariant now shows that u*r - v*n == 1 since r == 2 * alpha. */
#if BN_BITS2 == 64 && defined(BN_ULLONG)
assert(1 == ((BN_ULLONG)u * 2 * alpha) - ((BN_ULLONG)v * beta));
#endif
return v;
}

3
crypto/rsa/rsa_impl.c
View File

@ -681,8 +681,7 @@ static int mod_exp(BIGNUM *r0, const BIGNUM *I, RSA *rsa, BN_CTX *ctx) {
BIGNUM local_p, local_q;
BIGNUM *p = NULL, *q = NULL;
/* Make sure BN_mod_inverse in Montgomery intialization uses the
* BN_FLG_CONSTTIME flag. */
/* Make sure BN_mod in Montgomery initialization uses BN_FLG_CONSTTIME. */
BN_init(&local_p);
p = &local_p;
BN_with_flags(p, rsa->p, BN_FLG_CONSTTIME);