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>kris/onging/CECPQ3_patch15
@@ -57,6 +57,7 @@ add_library( | |||
gcd.c | |||
kronecker.c | |||
montgomery.c | |||
montgomery_inv.c | |||
mul.c | |||
prime.c | |||
random.c | |||
@@ -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 | |||
@@ -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; | |||
} | |||
R = &mont->RR; /* grab RR as a temp */ | |||
if (!BN_copy(&mont->N, mod)) { | |||
goto err; /* Set N */ | |||
} | |||
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; | |||
} | |||
tmod.top = 0; | |||
if ((buf[0] = mod->d[0])) { | |||
tmod.top = 1; | |||
if (!BN_is_odd(mod)) { | |||
OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS); | |||
return 0; | |||
} | |||
if ((buf[1] = mod->top > 1 ? mod->d[1] : 0)) { | |||
tmod.top = 2; | |||
if (BN_is_negative(mod)) { | |||
OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER); | |||
return 0; | |||
} | |||
if (BN_mod_inverse(Ri, R, &tmod, ctx) == NULL) { | |||
goto err; | |||
} | |||
if (!BN_lshift(Ri, Ri, 2 * BN_BITS2)) { | |||
goto err; /* R*Ri */ | |||
/* Save the modulus. */ | |||
if (!BN_copy(&mont->N, mod)) { | |||
OPENSSL_PUT_ERROR(BN, ERR_R_INTERNAL_ERROR); | |||
return 0; | |||
} | |||
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_get_flags(mod, BN_FLG_CONSTTIME)) { | |||
BN_set_flags(&mont->N, BN_FLG_CONSTTIME); | |||
} | |||
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, | |||
@@ -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; | |||
} |
@@ -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); | |||