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- /* 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 Henry S. Warren, Jr.'s "Montgomery
- * Multiplication" (http://www.hackersdelight.org/MontgomeryMultiplication.pdf).
- * It is very similar to the MODULAR-INVERSE function in Stephen R. Dussé's and
- * Burton S. Kaliski Jr.'s "A Cryptographic Library for the Motorola DSP56000"
- * (http://link.springer.com/chapter/10.1007%2F3-540-46877-3_21).
- *
- * 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;
- }
-
- /* bn_mod_exp_base_2_vartime calculates r = 2**p (mod n). |p| must be larger
- * than log_2(n); i.e. 2**p must be larger than |n|. |n| must be positive and
- * odd. */
- int bn_mod_exp_base_2_vartime(BIGNUM *r, unsigned p, const BIGNUM *n) {
- assert(!BN_is_zero(n));
- assert(!BN_is_negative(n));
- assert(BN_is_odd(n));
-
- BN_zero(r);
-
- unsigned n_bits = BN_num_bits(n);
- assert(n_bits != 0);
- if (n_bits == 1) {
- return 1;
- }
-
- /* Set |r| to the smallest power of two larger than |n|. */
- assert(p > n_bits);
- if (!BN_set_bit(r, n_bits)) {
- return 0;
- }
-
- /* Unconditionally reduce |r|. */
- assert(BN_cmp(r, n) > 0);
- if (!BN_usub(r, r, n)) {
- return 0;
- }
- assert(BN_cmp(r, n) < 0);
-
- for (unsigned i = n_bits; i < p; ++i) {
- /* This is like |BN_mod_lshift1_quick| except using |BN_usub|.
- *
- * TODO: Replace this with the use of a constant-time variant of
- * |BN_mod_lshift1_quick|. */
- if (!BN_lshift1(r, r)) {
- return 0;
- }
- if (BN_cmp(r, n) >= 0) {
- if (!BN_usub(r, r, n)) {
- return 0;
- }
- }
- }
-
- return 1;
- }
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