/* 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_grpc/bn.h> #include <assert.h> #include "internal.h" #include "../../internal.h" static uint64_t bn_neg_inv_mod_r_u64(uint64_t n); OPENSSL_STATIC_ASSERT(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2, "BN_MONT_CTX_N0_LIMBS value is invalid"); OPENSSL_STATIC_ASSERT(sizeof(BN_ULONG) * BN_MONT_CTX_N0_LIMBS == sizeof(uint64_t), "uint64_t is insufficient precision for n0"); // 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| or // |BN_MONT_CTX_new_consttime|. 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->width > 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; } int bn_mod_exp_base_2_consttime(BIGNUM *r, unsigned p, const BIGNUM *n, BN_CTX *ctx) { 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); assert(p > n_bits); if (n_bits == 1) { return 1; } // Set |r| to the larger power of two smaller than |n|, then shift with // reductions the rest of the way. if (!BN_set_bit(r, n_bits - 1) || !bn_mod_lshift_consttime(r, r, p - (n_bits - 1), n, ctx)) { return 0; } return 1; }