--- /dev/null
+/* 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_2);
+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->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;
+}
git://repos.xcallymotion.com / motion2.git / blobdiff