crypto/fipsmodule/bn/montgomery_inv.c - boringssl - Git at Google

 /* 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);

 static_assert(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2,
               "BN_MONT_CTX_N0_LIMBS value is invalid");
 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_mont_ctx_set_RR_consttime(BN_MONT_CTX *mont, BN_CTX *ctx) {
   assert(!BN_is_zero(&mont->N));
   assert(!BN_is_negative(&mont->N));
   assert(BN_is_odd(&mont->N));
   assert(bn_minimal_width(&mont->N) == mont->N.width);

   unsigned n_bits = BN_num_bits(&mont->N);
   assert(n_bits != 0);
   if (n_bits == 1) {
     BN_zero(&mont->RR);
     return bn_resize_words(&mont->RR, mont->N.width);
   }

   unsigned lgBigR = mont->N.width * BN_BITS2;
   assert(lgBigR >= n_bits);

   // RR is R, or 2^lgBigR, in the Montgomery domain. We can compute 2 in the
   // Montgomery domain, 2R or 2^(lgBigR+1), and then use Montgomery
   // square-and-multiply to exponentiate.
   //
   // The multiply steps take 2^n R to 2^(n+1) R. It is faster to double
   // the value instead. The square steps take 2^n R to 2^(2n) R. This is
   // equivalent to doubling n times. When n is below some threshold, doubling is
   // faster. When above, squaring is faster.
   //
   // We double to this threshold, then switch to Montgomery squaring. From
   // benchmarking various 32-bit and 64-bit architectures, the word count seems
   // to work well as a threshold. (Doubling scales linearly and Montgomery
   // reduction scales quadratically, so the threshold should scale roughly
   // linearly.)
   unsigned threshold = mont->N.width;
   unsigned iters;
   for (iters = 0; iters < sizeof(lgBigR) * 8; iters++) {
     if ((lgBigR >> iters) <= threshold) {
       break;
     }
   }

   // Compute 2^(lgBigR >> iters) R, or 2^((lgBigR >> iters) + lgBigR), by
   // doubling. The first n_bits - 1 doubles can be skipped because we don't need
   // to reduce.
   if (!BN_set_bit(&mont->RR, n_bits - 1) ||
       !bn_mod_lshift_consttime(&mont->RR, &mont->RR,
                                (lgBigR >> iters) + lgBigR - (n_bits - 1),
                                &mont->N, ctx)) {
     return 0;
   }

   for (unsigned i = iters - 1; i < iters; i--) {
     if (!BN_mod_mul_montgomery(&mont->RR, &mont->RR, &mont->RR, mont, ctx)) {
       return 0;
     }
     if ((lgBigR & (1u << i)) != 0 &&
         !bn_mod_lshift1_consttime(&mont->RR, &mont->RR, &mont->N, ctx)) {
       return 0;
     }
   }

   return bn_resize_words(&mont->RR, mont->N.width);
 }
	/* 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);

	static_assert(BN_MONT_CTX_N0_LIMBS == 1 \|\| BN_MONT_CTX_N0_LIMBS == 2,
	"BN_MONT_CTX_N0_LIMBS value is invalid");
	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 := t1n0n
	// 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 + t1n0n == 0 (mod r)
	// t1n0n == -t (mod r)
	// tn0n == -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 ur - vn == 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 \|ur + vn == 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) == u2alpha - vbeta.
	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 ur - vn == 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_mont_ctx_set_RR_consttime(BN_MONT_CTX mont, BN_CTX ctx) {
	assert(!BN_is_zero(&mont->N));
	assert(!BN_is_negative(&mont->N));
	assert(BN_is_odd(&mont->N));
	assert(bn_minimal_width(&mont->N) == mont->N.width);

	unsigned n_bits = BN_num_bits(&mont->N);
	assert(n_bits != 0);
	if (n_bits == 1) {
	BN_zero(&mont->RR);
	return bn_resize_words(&mont->RR, mont->N.width);
	}

	unsigned lgBigR = mont->N.width * BN_BITS2;
	assert(lgBigR >= n_bits);

	// RR is R, or 2^lgBigR, in the Montgomery domain. We can compute 2 in the
	// Montgomery domain, 2R or 2^(lgBigR+1), and then use Montgomery
	// square-and-multiply to exponentiate.
	//
	// The multiply steps take 2^n R to 2^(n+1) R. It is faster to double
	// the value instead. The square steps take 2^n R to 2^(2n) R. This is
	// equivalent to doubling n times. When n is below some threshold, doubling is
	// faster. When above, squaring is faster.
	//
	// We double to this threshold, then switch to Montgomery squaring. From
	// benchmarking various 32-bit and 64-bit architectures, the word count seems
	// to work well as a threshold. (Doubling scales linearly and Montgomery
	// reduction scales quadratically, so the threshold should scale roughly
	// linearly.)
	unsigned threshold = mont->N.width;
	unsigned iters;
	for (iters = 0; iters < sizeof(lgBigR) * 8; iters++) {
	if ((lgBigR >> iters) <= threshold) {
	break;
	}
	}

	// Compute 2^(lgBigR >> iters) R, or 2^((lgBigR >> iters) + lgBigR), by
	// doubling. The first n_bits - 1 doubles can be skipped because we don't need
	// to reduce.
	if (!BN_set_bit(&mont->RR, n_bits - 1) \|\|
	!bn_mod_lshift_consttime(&mont->RR, &mont->RR,
	(lgBigR >> iters) + lgBigR - (n_bits - 1),
	&mont->N, ctx)) {
	return 0;
	}

	for (unsigned i = iters - 1; i < iters; i--) {
	if (!BN_mod_mul_montgomery(&mont->RR, &mont->RR, &mont->RR, mont, ctx)) {
	return 0;
	}
	if ((lgBigR & (1u << i)) != 0 &&
	!bn_mod_lshift1_consttime(&mont->RR, &mont->RR, &mont->N, ctx)) {
	return 0;
	}
	}

	return bn_resize_words(&mont->RR, mont->N.width);
	}