1/* Copyright 2016 Brian Smith.
2 *
3 * Permission to use, copy, modify, and/or distribute this software for any
4 * purpose with or without fee is hereby granted, provided that the above
5 * copyright notice and this permission notice appear in all copies.
6 *
7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
14
15#include <openssl/bn.h>
16
17#include <assert.h>
18
19#include "internal.h"
20#include "../../internal.h"
21
22
23static uint64_t bn_neg_inv_mod_r_u64(uint64_t n);
24
25OPENSSL_COMPILE_ASSERT(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2,
26                       BN_MONT_CTX_N0_LIMBS_VALUE_INVALID_2);
27OPENSSL_COMPILE_ASSERT(sizeof(uint64_t) ==
28                       BN_MONT_CTX_N0_LIMBS * sizeof(BN_ULONG),
29                       BN_MONT_CTX_N0_LIMBS_DOES_NOT_MATCH_UINT64_T);
30
31// LG_LITTLE_R is log_2(r).
32#define LG_LITTLE_R (BN_MONT_CTX_N0_LIMBS * BN_BITS2)
33
34uint64_t bn_mont_n0(const BIGNUM *n) {
35  // These conditions are checked by the caller, |BN_MONT_CTX_set|.
36  assert(!BN_is_zero(n));
37  assert(!BN_is_negative(n));
38  assert(BN_is_odd(n));
39
40  // r == 2**(BN_MONT_CTX_N0_LIMBS * BN_BITS2) and LG_LITTLE_R == lg(r). This
41  // ensures that we can do integer division by |r| by simply ignoring
42  // |BN_MONT_CTX_N0_LIMBS| limbs. Similarly, we can calculate values modulo
43  // |r| by just looking at the lowest |BN_MONT_CTX_N0_LIMBS| limbs. This is
44  // what makes Montgomery multiplication efficient.
45  //
46  // As shown in Algorithm 1 of "Fast Prime Field Elliptic Curve Cryptography
47  // with 256 Bit Primes" by Shay Gueron and Vlad Krasnov, in the loop of a
48  // multi-limb Montgomery multiplication of |a * b (mod n)|, given the
49  // unreduced product |t == a * b|, we repeatedly calculate:
50  //
51  //    t1 := t % r         |t1| is |t|'s lowest limb (see previous paragraph).
52  //    t2 := t1*n0*n
53  //    t3 := t + t2
54  //    t := t3 / r         copy all limbs of |t3| except the lowest to |t|.
55  //
56  // In the last step, it would only make sense to ignore the lowest limb of
57  // |t3| if it were zero. The middle steps ensure that this is the case:
58  //
59  //                            t3 ==  0 (mod r)
60  //                        t + t2 ==  0 (mod r)
61  //                   t + t1*n0*n ==  0 (mod r)
62  //                       t1*n0*n == -t (mod r)
63  //                        t*n0*n == -t (mod r)
64  //                          n0*n == -1 (mod r)
65  //                            n0 == -1/n (mod r)
66  //
67  // Thus, in each iteration of the loop, we multiply by the constant factor
68  // |n0|, the negative inverse of n (mod r).
69
70  // n_mod_r = n % r. As explained above, this is done by taking the lowest
71  // |BN_MONT_CTX_N0_LIMBS| limbs of |n|.
72  uint64_t n_mod_r = n->d[0];
73#if BN_MONT_CTX_N0_LIMBS == 2
74  if (n->top > 1) {
75    n_mod_r |= (uint64_t)n->d[1] << BN_BITS2;
76  }
77#endif
78
79  return bn_neg_inv_mod_r_u64(n_mod_r);
80}
81
82// bn_neg_inv_r_mod_n_u64 calculates the -1/n mod r; i.e. it calculates |v|
83// such that u*r - v*n == 1. |r| is the constant defined in |bn_mont_n0|. |n|
84// must be odd.
85//
86// This is derived from |xbinGCD| in Henry S. Warren, Jr.'s "Montgomery
87// Multiplication" (http://www.hackersdelight.org/MontgomeryMultiplication.pdf).
88// It is very similar to the MODULAR-INVERSE function in Stephen R. Dussé's and
89// Burton S. Kaliski Jr.'s "A Cryptographic Library for the Motorola DSP56000"
90// (http://link.springer.com/chapter/10.1007%2F3-540-46877-3_21).
91//
92// This is inspired by Joppe W. Bos's "Constant Time Modular Inversion"
93// (http://www.joppebos.com/files/CTInversion.pdf) so that the inversion is
94// constant-time with respect to |n|. We assume uint64_t additions,
95// subtractions, shifts, and bitwise operations are all constant time, which
96// may be a large leap of faith on 32-bit targets. We avoid division and
97// multiplication, which tend to be the most problematic in terms of timing
98// leaks.
99//
100// Most GCD implementations return values such that |u*r + v*n == 1|, so the
101// caller would have to negate the resultant |v| for the purpose of Montgomery
102// multiplication. This implementation does the negation implicitly by doing
103// the computations as a difference instead of a sum.
104static uint64_t bn_neg_inv_mod_r_u64(uint64_t n) {
105  assert(n % 2 == 1);
106
107  // alpha == 2**(lg r - 1) == r / 2.
108  static const uint64_t alpha = UINT64_C(1) << (LG_LITTLE_R - 1);
109
110  const uint64_t beta = n;
111
112  uint64_t u = 1;
113  uint64_t v = 0;
114
115  // The invariant maintained from here on is:
116  // 2**(lg r - i) == u*2*alpha - v*beta.
117  for (size_t i = 0; i < LG_LITTLE_R; ++i) {
118#if BN_BITS2 == 64 && defined(BN_ULLONG)
119    assert((BN_ULLONG)(1) << (LG_LITTLE_R - i) ==
120           ((BN_ULLONG)u * 2 * alpha) - ((BN_ULLONG)v * beta));
121#endif
122
123    // Delete a common factor of 2 in u and v if |u| is even. Otherwise, set
124    // |u = (u + beta) / 2| and |v = (v / 2) + alpha|.
125
126    uint64_t u_is_odd = UINT64_C(0) - (u & 1);  // Either 0xff..ff or 0.
127
128    // The addition can overflow, so use Dietz's method for it.
129    //
130    // Dietz calculates (x+y)/2 by (x⊕y)>>1 + x&y. This is valid for all
131    // (unsigned) x and y, even when x+y overflows. Evidence for 32-bit values
132    // (embedded in 64 bits to so that overflow can be ignored):
133    //
134    // (declare-fun x () (_ BitVec 64))
135    // (declare-fun y () (_ BitVec 64))
136    // (assert (let (
137    //    (one (_ bv1 64))
138    //    (thirtyTwo (_ bv32 64)))
139    //    (and
140    //      (bvult x (bvshl one thirtyTwo))
141    //      (bvult y (bvshl one thirtyTwo))
142    //      (not (=
143    //        (bvadd (bvlshr (bvxor x y) one) (bvand x y))
144    //        (bvlshr (bvadd x y) one)))
145    // )))
146    // (check-sat)
147    uint64_t beta_if_u_is_odd = beta & u_is_odd;  // Either |beta| or 0.
148    u = ((u ^ beta_if_u_is_odd) >> 1) + (u & beta_if_u_is_odd);
149
150    uint64_t alpha_if_u_is_odd = alpha & u_is_odd;  // Either |alpha| or 0.
151    v = (v >> 1) + alpha_if_u_is_odd;
152  }
153
154  // The invariant now shows that u*r - v*n == 1 since r == 2 * alpha.
155#if BN_BITS2 == 64 && defined(BN_ULLONG)
156  assert(1 == ((BN_ULLONG)u * 2 * alpha) - ((BN_ULLONG)v * beta));
157#endif
158
159  return v;
160}
161
162// bn_mod_exp_base_2_vartime calculates r = 2**p (mod n). |p| must be larger
163// than log_2(n); i.e. 2**p must be larger than |n|. |n| must be positive and
164// odd.
165int bn_mod_exp_base_2_vartime(BIGNUM *r, unsigned p, const BIGNUM *n) {
166  assert(!BN_is_zero(n));
167  assert(!BN_is_negative(n));
168  assert(BN_is_odd(n));
169
170  BN_zero(r);
171
172  unsigned n_bits = BN_num_bits(n);
173  assert(n_bits != 0);
174  if (n_bits == 1) {
175    return 1;
176  }
177
178  // Set |r| to the smallest power of two larger than |n|.
179  assert(p > n_bits);
180  if (!BN_set_bit(r, n_bits)) {
181    return 0;
182  }
183
184  // Unconditionally reduce |r|.
185  assert(BN_cmp(r, n) > 0);
186  if (!BN_usub(r, r, n)) {
187    return 0;
188  }
189  assert(BN_cmp(r, n) < 0);
190
191  for (unsigned i = n_bits; i < p; ++i) {
192    // This is like |BN_mod_lshift1_quick| except using |BN_usub|.
193    //
194    // TODO: Replace this with the use of a constant-time variant of
195    // |BN_mod_lshift1_quick|.
196    if (!BN_lshift1(r, r)) {
197      return 0;
198    }
199    if (BN_cmp(r, n) >= 0) {
200      if (!BN_usub(r, r, n)) {
201        return 0;
202      }
203    }
204  }
205
206  return 1;
207}
208