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ecmult_const_impl.h
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1 /**********************************************************************
2  * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra *
3  * Distributed under the MIT software license, see the accompanying *
4  * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
5  **********************************************************************/
6 
7 #ifndef SECP256K1_ECMULT_CONST_IMPL_H
8 #define SECP256K1_ECMULT_CONST_IMPL_H
9 
10 #include "scalar.h"
11 #include "group.h"
12 #include "ecmult_const.h"
13 #include "ecmult_impl.h"
14 
15 /* This is like `ECMULT_TABLE_GET_GE` but is constant time */
16 #define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \
17  int m; \
18  int abs_n = (n) * (((n) > 0) * 2 - 1); \
19  int idx_n = abs_n / 2; \
20  secp256k1_fe neg_y; \
21  VERIFY_CHECK(((n) & 1) == 1); \
22  VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
23  VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \
24  VERIFY_SETUP(secp256k1_fe_clear(&(r)->x)); \
25  VERIFY_SETUP(secp256k1_fe_clear(&(r)->y)); \
26  for (m = 0; m < ECMULT_TABLE_SIZE(w); m++) { \
27  /* This loop is used to avoid secret data in array indices. See
28  * the comment in ecmult_gen_impl.h for rationale. */ \
29  secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == idx_n); \
30  secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \
31  } \
32  (r)->infinity = 0; \
33  secp256k1_fe_negate(&neg_y, &(r)->y, 1); \
34  secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \
35 } while(0)
36 
37 
51 static int secp256k1_wnaf_const(int *wnaf, secp256k1_scalar s, int w, int size) {
52  int global_sign;
53  int skew = 0;
54  int word = 0;
55 
56  /* 1 2 3 */
57  int u_last;
58  int u;
59 
60  int flip;
61  int bit;
62  secp256k1_scalar neg_s;
63  int not_neg_one;
64  /* Note that we cannot handle even numbers by negating them to be odd, as is
65  * done in other implementations, since if our scalars were specified to have
66  * width < 256 for performance reasons, their negations would have width 256
67  * and we'd lose any performance benefit. Instead, we use a technique from
68  * Section 4.2 of the Okeya/Tagaki paper, which is to add either 1 (for even)
69  * or 2 (for odd) to the number we are encoding, returning a skew value indicating
70  * this, and having the caller compensate after doing the multiplication.
71  *
72  * In fact, we _do_ want to negate numbers to minimize their bit-lengths (and in
73  * particular, to ensure that the outputs from the endomorphism-split fit into
74  * 128 bits). If we negate, the parity of our number flips, inverting which of
75  * {1, 2} we want to add to the scalar when ensuring that it's odd. Further
76  * complicating things, -1 interacts badly with `secp256k1_scalar_cadd_bit` and
77  * we need to special-case it in this logic. */
78  flip = secp256k1_scalar_is_high(&s);
79  /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
80  bit = flip ^ !secp256k1_scalar_is_even(&s);
81  /* We check for negative one, since adding 2 to it will cause an overflow */
82  secp256k1_scalar_negate(&neg_s, &s);
83  not_neg_one = !secp256k1_scalar_is_one(&neg_s);
84  secp256k1_scalar_cadd_bit(&s, bit, not_neg_one);
85  /* If we had negative one, flip == 1, s.d[0] == 0, bit == 1, so caller expects
86  * that we added two to it and flipped it. In fact for -1 these operations are
87  * identical. We only flipped, but since skewing is required (in the sense that
88  * the skew must be 1 or 2, never zero) and flipping is not, we need to change
89  * our flags to claim that we only skewed. */
90  global_sign = secp256k1_scalar_cond_negate(&s, flip);
91  global_sign *= not_neg_one * 2 - 1;
92  skew = 1 << bit;
93 
94  /* 4 */
95  u_last = secp256k1_scalar_shr_int(&s, w);
96  while (word * w < size) {
97  int sign;
98  int even;
99 
100  /* 4.1 4.4 */
101  u = secp256k1_scalar_shr_int(&s, w);
102  /* 4.2 */
103  even = ((u & 1) == 0);
104  sign = 2 * (u_last > 0) - 1;
105  u += sign * even;
106  u_last -= sign * even * (1 << w);
107 
108  /* 4.3, adapted for global sign change */
109  wnaf[word++] = u_last * global_sign;
110 
111  u_last = u;
112  }
113  wnaf[word] = u * global_sign;
114 
116  VERIFY_CHECK(word == WNAF_SIZE_BITS(size, w));
117  return skew;
118 }
119 
120 static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar, int size) {
122  secp256k1_ge tmpa;
123  secp256k1_fe Z;
124 
125  int skew_1;
126 #ifdef USE_ENDOMORPHISM
128  int wnaf_lam[1 + WNAF_SIZE(WINDOW_A - 1)];
129  int skew_lam;
130  secp256k1_scalar q_1, q_lam;
131 #endif
132  int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)];
133 
134  int i;
135  secp256k1_scalar sc = *scalar;
136 
137  /* build wnaf representation for q. */
138  int rsize = size;
139 #ifdef USE_ENDOMORPHISM
140  if (size > 128) {
141  rsize = 128;
142  /* split q into q_1 and q_lam (where q = q_1 + q_lam*lambda, and q_1 and q_lam are ~128 bit) */
143  secp256k1_scalar_split_lambda(&q_1, &q_lam, &sc);
144  skew_1 = secp256k1_wnaf_const(wnaf_1, q_1, WINDOW_A - 1, 128);
145  skew_lam = secp256k1_wnaf_const(wnaf_lam, q_lam, WINDOW_A - 1, 128);
146  } else
147 #endif
148  {
149  skew_1 = secp256k1_wnaf_const(wnaf_1, sc, WINDOW_A - 1, size);
150 #ifdef USE_ENDOMORPHISM
151  skew_lam = 0;
152 #endif
153  }
154 
155  /* Calculate odd multiples of a.
156  * All multiples are brought to the same Z 'denominator', which is stored
157  * in Z. Due to secp256k1' isomorphism we can do all operations pretending
158  * that the Z coordinate was 1, use affine addition formulae, and correct
159  * the Z coordinate of the result once at the end.
160  */
161  secp256k1_gej_set_ge(r, a);
163  for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
164  secp256k1_fe_normalize_weak(&pre_a[i].y);
165  }
166 #ifdef USE_ENDOMORPHISM
167  if (size > 128) {
168  for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
169  secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
170  }
171  }
172 #endif
173 
174  /* first loop iteration (separated out so we can directly set r, rather
175  * than having it start at infinity, get doubled several times, then have
176  * its new value added to it) */
177  i = wnaf_1[WNAF_SIZE_BITS(rsize, WINDOW_A - 1)];
178  VERIFY_CHECK(i != 0);
179  ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A);
180  secp256k1_gej_set_ge(r, &tmpa);
181 #ifdef USE_ENDOMORPHISM
182  if (size > 128) {
183  i = wnaf_lam[WNAF_SIZE_BITS(rsize, WINDOW_A - 1)];
184  VERIFY_CHECK(i != 0);
185  ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, i, WINDOW_A);
186  secp256k1_gej_add_ge(r, r, &tmpa);
187  }
188 #endif
189  /* remaining loop iterations */
190  for (i = WNAF_SIZE_BITS(rsize, WINDOW_A - 1) - 1; i >= 0; i--) {
191  int n;
192  int j;
193  for (j = 0; j < WINDOW_A - 1; ++j) {
194  secp256k1_gej_double_nonzero(r, r, NULL);
195  }
196 
197  n = wnaf_1[i];
198  ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A);
199  VERIFY_CHECK(n != 0);
200  secp256k1_gej_add_ge(r, r, &tmpa);
201 #ifdef USE_ENDOMORPHISM
202  if (size > 128) {
203  n = wnaf_lam[i];
204  ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A);
205  VERIFY_CHECK(n != 0);
206  secp256k1_gej_add_ge(r, r, &tmpa);
207  }
208 #endif
209  }
210 
211  secp256k1_fe_mul(&r->z, &r->z, &Z);
212 
213  {
214  /* Correct for wNAF skew */
215  secp256k1_ge correction = *a;
216  secp256k1_ge_storage correction_1_stor;
217 #ifdef USE_ENDOMORPHISM
218  secp256k1_ge_storage correction_lam_stor;
219 #endif
220  secp256k1_ge_storage a2_stor;
221  secp256k1_gej tmpj;
222  secp256k1_gej_set_ge(&tmpj, &correction);
223  secp256k1_gej_double_var(&tmpj, &tmpj, NULL);
224  secp256k1_ge_set_gej(&correction, &tmpj);
225  secp256k1_ge_to_storage(&correction_1_stor, a);
226 #ifdef USE_ENDOMORPHISM
227  if (size > 128) {
228  secp256k1_ge_to_storage(&correction_lam_stor, a);
229  }
230 #endif
231  secp256k1_ge_to_storage(&a2_stor, &correction);
232 
233  /* For odd numbers this is 2a (so replace it), for even ones a (so no-op) */
234  secp256k1_ge_storage_cmov(&correction_1_stor, &a2_stor, skew_1 == 2);
235 #ifdef USE_ENDOMORPHISM
236  if (size > 128) {
237  secp256k1_ge_storage_cmov(&correction_lam_stor, &a2_stor, skew_lam == 2);
238  }
239 #endif
240 
241  /* Apply the correction */
242  secp256k1_ge_from_storage(&correction, &correction_1_stor);
243  secp256k1_ge_neg(&correction, &correction);
244  secp256k1_gej_add_ge(r, r, &correction);
245 
246 #ifdef USE_ENDOMORPHISM
247  if (size > 128) {
248  secp256k1_ge_from_storage(&correction, &correction_lam_stor);
249  secp256k1_ge_neg(&correction, &correction);
250  secp256k1_ge_mul_lambda(&correction, &correction);
251  secp256k1_gej_add_ge(r, r, &correction);
252  }
253 #endif
254  }
255 }
256 
257 #endif /* SECP256K1_ECMULT_CONST_IMPL_H */
#define VERIFY_CHECK(cond)
Definition: util.h:67
static int secp256k1_scalar_is_even(const secp256k1_scalar *a)
Check whether a scalar, considered as an nonnegative integer, is even.
static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a)
static void secp256k1_fe_mul(secp256k1_fe *r, const secp256k1_fe *a, const secp256k1_fe *SECP256K1_RESTRICT b)
Sets a field element to be the product of two others.
static void secp256k1_gej_double_nonzero(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr)
Set r equal to the double of a.
static void secp256k1_scalar_negate(secp256k1_scalar *r, const secp256k1_scalar *a)
Compute the complement of a scalar (modulo the group order).
static int secp256k1_scalar_is_zero(const secp256k1_scalar *a)
Check whether a scalar equals zero.
#define ECMULT_TABLE_SIZE(w)
The number of entries a table with precomputed multiples needs to have.
Definition: ecmult_impl.h:54
static int secp256k1_scalar_shr_int(secp256k1_scalar *r, int n)
Shift a scalar right by some amount strictly between 0 and 16, returning the low bits that were shift...
A group element of the secp256k1 curve, in jacobian coordinates.
Definition: group.h:24
static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr)
Set r equal to the double of a.
static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a)
Fill a table &#39;pre&#39; with precomputed odd multiples of a.
Definition: ecmult_impl.h:130
static int secp256k1_wnaf_const(int *wnaf, secp256k1_scalar s, int w, int size)
Convert a number to WNAF notation.
static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar, int size)
#define ECMULT_CONST_TABLE_GET_GE(r, pre, n, w)
static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a)
Set a group element equal to another which is given in jacobian coordinates.
#define WNAF_SIZE_BITS(bits, w)
Definition: ecmult_impl.h:50
static int secp256k1_scalar_is_high(const secp256k1_scalar *a)
Check whether a scalar is higher than the group order divided by 2.
static void secp256k1_scalar_cadd_bit(secp256k1_scalar *r, unsigned int bit, int flag)
Conditionally add a power of two to a scalar.
A group element of the secp256k1 curve, in affine coordinates.
Definition: group.h:14
static void secp256k1_fe_normalize_weak(secp256k1_fe *r)
Weakly normalize a field element: reduce it magnitude to 1, but don&#39;t fully normalize.
static void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag)
If flag is true, set *r equal to *a; otherwise leave it.
A scalar modulo the group order of the secp256k1 curve.
Definition: scalar_4x64.h:13
static int secp256k1_scalar_cond_negate(secp256k1_scalar *a, int flag)
Conditionally negate a number, in constant time.
#define WINDOW_A
Definition: ecmult_impl.h:33
secp256k1_fe z
Definition: group.h:27
#define WNAF_SIZE(w)
Definition: ecmult_impl.h:51
static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b)
Set r equal to the sum of a and b (with b given in affine coordinates, and not infinity).
static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storage *a)
Convert a group element back from the storage type.
static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a)
Set a group element (jacobian) equal to another which is given in affine coordinates.
static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge *a)
Convert a group element to the storage type.
static int secp256k1_scalar_is_one(const secp256k1_scalar *a)
Check whether a scalar equals one.