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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 https://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 = 0; \
18 /* Extract the sign-bit for a constant time absolute-value. */ \
19 int mask = (n) >> (sizeof(n) * CHAR_BIT - 1); \
20 int abs_n = ((n) + mask) ^ mask; \
21 int idx_n = abs_n >> 1; \
22 secp256k1_fe neg_y; \
23 VERIFY_CHECK(((n) & 1) == 1); \
24 VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
25 VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \
26 VERIFY_SETUP(secp256k1_fe_clear(&(r)->x)); \
27 VERIFY_SETUP(secp256k1_fe_clear(&(r)->y)); \
28 /* Unconditionally set r->x = (pre)[m].x. r->y = (pre)[m].y. because it's either the correct one \
29 * or will get replaced in the later iterations, this is needed to make sure `r` is initialized. */ \
30 (r)->x = (pre)[m].x; \
31 (r)->y = (pre)[m].y; \
32 for (m = 1; m < ECMULT_TABLE_SIZE(w); m++) { \
33 /* This loop is used to avoid secret data in array indices. See
34 * the comment in ecmult_gen_impl.h for rationale. */ \
35 secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == idx_n); \
36 secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \
37 } \
38 (r)->infinity = 0; \
39 secp256k1_fe_negate(&neg_y, &(r)->y, 1); \
40 secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \
41} while(0)
42
43
57static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int w, int size) {
58 int global_sign;
59 int skew = 0;
60 int word = 0;
61
62 /* 1 2 3 */
63 int u_last;
64 int u;
65
66 int flip;
67 int bit;
69 int not_neg_one;
70
71 VERIFY_CHECK(w > 0);
72 VERIFY_CHECK(size > 0);
73
74 /* Note that we cannot handle even numbers by negating them to be odd, as is
75 * done in other implementations, since if our scalars were specified to have
76 * width < 256 for performance reasons, their negations would have width 256
77 * and we'd lose any performance benefit. Instead, we use a technique from
78 * Section 4.2 of the Okeya/Tagaki paper, which is to add either 1 (for even)
79 * or 2 (for odd) to the number we are encoding, returning a skew value indicating
80 * this, and having the caller compensate after doing the multiplication.
81 *
82 * In fact, we _do_ want to negate numbers to minimize their bit-lengths (and in
83 * particular, to ensure that the outputs from the endomorphism-split fit into
84 * 128 bits). If we negate, the parity of our number flips, inverting which of
85 * {1, 2} we want to add to the scalar when ensuring that it's odd. Further
86 * complicating things, -1 interacts badly with `secp256k1_scalar_cadd_bit` and
87 * we need to special-case it in this logic. */
89 /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
91 /* We check for negative one, since adding 2 to it will cause an overflow */
92 secp256k1_scalar_negate(&s, scalar);
94 s = *scalar;
96 /* If we had negative one, flip == 1, s.d[0] == 0, bit == 1, so caller expects
97 * that we added two to it and flipped it. In fact for -1 these operations are
98 * identical. We only flipped, but since skewing is required (in the sense that
99 * the skew must be 1 or 2, never zero) and flipping is not, we need to change
100 * our flags to claim that we only skewed. */
102 global_sign *= not_neg_one * 2 - 1;
103 skew = 1 << bit;
104
105 /* 4 */
107 do {
108 int even;
109
110 /* 4.1 4.4 */
111 u = secp256k1_scalar_shr_int(&s, w);
112 /* 4.2 */
113 even = ((u & 1) == 0);
114 /* In contrast to the original algorithm, u_last is always > 0 and
115 * therefore we do not need to check its sign. In particular, it's easy
116 * to see that u_last is never < 0 because u is never < 0. Moreover,
117 * u_last is never = 0 because u is never even after a loop
118 * iteration. The same holds analogously for the initial value of
119 * u_last (in the first loop iteration). */
120 VERIFY_CHECK(u_last > 0);
121 VERIFY_CHECK((u_last & 1) == 1);
122 u += even;
123 u_last -= even * (1 << w);
124
125 /* 4.3, adapted for global sign change */
126 wnaf[word++] = u_last * global_sign;
127
128 u_last = u;
129 } while (word * w < size);
130 wnaf[word] = u * global_sign;
131
133 VERIFY_CHECK(word == WNAF_SIZE_BITS(size, w));
134 return skew;
136
137static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar, int size) {
141
142 int skew_1;
144 int wnaf_lam[1 + WNAF_SIZE(WINDOW_A - 1)];
145 int skew_lam;
147 int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)];
148
149 int i;
150
151 /* build wnaf representation for q. */
152 int rsize = size;
153 if (size > 128) {
154 rsize = 128;
155 /* split q into q_1 and q_lam (where q = q_1 + q_lam*lambda, and q_1 and q_lam are ~128 bit) */
159 } else
160 {
161 skew_1 = secp256k1_wnaf_const(wnaf_1, scalar, WINDOW_A - 1, size);
162 skew_lam = 0;
163 }
164
165 /* Calculate odd multiples of a.
166 * All multiples are brought to the same Z 'denominator', which is stored
167 * in Z. Due to secp256k1' isomorphism we can do all operations pretending
168 * that the Z coordinate was 1, use affine addition formulae, and correct
169 * the Z coordinate of the result once at the end.
170 */
173 for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
174 secp256k1_fe_normalize_weak(&pre_a[i].y);
175 }
176 if (size > 128) {
177 for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
178 secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
179 }
180
181 }
182
183 /* first loop iteration (separated out so we can directly set r, rather
184 * than having it start at infinity, get doubled several times, then have
185 * its new value added to it) */
187 VERIFY_CHECK(i != 0);
190 if (size > 128) {
192 VERIFY_CHECK(i != 0);
193 ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, i, WINDOW_A);
195 }
196 /* remaining loop iterations */
197 for (i = WNAF_SIZE_BITS(rsize, WINDOW_A - 1) - 1; i >= 0; i--) {
198 int n;
199 int j;
200 for (j = 0; j < WINDOW_A - 1; ++j) {
202 }
203
204 n = wnaf_1[i];
206 VERIFY_CHECK(n != 0);
208 if (size > 128) {
209 n = wnaf_lam[i];
210 ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A);
211 VERIFY_CHECK(n != 0);
213 }
214 }
215
216 secp256k1_fe_mul(&r->z, &r->z, &Z);
217
218 {
219 /* Correct for wNAF skew */
229 if (size > 128) {
231 }
233
234 /* For odd numbers this is 2a (so replace it), for even ones a (so no-op) */
236 if (size > 128) {
238 }
239
240 /* Apply the correction */
244
245 if (size > 128) {
250 }
251 }
252}
253
254#endif /* SECP256K1_ECMULT_CONST_IMPL_H */
#define ECMULT_CONST_TABLE_GET_GE(r, pre, n, w)
static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar, int size)
static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int w, int size)
Convert a number to WNAF notation.
#define WNAF_SIZE(w)
Definition ecmult_impl.h:64
static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a)
Fill a table 'pre' with precomputed odd multiples of a.
#define WINDOW_A
Definition ecmult_impl.h:34
#define ECMULT_TABLE_SIZE(w)
The number of entries a table with precomputed multiples needs to have.
Definition ecmult_impl.h:67
#define WNAF_SIZE_BITS(bits, w)
Definition ecmult_impl.h:63
static void secp256k1_fe_normalize_weak(secp256k1_fe *r)
Weakly normalize a field element: reduce its magnitude to 1, but don't fully normalize.
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_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr)
Set r equal to the double of a.
static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a)
Set r to be equal to lambda times a, where lambda is chosen in a way such that this is very fast.
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_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.
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.
static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a)
Set r equal to the inverse of a (i.e., mirrored around the X axis)
static void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a)
Set r equal to the double of a.
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.
T GetRand(T nMax=std::numeric_limits< T >::max()) noexcept
Generate a uniform random integer of type T in the range [0..nMax) nMax defaults to std::numeric_limi...
Definition random.h:85
static int secp256k1_scalar_is_even(const secp256k1_scalar *a)
Check whether a scalar, considered as an nonnegative integer, is even.
static int secp256k1_scalar_is_zero(const secp256k1_scalar *a)
Check whether a scalar equals zero.
static int secp256k1_scalar_cond_negate(secp256k1_scalar *a, int flag)
Conditionally negate a number, in constant time.
static int secp256k1_scalar_is_one(const secp256k1_scalar *a)
Check whether a scalar equals one.
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_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.
static void secp256k1_scalar_split_lambda(secp256k1_scalar *r1, secp256k1_scalar *r2, const secp256k1_scalar *k)
Find r1 and r2 such that r1+r2*lambda = k, where r1 and r2 or their negations are maximum 128 bits lo...
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...
#define VERIFY_CHECK(cond)
Definition util.h:68
A group element of the secp256k1 curve, in affine coordinates.
Definition group.h:13
A group element of the secp256k1 curve, in jacobian coordinates.
Definition group.h:23
secp256k1_fe z
Definition group.h:26
A scalar modulo the group order of the secp256k1 curve.
Definition scalar_4x64.h:13