* file COPYING or http://www.opensource.org/licenses/mit-license.php.*
**********************************************************************/
-#ifndef _SECP256K1_ECMULT_CONST_IMPL_
-#define _SECP256K1_ECMULT_CONST_IMPL_
+#ifndef SECP256K1_ECMULT_CONST_IMPL_H
+#define SECP256K1_ECMULT_CONST_IMPL_H
#include "scalar.h"
#include "group.h"
#include "ecmult_const.h"
#include "ecmult_impl.h"
-#define WNAF_BITS 256
-#define WNAF_SIZE(w) ((WNAF_BITS + (w) - 1) / (w))
-
/* This is like `ECMULT_TABLE_GET_GE` but is constant time */
#define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \
- int m; \
- int abs_n = (n) * (((n) > 0) * 2 - 1); \
- secp256k1_fe_t neg_y; \
+ int m = 0; \
+ /* Extract the sign-bit for a constant time absolute-value. */ \
+ int mask = (n) >> (sizeof(n) * CHAR_BIT - 1); \
+ int abs_n = ((n) + mask) ^ mask; \
+ int idx_n = abs_n >> 1; \
+ secp256k1_fe neg_y; \
VERIFY_CHECK(((n) & 1) == 1); \
VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \
- for (m = 1; m < (1 << ((w) - 1)); m += 2) { \
+ VERIFY_SETUP(secp256k1_fe_clear(&(r)->x)); \
+ VERIFY_SETUP(secp256k1_fe_clear(&(r)->y)); \
+ /* Unconditionally set r->x = (pre)[m].x. r->y = (pre)[m].y. because it's either the correct one \
+ * or will get replaced in the later iterations, this is needed to make sure `r` is initialized. */ \
+ (r)->x = (pre)[m].x; \
+ (r)->y = (pre)[m].y; \
+ for (m = 1; m < ECMULT_TABLE_SIZE(w); m++) { \
/* This loop is used to avoid secret data in array indices. See
* the comment in ecmult_gen_impl.h for rationale. */ \
- secp256k1_fe_cmov(&(r)->x, &(pre)[(m - 1) / 2].x, m == abs_n); \
- secp256k1_fe_cmov(&(r)->y, &(pre)[(m - 1) / 2].y, m == abs_n); \
+ secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == idx_n); \
+ secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \
} \
(r)->infinity = 0; \
- secp256k1_fe_normalize_weak(&(r)->x); \
- secp256k1_fe_normalize_weak(&(r)->y); \
secp256k1_fe_negate(&neg_y, &(r)->y, 1); \
secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \
} while(0)
-/** Convert a number to WNAF notation. The number becomes represented by sum(2^{wi} * wnaf[i], i=0..return_val)
- * with the following guarantees:
+/** Convert a number to WNAF notation.
+ * The number becomes represented by sum(2^{wi} * wnaf[i], i=0..WNAF_SIZE(w)+1) - return_val.
+ * It has the following guarantees:
* - each wnaf[i] an odd integer between -(1 << w) and (1 << w)
* - each wnaf[i] is nonzero
- * - the number of words set is returned; this is always (WNAF_BITS + w - 1) / w
+ * - the number of words set is always WNAF_SIZE(w) + 1
*
* Adapted from `The Width-w NAF Method Provides Small Memory and Fast Elliptic Scalar
* Multiplications Secure against Side Channel Attacks`, Okeya and Tagaki. M. Joye (Ed.)
- * CT-RSA 2003, LNCS 2612, pp. 328-443, 2003. Springer-Verlagy Berlin Heidelberg 2003
+ * CT-RSA 2003, LNCS 2612, pp. 328-443, 2003. Springer-Verlag Berlin Heidelberg 2003
*
* Numbers reference steps of `Algorithm SPA-resistant Width-w NAF with Odd Scalar` on pp. 335
*/
-static void secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar_t *a, int w) {
- secp256k1_scalar_t s = *a;
- /* Negate to force oddness */
- int is_even = secp256k1_scalar_is_even(&s);
- int global_sign = secp256k1_scalar_cond_negate(&s, is_even);
-
+static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int w, int size) {
+ int global_sign;
+ int skew = 0;
int word = 0;
+
/* 1 2 3 */
- int u_last = secp256k1_scalar_shr_int(&s, w);
+ int u_last;
int u;
+
+ int flip;
+ int bit;
+ secp256k1_scalar s;
+ int not_neg_one;
+
+ VERIFY_CHECK(w > 0);
+ VERIFY_CHECK(size > 0);
+
+ /* Note that we cannot handle even numbers by negating them to be odd, as is
+ * done in other implementations, since if our scalars were specified to have
+ * width < 256 for performance reasons, their negations would have width 256
+ * and we'd lose any performance benefit. Instead, we use a technique from
+ * Section 4.2 of the Okeya/Tagaki paper, which is to add either 1 (for even)
+ * or 2 (for odd) to the number we are encoding, returning a skew value indicating
+ * this, and having the caller compensate after doing the multiplication.
+ *
+ * In fact, we _do_ want to negate numbers to minimize their bit-lengths (and in
+ * particular, to ensure that the outputs from the endomorphism-split fit into
+ * 128 bits). If we negate, the parity of our number flips, inverting which of
+ * {1, 2} we want to add to the scalar when ensuring that it's odd. Further
+ * complicating things, -1 interacts badly with `secp256k1_scalar_cadd_bit` and
+ * we need to special-case it in this logic. */
+ flip = secp256k1_scalar_is_high(scalar);
+ /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
+ bit = flip ^ !secp256k1_scalar_is_even(scalar);
+ /* We check for negative one, since adding 2 to it will cause an overflow */
+ secp256k1_scalar_negate(&s, scalar);
+ not_neg_one = !secp256k1_scalar_is_one(&s);
+ s = *scalar;
+ secp256k1_scalar_cadd_bit(&s, bit, not_neg_one);
+ /* If we had negative one, flip == 1, s.d[0] == 0, bit == 1, so caller expects
+ * that we added two to it and flipped it. In fact for -1 these operations are
+ * identical. We only flipped, but since skewing is required (in the sense that
+ * the skew must be 1 or 2, never zero) and flipping is not, we need to change
+ * our flags to claim that we only skewed. */
+ global_sign = secp256k1_scalar_cond_negate(&s, flip);
+ global_sign *= not_neg_one * 2 - 1;
+ skew = 1 << bit;
+
/* 4 */
- while (word * w < WNAF_BITS) {
- int sign;
+ u_last = secp256k1_scalar_shr_int(&s, w);
+ do {
int even;
/* 4.1 4.4 */
u = secp256k1_scalar_shr_int(&s, w);
/* 4.2 */
even = ((u & 1) == 0);
- sign = 2 * (u_last > 0) - 1;
- u += sign * even;
- u_last -= sign * even * (1 << w);
+ /* In contrast to the original algorithm, u_last is always > 0 and
+ * therefore we do not need to check its sign. In particular, it's easy
+ * to see that u_last is never < 0 because u is never < 0. Moreover,
+ * u_last is never = 0 because u is never even after a loop
+ * iteration. The same holds analogously for the initial value of
+ * u_last (in the first loop iteration). */
+ VERIFY_CHECK(u_last > 0);
+ VERIFY_CHECK((u_last & 1) == 1);
+ u += even;
+ u_last -= even * (1 << w);
/* 4.3, adapted for global sign change */
wnaf[word++] = u_last * global_sign;
u_last = u;
- }
+ } while (word * w < size);
wnaf[word] = u * global_sign;
VERIFY_CHECK(secp256k1_scalar_is_zero(&s));
- VERIFY_CHECK(word == WNAF_SIZE(w));
+ VERIFY_CHECK(word == WNAF_SIZE_BITS(size, w));
+ return skew;
}
+static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar, int size) {
+ secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
+ secp256k1_ge tmpa;
+ secp256k1_fe Z;
-static void secp256k1_ecmult_const(secp256k1_gej_t *r, const secp256k1_ge_t *a, const secp256k1_scalar_t *scalar) {
- secp256k1_ge_t pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
- secp256k1_ge_t tmpa;
- secp256k1_fe_t Z;
-
- int wnaf[1 + WNAF_SIZE(WINDOW_A - 1)];
+ int skew_1;
+#ifdef USE_ENDOMORPHISM
+ secp256k1_ge pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
+ int wnaf_lam[1 + WNAF_SIZE(WINDOW_A - 1)];
+ int skew_lam;
+ secp256k1_scalar q_1, q_lam;
+#endif
+ int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)];
int i;
- int is_zero = secp256k1_scalar_is_zero(scalar);
- secp256k1_scalar_t sc = *scalar;
- /* the wNAF ladder cannot handle zero, so bump this to one .. we will
- * correct the result after the fact */
- sc.d[0] += is_zero;
/* build wnaf representation for q. */
- secp256k1_wnaf_const(wnaf, &sc, WINDOW_A - 1);
+ int rsize = size;
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ rsize = 128;
+ /* split q into q_1 and q_lam (where q = q_1 + q_lam*lambda, and q_1 and q_lam are ~128 bit) */
+ secp256k1_scalar_split_lambda(&q_1, &q_lam, scalar);
+ skew_1 = secp256k1_wnaf_const(wnaf_1, &q_1, WINDOW_A - 1, 128);
+ skew_lam = secp256k1_wnaf_const(wnaf_lam, &q_lam, WINDOW_A - 1, 128);
+ } else
+#endif
+ {
+ skew_1 = secp256k1_wnaf_const(wnaf_1, scalar, WINDOW_A - 1, size);
+#ifdef USE_ENDOMORPHISM
+ skew_lam = 0;
+#endif
+ }
/* Calculate odd multiples of a.
* All multiples are brought to the same Z 'denominator', which is stored
*/
secp256k1_gej_set_ge(r, a);
secp256k1_ecmult_odd_multiples_table_globalz_windowa(pre_a, &Z, r);
+ for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
+ secp256k1_fe_normalize_weak(&pre_a[i].y);
+ }
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
+ secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
+ }
+
+ }
+#endif
/* first loop iteration (separated out so we can directly set r, rather
* than having it start at infinity, get doubled several times, then have
* its new value added to it) */
- i = wnaf[WNAF_SIZE(WINDOW_A - 1)];
+ i = wnaf_1[WNAF_SIZE_BITS(rsize, WINDOW_A - 1)];
VERIFY_CHECK(i != 0);
ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A);
secp256k1_gej_set_ge(r, &tmpa);
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ i = wnaf_lam[WNAF_SIZE_BITS(rsize, WINDOW_A - 1)];
+ VERIFY_CHECK(i != 0);
+ ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, i, WINDOW_A);
+ secp256k1_gej_add_ge(r, r, &tmpa);
+ }
+#endif
/* remaining loop iterations */
- for (i = WNAF_SIZE(WINDOW_A - 1) - 1; i >= 0; i--) {
+ for (i = WNAF_SIZE_BITS(rsize, WINDOW_A - 1) - 1; i >= 0; i--) {
int n;
int j;
for (j = 0; j < WINDOW_A - 1; ++j) {
- secp256k1_gej_double_nonzero(r, r, NULL);
+ secp256k1_gej_double(r, r);
}
- n = wnaf[i];
- VERIFY_CHECK(n != 0);
+
+ n = wnaf_1[i];
ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A);
+ VERIFY_CHECK(n != 0);
secp256k1_gej_add_ge(r, r, &tmpa);
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ n = wnaf_lam[i];
+ ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A);
+ VERIFY_CHECK(n != 0);
+ secp256k1_gej_add_ge(r, r, &tmpa);
+ }
+#endif
}
secp256k1_fe_mul(&r->z, &r->z, &Z);
- /* correct for zero */
- r->infinity |= is_zero;
-}
+ {
+ /* Correct for wNAF skew */
+ secp256k1_ge correction = *a;
+ secp256k1_ge_storage correction_1_stor;
+#ifdef USE_ENDOMORPHISM
+ secp256k1_ge_storage correction_lam_stor;
+#endif
+ secp256k1_ge_storage a2_stor;
+ secp256k1_gej tmpj;
+ secp256k1_gej_set_ge(&tmpj, &correction);
+ secp256k1_gej_double_var(&tmpj, &tmpj, NULL);
+ secp256k1_ge_set_gej(&correction, &tmpj);
+ secp256k1_ge_to_storage(&correction_1_stor, a);
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ secp256k1_ge_to_storage(&correction_lam_stor, a);
+ }
+#endif
+ secp256k1_ge_to_storage(&a2_stor, &correction);
+ /* For odd numbers this is 2a (so replace it), for even ones a (so no-op) */
+ secp256k1_ge_storage_cmov(&correction_1_stor, &a2_stor, skew_1 == 2);
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ secp256k1_ge_storage_cmov(&correction_lam_stor, &a2_stor, skew_lam == 2);
+ }
#endif
+
+ /* Apply the correction */
+ secp256k1_ge_from_storage(&correction, &correction_1_stor);
+ secp256k1_ge_neg(&correction, &correction);
+ secp256k1_gej_add_ge(r, r, &correction);
+
+#ifdef USE_ENDOMORPHISM
+ if (size > 128) {
+ secp256k1_ge_from_storage(&correction, &correction_lam_stor);
+ secp256k1_ge_neg(&correction, &correction);
+ secp256k1_ge_mul_lambda(&correction, &correction);
+ secp256k1_gej_add_ge(r, r, &correction);
+ }
+#endif
+ }
+}
+
+#endif /* SECP256K1_ECMULT_CONST_IMPL_H */
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