/********************************************************************** * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra * * Distributed under the MIT software license, see the accompanying * * file COPYING or http://www.opensource.org/licenses/mit-license.php.* **********************************************************************/ #ifndef _SECP256K1_ECMULT_CONST_IMPL_ #define _SECP256K1_ECMULT_CONST_IMPL_ #include "scalar.h" #include "group.h" #include "ecmult_const.h" #include "ecmult_impl.h" #ifdef USE_ENDOMORPHISM #define WNAF_BITS 128 #else #define WNAF_BITS 256 #endif #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); \ int idx_n = abs_n / 2; \ secp256k1_fe neg_y; \ VERIFY_CHECK(((n) & 1) == 1); \ VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \ VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \ VERIFY_SETUP(secp256k1_fe_clear(&(r)->x)); \ VERIFY_SETUP(secp256k1_fe_clear(&(r)->y)); \ for (m = 0; 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].x, m == idx_n); \ secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \ } \ (r)->infinity = 0; \ 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: * - 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 * * 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 * * Numbers reference steps of `Algorithm SPA-resistant Width-w NAF with Odd Scalar` on pp. 335 */ static int secp256k1_wnaf_const(int *wnaf, secp256k1_scalar s, int w) { int global_sign; int skew = 0; int word = 0; /* 1 2 3 */ int u_last; int u; int flip; int bit; secp256k1_scalar neg_s; int not_neg_one; /* 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. */ /* Negative numbers will be negated to keep their bit representation below the maximum width */ flip = secp256k1_scalar_is_high(&s); /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */ bit = flip ^ (s.d[0] & 1); /* We check for negative one, since adding 2 to it will cause an overflow */ secp256k1_scalar_negate(&neg_s, &s); not_neg_one = !secp256k1_scalar_is_one(&neg_s); 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 */ u_last = secp256k1_scalar_shr_int(&s, w); while (word * w < WNAF_BITS) { int sign; 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); /* 4.3, adapted for global sign change */ wnaf[word++] = u_last * global_sign; u_last = u; } wnaf[word] = u * global_sign; VERIFY_CHECK(secp256k1_scalar_is_zero(&s)); VERIFY_CHECK(word == WNAF_SIZE(w)); return skew; } static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar) { secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)]; secp256k1_ge tmpa; secp256k1_fe Z; int skew_1; int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 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 i; secp256k1_scalar sc = *scalar; /* build wnaf representation for q. */ #ifdef USE_ENDOMORPHISM /* 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, &sc); skew_1 = secp256k1_wnaf_const(wnaf_1, q_1, WINDOW_A - 1); skew_lam = secp256k1_wnaf_const(wnaf_lam, q_lam, WINDOW_A - 1); #else skew_1 = secp256k1_wnaf_const(wnaf_1, sc, WINDOW_A - 1); #endif /* Calculate odd multiples of a. * All multiples are brought to the same Z 'denominator', which is stored * in Z. Due to secp256k1' isomorphism we can do all operations pretending * that the Z coordinate was 1, use affine addition formulae, and correct * the Z coordinate of the result once at the end. */ 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 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_1[WNAF_SIZE(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 i = wnaf_lam[WNAF_SIZE(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--) { int n; int j; for (j = 0; j < WINDOW_A - 1; ++j) { secp256k1_gej_double_nonzero(r, r, NULL); } 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 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 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 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 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 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