From 4d16e90111c050de3b7e25ac451d87cd4e3f874e Mon Sep 17 00:00:00 2001 From: Pieter Wuille Date: Fri, 30 Dec 2022 14:43:37 -0500 Subject: [PATCH] Signed-digit based ecmult_const algorithm --- src/ecmult_const_impl.h | 302 ++++++++++++++++++++++++++++------------ src/scalar.h | 2 +- 2 files changed, 217 insertions(+), 87 deletions(-) diff --git a/src/ecmult_const_impl.h b/src/ecmult_const_impl.h index 06f9e53ffdbd0..f166628ca825c 100644 --- a/src/ecmult_const_impl.h +++ b/src/ecmult_const_impl.h @@ -1,5 +1,5 @@ /*********************************************************************** - * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra * + * Copyright (c) 2015, 2022 Pieter Wuille, Andrew Poelstra * * Distributed under the MIT software license, see the accompanying * * file COPYING or https://www.opensource.org/licenses/mit-license.php.* ***********************************************************************/ @@ -12,45 +12,96 @@ #include "ecmult_const.h" #include "ecmult_impl.h" +#if defined(EXHAUSTIVE_TEST_ORDER) +/* We need 2^ECMULT_CONST_GROUP_SIZE - 1 to be less than EXHAUSTIVE_TEST_ORDER, because + * the tables cannot have infinities in them (this breaks the effective-affine technique's + * z-ratio tracking) */ +# if EXHAUSTIVE_TEST_ORDER == 199 +# define ECMULT_CONST_GROUP_SIZE 4 +# elif EXHAUSTIVE_TEST_ORDER == 13 +# define ECMULT_CONST_GROUP_SIZE 3 +# elif EXHAUSTIVE_TEST_ORDER == 7 +# define ECMULT_CONST_GROUP_SIZE 2 +# else +# error "Unknown EXHAUSTIVE_TEST_ORDER" +# endif +#else +/* Group size 4 or 5 appears optimal. */ +# define ECMULT_CONST_GROUP_SIZE 5 +#endif + +#define ECMULT_CONST_TABLE_SIZE (1L << (ECMULT_CONST_GROUP_SIZE - 1)) +#define ECMULT_CONST_GROUPS ((129 + ECMULT_CONST_GROUP_SIZE - 1) / ECMULT_CONST_GROUP_SIZE) +#define ECMULT_CONST_BITS (ECMULT_CONST_GROUPS * ECMULT_CONST_GROUP_SIZE) + /** Fill a table 'pre' with precomputed odd multiples of a. * * The resulting point set is brought to a single constant Z denominator, stores the X and Y - * coordinates as ge_storage points in pre, and stores the global Z in globalz. - * It only operates on tables sized for WINDOW_A wnaf multiples. + * coordinates as ge points in pre, and stores the global Z in globalz. + * + * 'pre' must be an array of size ECMULT_CONST_TABLE_SIZE. */ -static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) { - secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)]; +static void secp256k1_ecmult_const_odd_multiples_table_globalz(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) { + secp256k1_fe zr[ECMULT_CONST_TABLE_SIZE]; - secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr, globalz, a); - secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr); + secp256k1_ecmult_odd_multiples_table(ECMULT_CONST_TABLE_SIZE, pre, zr, globalz, a); + secp256k1_ge_table_set_globalz(ECMULT_CONST_TABLE_SIZE, pre, zr); } -/* This is like `ECMULT_TABLE_GET_GE` but is constant time */ -#define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \ - int m = 0; \ - /* Extract the sign-bit for a constant time absolute-value. */ \ - int volatile mask = (n) >> (sizeof(n) * CHAR_BIT - 1); \ - int abs_n = ((n) + mask) ^ mask; \ - int idx_n = abs_n >> 1; \ +/* Given a table 'pre' with odd multiples of a point, put in r the signed-bit multiplication of n with that point. + * + * For example, if ECMULT_CONST_GROUP_SIZE is 4, then pre is expected to contain 8 entries: + * [1*P, 3*P, 5*P, 7*P, 9*P, 11*P, 13*P, 15*P]. n is then expected to be a 4-bit integer (range 0-15), and its + * bits are interpreted as signs of powers of two to look up. + * + * For example, if n=4, which is 0100 in binary, which is interpreted as [- + - -], so the looked up value is + * [ -(2^3) + (2^2) - (2^1) - (2^0) ]*P = -7*P. Every valid n translates to an odd number in range [-15,15], + * which means we just need to look up one of the precomputed values, and optionally negate it. + */ +#define ECMULT_CONST_TABLE_GET_GE(r,pre,n) do { \ + unsigned int m = 0; \ + /* If the top bit of n is 0, we want the negation. */ \ + volatile unsigned int negative = ((n) >> (ECMULT_CONST_GROUP_SIZE - 1)) ^ 1; \ + /* Let n[i] be the i-th bit of n, then the index is + * sum(cnot(n[i]) * 2^i, i=0..l-2) + * where cnot(b) = b if n[l-1] = 1 and 1 - b otherwise. + * For example, if n = 4, in binary 0100, the index is 3, in binary 011. + * + * Proof: + * Let + * x = sum((2*n[i] - 1)*2^i, i=0..l-1) + * = 2*sum(n[i] * 2^i, i=0..l-1) - 2^l + 1 + * be the value represented by n. + * The index is (x - 1)/2 if x > 0 and -(x + 1)/2 otherwise. + * Case x > 0: + * n[l-1] = 1 + * index = sum(n[i] * 2^i, i=0..l-1) - 2^(l-1) + * = sum(n[i] * 2^i, i=0..l-2) + * Case x <= 0: + * n[l-1] = 0 + * index = -(2*sum(n[i] * 2^i, i=0..l-1) - 2^l + 2)/2 + * = 2^(l-1) - 1 - sum(n[i] * 2^i, i=0..l-1) + * = sum((1 - n[i]) * 2^i, i=0..l-2) + */ \ + unsigned int index = ((unsigned int)(-negative) ^ n) & ((1U << (ECMULT_CONST_GROUP_SIZE - 1)) - 1U); \ secp256k1_fe neg_y; \ - VERIFY_CHECK(((n) & 1) == 1); \ - VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \ - VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \ + VERIFY_CHECK((n) < (1U << ECMULT_CONST_GROUP_SIZE)); \ + VERIFY_CHECK(index < (1U << (ECMULT_CONST_GROUP_SIZE - 1))); \ 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 \ + /* 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++) { \ + for (m = 1; m < ECMULT_CONST_TABLE_SIZE; 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); \ + secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == index); \ + secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == index); \ } \ (r)->infinity = 0; \ secp256k1_fe_negate(&neg_y, &(r)->y, 1); \ - secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \ + secp256k1_fe_cmov(&(r)->y, &neg_y, negative); \ } while(0) /** Convert a number to WNAF notation. @@ -130,90 +181,169 @@ static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int 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; - 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; - int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)]; +/* For K as defined in the comment of secp256k1_ecmult_const, we have several precomputed + * formulas/constants. + * - in exhaustive test mode, we give an explicit expression to compute it at compile time: */ +#ifdef EXHAUSTIVE_TEST_ORDER +static const secp256k1_scalar secp256k1_ecmult_const_K = ((SECP256K1_SCALAR_CONST(0, 0, 0, (1U << (ECMULT_CONST_BITS - 128)) - 2U, 0, 0, 0, 0) + EXHAUSTIVE_TEST_ORDER - 1U) * (1U + EXHAUSTIVE_TEST_LAMBDA)) % EXHAUSTIVE_TEST_ORDER; +/* - for the real secp256k1 group we have constants for various ECMULT_CONST_BITS values. */ +#elif ECMULT_CONST_BITS == 129 +/* For GROUP_SIZE = 1,3. */ +static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0xac9c52b3ul, 0x3fa3cf1ful, 0x5ad9e3fdul, 0x77ed9ba4ul, 0xa880b9fcul, 0x8ec739c2ul, 0xe0cfc810ul, 0xb51283ceul); +#elif ECMULT_CONST_BITS == 130 +/* For GROUP_SIZE = 2,5. */ +static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0xa4e88a7dul, 0xcb13034eul, 0xc2bdd6bful, 0x7c118d6bul, 0x589ae848ul, 0x26ba29e4ul, 0xb5c2c1dcul, 0xde9798d9ul); +#elif ECMULT_CONST_BITS == 132 +/* For GROUP_SIZE = 4,6 */ +static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0x76b1d93dul, 0x0fae3c6bul, 0x3215874bul, 0x94e93813ul, 0x7937fe0dul, 0xb66bcaaful, 0xb3749ca5ul, 0xd7b6171bul); +#else +# error "Unknown ECMULT_CONST_BITS" +#endif - int i; +static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q) { + /* The approach below combines the signed-digit logic from Mike Hamburg's + * "Fast and compact elliptic-curve cryptography" (https://eprint.iacr.org/2012/309) + * Section 3.3, with the GLV endomorphism. + * + * The idea there is to interpret the bits of a scalar as signs (1 = +, 0 = -), and compute a + * point multiplication in that fashion. Let v be an n-bit non-negative integer (0 <= v < 2^n), + * and v[i] its i'th bit (so v = sum(v[i] * 2^i, i=0..n-1)). Then define: + * + * C_l(v, A) = sum((2*v[i] - 1) * 2^i*A, i=0..l-1) + * + * Then it holds that C_l(v, A) = sum((2*v[i] - 1) * 2^i*A, i=0..l-1) + * = (2*sum(v[i] * 2^i, i=0..l-1) + 1 - 2^l) * A + * = (2*v + 1 - 2^l) * A + * + * Thus, one can compute q*A as C_256((q + 2^256 - 1) / 2, A). This is the basis for the + * paper's signed-digit multi-comb algorithm for multiplication using a precomputed table. + * + * It is appealing to try to combine this with the GLV optimization: the idea that a scalar + * s can be written as s1 + lambda*s2, where lambda is a curve-specific constant such that + * lambda*A is easy to compute, and where s1 and s2 are small. In particular we have the + * secp256k1_scalar_split_lambda function which performs such a split with the resulting s1 + * and s2 in range (-2^128, 2^128) mod n. This does work, but is uninteresting: + * + * To compute q*A: + * - Let s1, s2 = split_lambda(q) + * - Let R1 = C_256((s1 + 2^256 - 1) / 2, A) + * - Let R2 = C_256((s2 + 2^256 - 1) / 2, lambda*A) + * - Return R1 + R2 + * + * The issue is that while s1 and s2 are small-range numbers, (s1 + 2^256 - 1) / 2 (mod n) + * and (s2 + 2^256 - 1) / 2 (mod n) are not, undoing the benefit of the splitting. + * + * To make it work, we want to modify the input scalar q first, before splitting, and then only + * add a 2^128 offset of the split results (so that they end up in the single 129-bit range + * [0,2^129]). A slightly smaller offset would work due to the bounds on the split, but we pick + * 2^128 for simplicity. Let s be the scalar fed to split_lambda, and f(q) the function to + * compute it from q: + * + * To compute q*A: + * - Compute s = f(q) + * - Let s1, s2 = split_lambda(s) + * - Let v1 = s1 + 2^128 (mod n) + * - Let v2 = s2 + 2^128 (mod n) + * - Let R1 = C_l(v1, A) + * - Let R2 = C_l(v2, lambda*A) + * - Return R1 + R2 + * + * l will thus need to be at least 129, but we may overshoot by a few bits (see + * further), so keep it as a variable. + * + * To solve for s, we reason: + * q*A = R1 + R2 + * <=> q*A = C_l(s1 + 2^128, A) + C_l(s2 + 2^128, lambda*A) + * <=> q*A = (2*(s1 + 2^128) + 1 - 2^l) * A + (2*(s2 + 2^128) + 1 - 2^l) * lambda*A + * <=> q*A = (2*(s1 + s2*lambda) + (2^129 + 1 - 2^l) * (1 + lambda)) * A + * <=> q = 2*(s1 + s2*lambda) + (2^129 + 1 - 2^l) * (1 + lambda) (mod n) + * <=> q = 2*s + (2^129 + 1 - 2^l) * (1 + lambda) (mod n) + * <=> s = (q + (2^l - 2^129 - 1) * (1 + lambda)) / 2 (mod n) + * <=> f(q) = (q + K) / 2 (mod n) + * where K = (2^l - 2^129 - 1)*(1 + lambda) (mod n) + * + * We will process the computation of C_l(v1, A) and C_l(v2, lambda*A) in groups of + * ECMULT_CONST_GROUP_SIZE, so we set l to the smallest multiple of ECMULT_CONST_GROUP_SIZE + * that is not less than 129; this equals ECMULT_CONST_BITS. + */ + /* The offset to add to s1 and s2 to make them non-negative. Equal to 2^128. */ + static const secp256k1_scalar S_OFFSET = SECP256K1_SCALAR_CONST(0, 0, 0, 1, 0, 0, 0, 0); + secp256k1_scalar s, v1, v2; + secp256k1_ge pre_a[ECMULT_CONST_TABLE_SIZE]; + secp256k1_ge pre_a_lam[ECMULT_CONST_TABLE_SIZE]; + secp256k1_fe global_z; + int group, i; + + /* We're allowed to be non-constant time in the point, and the code below (in particular, + * secp256k1_ecmult_const_odd_multiples_table_globalz) cannot deal with infinity in a + * constant-time manner anyway. */ if (secp256k1_ge_is_infinity(a)) { secp256k1_gej_set_infinity(r); return; } - /* build wnaf representation for q. */ - /* 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); + /* Compute v1 and v2. */ + secp256k1_scalar_add(&s, q, &secp256k1_ecmult_const_K); + secp256k1_scalar_half(&s, &s); + secp256k1_scalar_split_lambda(&v1, &v2, &s); + secp256k1_scalar_add(&v1, &v1, &S_OFFSET); + secp256k1_scalar_add(&v2, &v2, &S_OFFSET); - /* Calculate odd multiples of a. +#ifdef VERIFY + /* Verify that v1 and v2 are in range [0, 2^129-1]. */ + for (i = 129; i < 256; ++i) { + VERIFY_CHECK(secp256k1_scalar_get_bits(&v1, i, 1) == 0); + VERIFY_CHECK(secp256k1_scalar_get_bits(&v2, i, 1) == 0); + } +#endif + + /* Calculate odd multiples of A and A*lambda. * All multiples are brought to the same Z 'denominator', which is stored - * in Z. Due to secp256k1' isomorphism we can do all operations pretending + * in global_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. */ - VERIFY_CHECK(!a->infinity); 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); - } - for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) { + secp256k1_ecmult_const_odd_multiples_table_globalz(pre_a, &global_z, r); + for (i = 0; i < ECMULT_CONST_TABLE_SIZE; i++) { secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]); } - /* 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_BITS(128, WINDOW_A - 1)]; - VERIFY_CHECK(i != 0); - ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A); - secp256k1_gej_set_ge(r, &tmpa); - i = wnaf_lam[WNAF_SIZE_BITS(128, 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); - /* remaining loop iterations */ - for (i = WNAF_SIZE_BITS(128, WINDOW_A - 1) - 1; i >= 0; i--) { - int n; + /* Next, we compute r = C_l(v1, A) + C_l(v2, lambda*A). + * + * We proceed in groups of ECMULT_CONST_GROUP_SIZE bits, operating on that many bits + * at a time, from high in v1, v2 to low. Call these bits1 (from v1) and bits2 (from v2). + * + * Now note that ECMULT_CONST_TABLE_GET_GE(&t, pre_a, bits1) loads into t a point equal + * to C_{ECMULT_CONST_GROUP_SIZE}(bits1, A), and analogously for pre_lam_a / bits2. + * This means that all we need to do is add these looked up values together, multiplied + * by 2^(ECMULT_GROUP_SIZE * group). + */ + for (group = ECMULT_CONST_GROUPS - 1; group >= 0; --group) { + /* Using the _var get_bits function is ok here, since it's only variable in offset and count, not in the scalar. */ + unsigned int bits1 = secp256k1_scalar_get_bits_var(&v1, group * ECMULT_CONST_GROUP_SIZE, ECMULT_CONST_GROUP_SIZE); + unsigned int bits2 = secp256k1_scalar_get_bits_var(&v2, group * ECMULT_CONST_GROUP_SIZE, ECMULT_CONST_GROUP_SIZE); + secp256k1_ge t; int j; - for (j = 0; j < WINDOW_A - 1; ++j) { - secp256k1_gej_double(r, r); - } - 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); - 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); - } - - { - /* Correct for wNAF skew */ - secp256k1_gej tmpj; - - secp256k1_ge_neg(&tmpa, &pre_a[0]); - secp256k1_gej_add_ge(&tmpj, r, &tmpa); - secp256k1_gej_cmov(r, &tmpj, skew_1); - - secp256k1_ge_neg(&tmpa, &pre_a_lam[0]); - secp256k1_gej_add_ge(&tmpj, r, &tmpa); - secp256k1_gej_cmov(r, &tmpj, skew_lam); + ECMULT_CONST_TABLE_GET_GE(&t, pre_a, bits1); + if (group == ECMULT_CONST_GROUPS - 1) { + /* Directly set r in the first iteration. */ + secp256k1_gej_set_ge(r, &t); + } else { + /* Shift the result so far up. */ + for (j = 0; j < ECMULT_CONST_GROUP_SIZE; ++j) { + secp256k1_gej_double(r, r); + } + secp256k1_gej_add_ge(r, r, &t); + } + ECMULT_CONST_TABLE_GET_GE(&t, pre_a_lam, bits2); + secp256k1_gej_add_ge(r, r, &t); } - secp256k1_fe_mul(&r->z, &r->z, &Z); + /* Map the result back to the secp256k1 curve from the isomorphic curve. */ + secp256k1_fe_mul(&r->z, &r->z, &global_z); } static int secp256k1_ecmult_const_xonly(secp256k1_fe* r, const secp256k1_fe *n, const secp256k1_fe *d, const secp256k1_scalar *q, int known_on_curve) { diff --git a/src/scalar.h b/src/scalar.h index a188c1eb062d2..ed67b6fdd1ad3 100644 --- a/src/scalar.h +++ b/src/scalar.h @@ -25,7 +25,7 @@ static void secp256k1_scalar_clear(secp256k1_scalar *r); /** Access bits from a scalar. All requested bits must belong to the same 32-bit limb. */ static unsigned int secp256k1_scalar_get_bits(const secp256k1_scalar *a, unsigned int offset, unsigned int count); -/** Access bits from a scalar. Not constant time. */ +/** Access bits from a scalar. Not constant time in offset and count. */ static unsigned int secp256k1_scalar_get_bits_var(const secp256k1_scalar *a, unsigned int offset, unsigned int count); /** Set a scalar from a big endian byte array. The scalar will be reduced modulo group order `n`.