secp256k1

src/ecmult_const_impl.h

/***********************************************************************
* 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.*
***********************************************************************/
#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"
#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 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_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_CONST_TABLE_SIZE, pre, zr, globalz, a);
secp256k1_ge_table_set_globalz(ECMULT_CONST_TABLE_SIZE, pre, zr);
}
/* 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) < (1U << ECMULT_CONST_GROUP_SIZE)); \
VERIFY_CHECK(index < (1U << (ECMULT_CONST_GROUP_SIZE - 1))); \
/* 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_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 == 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, negative); \
} while(0)
/* 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
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;
}
/* 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);
#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_limb32(&v1, i, 1) == 0);
VERIFY_CHECK(secp256k1_scalar_get_bits_limb32(&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 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.
*/
secp256k1_gej_set_ge(r, a);
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]);
}
/* 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;
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);
}
/* 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) {
/* This algorithm is a generalization of Peter Dettman's technique for
* avoiding the square root in a random-basepoint x-only multiplication
* on a Weierstrass curve:
* https://mailarchive.ietf.org/arch/msg/cfrg/7DyYY6gg32wDgHAhgSb6XxMDlJA/
*
*
* === Background: the effective affine technique ===
*
* Let phi_u be the isomorphism that maps (x, y) on secp256k1 curve y^2 = x^3 + 7 to
* x' = u^2*x, y' = u^3*y on curve y'^2 = x'^3 + u^6*7. This new curve has the same order as
* the original (it is isomorphic), but moreover, has the same addition/doubling formulas, as
* the curve b=7 coefficient does not appear in those formulas (or at least does not appear in
* the formulas implemented in this codebase, both affine and Jacobian). See also Example 9.5.2
* in https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf.
*
* This means any linear combination of secp256k1 points can be computed by applying phi_u
* (with non-zero u) on all input points (including the generator, if used), computing the
* linear combination on the isomorphic curve (using the same group laws), and then applying
* phi_u^{-1} to get back to secp256k1.
*
* Switching to Jacobian coordinates, note that phi_u applied to (X, Y, Z) is simply
* (X, Y, Z/u). Thus, if we want to compute (X1, Y1, Z) + (X2, Y2, Z), with identical Z
* coordinates, we can use phi_Z to transform it to (X1, Y1, 1) + (X2, Y2, 1) on an isomorphic
* curve where the affine addition formula can be used instead.
* If (X3, Y3, Z3) = (X1, Y1) + (X2, Y2) on that curve, then our answer on secp256k1 is
* (X3, Y3, Z3*Z).
*
* This is the effective affine technique: if we have a linear combination of group elements
* to compute, and all those group elements have the same Z coordinate, we can simply pretend
* that all those Z coordinates are 1, perform the computation that way, and then multiply the
* original Z coordinate back in.
*
* The technique works on any a=0 short Weierstrass curve. It is possible to generalize it to
* other curves too, but there the isomorphic curves will have different 'a' coefficients,
* which typically does affect the group laws.
*
*
* === Avoiding the square root for x-only point multiplication ===
*
* In this function, we want to compute the X coordinate of q*(n/d, y), for
* y = sqrt((n/d)^3 + 7). Its negation would also be a valid Y coordinate, but by convention
* we pick whatever sqrt returns (which we assume to be a deterministic function).
*
* Let g = y^2*d^3 = n^3 + 7*d^3. This also means y = sqrt(g/d^3).
* Further let v = sqrt(d*g), which must exist as d*g = y^2*d^4 = (y*d^2)^2.
*
* The input point (n/d, y) also has Jacobian coordinates:
*
* (n/d, y, 1)
* = (n/d * v^2, y * v^3, v)
* = (n/d * d*g, y * sqrt(d^3*g^3), v)
* = (n/d * d*g, sqrt(y^2 * d^3*g^3), v)
* = (n*g, sqrt(g/d^3 * d^3*g^3), v)
* = (n*g, sqrt(g^4), v)
* = (n*g, g^2, v)
*
* It is easy to verify that both (n*g, g^2, v) and its negation (n*g, -g^2, v) have affine X
* coordinate n/d, and this holds even when the square root function doesn't have a
* deterministic sign. We choose the (n*g, g^2, v) version.
*
* Now switch to the effective affine curve using phi_v, where the input point has coordinates
* (n*g, g^2). Compute (X, Y, Z) = q * (n*g, g^2) there.
*
* Back on secp256k1, that means q * (n*g, g^2, v) = (X, Y, v*Z). This last point has affine X
* coordinate X / (v^2*Z^2) = X / (d*g*Z^2). Determining the affine Y coordinate would involve
* a square root, but as long as we only care about the resulting X coordinate, no square root
* is needed anywhere in this computation.
*/
secp256k1_fe g, i;
secp256k1_ge p;
secp256k1_gej rj;
/* Compute g = (n^3 + B*d^3). */
secp256k1_fe_sqr(&g, n);
secp256k1_fe_mul(&g, &g, n);
if (d) {
secp256k1_fe b;
VERIFY_CHECK(!secp256k1_fe_normalizes_to_zero(d));
secp256k1_fe_sqr(&b, d);
VERIFY_CHECK(SECP256K1_B <= 8); /* magnitude of b will be <= 8 after the next call */
secp256k1_fe_mul_int(&b, SECP256K1_B);
secp256k1_fe_mul(&b, &b, d);
secp256k1_fe_add(&g, &b);
if (!known_on_curve) {
/* We need to determine whether (n/d)^3 + 7 is square.
*
* is_square((n/d)^3 + 7)
* <=> is_square(((n/d)^3 + 7) * d^4)
* <=> is_square((n^3 + 7*d^3) * d)
* <=> is_square(g * d)
*/
secp256k1_fe c;
secp256k1_fe_mul(&c, &g, d);
if (!secp256k1_fe_is_square_var(&c)) return 0;
}
} else {
secp256k1_fe_add_int(&g, SECP256K1_B);
if (!known_on_curve) {
/* g at this point equals x^3 + 7. Test if it is square. */
if (!secp256k1_fe_is_square_var(&g)) return 0;
}
}
SECP256K1_FE_VERIFY_MAGNITUDE(&g, 2);
/* Compute base point P = (n*g, g^2), the effective affine version of (n*g, g^2, v), which has
* corresponding affine X coordinate n/d. */
secp256k1_fe_mul(&p.x, &g, n);
secp256k1_fe_sqr(&p.y, &g);
p.infinity = 0;
/* Perform x-only EC multiplication of P with q. */
VERIFY_CHECK(!secp256k1_scalar_is_zero(q));
secp256k1_ecmult_const(&rj, &p, q);
VERIFY_CHECK(!secp256k1_gej_is_infinity(&rj));
/* The resulting (X, Y, Z) point on the effective-affine isomorphic curve corresponds to
* (X, Y, Z*v) on the secp256k1 curve. The affine version of that has X coordinate
* (X / (Z^2*d*g)). */
secp256k1_fe_sqr(&i, &rj.z);
secp256k1_fe_mul(&i, &i, &g);
if (d) secp256k1_fe_mul(&i, &i, d);
secp256k1_fe_inv(&i, &i);
secp256k1_fe_mul(r, &rj.x, &i);
return 1;
}
#endif /* SECP256K1_ECMULT_CONST_IMPL_H */