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nostril/ext/secp256k1/src/ecmult_const_impl.h

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/***********************************************************************
* Copyright (c) 2015 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"
/** 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.
*/
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)];
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);
}
/* 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; \
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)); \
/* 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].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..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 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-Verlag 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, const secp256k1_scalar *scalar, int w, int size) {
int global_sign;
int skew;
int word = 0;
/* 1 2 3 */
int u_last;
int u;
int flip;
secp256k1_scalar s = *scalar;
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 variation of a
* technique from Section 4.2 of the Okeya/Tagaki paper, which is to add 1 to the
* number we are encoding when it is even, 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, affecting whether
* we want to add to the scalar to ensure that it's odd. */
flip = secp256k1_scalar_is_high(&s);
skew = flip ^ secp256k1_scalar_is_even(&s);
secp256k1_scalar_cadd_bit(&s, 0, skew);
global_sign = secp256k1_scalar_cond_negate(&s, flip);
/* 4 */
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);
/* 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_BITS(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;
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)];
int i;
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);
/* 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.
*/
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_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;
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);
}
secp256k1_fe_mul(&r->z, &r->z, &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
* determinstic 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;
#ifdef VERIFY
VERIFY_CHECK(!secp256k1_fe_normalizes_to_zero(d));
#endif
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;
}
}
/* 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. */
#ifdef VERIFY
VERIFY_CHECK(!secp256k1_scalar_is_zero(q));
#endif
secp256k1_ecmult_const(&rj, &p, q);
#ifdef VERIFY
VERIFY_CHECK(!secp256k1_gej_is_infinity(&rj));
#endif
/* 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 */
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