Add x-only ecmult_const version for x=n/d
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Pieter Wuille
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@ -18,4 +18,25 @@
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*/
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*/
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static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q, int bits);
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static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q, int bits);
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/**
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* Same as secp256k1_ecmult_const, but takes in an x coordinate of the base point
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* only, specified as fraction n/d (numerator/denominator). Only the x coordinate of the result is
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* returned.
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*
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* If known_on_curve is 0, a verification is performed that n/d is a valid X
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* coordinate, and 0 is returned if not. Otherwise, 1 is returned.
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*
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* d being NULL is interpreted as d=1. If non-NULL, d must not be zero. q must not be zero.
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*
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* Constant time in the value of q, but not any other inputs.
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*/
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static int secp256k1_ecmult_const_xonly(
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secp256k1_fe *r,
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const secp256k1_fe *n,
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const secp256k1_fe *d,
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const secp256k1_scalar *q,
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int bits,
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int known_on_curve
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);
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#endif /* SECP256K1_ECMULT_CONST_H */
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#endif /* SECP256K1_ECMULT_CONST_H */
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@ -228,4 +228,139 @@ static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, cons
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secp256k1_fe_mul(&r->z, &r->z, &Z);
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secp256k1_fe_mul(&r->z, &r->z, &Z);
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}
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}
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static int secp256k1_ecmult_const_xonly(secp256k1_fe* r, const secp256k1_fe *n, const secp256k1_fe *d, const secp256k1_scalar *q, int bits, int known_on_curve) {
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/* This algorithm is a generalization of Peter Dettman's technique for
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* avoiding the square root in a random-basepoint x-only multiplication
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* on a Weierstrass curve:
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* https://mailarchive.ietf.org/arch/msg/cfrg/7DyYY6gg32wDgHAhgSb6XxMDlJA/
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*
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*
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* === Background: the effective affine technique ===
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*
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* Let phi_u be the isomorphism that maps (x, y) on secp256k1 curve y^2 = x^3 + 7 to
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* 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
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* the original (it is isomorphic), but moreover, has the same addition/doubling formulas, as
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* the curve b=7 coefficient does not appear in those formulas (or at least does not appear in
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* the formulas implemented in this codebase, both affine and Jacobian). See also Example 9.5.2
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* in https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf.
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*
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* This means any linear combination of secp256k1 points can be computed by applying phi_u
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* (with non-zero u) on all input points (including the generator, if used), computing the
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* linear combination on the isomorphic curve (using the same group laws), and then applying
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* phi_u^{-1} to get back to secp256k1.
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*
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* Switching to Jacobian coordinates, note that phi_u applied to (X, Y, Z) is simply
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* (X, Y, Z/u). Thus, if we want to compute (X1, Y1, Z) + (X2, Y2, Z), with identical Z
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* coordinates, we can use phi_Z to transform it to (X1, Y1, 1) + (X2, Y2, 1) on an isomorphic
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* curve where the affine addition formula can be used instead.
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* If (X3, Y3, Z3) = (X1, Y1) + (X2, Y2) on that curve, then our answer on secp256k1 is
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* (X3, Y3, Z3*Z).
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*
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* This is the effective affine technique: if we have a linear combination of group elements
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* to compute, and all those group elements have the same Z coordinate, we can simply pretend
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* that all those Z coordinates are 1, perform the computation that way, and then multiply the
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* original Z coordinate back in.
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*
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* The technique works on any a=0 short Weierstrass curve. It is possible to generalize it to
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* other curves too, but there the isomorphic curves will have different 'a' coefficients,
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* which typically does affect the group laws.
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*
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*
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* === Avoiding the square root for x-only point multiplication ===
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*
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* In this function, we want to compute the X coordinate of q*(n/d, y), for
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* y = sqrt((n/d)^3 + 7). Its negation would also be a valid Y coordinate, but by convention
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* we pick whatever sqrt returns (which we assume to be a deterministic function).
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*
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* Let g = y^2*d^3 = n^3 + 7*d^3. This also means y = sqrt(g/d^3).
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* Further let v = sqrt(d*g), which must exist as d*g = y^2*d^4 = (y*d^2)^2.
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*
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* The input point (n/d, y) also has Jacobian coordinates:
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*
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* (n/d, y, 1)
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* = (n/d * v^2, y * v^3, v)
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* = (n/d * d*g, y * sqrt(d^3*g^3), v)
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* = (n/d * d*g, sqrt(y^2 * d^3*g^3), v)
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* = (n*g, sqrt(g/d^3 * d^3*g^3), v)
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* = (n*g, sqrt(g^4), v)
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* = (n*g, g^2, v)
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*
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* It is easy to verify that both (n*g, g^2, v) and its negation (n*g, -g^2, v) have affine X
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* coordinate n/d, and this holds even when the square root function doesn't have a
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* determinstic sign. We choose the (n*g, g^2, v) version.
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*
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* Now switch to the effective affine curve using phi_v, where the input point has coordinates
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* (n*g, g^2). Compute (X, Y, Z) = q * (n*g, g^2) there.
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*
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* Back on secp256k1, that means q * (n*g, g^2, v) = (X, Y, v*Z). This last point has affine X
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* coordinate X / (v^2*Z^2) = X / (d*g*Z^2). Determining the affine Y coordinate would involve
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* a square root, but as long as we only care about the resulting X coordinate, no square root
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* is needed anywhere in this computation.
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*/
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secp256k1_fe g, i;
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secp256k1_ge p;
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secp256k1_gej rj;
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/* Compute g = (n^3 + B*d^3). */
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secp256k1_fe_sqr(&g, n);
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secp256k1_fe_mul(&g, &g, n);
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if (d) {
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secp256k1_fe b;
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#ifdef VERIFY
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VERIFY_CHECK(!secp256k1_fe_normalizes_to_zero(d));
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#endif
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secp256k1_fe_sqr(&b, d);
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VERIFY_CHECK(SECP256K1_B <= 8); /* magnitude of b will be <= 8 after the next call */
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secp256k1_fe_mul_int(&b, SECP256K1_B);
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secp256k1_fe_mul(&b, &b, d);
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secp256k1_fe_add(&g, &b);
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if (!known_on_curve) {
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/* We need to determine whether (n/d)^3 + 7 is square.
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*
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* is_square((n/d)^3 + 7)
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* <=> is_square(((n/d)^3 + 7) * d^4)
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* <=> is_square((n^3 + 7*d^3) * d)
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* <=> is_square(g * d)
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*/
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secp256k1_fe c;
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secp256k1_fe_mul(&c, &g, d);
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if (!secp256k1_fe_is_square_var(&c)) return 0;
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}
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} else {
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secp256k1_fe_add_int(&g, SECP256K1_B);
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if (!known_on_curve) {
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/* g at this point equals x^3 + 7. Test if it is square. */
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if (!secp256k1_fe_is_square_var(&g)) return 0;
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}
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}
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/* Compute base point P = (n*g, g^2), the effective affine version of (n*g, g^2, v), which has
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* corresponding affine X coordinate n/d. */
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secp256k1_fe_mul(&p.x, &g, n);
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secp256k1_fe_sqr(&p.y, &g);
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p.infinity = 0;
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/* Perform x-only EC multiplication of P with q. */
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#ifdef VERIFY
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VERIFY_CHECK(!secp256k1_scalar_is_zero(q));
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#endif
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secp256k1_ecmult_const(&rj, &p, q, bits);
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#ifdef VERIFY
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VERIFY_CHECK(!secp256k1_gej_is_infinity(&rj));
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#endif
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/* The resulting (X, Y, Z) point on the effective-affine isomorphic curve corresponds to
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* (X, Y, Z*v) on the secp256k1 curve. The affine version of that has X coordinate
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* (X / (Z^2*d*g)). */
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secp256k1_fe_sqr(&i, &rj.z);
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secp256k1_fe_mul(&i, &i, &g);
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if (d) secp256k1_fe_mul(&i, &i, d);
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secp256k1_fe_inv(&i, &i);
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secp256k1_fe_mul(r, &rj.x, &i);
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return 1;
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}
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#endif /* SECP256K1_ECMULT_CONST_IMPL_H */
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#endif /* SECP256K1_ECMULT_CONST_IMPL_H */
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63
src/tests.c
63
src/tests.c
@ -4452,6 +4452,68 @@ static void ecmult_const_mult_zero_one(void) {
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ge_equals_ge(&res2, &point);
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ge_equals_ge(&res2, &point);
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}
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}
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static void ecmult_const_mult_xonly(void) {
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int i;
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/* Test correspondence between secp256k1_ecmult_const and secp256k1_ecmult_const_xonly. */
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for (i = 0; i < 2*COUNT; ++i) {
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secp256k1_ge base;
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secp256k1_gej basej, resj;
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secp256k1_fe n, d, resx, v;
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secp256k1_scalar q;
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int res;
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/* Random base point. */
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random_group_element_test(&base);
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/* Random scalar to multiply it with. */
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random_scalar_order_test(&q);
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/* If i is odd, n=d*base.x for random non-zero d */
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if (i & 1) {
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do {
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random_field_element_test(&d);
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} while (secp256k1_fe_normalizes_to_zero_var(&d));
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secp256k1_fe_mul(&n, &base.x, &d);
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} else {
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n = base.x;
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}
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/* Perform x-only multiplication. */
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res = secp256k1_ecmult_const_xonly(&resx, &n, (i & 1) ? &d : NULL, &q, 256, i & 2);
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CHECK(res);
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/* Perform normal multiplication. */
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secp256k1_gej_set_ge(&basej, &base);
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secp256k1_ecmult(&resj, &basej, &q, NULL);
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/* Check that resj's X coordinate corresponds with resx. */
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secp256k1_fe_sqr(&v, &resj.z);
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secp256k1_fe_mul(&v, &v, &resx);
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CHECK(check_fe_equal(&v, &resj.x));
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}
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/* Test that secp256k1_ecmult_const_xonly correctly rejects X coordinates not on curve. */
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for (i = 0; i < 2*COUNT; ++i) {
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secp256k1_fe x, n, d, c, r;
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int res;
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secp256k1_scalar q;
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random_scalar_order_test(&q);
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/* Generate random X coordinate not on the curve. */
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do {
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random_field_element_test(&x);
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secp256k1_fe_sqr(&c, &x);
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secp256k1_fe_mul(&c, &c, &x);
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secp256k1_fe_add(&c, &secp256k1_fe_const_b);
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} while (secp256k1_fe_is_square_var(&c));
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/* If i is odd, n=d*x for random non-zero d. */
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if (i & 1) {
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do {
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random_field_element_test(&d);
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} while (secp256k1_fe_normalizes_to_zero_var(&d));
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secp256k1_fe_mul(&n, &x, &d);
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} else {
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n = x;
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}
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res = secp256k1_ecmult_const_xonly(&r, &n, (i & 1) ? &d : NULL, &q, 256, 0);
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CHECK(res == 0);
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}
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}
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static void ecmult_const_chain_multiply(void) {
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static void ecmult_const_chain_multiply(void) {
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/* Check known result (randomly generated test problem from sage) */
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/* Check known result (randomly generated test problem from sage) */
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const secp256k1_scalar scalar = SECP256K1_SCALAR_CONST(
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const secp256k1_scalar scalar = SECP256K1_SCALAR_CONST(
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@ -4483,6 +4545,7 @@ static void run_ecmult_const_tests(void) {
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ecmult_const_random_mult();
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ecmult_const_random_mult();
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ecmult_const_commutativity();
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ecmult_const_commutativity();
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ecmult_const_chain_multiply();
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ecmult_const_chain_multiply();
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ecmult_const_mult_xonly();
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}
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}
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typedef struct {
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typedef struct {
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