Merge bitcoin-core/secp256k1#1118: Add x-only ecmult_const version with x specified as n/d

0f8642079b0f2e4874393792f5854e3c33742cbd Add exhaustive tests for ecmult_const_xonly (Pieter Wuille)
4485926ace489d87929be5218ae1ff3aa8591006 Add x-only ecmult_const version for x=n/d (Pieter Wuille)

Pull request description:

  This implements a generalization of Peter Dettman's sqrt-less x-only random-base multiplication algorithm from #262, using the Jacobi symbol algorithm from #979. The generalization is to permit the X coordinate of the base point to be specified as a fraction $n/d$:

  To compute $x(q \cdot P)$, where $x(P) = n/d$:
  * Compute $g=n^3 + 7d^3$.
  * Let $P' = (ng, g^2, 1)$ (the Jacobian coordinates of $P$ mapped to the isomorphic curve $y^2 = x^3 + 7(dg)^3$).
  * Compute the Jacobian coordinates $(X',Y',Z') = q \cdot P'$ on the isomorphic curve.
  * Return $X'/(dgZ'^2)$, which is the affine x coordinate on the isomorphic curve $X/Z'^2$ mapped back to secp256k1.

  This ability to specify the X coordinate as a fraction is useful in the context of x-only [Elligator Swift](https://eprint.iacr.org/2022/759), which can decode to X coordinates on the curve without inversions this way.

ACKs for top commit:
  jonasnick:
    ACK 0f8642079b0f2e4874393792f5854e3c33742cbd
  real-or-random:
    ACK 0f8642079b0f2e4874393792f5854e3c33742cbd

Tree-SHA512: eeedb3045bfabcb4bcaf3a1738067c83a5ea9a79b150b8fd1c00dc3f68505d34c19654885a90e2292ae40ddf40a58dfb27197d98eebcf5d6d9e25897e07ae595

src/ecmult_const.h

@@ -18,4 +18,25 @@
  */
 static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q, int bits);
 
+/**
+ * Same as secp256k1_ecmult_const, but takes in an x coordinate of the base point
+ * only, specified as fraction n/d (numerator/denominator). Only the x coordinate of the result is
+ * returned.
+ *
+ * If known_on_curve is 0, a verification is performed that n/d is a valid X
+ * coordinate, and 0 is returned if not. Otherwise, 1 is returned.
+ *
+ * d being NULL is interpreted as d=1. If non-NULL, d must not be zero. q must not be zero.
+ *
+ * Constant time in the value of q, but not any other inputs.
+ */
+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
+);
+
 #endif /* SECP256K1_ECMULT_CONST_H */

src/ecmult_const_impl.h

@@ -228,4 +228,139 @@ static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, cons
     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 bits, 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, bits);
+#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 */

src/tests.c

@@ -4452,6 +4452,68 @@ static void ecmult_const_mult_zero_one(void) {
     ge_equals_ge(&res2, &point);
 }
 
+static void ecmult_const_mult_xonly(void) {
+    int i;
+
+    /* Test correspondence between secp256k1_ecmult_const and secp256k1_ecmult_const_xonly. */
+    for (i = 0; i < 2*COUNT; ++i) {
+        secp256k1_ge base;
+        secp256k1_gej basej, resj;
+        secp256k1_fe n, d, resx, v;
+        secp256k1_scalar q;
+        int res;
+        /* Random base point. */
+        random_group_element_test(&base);
+        /* Random scalar to multiply it with. */
+        random_scalar_order_test(&q);
+        /* If i is odd, n=d*base.x for random non-zero d */
+        if (i & 1) {
+            do {
+                random_field_element_test(&d);
+            } while (secp256k1_fe_normalizes_to_zero_var(&d));
+            secp256k1_fe_mul(&n, &base.x, &d);
+        } else {
+            n = base.x;
+        }
+        /* Perform x-only multiplication. */
+        res = secp256k1_ecmult_const_xonly(&resx, &n, (i & 1) ? &d : NULL, &q, 256, i & 2);
+        CHECK(res);
+        /* Perform normal multiplication. */
+        secp256k1_gej_set_ge(&basej, &base);
+        secp256k1_ecmult(&resj, &basej, &q, NULL);
+        /* Check that resj's X coordinate corresponds with resx. */
+        secp256k1_fe_sqr(&v, &resj.z);
+        secp256k1_fe_mul(&v, &v, &resx);
+        CHECK(check_fe_equal(&v, &resj.x));
+    }
+
+    /* Test that secp256k1_ecmult_const_xonly correctly rejects X coordinates not on curve. */
+    for (i = 0; i < 2*COUNT; ++i) {
+        secp256k1_fe x, n, d, c, r;
+        int res;
+        secp256k1_scalar q;
+        random_scalar_order_test(&q);
+        /* Generate random X coordinate not on the curve. */
+        do {
+            random_field_element_test(&x);
+            secp256k1_fe_sqr(&c, &x);
+            secp256k1_fe_mul(&c, &c, &x);
+            secp256k1_fe_add(&c, &secp256k1_fe_const_b);
+        } while (secp256k1_fe_is_square_var(&c));
+        /* If i is odd, n=d*x for random non-zero d. */
+        if (i & 1) {
+            do {
+                random_field_element_test(&d);
+            } while (secp256k1_fe_normalizes_to_zero_var(&d));
+            secp256k1_fe_mul(&n, &x, &d);
+        } else {
+            n = x;
+        }
+        res = secp256k1_ecmult_const_xonly(&r, &n, (i & 1) ? &d : NULL, &q, 256, 0);
+        CHECK(res == 0);
+    }
+}
+
 static void ecmult_const_chain_multiply(void) {
     /* Check known result (randomly generated test problem from sage) */
     const secp256k1_scalar scalar = SECP256K1_SCALAR_CONST(
@@ -4483,6 +4545,7 @@ static void run_ecmult_const_tests(void) {
     ecmult_const_random_mult();
     ecmult_const_commutativity();
     ecmult_const_chain_multiply();
+    ecmult_const_mult_xonly();
 }
 
 typedef struct {

src/tests_exhaustive.c

@@ -59,6 +59,19 @@ static void random_fe(secp256k1_fe *x) {
         }
     } while(1);
 }
+
+static void random_fe_non_zero(secp256k1_fe *nz) {
+    int tries = 10;
+    while (--tries >= 0) {
+        random_fe(nz);
+        secp256k1_fe_normalize(nz);
+        if (!secp256k1_fe_is_zero(nz)) {
+            break;
+        }
+    }
+    /* Infinitesimal probability of spurious failure here */
+    CHECK(tries >= 0);
+}
 /** END stolen from tests.c */
 
 static uint32_t num_cores = 1;
@@ -174,10 +187,37 @@ static void test_exhaustive_ecmult(const secp256k1_ge *group, const secp256k1_ge
                 secp256k1_ecmult(&tmp, &groupj[r_log], &na, &ng);
                 ge_equals_gej(&group[(i * r_log + j) % EXHAUSTIVE_TEST_ORDER], &tmp);
 
-                if (i > 0) {
+            }
-                    secp256k1_ecmult_const(&tmp, &group[i], &ng, 256);
+        }
-                    ge_equals_gej(&group[(i * j) % EXHAUSTIVE_TEST_ORDER], &tmp);
+    }
-                }
+
+    for (j = 0; j < EXHAUSTIVE_TEST_ORDER; j++) {
+        for (i = 1; i < EXHAUSTIVE_TEST_ORDER; i++) {
+            int ret;
+            secp256k1_gej tmp;
+            secp256k1_fe xn, xd, tmpf;
+            secp256k1_scalar ng;
+
+            if (skip_section(&iter)) continue;
+
+            secp256k1_scalar_set_int(&ng, j);
+
+            /* Test secp256k1_ecmult_const. */
+            secp256k1_ecmult_const(&tmp, &group[i], &ng, 256);
+            ge_equals_gej(&group[(i * j) % EXHAUSTIVE_TEST_ORDER], &tmp);
+
+            if (j != 0) {
+                /* Test secp256k1_ecmult_const_xonly with all curve X coordinates, and xd=NULL. */
+                ret = secp256k1_ecmult_const_xonly(&tmpf, &group[i].x, NULL, &ng, 256, 0);
+                CHECK(ret);
+                CHECK(secp256k1_fe_equal_var(&tmpf, &group[(i * j) % EXHAUSTIVE_TEST_ORDER].x));
+
+                /* Test secp256k1_ecmult_const_xonly with all curve X coordinates, with random xd. */
+                random_fe_non_zero(&xd);
+                secp256k1_fe_mul(&xn, &xd, &group[i].x);
+                ret = secp256k1_ecmult_const_xonly(&tmpf, &xn, &xd, &ng, 256, 0);
+                CHECK(ret);
+                CHECK(secp256k1_fe_equal_var(&tmpf, &group[(i * j) % EXHAUSTIVE_TEST_ORDER].x));
             }
         }
     }