Merge pull request #261

7657420 Add tests for adding P+Q with P.x!=Q.x and P.y=-Q.y (Pieter Wuille)
8c5d5f7 tests: Add failing unit test for #257 (bad addition formula) (Andrew Poelstra)
5de4c5d gej_add_ge: fix degenerate case when computing P + (-lambda)P (Andrew Poelstra)
bcf2fcf gej_add_ge: rearrange algebra (Andrew Poelstra)

Author: sipa

src/group_impl.h

@@ -463,8 +463,9 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej_t *r, const secp256k1_gej_t
 static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b) {
     /* Operations: 7 mul, 5 sqr, 5 normalize, 17 mul_int/add/negate/cmov */
     static const secp256k1_fe_t fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
-    secp256k1_fe_t zz, u1, u2, s1, s2, z, t, m, n, q, rr;
-    int infinity;
+    secp256k1_fe_t zz, u1, u2, s1, s2, z, t, tt, m, n, q, rr;
+    secp256k1_fe_t m_alt, rr_alt;
+    int infinity, degenerate;
     VERIFY_CHECK(!b->infinity);
     VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
 
@@ -488,6 +489,34 @@ static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, c
      *    Y3 = 4*(R*(3*Q-2*R^2)-M^4)
      *    Z3 = 2*M*Z
      *  (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
+     *
+     *  This formula has the benefit of being the same for both addition
+     *  of distinct points and doubling. However, it breaks down in the
+     *  case that either point is infinity, or that y1 = -y2. We handle
+     *  these cases in the following ways:
+     *
+     *    - If b is infinity we simply bail by means of a VERIFY_CHECK.
+     *
+     *    - If a is infinity, we detect this, and at the end of the
+     *      computation replace the result (which will be meaningless,
+     *      but we compute to be constant-time) with b.x : b.y : 1.
+     *
+     *    - If a = -b, we have y1 = -y2, which is a degenerate case.
+     *      But here the answer is infinity, so we simply set the
+     *      infinity flag of the result, overriding the computed values
+     *      without even needing to cmov.
+     *
+     *    - If y1 = -y2 but x1 != x2, which does occur thanks to certain
+     *      properties of our curve (specifically, 1 has nontrivial cube
+     *      roots in our field, and the curve equation has no x coefficient)
+     *      then the answer is not infinity but also not given by the above
+     *      equation. In this case, we cmov in place an alternate expression
+     *      for lambda. Specifically (y1 - y2)/(x1 - x2). Where both these
+     *      expressions for lambda are defined, they are equal, and can be
+     *      obtained from each other by multiplication by (y1 + y2)/(y1 + y2)
+     *      then substitution of x^3 + 7 for y^2 (using the curve equation).
+     *      For all pairs of nonzero points (a, b) at least one is defined,
+     *      so this covers everything.
      */
 
     secp256k1_fe_sqr(&zz, &a->z);                       /* z = Z1^2 */
@@ -499,29 +528,55 @@ static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, c
     z = a->z;                                           /* z = Z = Z1*Z2 (8) */
     t = u1; secp256k1_fe_add(&t, &u2);                  /* t = T = U1+U2 (2) */
     m = s1; secp256k1_fe_add(&m, &s2);                  /* m = M = S1+S2 (2) */
-    secp256k1_fe_sqr(&n, &m);                           /* n = M^2 (1) */
-    secp256k1_fe_mul(&q, &n, &t);                       /* q = Q = T*M^2 (1) */
-    secp256k1_fe_sqr(&n, &n);                           /* n = M^4 (1) */
     secp256k1_fe_sqr(&rr, &t);                          /* rr = T^2 (1) */
-    secp256k1_fe_mul(&t, &u1, &u2); secp256k1_fe_negate(&t, &t, 1); /* t = -U1*U2 (2) */
-    secp256k1_fe_add(&rr, &t);                                      /* rr = R = T^2-U1*U2 (3) */
-    secp256k1_fe_sqr(&t, &rr);                                      /* t = R^2 (1) */
-    secp256k1_fe_mul(&r->z, &m, &z);                                /* r->z = M*Z (1) */
+    secp256k1_fe_mul(&tt, &u1, &u2); secp256k1_fe_negate(&tt, &tt, 1); /* tt = -U1*U2 (2) */
+    secp256k1_fe_add(&rr, &tt);                                        /* rr = R = T^2-U1*U2 (3) */
+    /** If lambda = R/M = 0/0 we have a problem (except in the "trivial"
+     *  case that Z = z1z2 = 0, and this is special-cased later on). */
+    degenerate = secp256k1_fe_normalizes_to_zero(&m) &
+                 secp256k1_fe_normalizes_to_zero(&rr);
+    /* This only occurs when y1 == -y2 and x1^3 == x2^3, but x1 != x2.
+     * This means either x1 == beta*x2 or beta*x1 == x2, where beta is
+     * a nontrivial cube root of one. In either case, an alternate
+     * non-indeterminate expression for lambda is (y1 - y2)/(x1 - x2),
+     * so we set R/M equal to this. */
+    secp256k1_fe_negate(&rr_alt, &s2, 1);   /* rr = -Y2*Z1^3  */
+    secp256k1_fe_add(&rr_alt, &s1);         /* rr = Y1*Z2^3 - Y2*Z1^3 */
+    secp256k1_fe_negate(&m_alt, &u2, 1);    /* m = -X2*Z1^2 */
+    secp256k1_fe_add(&m_alt, &u1);          /* m = X1*Z2^2 - X2*Z1^2 */
+
+    secp256k1_fe_cmov(&rr_alt, &rr, !degenerate);
+    secp256k1_fe_cmov(&m_alt, &m, !degenerate);
+    /* Now Ralt / Malt = lambda and is guaranteed not to be 0/0.
+     * From here on out Ralt and Malt represent the numerator
+     * and denominator of lambda; R and M represent the explicit
+     * expressions x1^2 + x2^2 + x1x2 and y1 + y2. */
+    secp256k1_fe_sqr(&n, &m_alt);                       /* n = Malt^2 (1) */
+    secp256k1_fe_mul(&q, &n, &t);                       /* q = Q = T*Malt^2 (1) */
+    /* These two lines use the observation that either M == Malt or M == 0,
+     * so M^3 * Malt is either Malt^4 (which is computed by squaring), or
+     * zero (which is "computed" by cmov). So the cost is one squaring
+     * versus two multiplications. */
+    secp256k1_fe_sqr(&n, &n);                           /* n = M^3 * Malt (1) */
+    secp256k1_fe_cmov(&n, &m, degenerate);
+    secp256k1_fe_normalize_weak(&n);
+    secp256k1_fe_sqr(&t, &rr_alt);                      /* t = Ralt^2 (1) */
+    secp256k1_fe_mul(&r->z, &m_alt, &z);                /* r->z = Malt*Z (1) */
     infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
-    secp256k1_fe_mul_int(&r->z, 2);                     /* r->z = Z3 = 2*M*Z (2) */
-    r->x = t;                                           /* r->x = R^2 (1) */
+    secp256k1_fe_mul_int(&r->z, 2);                     /* r->z = Z3 = 2*Malt*Z (2) */
+    r->x = t;                                           /* r->x = Ralt^2 (1) */
     secp256k1_fe_negate(&q, &q, 1);                     /* q = -Q (2) */
-    secp256k1_fe_add(&r->x, &q);                        /* r->x = R^2-Q (3) */
+    secp256k1_fe_add(&r->x, &q);                        /* r->x = Ralt^2-Q (3) */
     secp256k1_fe_normalize(&r->x);
-    secp256k1_fe_mul_int(&q, 3);                        /* q = -3*Q (6) */
-    secp256k1_fe_mul_int(&t, 2);                        /* t = 2*R^2 (2) */
-    secp256k1_fe_add(&t, &q);                           /* t = 2*R^2-3*Q (8) */
-    secp256k1_fe_mul(&t, &t, &rr);                      /* t = R*(2*R^2-3*Q) (1) */
-    secp256k1_fe_add(&t, &n);                           /* t = R*(2*R^2-3*Q)+M^4 (2) */
-    secp256k1_fe_negate(&r->y, &t, 2);                  /* r->y = R*(3*Q-2*R^2)-M^4 (3) */
+    t = r->x;
+    secp256k1_fe_mul_int(&t, 2);                        /* t = 2*x3 (2) */
+    secp256k1_fe_add(&t, &q);                           /* t = 2*x3 - Q: (8) */
+    secp256k1_fe_mul(&t, &t, &rr_alt);                  /* t = Ralt*(2*x3 - Q) (1) */
+    secp256k1_fe_add(&t, &n);                           /* t = Ralt*(2*x3 - Q) + M^3*Malt (2) */
+    secp256k1_fe_negate(&r->y, &t, 2);                  /* r->y = Ralt*(Q - 2x3) - M^3*Malt (3) */
     secp256k1_fe_normalize_weak(&r->y);
-    secp256k1_fe_mul_int(&r->x, 4);                     /* r->x = X3 = 4*(R^2-Q) */
-    secp256k1_fe_mul_int(&r->y, 4);                     /* r->y = Y3 = 4*R*(3*Q-2*R^2)-4*M^4 (4) */
+    secp256k1_fe_mul_int(&r->x, 4);                     /* r->x = X3 = 4*(Ralt^2-Q) */
+    secp256k1_fe_mul_int(&r->y, 4);                     /* r->y = Y3 = 4*Ralt*(Q - 2x3) - 4*M^3*Malt (4) */
 
     /** In case a->infinity == 1, replace r with (b->x, b->y, 1). */
     secp256k1_fe_cmov(&r->x, &b->x, a->infinity);

src/tests.c

@@ -958,11 +958,17 @@ void ge_equals_gej(const secp256k1_ge_t *a, const secp256k1_gej_t *b) {
 
 void test_ge(void) {
     int i, i1;
+#ifdef USE_ENDOMORPHISM
+    int runs = 6;
+#else
     int runs = 4;
+#endif
     /* Points: (infinity, p1, p1, -p1, -p1, p2, p2, -p2, -p2, p3, p3, -p3, -p3, p4, p4, -p4, -p4).
      * The second in each pair of identical points uses a random Z coordinate in the Jacobian form.
      * All magnitudes are randomized.
      * All 17*17 combinations of points are added to eachother, using all applicable methods.
+     *
+     * When the endomorphism code is compiled in, p5 = lambda*p1 and p6 = lambda^2*p1 are added as well.
      */
     secp256k1_ge_t *ge = (secp256k1_ge_t *)malloc(sizeof(secp256k1_ge_t) * (1 + 4 * runs));
     secp256k1_gej_t *gej = (secp256k1_gej_t *)malloc(sizeof(secp256k1_gej_t) * (1 + 4 * runs));
@@ -977,6 +983,14 @@ void test_ge(void) {
         int j;
         secp256k1_ge_t g;
         random_group_element_test(&g);
+#ifdef USE_ENDOMORPHISM
+        if (i >= runs - 2) {
+            secp256k1_ge_mul_lambda(&g, &ge[1]);
+        }
+        if (i >= runs - 1) {
+            secp256k1_ge_mul_lambda(&g, &g);
+        }
+#endif
         ge[1 + 4 * i] = g;
         ge[2 + 4 * i] = g;
         secp256k1_ge_neg(&ge[3 + 4 * i], &g);
@@ -1146,11 +1160,79 @@ void test_ge(void) {
     free(zinv);
 }
 
+void test_add_neg_y_diff_x(void) {
+    /* The point of this test is to check that we can add two points
+     * whose y-coordinates are negatives of each other but whose x
+     * coordinates differ. If the x-coordinates were the same, these
+     * points would be negatives of each other and their sum is
+     * infinity. This is cool because it "covers up" any degeneracy
+     * in the addition algorithm that would cause the xy coordinates
+     * of the sum to be wrong (since infinity has no xy coordinates).
+     * HOWEVER, if the x-coordinates are different, infinity is the
+     * wrong answer, and such degeneracies are exposed. This is the
+     * root of https://github.com/bitcoin/secp256k1/issues/257 which
+     * this test is a regression test for.
+     *
+     * These points were generated in sage as
+     * # secp256k1 params
+     * F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
+     * C = EllipticCurve ([F (0), F (7)])
+     * G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
+     * N = FiniteField(G.order())
+     *
+     * # endomorphism values (lambda is 1^{1/3} in N, beta is 1^{1/3} in F)
+     * x = polygen(N)
+     * lam  = (1 - x^3).roots()[1][0]
+     *
+     * # random "bad pair"
+     * P = C.random_element()
+     * Q = -int(lam) * P
+     * print "    P: %x %x" % P.xy()
+     * print "    Q: %x %x" % Q.xy()
+     * print "P + Q: %x %x" % (P + Q).xy()
+     */
+    secp256k1_gej_t aj = SECP256K1_GEJ_CONST(
+        0x8d24cd95, 0x0a355af1, 0x3c543505, 0x44238d30,
+        0x0643d79f, 0x05a59614, 0x2f8ec030, 0xd58977cb,
+        0x001e337a, 0x38093dcd, 0x6c0f386d, 0x0b1293a8,
+        0x4d72c879, 0xd7681924, 0x44e6d2f3, 0x9190117d
+    );
+    secp256k1_gej_t bj = SECP256K1_GEJ_CONST(
+        0xc7b74206, 0x1f788cd9, 0xabd0937d, 0x164a0d86,
+        0x95f6ff75, 0xf19a4ce9, 0xd013bd7b, 0xbf92d2a7,
+        0xffe1cc85, 0xc7f6c232, 0x93f0c792, 0xf4ed6c57,
+        0xb28d3786, 0x2897e6db, 0xbb192d0b, 0x6e6feab2
+    );
+    secp256k1_gej_t sumj = SECP256K1_GEJ_CONST(
+        0x671a63c0, 0x3efdad4c, 0x389a7798, 0x24356027,
+        0xb3d69010, 0x278625c3, 0x5c86d390, 0x184a8f7a,
+        0x5f6409c2, 0x2ce01f2b, 0x511fd375, 0x25071d08,
+        0xda651801, 0x70e95caf, 0x8f0d893c, 0xbed8fbbe
+    );
+    secp256k1_ge_t b;
+    secp256k1_gej_t resj;
+    secp256k1_ge_t res;
+    secp256k1_ge_set_gej(&b, &bj);
+
+    secp256k1_gej_add_var(&resj, &aj, &bj, NULL);
+    secp256k1_ge_set_gej(&res, &resj);
+    ge_equals_gej(&res, &sumj);
+
+    secp256k1_gej_add_ge(&resj, &aj, &b);
+    secp256k1_ge_set_gej(&res, &resj);
+    ge_equals_gej(&res, &sumj);
+
+    secp256k1_gej_add_ge_var(&resj, &aj, &b, NULL);
+    secp256k1_ge_set_gej(&res, &resj);
+    ge_equals_gej(&res, &sumj);
+}
+
 void run_ge(void) {
     int i;
     for (i = 0; i < count * 32; i++) {
         test_ge();
     }
+    test_add_neg_y_diff_x();
 }
 
 /***** ECMULT TESTS *****/