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 */
+ /* Operations: 7 mul, 5 sqr, 4 normalize, 21 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, tt, m, n, q, rr;
- int infinity;
+ secp256k1_fe_t zz, u1, u2, s1, s2, 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);
* 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 */
s1 = a->y; secp256k1_fe_normalize_weak(&s1); /* s1 = S1 = Y1*Z2^3 (1) */
secp256k1_fe_mul(&s2, &b->y, &zz); /* s2 = Y2*Z2^2 (1) */
secp256k1_fe_mul(&s2, &s2, &a->z); /* s2 = S2 = Y2*Z1^3 (1) */
- 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(&rr, &t); /* rr = T^2 (1) */
- secp256k1_fe_mul(&tt, &u1, &u2); secp256k1_fe_negate(&tt, &tt, 1); /* t = -U1*U2 (2) */
- secp256k1_fe_add(&rr, &tt); /* rr = R = T^2-U1*U2 (3) */
- 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(&t, &rr); /* t = R^2 (1) */
- secp256k1_fe_mul(&r->z, &m, &z); /* r->z = M*Z (1) */
+ secp256k1_fe_negate(&m_alt, &u2, 1); /* Malt = -X2*Z1^2 */
+ secp256k1_fe_mul(&tt, &u1, &m_alt); /* 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. */
+ rr_alt = s1;
+ secp256k1_fe_mul_int(&rr_alt, 2); /* rr = Y1*Z2^3 - Y2*Z1^3 (2) */
+ secp256k1_fe_add(&m_alt, &u1); /* Malt = 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);
+ secp256k1_fe_cmov(&n, &m, degenerate); /* n = M^3 * Malt (2) */
+ secp256k1_fe_sqr(&t, &rr_alt); /* t = Ralt^2 (1) */
+ secp256k1_fe_mul(&r->z, &a->z, &m_alt); /* 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) */
secp256k1_fe_negate(&q, &q, 1); /* q = -Q (2) */
- secp256k1_fe_add(&r->x, &q); /* r->x = R^2-Q (3) */
- secp256k1_fe_normalize(&r->x);
- t = r->x;
+ secp256k1_fe_add(&t, &q); /* t = Ralt^2-Q (3) */
+ secp256k1_fe_normalize_weak(&t);
+ r->x = t; /* r->x = Ralt^2-Q (1) */
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); /* t = R*(2*x3 - 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) */
+ secp256k1_fe_add(&t, &q); /* t = 2*x3 - Q: (4) */
+ 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 (3) */
+ secp256k1_fe_negate(&r->y, &t, 3); /* r->y = Ralt*(Q - 2x3) - M^3*Malt (4) */
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);
projects / secp256k1.git / blobdiff