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libsecp256k1/src/group_impl.h  view on Meta::CPAN

    /** We need to calculate (rx,ry,rz) = (ax,ay,az) + (bx,by,1/bzinv). Due to
     *  secp256k1's isomorphism we can multiply the Z coordinates on both sides
     *  by bzinv, and get: (rx,ry,rz*bzinv) = (ax,ay,az*bzinv) + (bx,by,1).
     *  This means that (rx,ry,rz) can be calculated as
     *  (ax,ay,az*bzinv) + (bx,by,1), when not applying the bzinv factor to rz.
     *  The variable az below holds the modified Z coordinate for a, which is used
     *  for the computation of rx and ry, but not for rz.
     */
    secp256k1_fe_mul(&az, &a->z, bzinv);

    secp256k1_fe_sqr(&z12, &az);
    u1 = a->x;
    secp256k1_fe_mul(&u2, &b->x, &z12);
    s1 = a->y;
    secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &az);
    secp256k1_fe_negate(&h, &u1, SECP256K1_GEJ_X_MAGNITUDE_MAX); secp256k1_fe_add(&h, &u2);
    secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
    if (secp256k1_fe_normalizes_to_zero_var(&h)) {
        if (secp256k1_fe_normalizes_to_zero_var(&i)) {
            secp256k1_gej_double_var(r, a, NULL);
        } else {
            secp256k1_gej_set_infinity(r);
        }
        return;
    }

    r->infinity = 0;
    secp256k1_fe_mul(&r->z, &a->z, &h);

    secp256k1_fe_sqr(&h2, &h);
    secp256k1_fe_negate(&h2, &h2, 1);
    secp256k1_fe_mul(&h3, &h2, &h);
    secp256k1_fe_mul(&t, &u1, &h2);

    secp256k1_fe_sqr(&r->x, &i);
    secp256k1_fe_add(&r->x, &h3);
    secp256k1_fe_add(&r->x, &t);
    secp256k1_fe_add(&r->x, &t);

    secp256k1_fe_add(&t, &r->x);
    secp256k1_fe_mul(&r->y, &t, &i);
    secp256k1_fe_mul(&h3, &h3, &s1);
    secp256k1_fe_add(&r->y, &h3);

    SECP256K1_GEJ_VERIFY(r);
}


static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b) {
    /* Operations: 7 mul, 5 sqr, 21 add/cmov/half/mul_int/negate/normalizes_to_zero */
    secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
    secp256k1_fe m_alt, rr_alt;
    int degenerate;
    SECP256K1_GEJ_VERIFY(a);
    SECP256K1_GE_VERIFY(b);
    VERIFY_CHECK(!b->infinity);

    /*  In:
     *    Eric Brier and Marc Joye, Weierstrass Elliptic Curves and Side-Channel Attacks.
     *    In D. Naccache and P. Paillier, Eds., Public Key Cryptography, vol. 2274 of Lecture Notes in Computer Science, pages 335-345. Springer-Verlag, 2002.
     *  we find as solution for a unified addition/doubling formula:
     *    lambda = ((x1 + x2)^2 - x1 * x2 + a) / (y1 + y2), with a = 0 for secp256k1's curve equation.
     *    x3 = lambda^2 - (x1 + x2)
     *    2*y3 = lambda * (x1 + x2 - 2 * x3) - (y1 + y2).
     *
     *  Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives:
     *    U1 = X1*Z2^2, U2 = X2*Z1^2
     *    S1 = Y1*Z2^3, S2 = Y2*Z1^3
     *    Z = Z1*Z2
     *    T = U1+U2
     *    M = S1+S2
     *    Q = -T*M^2
     *    R = T^2-U1*U2
     *    X3 = R^2+Q
     *    Y3 = -(R*(2*X3+Q)+M^4)/2
     *    Z3 = 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 */
    u1 = a->x;                                          /* u1 = U1 = X1*Z2^2 (GEJ_X_M) */
    secp256k1_fe_mul(&u2, &b->x, &zz);                  /* u2 = U2 = X2*Z1^2 (1) */
    s1 = a->y;                                          /* s1 = S1 = Y1*Z2^3 (GEJ_Y_M) */
    secp256k1_fe_mul(&s2, &b->y, &zz);                  /* s2 = Y2*Z1^2 (1) */
    secp256k1_fe_mul(&s2, &s2, &a->z);                  /* s2 = S2 = Y2*Z1^3 (1) */
    t = u1; secp256k1_fe_add(&t, &u2);                  /* t = T = U1+U2 (GEJ_X_M+1) */
    m = s1; secp256k1_fe_add(&m, &s2);                  /* m = M = S1+S2 (GEJ_Y_M+1) */
    secp256k1_fe_sqr(&rr, &t);                          /* rr = T^2 (1) */
    secp256k1_fe_negate(&m_alt, &u2, 1);                /* Malt = -X2*Z1^2 (2) */
    secp256k1_fe_mul(&tt, &u1, &m_alt);                 /* tt = -U1*U2 (1) */
    secp256k1_fe_add(&rr, &tt);                         /* rr = R = T^2-U1*U2 (2) */
    /* If lambda = R/M = R/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);
    /* 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_alt = Y1*Z2^3 - Y2*Z1^3 (GEJ_Y_M*2) */
    secp256k1_fe_add(&m_alt, &u1);          /* Malt = X1*Z2^2 - X2*Z1^2 (GEJ_X_M+2) */

    secp256k1_fe_cmov(&rr_alt, &rr, !degenerate);       /* rr_alt (GEJ_Y_M*2) */
    secp256k1_fe_cmov(&m_alt, &m, !degenerate);         /* m_alt (GEJ_X_M+2) */
    /* Now Ralt / Malt = lambda and is guaranteed not to be Ralt / 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_negate(&q, &t,