コード例 #1
0
        private static Precomputed FlippedMillerLoopDoubling(Bn128Fp2 g2)
        {
            Fp2 x = g2.X, y = g2.Y, z = g2.Z;

            Fp2 a  = Fp.InverseOf2.Mul(x.Mul(y));         // a = x * y / 2
            Fp2 b  = y.Squared();                         // b = y^2
            Fp2 c  = z.Squared();                         // c = z^2
            Fp2 d  = c.Add(c).Add(c);                     // d = 3 * c
            Fp2 e  = Parameters.Fp2B.Mul(d);              // e = twist_b * d
            Fp2 f  = e.Add(e).Add(e);                     // f = 3 * e
            Fp2 g  = Fp.InverseOf2.Mul(b.Add(f));         // g = (b + f) / 2
            Fp2 h  = y.Add(z).Squared().Sub(b.Add(c));    // h = (y + z)^2 - (b + c)
            Fp2 i  = e.Sub(b);                            // i = e - b
            Fp2 j  = x.Squared();                         // j = x^2
            Fp2 e2 = e.Squared();                         // e2 = e^2

            Fp2 rx = a.Mul(b.Sub(f));                     // rx = a * (b - f)
            Fp2 ry = g.Squared().Sub(e2.Add(e2).Add(e2)); // ry = g^2 - 3 * e^2
            Fp2 rz = b.Mul(h);                            // rz = b * h

            Fp2 ell0  = Parameters.Twist.Mul(i);          // ell_0 = twist * i
            Fp2 ellVw = h.Negate();                       // ell_VW = -h
            Fp2 ellVv = j.Add(j).Add(j);                  // ell_VV = 3 * j

            return(new Precomputed(
                       new Bn128Fp2(rx, ry, rz),
                       new EllCoeffs(ell0, ellVw, ellVv)
                       ));
        }
コード例 #2
0
        private static Precomputed FlippedMillerLoopMixedAddition(Bn128Fp2 baseElement, Bn128Fp2 addend)
        {
            Fp2 x1 = addend.X, y1 = addend.Y, z1 = addend.Z;
            Fp2 x2 = baseElement.X, y2 = baseElement.Y;

            Fp2 d = x1.Sub(x2.Mul(z1));                                 // d = x1 - x2 * z1
            Fp2 e = y1.Sub(y2.Mul(z1));                                 // e = y1 - y2 * z1
            Fp2 f = d.Squared();                                        // f = d^2
            Fp2 g = e.Squared();                                        // g = e^2
            Fp2 h = d.Mul(f);                                           // h = d * f
            Fp2 i = x1.Mul(f);                                          // i = x1 * f
            Fp2 j = h.Add(z1.Mul(g)).Sub(i.Double());                   // j = h + z1 * g - 2 * i

            Fp2 x3 = d.Mul(j);                                          // x3 = d * j
            Fp2 y3 = e.Mul(i.Sub(j)).Sub(h.Mul(y1));                    // y3 = e * (i - j) - h * y1)
            Fp2 z3 = z1.Mul(h);                                         // z3 = Z1*H

            Fp2 ell0  = Parameters.Twist.Mul(e.Mul(x2).Sub(d.Mul(y2))); // ell_0 = TWIST * (e * x2 - d * y2)
            Fp2 ellVv = e.Negate();                                     // ell_VV = -e
            Fp2 ellVw = d;                                              // ell_VW = d

            return(new Precomputed(
                       new Bn128Fp2(x3, y3, z3),
                       new EllCoeffs(ell0, ellVw, ellVv)
                       ));
        }