/**
* Runs affine transformation and encodes point at infinity as (0; 0; 0)
*/
public BN128 <T> ToEthNotation()
{
BN128 <T> affine = ToAffine();
// affine zero is (0; 1; 0), convert to Ethereum zero: (0; 0; 0)
if (affine.IsZero())
{
return(Zero());
}
else
{
return(affine);
}
}
public BN128 <T> Add(BN128 <T> o)
{
if (this.IsZero())
{
return(o); // 0 + P = P
}
if (o.IsZero())
{
return(this); // P + 0 = P
}
T x1 = this.x, y1 = this.y, z1 = this.z;
T x2 = o.x, y2 = o.y, z2 = o.z;
// ported code is started from here
// next calculations are done in Jacobian coordinates
T z1z1 = z1.Squared();
T z2z2 = z2.Squared();
T u1 = x1.Mul(z2z2);
T u2 = x2.Mul(z1z1);
T z1Cubed = z1.Mul(z1z1);
T z2Cubed = z2.Mul(z2z2);
T s1 = y1.Mul(z2Cubed); // s1 = y1 * Z2^3
T s2 = y2.Mul(z1Cubed); // s2 = y2 * Z1^3
if (u1.Equals(u2) && s1.Equals(s2))
{
return(Dbl()); // P + P = 2P
}
T h = u2.Sub(u1); // h = u2 - u1
T i = h.Dbl().Squared(); // i = (2 * h)^2
T j = h.Mul(i); // j = h * i
T r = s2.Sub(s1).Dbl(); // r = 2 * (s2 - s1)
T v = u1.Mul(i); // v = u1 * i
T zz = z1.Add(z2).Squared()
.Sub(z1.Squared()).Sub(z2.Squared());
T x3 = r.Squared().Sub(j).Sub(v.Dbl()); // x3 = r^2 - j - 2 * v
T y3 = v.Sub(x3).Mul(r).Sub(s1.Mul(j).Dbl()); // y3 = r * (v - x3) - 2 * (s1 * j)
T z3 = zz.Mul(h); // z3 = ((z1+z2)^2 - z1^2 - z2^2) * h = zz * h
return(Instance(x3, y3, z3));
}
private static bool IsGroupMember(BN128 <Fp2> p)
{
BN128 <Fp2> left = p.Mul(FR_NEG_ONE).Add(p);
return(left.IsZero()); // should satisfy condition: -1 * p + p == 0, where -1 belongs to F_r
}