private static (Fp12 numerator, Fp12 denominator) LineFunction(FpVector3 <Fp12> p1, FpVector3 <Fp12> p2, FpVector3 <Fp12> t)
{
// As mentioned in py_ecc: the projective coordinates given are (x / z, y / z).
// For slope of a line m, we usually have delta_y / delta_x. In this case it would be
// m = ((p2.y / p2.z) - (p1.y / p1.z)) / ((p2.x / p2.z) - (p1.x / p1.z))
// So to eliminate these fractions, we multiply both numerator and denominator by p1.z * p2.z,
// which yields the values below. This only affects scale but keeps the same m result.
Fp12 slopeNumerator = (p2.Y * p1.Z) - (p1.Y * p2.Z);
Fp12 slopeDenominator = (p2.X * p1.Z) - (p1.X * p2.Z);
// Determine how to compute our data.
bool denominatorIsZero = slopeDenominator == Fp12.ZeroValue;
bool numeratorIsZero = slopeNumerator == Fp12.ZeroValue;
if (denominatorIsZero && !numeratorIsZero)
{
// Slope is undefined.
return((t.X * p1.Z) - (p1.X * t.Z), p1.Z *t.Z);
}
else if (numeratorIsZero)
{
slopeNumerator = 3 * p1.X * p1.X;
slopeDenominator = 2 * p1.Y * p1.Z;
}
Fp12 resultNumerator = (slopeNumerator * ((t.X * p1.Z) - (p1.X * t.Z))) - (slopeDenominator * ((t.Y * p1.Z) - (p1.Y * t.Z)));
Fp12 resultDenominator = slopeDenominator * t.Z * p1.Z;
return(resultNumerator, resultDenominator);
}
public static FpVector3 <Fp12> Twist(FpVector3 <Fp2> point)
{
// If the point isn't properly initialize
if (point == null)
{
return(null);
}
// Place the items at the start and half way point
BigInteger[] xcoefficients = new BigInteger[12] {
point.X.Coefficients.ElementAt(0) - (point.X.Coefficients.ElementAt(1) * 9), 0, 0, 0, 0, 0, point.X.Coefficients.ElementAt(1), 0, 0, 0, 0, 0
};
BigInteger[] ycoefficients = new BigInteger[12] {
point.Y.Coefficients.ElementAt(0) - (point.Y.Coefficients.ElementAt(1) * 9), 0, 0, 0, 0, 0, point.Y.Coefficients.ElementAt(1), 0, 0, 0, 0, 0
};
BigInteger[] zcoefficients = new BigInteger[12] {
point.Z.Coefficients.ElementAt(0) - (point.Z.Coefficients.ElementAt(1) * 9), 0, 0, 0, 0, 0, point.Z.Coefficients.ElementAt(1), 0, 0, 0, 0, 0
};
// Instantiate fp12s from the extended fp2
Fp12 newx = new Fp12(xcoefficients);
Fp12 newy = new Fp12(ycoefficients);
Fp12 newz = new Fp12(zcoefficients);
// Return a twisted point from fp2 to fp12.
return(new FpVector3 <Fp12>(newx * _twistPoint.Pow(2), newy * _twistPoint.Pow(3), newz));
}
public static Fp12 Pair(FpVector3 <Fp2> q, FpVector3 <Fp> p, bool finalExponentiate = true)
{
// Check z's for zero.
if (p.Z == Fp.ZeroValue || q.Z == Fp2.ZeroValue)
{
return(Fp12.OneValue);
}
// Run the miller loop on twist(q) and fq12(p)
return(MillerLoop(Bn128Curve.Twist(q), CastFpPointToFp12Point(p), finalExponentiate));
}
private static Fp12 MillerLoop(FpVector3 <Fp12> q, FpVector3 <Fp12> p, bool finalExponentiate = true)
{
if (q == null || p == null)
{
return(Fp12.OneValue);
}
FpVector3 <Fp12> nq = q.Negate();
FpVector3 <Fp12> r = (FpVector3 <Fp12>)q.Clone();
Fp12 fNumerator = Fp12.OneValue;
Fp12 fDenominator = Fp12.OneValue;
for (int i = LogAteLoopCount; i >= 0; i--)
{
(Fp12 num, Fp12 denom) = LineFunction(r, r, p);
fNumerator = fNumerator * fNumerator * num;
fDenominator = fDenominator * fDenominator * denom;
r = r.Double();
int v = PseudoBinaryEncoding[i];
if (v == 1)
{
(num, denom) = LineFunction(r, q, p);
fNumerator *= num;
fDenominator *= denom;
r = r.Add(q);
}
else if (v == -1)
{
(num, denom) = LineFunction(r, nq, p);
fNumerator *= num;
fDenominator *= denom;
r = r.Add(nq);
}
}
FpVector3 <Fp12> q1 = new FpVector3 <Fp12>(q.X.Pow(Bn128Curve.P), q.Y.Pow(Bn128Curve.P), q.Z.Pow(Bn128Curve.P));
FpVector3 <Fp12> nq2 = new FpVector3 <Fp12>(q1.X.Pow(Bn128Curve.P), -q1.Y.Pow(Bn128Curve.P), q1.Z.Pow(Bn128Curve.P));
(Fp12 num1, Fp12 denom1) = LineFunction(r, q1, p);
r = r.Add(q1);
(Fp12 num2, Fp12 denom2) = LineFunction(r, nq2, p);
Fp12 f = (fNumerator * num1 * num2) / (fDenominator * denom1 * denom2);
return(finalExponentiate ? FinalExponentiate(f) : f);
}
/// <summary>
/// Our default static constructor, sets static read only variables.
/// </summary>
static Bn128Curve()
{
N = BigInteger.Parse("21888242871839275222246405745257275088548364400416034343698204186575808495617", CultureInfo.InvariantCulture);
P = BigInteger.Parse("21888242871839275222246405745257275088696311157297823662689037894645226208583", CultureInfo.InvariantCulture);
_twistPoint = new Fp12(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
G1 = new FpVector3 <Fp>(Fp.OneValue, new Fp(2), Fp.OneValue);
G2 = new FpVector3 <Fp2>(
new Fp2(
BigInteger.Parse("10857046999023057135944570762232829481370756359578518086990519993285655852781", CultureInfo.InvariantCulture),
BigInteger.Parse("11559732032986387107991004021392285783925812861821192530917403151452391805634", CultureInfo.InvariantCulture)),
new Fp2(
BigInteger.Parse("8495653923123431417604973247489272438418190587263600148770280649306958101930", CultureInfo.InvariantCulture),
BigInteger.Parse("4082367875863433681332203403145435568316851327593401208105741076214120093531", CultureInfo.InvariantCulture)),
Fp2.OneValue);
B = new Fp(3);
B2 = new Fp2(3, 0) / new Fp2(9, 1);
B12 = new Fp12(3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
}
private static FpVector3 <Fp12> CastFpPointToFp12Point(FpVector3 <Fp> point)
{
// If our point isn't initialized, return null.
if (point == null)
{
return(null);
}
// Create our data for our fq
BigInteger[] xData = new BigInteger[12];
xData[0] = point.X.N;
BigInteger[] yData = new BigInteger[12];
yData[0] = point.Y.N;
BigInteger[] zData = new BigInteger[12];
zData[0] = point.Z.N;
// Return our Fp12 vector.
return(new FpVector3 <Fp12>(new Fp12(xData), new Fp12(yData), new Fp12(zData)));
}
public FpVector3 <T> Add(FpVector3 <T> p2)
{
// Verify our z coordinates aren't zero
if (p2.Z.Equals(X.Zero))
{
return(this);
}
else if (Z.Equals(X.Zero))
{
return(p2);
}
T u1 = p2.Y.Multiply(Z);
T u2 = Y.Multiply(p2.Z);
T v1 = p2.X.Multiply(Z);
T v2 = X.Multiply(p2.Z);
if (v1.Equals(v2) && u1.Equals(u2))
{
return(Double());
}
else if (v1.Equals(v2))
{
return(new FpVector3 <T>(X.One, X.One, X.Zero));
}
T u = u1.Subtract(u2);
T v = v1.Subtract(v2);
T v_sq = v.Multiply(v);
T v_sq_v2 = v_sq.Multiply(v2);
T v_cu = v_sq.Multiply(v);
T w = Z.Multiply(p2.Z);
T a = u.Multiply(u).Multiply(w).Subtract(v_cu).Subtract(v_sq_v2.Multiply(2));
T newx = v.Multiply(a);
T newy = u.Multiply(v_sq_v2.Subtract(a)).Subtract(v_cu.Multiply(u2));
T newz = v_cu.Multiply(w);
return(new FpVector3 <T>(newx, newy, newz));
}