/// <summary>
/// https://eprint.iacr.org/2008/522.pdf uses extended twisted coordinates
/// </summary>
private ExtendedEdPoint Double(ExtendedEdPoint point)
{
var A = _operator.Multiply(point.X, point.X);
var B = _operator.Multiply(point.Y, point.Y);
var C = _operator.Multiply(2, _operator.Multiply(point.Z, point.Z));
var D = _operator.Multiply(CoefficientA, A);
var Esub = _operator.Add(point.X, point.Y);
var E = _operator.Subtract(_operator.Subtract(_operator.Multiply(Esub, Esub), A), B);
var G = _operator.Add(D, B);
var F = _operator.Subtract(G, C);
var H = _operator.Subtract(D, B);
var X3 = _operator.Multiply(E, F);
var Y3 = _operator.Multiply(G, H);
var T3 = _operator.Multiply(E, H);
var Z3 = _operator.Multiply(F, G);
return(new ExtendedEdPoint(X3, Y3, T3, Z3));
}
/// <summary>
/// https://eprint.iacr.org/2008/522.pdf uses extended twisted coordinates
/// </summary>
private ExtendedEdPoint Add(ExtendedEdPoint pointA, ExtendedEdPoint pointB)
{
var A = _operator.Multiply(pointA.X, pointB.X);
var B = _operator.Multiply(pointA.Y, pointB.Y);
var C = _operator.Multiply(pointA.Z, pointB.T);
var D = _operator.Multiply(pointA.T, pointB.Z);
var E = _operator.Add(D, C);
var F = _operator.Multiply(_operator.Subtract(pointA.X, pointA.Y), _operator.Add(pointB.X, pointB.Y));
F = _operator.Subtract(_operator.Add(F, B), A);
var G = _operator.Add(B, _operator.Multiply(CoefficientA, A));
var H = _operator.Subtract(D, C);
var X3 = _operator.Multiply(E, F);
var Y3 = _operator.Multiply(G, H);
var T3 = _operator.Multiply(E, H);
var Z3 = _operator.Multiply(F, G);
return(new ExtendedEdPoint(X3, Y3, T3, Z3));
}
private EdPoint Multiply(ExtendedEdPoint startPoint, NonAdjacentBitString nafBs)
{
var point = new ExtendedEdPoint(0, 1, 0, 1);
var naBits = nafBs.Bits;
for (var i = naBits.Length - 1; i >= 0; i--)
{
point = Double(point);
if (naBits[i] == 1)
{
point = Add(point, startPoint);
}
else if (naBits[i] == -1)
{
point = Subtract(point, startPoint);
}
}
return(ExtendedToEdPoint(point));
}
private EdPoint ExtendedToEdPoint(ExtendedEdPoint point)
{
return(new EdPoint(_operator.Divide(point.X, point.Z), _operator.Divide(point.Y, point.Z)));
}
private ExtendedEdPoint Subtract(ExtendedEdPoint pointA, ExtendedEdPoint pointB)
{
return(Add(pointA, Negate(pointB)));
}
private ExtendedEdPoint Negate(ExtendedEdPoint point)
{
return(new ExtendedEdPoint(_operator.Negate(point.X), point.Y, _operator.Negate(point.T), point.Z));
}