EdwardsTwistedCurve.cs

using System;
using System.Numerics;
 
namespace Waher.Security.EllipticCurves
{
    public abstract class EdwardsTwistedCurve : EdwardsCurveBase
    {
        private readonly BigInteger p58;
        private readonly BigInteger twoP14;
 
        public EdwardsTwistedCurve(BigInteger Prime, PointOnCurve BasePoint,
            BigInteger d, BigInteger Order, int Cofactor)
            : this(Prime, BasePoint, d, Order, Cofactor, null)
        {
        }
 
        public EdwardsTwistedCurve(BigInteger Prime, PointOnCurve BasePoint, BigInteger d,
            BigInteger Order, int Cofactor, byte[] Secret)
            : base(Prime, BasePoint, d, Order, Cofactor, Secret)
        {
            this.p58 = (this.p - 5) / 8;
            this.twoP14 = BigInteger.ModPow(Two, (this.p - 1) / 4, this.p);
        }
 
        public override void AddTo(ref PointOnCurve P, PointOnCurve Q)
        {
            if (!P.IsHomogeneous)
                P.T = P.X * P.Y;
 
            if (!Q.IsHomogeneous)
                Q.T = Q.X * Q.Y;
 
            BigInteger A = this.modP.Multiply(P.Y - P.X, Q.Y - Q.X);
            BigInteger B = this.modP.Multiply(P.Y + P.X, Q.Y + Q.X);
            BigInteger C = this.modP.Multiply(this.modP.Multiply(this.d2, P.T), Q.T);
            BigInteger D = this.modP.Multiply(P.Z << 1, Q.Z);
            BigInteger E = this.modP.Subtract(B, A);
            BigInteger F = this.modP.Subtract(D, C);
            BigInteger G = this.modP.Add(D, C);
            BigInteger H = this.modP.Add(B, A);
 
            P.X = this.modP.Multiply(E, F);
            P.Y = this.modP.Multiply(G, H);
            P.T = this.modP.Multiply(E, H);
            P.Z = this.modP.Multiply(F, G);
        }
 
        public override void Double(ref PointOnCurve P)
        {
            if (!P.IsHomogeneous)
                P.T = P.X * P.Y;
 
            BigInteger A = P.Y - P.X;
            A = this.modP.Multiply(A, A);
 
            BigInteger B = P.Y + P.X;
            B = this.modP.Multiply(B, B);
 
            BigInteger C = this.modP.Multiply(this.modP.Multiply(this.d2, P.T), P.T);
            BigInteger D = this.modP.Multiply(P.Z << 1, P.Z);
            BigInteger E = this.modP.Subtract(B, A);
            BigInteger F = this.modP.Subtract(D, C);
            BigInteger G = this.modP.Add(D, C);
            BigInteger H = this.modP.Add(B, A);
 
            P.X = this.modP.Multiply(E, F);
            P.Y = this.modP.Multiply(G, H);
            P.T = this.modP.Multiply(E, H);
            P.Z = this.modP.Multiply(F, G);
        }
 
        public override BigInteger GetX(BigInteger Y, bool X0)
        {
            BigInteger y2 = this.modP.Multiply(Y, Y);
            BigInteger u = y2 - BigInteger.One;
            if (u.Sign < 0)
                u += this.p;
 
            BigInteger v = this.modP.Multiply(this.d, y2) + BigInteger.One;
            BigInteger v2 = this.modP.Multiply(v, v);
            BigInteger v3 = this.modP.Multiply(v, v2);
            BigInteger v4 = this.modP.Multiply(v2, v2);
            BigInteger v7 = this.modP.Multiply(v3, v4);
            BigInteger x = this.modP.Multiply(this.modP.Multiply(u, v3),
                BigInteger.ModPow(this.modP.Multiply(u, v7), this.p58, this.Prime));
 
            BigInteger x2 = this.modP.Multiply(x, x);
            BigInteger Test = this.modP.Multiply(v, x2);
            if (Test.Sign < 0)
                Test += this.Prime;
 
            if (Test != u)
            {
                if (Test == this.Prime - u)
                    x = this.modP.Multiply(x, this.twoP14);
                else
                    throw new ArgumentException("Not a valid point.", nameof(Y));
            }
 
            if (X0)
            {
                if (x.IsZero)
                    throw new ArgumentException("Not a valid point.", nameof(Y));
 
                if (x.IsEven)
                    x = this.Prime - x;
            }
            else if (!x.IsEven)
                x = this.Prime - x;
 
            return x;
        }
 
        public override bool IsPoint(PointOnCurve Point)
        {
            // Check if point matches -x²+y²=1+dx²y²
 
            BigInteger X2 = this.modP.Multiply(Point.X, Point.X);
            BigInteger Y2 = this.modP.Multiply(Point.Y, Point.Y);
 
            BigInteger Left = this.modP.Subtract(Y2, X2);
 
            BigInteger Right = this.modP.Add(
                1,
                this.modP.Multiply(
                    this.modP.Multiply(this.d, X2),
                    Y2));
 
            return Left == Right;
        }
    }
}