public override ECFieldElement Multiply(ECFieldElement b)
 {
     uint[] z = Nat256.Create();
     Curve25519Field.Multiply(x, ((Curve25519FieldElement)b).x, z);
     return(new Curve25519FieldElement(z));
 }
 public override ECFieldElement Subtract(ECFieldElement b)
 {
     uint[] z = Nat256.Create();
     Curve25519Field.Subtract(x, ((Curve25519FieldElement)b).x, z);
     return(new Curve25519FieldElement(z));
 }
        /**
         * return a sqrt root - the routine verifies that the calculation returns the right value - if
         * none exists it returns null.
         */
        public override ECFieldElement Sqrt()
        {
            /*
             * Q == 8m + 5, so we use Pocklington's method for this case.
             *
             * First, raise this element to the exponent 2^252 - 2^1 (i.e. m + 1)
             *
             * Breaking up the exponent's binary representation into "repunits", we get:
             * { 251 1s } { 1 0s }
             *
             * Therefore we need an addition chain containing 251 (the lengths of the repunits)
             * We use: 1, 2, 3, 4, 7, 11, 15, 30, 60, 120, 131, [251]
             */

            uint[] x1 = this.x;
            if (Nat256.IsZero(x1) || Nat256.IsOne(x1))
            {
                return(this);
            }

            uint[] x2 = Nat256.Create();
            Curve25519Field.Square(x1, x2);
            Curve25519Field.Multiply(x2, x1, x2);
            uint[] x3 = x2;
            Curve25519Field.Square(x2, x3);
            Curve25519Field.Multiply(x3, x1, x3);
            uint[] x4 = Nat256.Create();
            Curve25519Field.Square(x3, x4);
            Curve25519Field.Multiply(x4, x1, x4);
            uint[] x7 = Nat256.Create();
            Curve25519Field.SquareN(x4, 3, x7);
            Curve25519Field.Multiply(x7, x3, x7);
            uint[] x11 = x3;
            Curve25519Field.SquareN(x7, 4, x11);
            Curve25519Field.Multiply(x11, x4, x11);
            uint[] x15 = x7;
            Curve25519Field.SquareN(x11, 4, x15);
            Curve25519Field.Multiply(x15, x4, x15);
            uint[] x30 = x4;
            Curve25519Field.SquareN(x15, 15, x30);
            Curve25519Field.Multiply(x30, x15, x30);
            uint[] x60 = x15;
            Curve25519Field.SquareN(x30, 30, x60);
            Curve25519Field.Multiply(x60, x30, x60);
            uint[] x120 = x30;
            Curve25519Field.SquareN(x60, 60, x120);
            Curve25519Field.Multiply(x120, x60, x120);
            uint[] x131 = x60;
            Curve25519Field.SquareN(x120, 11, x131);
            Curve25519Field.Multiply(x131, x11, x131);
            uint[] x251 = x11;
            Curve25519Field.SquareN(x131, 120, x251);
            Curve25519Field.Multiply(x251, x120, x251);

            uint[] t1 = x251;
            Curve25519Field.Square(t1, t1);

            uint[] t2 = x120;
            Curve25519Field.Square(t1, t2);

            if (Nat256.Eq(x1, t2))
            {
                return(new Curve25519FieldElement(t1));
            }

            /*
             * If the first guess is incorrect, we multiply by a precomputed power of 2 to get the second guess,
             * which is ((4x)^(m + 1))/2 mod Q
             */
            Curve25519Field.Multiply(t1, PRECOMP_POW2, t1);

            Curve25519Field.Square(t1, t2);

            if (Nat256.Eq(x1, t2))
            {
                return(new Curve25519FieldElement(t1));
            }

            return(null);
        }
 public override ECFieldElement AddOne()
 {
     uint[] z = Nat256.Create();
     Curve25519Field.AddOne(x, z);
     return(new Curve25519FieldElement(z));
 }
Beispiel #5
0
        public override ECPoint Add(ECPoint b)
        {
            if (this.IsInfinity)
            {
                return(b);
            }
            if (b.IsInfinity)
            {
                return(this);
            }
            if (this == b)
            {
                return(Twice());
            }

            ECCurve curve = this.Curve;

            Curve25519FieldElement X1 = (Curve25519FieldElement)this.RawXCoord, Y1 = (Curve25519FieldElement)this.RawYCoord,
                                   Z1 = (Curve25519FieldElement)this.RawZCoords[0];
            Curve25519FieldElement X2 = (Curve25519FieldElement)b.RawXCoord, Y2 = (Curve25519FieldElement)b.RawYCoord,
                                   Z2 = (Curve25519FieldElement)b.RawZCoords[0];

            uint c;

            uint[] tt1 = Nat256.CreateExt();
            uint[] t2  = Nat256.Create();
            uint[] t3  = Nat256.Create();
            uint[] t4  = Nat256.Create();

            bool Z1IsOne = Z1.IsOne;

            uint[] U2, S2;
            if (Z1IsOne)
            {
                U2 = X2.x;
                S2 = Y2.x;
            }
            else
            {
                S2 = t3;
                Curve25519Field.Square(Z1.x, S2);

                U2 = t2;
                Curve25519Field.Multiply(S2, X2.x, U2);

                Curve25519Field.Multiply(S2, Z1.x, S2);
                Curve25519Field.Multiply(S2, Y2.x, S2);
            }

            bool Z2IsOne = Z2.IsOne;

            uint[] U1, S1;
            if (Z2IsOne)
            {
                U1 = X1.x;
                S1 = Y1.x;
            }
            else
            {
                S1 = t4;
                Curve25519Field.Square(Z2.x, S1);

                U1 = tt1;
                Curve25519Field.Multiply(S1, X1.x, U1);

                Curve25519Field.Multiply(S1, Z2.x, S1);
                Curve25519Field.Multiply(S1, Y1.x, S1);
            }

            uint[] H = Nat256.Create();
            Curve25519Field.Subtract(U1, U2, H);

            uint[] R = t2;
            Curve25519Field.Subtract(S1, S2, R);

            // Check if b == this or b == -this
            if (Nat256.IsZero(H))
            {
                if (Nat256.IsZero(R))
                {
                    // this == b, i.e. this must be doubled
                    return(this.Twice());
                }

                // this == -b, i.e. the result is the point at infinity
                return(curve.Infinity);
            }

            uint[] HSquared = Nat256.Create();
            Curve25519Field.Square(H, HSquared);

            uint[] G = Nat256.Create();
            Curve25519Field.Multiply(HSquared, H, G);

            uint[] V = t3;
            Curve25519Field.Multiply(HSquared, U1, V);

            Curve25519Field.Negate(G, G);
            Nat256.Mul(S1, G, tt1);

            c = Nat256.AddBothTo(V, V, G);
            Curve25519Field.Reduce27(c, G);

            Curve25519FieldElement X3 = new Curve25519FieldElement(t4);

            Curve25519Field.Square(R, X3.x);
            Curve25519Field.Subtract(X3.x, G, X3.x);

            Curve25519FieldElement Y3 = new Curve25519FieldElement(G);

            Curve25519Field.Subtract(V, X3.x, Y3.x);
            Curve25519Field.MultiplyAddToExt(Y3.x, R, tt1);
            Curve25519Field.Reduce(tt1, Y3.x);

            Curve25519FieldElement Z3 = new Curve25519FieldElement(H);

            if (!Z1IsOne)
            {
                Curve25519Field.Multiply(Z3.x, Z1.x, Z3.x);
            }
            if (!Z2IsOne)
            {
                Curve25519Field.Multiply(Z3.x, Z2.x, Z3.x);
            }

            uint[] Z3Squared = (Z1IsOne && Z2IsOne) ? HSquared : null;

            // TODO If the result will only be used in a subsequent addition, we don't need W3
            Curve25519FieldElement W3 = CalculateJacobianModifiedW((Curve25519FieldElement)Z3, Z3Squared);

            ECFieldElement[] zs = new ECFieldElement[] { Z3, W3 };

            return(new Curve25519Point(curve, X3, Y3, zs, IsCompressed));
        }