Esempio n. 1
0
        /**
         * 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 2^221 - 2^29 - 2^9 - 2^8 - 2^6 - 2^4 - 2^1 (i.e. m + 1)
             *
             * Breaking up the exponent's binary representation into "repunits", we get:
             * { 191 1s } { 1 0s } { 19 1s } { 2 0s } { 1 1s } { 1 0s} { 1 1s } { 1 0s} { 3 1s } { 1 0s}
             *
             * Therefore we need an addition chain containing 1, 3, 19, 191 (the lengths of the repunits)
             * We use: [1], 2, [3], 4, 8, 11, [19], 23, 42, 84, 107, [191]
             */

            uint[] x1 = this.x;
            if (Nat224.IsZero(x1) || Nat224.IsOne(x1))
            {
                return(this);
            }

            uint[] x2 = Nat224.Create();
            SecP224K1Field.Square(x1, x2);
            SecP224K1Field.Multiply(x2, x1, x2);
            uint[] x3 = x2;
            SecP224K1Field.Square(x2, x3);
            SecP224K1Field.Multiply(x3, x1, x3);
            uint[] x4 = Nat224.Create();
            SecP224K1Field.Square(x3, x4);
            SecP224K1Field.Multiply(x4, x1, x4);
            uint[] x8 = Nat224.Create();
            SecP224K1Field.SquareN(x4, 4, x8);
            SecP224K1Field.Multiply(x8, x4, x8);
            uint[] x11 = Nat224.Create();
            SecP224K1Field.SquareN(x8, 3, x11);
            SecP224K1Field.Multiply(x11, x3, x11);
            uint[] x19 = x11;
            SecP224K1Field.SquareN(x11, 8, x19);
            SecP224K1Field.Multiply(x19, x8, x19);
            uint[] x23 = x8;
            SecP224K1Field.SquareN(x19, 4, x23);
            SecP224K1Field.Multiply(x23, x4, x23);
            uint[] x42 = x4;
            SecP224K1Field.SquareN(x23, 19, x42);
            SecP224K1Field.Multiply(x42, x19, x42);
            uint[] x84 = Nat224.Create();
            SecP224K1Field.SquareN(x42, 42, x84);
            SecP224K1Field.Multiply(x84, x42, x84);
            uint[] x107 = x42;
            SecP224K1Field.SquareN(x84, 23, x107);
            SecP224K1Field.Multiply(x107, x23, x107);
            uint[] x191 = x23;
            SecP224K1Field.SquareN(x107, 84, x191);
            SecP224K1Field.Multiply(x191, x84, x191);

            uint[] t1 = x191;
            SecP224K1Field.SquareN(t1, 20, t1);
            SecP224K1Field.Multiply(t1, x19, t1);
            SecP224K1Field.SquareN(t1, 3, t1);
            SecP224K1Field.Multiply(t1, x1, t1);
            SecP224K1Field.SquareN(t1, 2, t1);
            SecP224K1Field.Multiply(t1, x1, t1);
            SecP224K1Field.SquareN(t1, 4, t1);
            SecP224K1Field.Multiply(t1, x3, t1);
            SecP224K1Field.Square(t1, t1);

            uint[] t2 = x84;
            SecP224K1Field.Square(t1, t2);

            if (Nat224.Eq(x1, t2))
            {
                return(new SecP224K1FieldElement(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
             */
            SecP224K1Field.Multiply(t1, PRECOMP_POW2, t1);

            SecP224K1Field.Square(t1, t2);

            if (Nat224.Eq(x1, t2))
            {
                return(new SecP224K1FieldElement(t1));
            }

            return(null);
        }
Esempio n. 2
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;

            SecP224K1FieldElement X1 = (SecP224K1FieldElement)this.RawXCoord, Y1 = (SecP224K1FieldElement)this.RawYCoord;
            SecP224K1FieldElement X2 = (SecP224K1FieldElement)b.RawXCoord, Y2 = (SecP224K1FieldElement)b.RawYCoord;

            SecP224K1FieldElement Z1 = (SecP224K1FieldElement)this.RawZCoords[0];
            SecP224K1FieldElement Z2 = (SecP224K1FieldElement)b.RawZCoords[0];

            uint c;

            uint[] tt1 = Nat224.CreateExt();
            uint[] t2  = Nat224.Create();
            uint[] t3  = Nat224.Create();
            uint[] t4  = Nat224.Create();

            bool Z1IsOne = Z1.IsOne;

            uint[] U2, S2;
            if (Z1IsOne)
            {
                U2 = X2.x;
                S2 = Y2.x;
            }
            else
            {
                S2 = t3;
                SecP224K1Field.Square(Z1.x, S2);

                U2 = t2;
                SecP224K1Field.Multiply(S2, X2.x, U2);

                SecP224K1Field.Multiply(S2, Z1.x, S2);
                SecP224K1Field.Multiply(S2, Y2.x, S2);
            }

            bool Z2IsOne = Z2.IsOne;

            uint[] U1, S1;
            if (Z2IsOne)
            {
                U1 = X1.x;
                S1 = Y1.x;
            }
            else
            {
                S1 = t4;
                SecP224K1Field.Square(Z2.x, S1);

                U1 = tt1;
                SecP224K1Field.Multiply(S1, X1.x, U1);

                SecP224K1Field.Multiply(S1, Z2.x, S1);
                SecP224K1Field.Multiply(S1, Y1.x, S1);
            }

            uint[] H = Nat224.Create();
            SecP224K1Field.Subtract(U1, U2, H);

            uint[] R = t2;
            SecP224K1Field.Subtract(S1, S2, R);

            // Check if b == this or b == -this
            if (Nat224.IsZero(H))
            {
                if (Nat224.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 = t3;
            SecP224K1Field.Square(H, HSquared);

            uint[] G = Nat224.Create();
            SecP224K1Field.Multiply(HSquared, H, G);

            uint[] V = t3;
            SecP224K1Field.Multiply(HSquared, U1, V);

            SecP224K1Field.Negate(G, G);
            Nat224.Mul(S1, G, tt1);

            c = Nat224.AddBothTo(V, V, G);
            SecP224K1Field.Reduce32(c, G);

            SecP224K1FieldElement X3 = new SecP224K1FieldElement(t4);

            SecP224K1Field.Square(R, X3.x);
            SecP224K1Field.Subtract(X3.x, G, X3.x);

            SecP224K1FieldElement Y3 = new SecP224K1FieldElement(G);

            SecP224K1Field.Subtract(V, X3.x, Y3.x);
            SecP224K1Field.MultiplyAddToExt(Y3.x, R, tt1);
            SecP224K1Field.Reduce(tt1, Y3.x);

            SecP224K1FieldElement Z3 = new SecP224K1FieldElement(H);

            if (!Z1IsOne)
            {
                SecP224K1Field.Multiply(Z3.x, Z1.x, Z3.x);
            }
            if (!Z2IsOne)
            {
                SecP224K1Field.Multiply(Z3.x, Z2.x, Z3.x);
            }

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

            return(new SecP224K1Point(curve, X3, Y3, zs, IsCompressed));
        }
Esempio n. 3
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 public override ECFieldElement Square()
 {
     uint[] z = Nat224.Create();
     SecP224K1Field.Square(x, z);
     return(new SecP224K1FieldElement(z));
 }
Esempio n. 4
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        public override ECPoint Twice()
        {
            if (this.IsInfinity)
            {
                return(this);
            }

            ECCurve curve = this.Curve;

            SecP224K1FieldElement Y1 = (SecP224K1FieldElement)this.RawYCoord;

            if (Y1.IsZero)
            {
                return(curve.Infinity);
            }

            SecP224K1FieldElement X1 = (SecP224K1FieldElement)this.RawXCoord, Z1 = (SecP224K1FieldElement)this.RawZCoords[0];

            uint c;

            uint[] Y1Squared = Nat224.Create();
            SecP224K1Field.Square(Y1.x, Y1Squared);

            uint[] T = Nat224.Create();
            SecP224K1Field.Square(Y1Squared, T);

            uint[] M = Nat224.Create();
            SecP224K1Field.Square(X1.x, M);
            c = Nat224.AddBothTo(M, M, M);
            SecP224K1Field.Reduce32(c, M);

            uint[] S = Y1Squared;
            SecP224K1Field.Multiply(Y1Squared, X1.x, S);
            c = Nat.ShiftUpBits(7, S, 2, 0);
            SecP224K1Field.Reduce32(c, S);

            uint[] t1 = Nat224.Create();
            c = Nat.ShiftUpBits(7, T, 3, 0, t1);
            SecP224K1Field.Reduce32(c, t1);

            SecP224K1FieldElement X3 = new SecP224K1FieldElement(T);

            SecP224K1Field.Square(M, X3.x);
            SecP224K1Field.Subtract(X3.x, S, X3.x);
            SecP224K1Field.Subtract(X3.x, S, X3.x);

            SecP224K1FieldElement Y3 = new SecP224K1FieldElement(S);

            SecP224K1Field.Subtract(S, X3.x, Y3.x);
            SecP224K1Field.Multiply(Y3.x, M, Y3.x);
            SecP224K1Field.Subtract(Y3.x, t1, Y3.x);

            SecP224K1FieldElement Z3 = new SecP224K1FieldElement(M);

            SecP224K1Field.Twice(Y1.x, Z3.x);
            if (!Z1.IsOne)
            {
                SecP224K1Field.Multiply(Z3.x, Z1.x, Z3.x);
            }

            return(new SecP224K1Point(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed));
        }