/**
* Subtracts another <code>ECPoints.F2m</code> from <code>this</code>
* without checking if both points are on the same curve. Used by
* multiplication algorithms, because there all points are a multiple
* of the same point and hence the checks can be omitted.
* @param b The other <code>ECPoints.F2m</code> to subtract from
* <code>this</code>.
* @return <code>this - b</code>
*/
internal F2mPoint SubtractSimple(
F2mPoint b)
{
if (b.IsInfinity)
return this;
// Add -b
return AddSimple((F2mPoint) b.Negate());
}
protected F2mCurve(int m, int k1, int k2, int k3, ECFieldElement a, ECFieldElement b, BigInteger order, BigInteger cofactor)
: base(m, k1, k2, k3)
{
this.m = m;
this.k1 = k1;
this.k2 = k2;
this.k3 = k3;
this.m_order = order;
this.m_cofactor = cofactor;
this.m_infinity = new F2mPoint(this, null, null);
this.m_a = a;
this.m_b = b;
this.m_coord = F2M_DEFAULT_COORDS;
}
/**
* Adds another <code>ECPoints.F2m</code> to <code>this</code> without
* checking if both points are on the same curve. Used by multiplication
* algorithms, because there all points are a multiple of the same point
* and hence the checks can be omitted.
* @param b The other <code>ECPoints.F2m</code> to add to
* <code>this</code>.
* @return <code>this + b</code>
*/
internal F2mPoint AddSimple(F2mPoint b)
{
if (this.IsInfinity)
return b;
if (b.IsInfinity)
return this;
ECCurve curve = this.Curve;
int coord = curve.CoordinateSystem;
ECFieldElement X1 = this.RawXCoord;
ECFieldElement X2 = b.RawXCoord;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement Y1 = this.RawYCoord;
ECFieldElement Y2 = b.RawYCoord;
ECFieldElement dx = X1.Add(X2), dy = Y1.Add(Y2);
if (dx.IsZero)
{
if (dy.IsZero)
{
return (F2mPoint)Twice();
}
return (F2mPoint)curve.Infinity;
}
ECFieldElement L = dy.Divide(dx);
ECFieldElement X3 = L.Square().Add(L).Add(dx).Add(curve.A);
ECFieldElement Y3 = L.Multiply(X1.Add(X3)).Add(X3).Add(Y1);
return new F2mPoint(curve, X3, Y3, IsCompressed);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Y1 = this.RawYCoord, Z1 = this.RawZCoords[0];
ECFieldElement Y2 = b.RawYCoord, Z2 = b.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement U1 = Y2, V1 = X2;
if (!Z1IsOne)
{
U1 = U1.Multiply(Z1);
V1 = V1.Multiply(Z1);
}
bool Z2IsOne = Z2.IsOne;
ECFieldElement U2 = Y1, V2 = X1;
if (!Z2IsOne)
{
U2 = U2.Multiply(Z2);
V2 = V2.Multiply(Z2);
}
ECFieldElement U = U1.Add(U2);
ECFieldElement V = V1.Add(V2);
if (V.IsZero)
{
if (U.IsZero)
{
return (F2mPoint)Twice();
}
return (F2mPoint)curve.Infinity;
}
ECFieldElement VSq = V.Square();
ECFieldElement VCu = VSq.Multiply(V);
ECFieldElement W = Z1IsOne ? Z2 : Z2IsOne ? Z1 : Z1.Multiply(Z2);
ECFieldElement uv = U.Add(V);
ECFieldElement A = uv.MultiplyPlusProduct(U, VSq, curve.A).Multiply(W).Add(VCu);
ECFieldElement X3 = V.Multiply(A);
ECFieldElement VSqZ2 = Z2IsOne ? VSq : VSq.Multiply(Z2);
ECFieldElement Y3 = U.MultiplyPlusProduct(X1, V, Y1).MultiplyPlusProduct(VSqZ2, uv, A);
ECFieldElement Z3 = VCu.Multiply(W);
return new F2mPoint(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed);
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
if (X1.IsZero)
{
if (X2.IsZero)
return (F2mPoint)curve.Infinity;
return b.AddSimple(this);
}
ECFieldElement L1 = this.RawYCoord, Z1 = this.RawZCoords[0];
ECFieldElement L2 = b.RawYCoord, Z2 = b.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement U2 = X2, S2 = L2;
if (!Z1IsOne)
{
U2 = U2.Multiply(Z1);
S2 = S2.Multiply(Z1);
}
bool Z2IsOne = Z2.IsOne;
ECFieldElement U1 = X1, S1 = L1;
if (!Z2IsOne)
{
U1 = U1.Multiply(Z2);
S1 = S1.Multiply(Z2);
}
ECFieldElement A = S1.Add(S2);
ECFieldElement B = U1.Add(U2);
if (B.IsZero)
{
if (A.IsZero)
{
return (F2mPoint)Twice();
}
return (F2mPoint)curve.Infinity;
}
ECFieldElement X3, L3, Z3;
if (X2.IsZero)
{
// TODO This can probably be optimized quite a bit
ECPoint p = this.Normalize();
X1 = p.RawXCoord;
ECFieldElement Y1 = p.YCoord;
ECFieldElement Y2 = L2;
ECFieldElement L = Y1.Add(Y2).Divide(X1);
X3 = L.Square().Add(L).Add(X1).Add(curve.A);
if (X3.IsZero)
{
return new F2mPoint(curve, X3, curve.B.Sqrt(), IsCompressed);
}
ECFieldElement Y3 = L.Multiply(X1.Add(X3)).Add(X3).Add(Y1);
L3 = Y3.Divide(X3).Add(X3);
Z3 = curve.FromBigInteger(BigInteger.One);
}
else
{
B = B.Square();
ECFieldElement AU1 = A.Multiply(U1);
ECFieldElement AU2 = A.Multiply(U2);
X3 = AU1.Multiply(AU2);
if (X3.IsZero)
{
return new F2mPoint(curve, X3, curve.B.Sqrt(), IsCompressed);
}
ECFieldElement ABZ2 = A.Multiply(B);
if (!Z2IsOne)
{
ABZ2 = ABZ2.Multiply(Z2);
}
L3 = AU2.Add(B).SquarePlusProduct(ABZ2, L1.Add(Z1));
Z3 = ABZ2;
if (!Z1IsOne)
{
Z3 = Z3.Multiply(Z1);
}
}
return new F2mPoint(curve, X3, L3, new ECFieldElement[] { Z3 }, IsCompressed);
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}
/**
* Constructor for Pentanomial Polynomial Basis (PPB).
* @param m The exponent <code>m</code> of
* <code>F<sub>2<sup>m</sup></sub></code>.
* @param k1 The integer <code>k1</code> where <code>x<sup>m</sup> +
* x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
* represents the reduction polynomial <code>f(z)</code>.
* @param k2 The integer <code>k2</code> where <code>x<sup>m</sup> +
* x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
* represents the reduction polynomial <code>f(z)</code>.
* @param k3 The integer <code>k3</code> where <code>x<sup>m</sup> +
* x<sup>k3</sup> + x<sup>k2</sup> + x<sup>k1</sup> + 1</code>
* represents the reduction polynomial <code>f(z)</code>.
* @param a The coefficient <code>a</code> in the Weierstrass equation
* for non-supersingular elliptic curves over
* <code>F<sub>2<sup>m</sup></sub></code>.
* @param b The coefficient <code>b</code> in the Weierstrass equation
* for non-supersingular elliptic curves over
* <code>F<sub>2<sup>m</sup></sub></code>.
* @param order The order of the main subgroup of the elliptic curve.
* @param cofactor The cofactor of the elliptic curve, i.e.
* <code>#E<sub>a</sub>(F<sub>2<sup>m</sup></sub>) = h * n</code>.
*/
public F2mCurve(
int m,
int k1,
int k2,
int k3,
BigInteger a,
BigInteger b,
BigInteger order,
BigInteger cofactor)
: base(m, k1, k2, k3)
{
this.m = m;
this.k1 = k1;
this.k2 = k2;
this.k3 = k3;
this.m_order = order;
this.m_cofactor = cofactor;
this.m_infinity = new F2mPoint(this, null, null);
if (k1 == 0)
throw new ArgumentException("k1 must be > 0");
if (k2 == 0)
{
if (k3 != 0)
throw new ArgumentException("k3 must be 0 if k2 == 0");
}
else
{
if (k2 <= k1)
throw new ArgumentException("k2 must be > k1");
if (k3 <= k2)
throw new ArgumentException("k3 must be > k2");
}
this.m_a = FromBigInteger(a);
this.m_b = FromBigInteger(b);
this.m_coord = F2M_DEFAULT_COORDS;
}
