using System;
using System.Collections;
using System.Diagnostics;
using System.Text;
using Org.BouncyCastle.Math.EC.Multiplier;
namespace Org.BouncyCastle.Math.EC
{
/**
* base class for points on elliptic curves.
*/
public abstract class ECPoint
{
protected static ECFieldElement[] EMPTY_ZS = new ECFieldElement[0];
protected static ECFieldElement[] GetInitialZCoords(ECCurve curve)
{
// Cope with null curve, most commonly used by implicitlyCa
int coord = null == curve ? ECCurve.COORD_AFFINE : curve.CoordinateSystem;
switch (coord)
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
return EMPTY_ZS;
default:
break;
}
ECFieldElement one = curve.FromBigInteger(BigInteger.One);
switch (coord)
{
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
return new ECFieldElement[] { one };
case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
return new ECFieldElement[] { one, one, one };
case ECCurve.COORD_JACOBIAN_MODIFIED:
return new ECFieldElement[] { one, curve.A };
default:
throw new ArgumentException("unknown coordinate system");
}
}
protected internal readonly ECCurve m_curve;
protected internal readonly ECFieldElement m_x, m_y;
protected internal readonly ECFieldElement[] m_zs;
protected internal readonly bool m_withCompression;
protected internal PreCompInfo m_preCompInfo = null;
protected ECPoint(ECCurve curve, ECFieldElement x, ECFieldElement y, bool withCompression)
: this(curve, x, y, GetInitialZCoords(curve), withCompression)
{
}
internal ECPoint(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs, bool withCompression)
{
this.m_curve = curve;
this.m_x = x;
this.m_y = y;
this.m_zs = zs;
this.m_withCompression = withCompression;
}
public ECPoint GetDetachedPoint()
{
return Normalize().Detach();
}
public virtual ECCurve Curve
{
get { return m_curve; }
}
protected abstract ECPoint Detach();
protected virtual int CurveCoordinateSystem
{
get
{
// Cope with null curve, most commonly used by implicitlyCa
return null == m_curve ? ECCurve.COORD_AFFINE : m_curve.CoordinateSystem;
}
}
/**
* Normalizes this point, and then returns the affine x-coordinate.
*
* Note: normalization can be expensive, this method is deprecated in favour
* of caller-controlled normalization.
*/
[Obsolete("Use AffineXCoord, or Normalize() and XCoord, instead")]
public virtual ECFieldElement X
{
get { return Normalize().XCoord; }
}
/**
* Normalizes this point, and then returns the affine y-coordinate.
*
* Note: normalization can be expensive, this method is deprecated in favour
* of caller-controlled normalization.
*/
[Obsolete("Use AffineYCoord, or Normalize() and YCoord, instead")]
public virtual ECFieldElement Y
{
get { return Normalize().YCoord; }
}
/**
* Returns the affine x-coordinate after checking that this point is normalized.
*
* @return The affine x-coordinate of this point
* @throws IllegalStateException if the point is not normalized
*/
public virtual ECFieldElement AffineXCoord
{
get
{
CheckNormalized();
return XCoord;
}
}
/**
* Returns the affine y-coordinate after checking that this point is normalized
*
* @return The affine y-coordinate of this point
* @throws IllegalStateException if the point is not normalized
*/
public virtual ECFieldElement AffineYCoord
{
get
{
CheckNormalized();
return YCoord;
}
}
/**
* Returns the x-coordinate.
*
* Caution: depending on the curve's coordinate system, this may not be the same value as in an
* affine coordinate system; use Normalize() to get a point where the coordinates have their
* affine values, or use AffineXCoord if you expect the point to already have been normalized.
*
* @return the x-coordinate of this point
*/
public virtual ECFieldElement XCoord
{
get { return m_x; }
}
/**
* Returns the y-coordinate.
*
* Caution: depending on the curve's coordinate system, this may not be the same value as in an
* affine coordinate system; use Normalize() to get a point where the coordinates have their
* affine values, or use AffineYCoord if you expect the point to already have been normalized.
*
* @return the y-coordinate of this point
*/
public virtual ECFieldElement YCoord
{
get { return m_y; }
}
public virtual ECFieldElement GetZCoord(int index)
{
return (index < 0 || index >= m_zs.Length) ? null : m_zs[index];
}
public virtual ECFieldElement[] GetZCoords()
{
int zsLen = m_zs.Length;
if (zsLen == 0)
{
return m_zs;
}
ECFieldElement[] copy = new ECFieldElement[zsLen];
Array.Copy(m_zs, 0, copy, 0, zsLen);
return copy;
}
protected internal ECFieldElement RawXCoord
{
get { return m_x; }
}
protected internal ECFieldElement RawYCoord
{
get { return m_y; }
}
protected internal ECFieldElement[] RawZCoords
{
get { return m_zs; }
}
protected virtual void CheckNormalized()
{
if (!IsNormalized())
throw new InvalidOperationException("point not in normal form");
}
public virtual bool IsNormalized()
{
int coord = this.CurveCoordinateSystem;
return coord == ECCurve.COORD_AFFINE
|| coord == ECCurve.COORD_LAMBDA_AFFINE
|| IsInfinity
|| RawZCoords[0].IsOne;
}
/**
* Normalization ensures that any projective coordinate is 1, and therefore that the x, y
* coordinates reflect those of the equivalent point in an affine coordinate system.
*
* @return a new ECPoint instance representing the same point, but with normalized coordinates
*/
public virtual ECPoint Normalize()
{
if (this.IsInfinity)
{
return this;
}
switch (this.CurveCoordinateSystem)
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
{
return this;
}
default:
{
ECFieldElement Z1 = RawZCoords[0];
if (Z1.IsOne)
{
return this;
}
return Normalize(Z1.Invert());
}
}
}
internal virtual ECPoint Normalize(ECFieldElement zInv)
{
switch (this.CurveCoordinateSystem)
{
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
return CreateScaledPoint(zInv, zInv);
}
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_CHUDNOVSKY:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
ECFieldElement zInv2 = zInv.Square(), zInv3 = zInv2.Multiply(zInv);
return CreateScaledPoint(zInv2, zInv3);
}
default:
{
throw new InvalidOperationException("not a projective coordinate system");
}
}
}
protected virtual ECPoint CreateScaledPoint(ECFieldElement sx, ECFieldElement sy)
{
return Curve.CreateRawPoint(RawXCoord.Multiply(sx), RawYCoord.Multiply(sy), IsCompressed);
}
public bool IsInfinity
{
get { return m_x == null && m_y == null; }
}
public bool IsCompressed
{
get { return m_withCompression; }
}
public override bool Equals(object obj)
{
return Equals(obj as ECPoint);
}
public virtual bool Equals(ECPoint other)
{
if (this == other)
return true;
if (null == other)
return false;
ECCurve c1 = this.Curve, c2 = other.Curve;
bool n1 = (null == c1), n2 = (null == c2);
bool i1 = IsInfinity, i2 = other.IsInfinity;
if (i1 || i2)
{
return (i1 && i2) && (n1 || n2 || c1.Equals(c2));
}
ECPoint p1 = this, p2 = other;
if (n1 && n2)
{
// Points with null curve are in affine form, so already normalized
}
else if (n1)
{
p2 = p2.Normalize();
}
else if (n2)
{
p1 = p1.Normalize();
}
else if (!c1.Equals(c2))
{
return false;
}
else
{
// TODO Consider just requiring already normalized, to avoid silent performance degradation
ECPoint[] points = new ECPoint[] { this, c1.ImportPoint(p2) };
// TODO This is a little strong, really only requires coZNormalizeAll to get Zs equal
c1.NormalizeAll(points);
p1 = points[0];
p2 = points[1];
}
return p1.XCoord.Equals(p2.XCoord) && p1.YCoord.Equals(p2.YCoord);
}
public override int GetHashCode()
{
ECCurve c = this.Curve;
int hc = (null == c) ? 0 : ~c.GetHashCode();
if (!this.IsInfinity)
{
// TODO Consider just requiring already normalized, to avoid silent performance degradation
ECPoint p = Normalize();
hc ^= p.XCoord.GetHashCode() * 17;
hc ^= p.YCoord.GetHashCode() * 257;
}
return hc;
}
public override string ToString()
{
if (this.IsInfinity)
{
return "INF";
}
StringBuilder sb = new StringBuilder();
sb.Append('(');
sb.Append(RawXCoord);
sb.Append(',');
sb.Append(RawYCoord);
for (int i = 0; i < m_zs.Length; ++i)
{
sb.Append(',');
sb.Append(m_zs[i]);
}
sb.Append(')');
return sb.ToString();
}
public virtual byte[] GetEncoded()
{
return GetEncoded(m_withCompression);
}
public abstract byte[] GetEncoded(bool compressed);
protected internal abstract bool CompressionYTilde { get; }
public abstract ECPoint Add(ECPoint b);
public abstract ECPoint Subtract(ECPoint b);
public abstract ECPoint Negate();
public virtual ECPoint TimesPow2(int e)
{
if (e < 0)
throw new ArgumentException("cannot be negative", "e");
ECPoint p = this;
while (--e >= 0)
{
p = p.Twice();
}
return p;
}
public abstract ECPoint Twice();
public abstract ECPoint Multiply(BigInteger b);
public virtual ECPoint TwicePlus(ECPoint b)
{
return Twice().Add(b);
}
public virtual ECPoint ThreeTimes()
{
return TwicePlus(this);
}
}
public abstract class ECPointBase
: ECPoint
{
protected internal ECPointBase(
ECCurve curve,
ECFieldElement x,
ECFieldElement y,
bool withCompression)
: base(curve, x, y, withCompression)
{
}
protected internal ECPointBase(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs, bool withCompression)
: base(curve, x, y, zs, withCompression)
{
}
/**
* return the field element encoded with point compression. (S 4.3.6)
*/
public override byte[] GetEncoded(bool compressed)
{
if (this.IsInfinity)
{
return new byte[1];
}
ECPoint normed = Normalize();
byte[] X = normed.XCoord.GetEncoded();
if (compressed)
{
byte[] PO = new byte[X.Length + 1];
PO[0] = (byte)(normed.CompressionYTilde ? 0x03 : 0x02);
Array.Copy(X, 0, PO, 1, X.Length);
return PO;
}
byte[] Y = normed.YCoord.GetEncoded();
{
byte[] PO = new byte[X.Length + Y.Length + 1];
PO[0] = 0x04;
Array.Copy(X, 0, PO, 1, X.Length);
Array.Copy(Y, 0, PO, X.Length + 1, Y.Length);
return PO;
}
}
/**
* Multiplies this ECPoint by the given number.
* @param k The multiplicator.
* @return k * this.
*/
public override ECPoint Multiply(BigInteger k)
{
return this.Curve.GetMultiplier().Multiply(this, k);
}
}
/**
* Elliptic curve points over Fp
*/
public class FpPoint
: ECPointBase
{
/**
* Create a point which encodes with point compression.
*
* @param curve the curve to use
* @param x affine x co-ordinate
* @param y affine y co-ordinate
*/
public FpPoint(ECCurve curve, ECFieldElement x, ECFieldElement y)
: this(curve, x, y, false)
{
}
/**
* Create a point that encodes with or without point compresion.
*
* @param curve the curve to use
* @param x affine x co-ordinate
* @param y affine y co-ordinate
* @param withCompression if true encode with point compression
*/
public FpPoint(ECCurve curve, ECFieldElement x, ECFieldElement y, bool withCompression)
: base(curve, x, y, withCompression)
{
if ((x == null) != (y == null))
throw new ArgumentException("Exactly one of the field elements is null");
}
internal FpPoint(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs, bool withCompression)
: base(curve, x, y, zs, withCompression)
{
}
protected override ECPoint Detach()
{
return new FpPoint(null, AffineXCoord, AffineYCoord);
}
protected internal override bool CompressionYTilde
{
get { return this.AffineYCoord.TestBitZero(); }
}
public override ECFieldElement GetZCoord(int index)
{
if (index == 1 && ECCurve.COORD_JACOBIAN_MODIFIED == this.CurveCoordinateSystem)
{
return GetJacobianModifiedW();
}
return base.GetZCoord(index);
}
// B.3 pg 62
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;
int coord = curve.CoordinateSystem;
ECFieldElement X1 = this.RawXCoord, Y1 = this.RawYCoord;
ECFieldElement X2 = b.RawXCoord, Y2 = b.RawYCoord;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement dx = X2.Subtract(X1), dy = Y2.Subtract(Y1);
if (dx.IsZero)
{
if (dy.IsZero)
{
// this == b, i.e. this must be doubled
return Twice();
}
// this == -b, i.e. the result is the point at infinity
return Curve.Infinity;
}
ECFieldElement gamma = dy.Divide(dx);
ECFieldElement X3 = gamma.Square().Subtract(X1).Subtract(X2);
ECFieldElement Y3 = gamma.Multiply(X1.Subtract(X3)).Subtract(Y1);
return new FpPoint(Curve, X3, Y3, IsCompressed);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z1 = this.RawZCoords[0];
ECFieldElement Z2 = b.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
bool Z2IsOne = Z2.IsOne;
ECFieldElement u1 = Z1IsOne ? Y2 : Y2.Multiply(Z1);
ECFieldElement u2 = Z2IsOne ? Y1 : Y1.Multiply(Z2);
ECFieldElement u = u1.Subtract(u2);
ECFieldElement v1 = Z1IsOne ? X2 : X2.Multiply(Z1);
ECFieldElement v2 = Z2IsOne ? X1 : X1.Multiply(Z2);
ECFieldElement v = v1.Subtract(v2);
// Check if b == this or b == -this
if (v.IsZero)
{
if (u.IsZero)
{
// 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;
}
// TODO Optimize for when w == 1
ECFieldElement w = Z1IsOne ? Z2 : Z2IsOne ? Z1 : Z1.Multiply(Z2);
ECFieldElement vSquared = v.Square();
ECFieldElement vCubed = vSquared.Multiply(v);
ECFieldElement vSquaredV2 = vSquared.Multiply(v2);
ECFieldElement A = u.Square().Multiply(w).Subtract(vCubed).Subtract(Two(vSquaredV2));
ECFieldElement X3 = v.Multiply(A);
ECFieldElement Y3 = vSquaredV2.Subtract(A).Multiply(u).Subtract(vCubed.Multiply(u2));
ECFieldElement Z3 = vCubed.Multiply(w);
return new FpPoint(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed);
}
case ECCurve.COORD_JACOBIAN:
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
ECFieldElement Z1 = this.RawZCoords[0];
ECFieldElement Z2 = b.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement X3, Y3, Z3, Z3Squared = null;
if (!Z1IsOne && Z1.Equals(Z2))
{
// TODO Make this available as public method coZAdd?
ECFieldElement dx = X1.Subtract(X2), dy = Y1.Subtract(Y2);
if (dx.IsZero)
{
if (dy.IsZero)
{
return Twice();
}
return curve.Infinity;
}
ECFieldElement C = dx.Square();
ECFieldElement W1 = X1.Multiply(C), W2 = X2.Multiply(C);
ECFieldElement A1 = W1.Subtract(W2).Multiply(Y1);
X3 = dy.Square().Subtract(W1).Subtract(W2);
Y3 = W1.Subtract(X3).Multiply(dy).Subtract(A1);
Z3 = dx;
if (Z1IsOne)
{
Z3Squared = C;
}
else
{
Z3 = Z3.Multiply(Z1);
}
}
else
{
ECFieldElement Z1Squared, U2, S2;
if (Z1IsOne)
{
Z1Squared = Z1; U2 = X2; S2 = Y2;
}
else
{
Z1Squared = Z1.Square();
U2 = Z1Squared.Multiply(X2);
ECFieldElement Z1Cubed = Z1Squared.Multiply(Z1);
S2 = Z1Cubed.Multiply(Y2);
}
bool Z2IsOne = Z2.IsOne;
ECFieldElement Z2Squared, U1, S1;
if (Z2IsOne)
{
Z2Squared = Z2; U1 = X1; S1 = Y1;
}
else
{
Z2Squared = Z2.Square();
U1 = Z2Squared.Multiply(X1);
ECFieldElement Z2Cubed = Z2Squared.Multiply(Z2);
S1 = Z2Cubed.Multiply(Y1);
}
ECFieldElement H = U1.Subtract(U2);
ECFieldElement R = S1.Subtract(S2);
// Check if b == this or b == -this
if (H.IsZero)
{
if (R.IsZero)
{
// 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;
}
ECFieldElement HSquared = H.Square();
ECFieldElement G = HSquared.Multiply(H);
ECFieldElement V = HSquared.Multiply(U1);
X3 = R.Square().Add(G).Subtract(Two(V));
Y3 = V.Subtract(X3).Multiply(R).Subtract(S1.Multiply(G));
Z3 = H;
if (!Z1IsOne)
{
Z3 = Z3.Multiply(Z1);
}
if (!Z2IsOne)
{
Z3 = Z3.Multiply(Z2);
}
// Alternative calculation of Z3 using fast square
//X3 = four(X3);
//Y3 = eight(Y3);
//Z3 = doubleProductFromSquares(Z1, Z2, Z1Squared, Z2Squared).multiply(H);
if (Z3 == H)
{
Z3Squared = HSquared;
}
}
ECFieldElement[] zs;
if (coord == ECCurve.COORD_JACOBIAN_MODIFIED)
{
// TODO If the result will only be used in a subsequent addition, we don't need W3
ECFieldElement W3 = CalculateJacobianModifiedW(Z3, Z3Squared);
zs = new ECFieldElement[] { Z3, W3 };
}
else
{
zs = new ECFieldElement[] { Z3 };
}
return new FpPoint(curve, X3, Y3, zs, IsCompressed);
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}
// B.3 pg 62
public override ECPoint Twice()
{
if (this.IsInfinity)
return this;
ECCurve curve = this.Curve;
ECFieldElement Y1 = this.RawYCoord;
if (Y1.IsZero)
return curve.Infinity;
int coord = curve.CoordinateSystem;
ECFieldElement X1 = this.RawXCoord;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement X1Squared = X1.Square();
ECFieldElement gamma = Three(X1Squared).Add(this.Curve.A).Divide(Two(Y1));
ECFieldElement X3 = gamma.Square().Subtract(Two(X1));
ECFieldElement Y3 = gamma.Multiply(X1.Subtract(X3)).Subtract(Y1);
return new FpPoint(Curve, X3, Y3, IsCompressed);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Z1 = this.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
// TODO Optimize for small negative a4 and -3
ECFieldElement w = curve.A;
if (!w.IsZero && !Z1IsOne)
{
w = w.Multiply(Z1.Square());
}
w = w.Add(Three(X1.Square()));
ECFieldElement s = Z1IsOne ? Y1 : Y1.Multiply(Z1);
ECFieldElement t = Z1IsOne ? Y1.Square() : s.Multiply(Y1);
ECFieldElement B = X1.Multiply(t);
ECFieldElement _4B = Four(B);
ECFieldElement h = w.Square().Subtract(Two(_4B));
ECFieldElement _2s = Two(s);
ECFieldElement X3 = h.Multiply(_2s);
ECFieldElement _2t = Two(t);
ECFieldElement Y3 = _4B.Subtract(h).Multiply(w).Subtract(Two(_2t.Square()));
ECFieldElement _4sSquared = Z1IsOne ? Two(_2t) : _2s.Square();
ECFieldElement Z3 = Two(_4sSquared).Multiply(s);
return new FpPoint(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed);
}
case ECCurve.COORD_JACOBIAN:
{
ECFieldElement Z1 = this.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement Y1Squared = Y1.Square();
ECFieldElement T = Y1Squared.Square();
ECFieldElement a4 = curve.A;
ECFieldElement a4Neg = a4.Negate();
ECFieldElement M, S;
if (a4Neg.ToBigInteger().Equals(BigInteger.ValueOf(3)))
{
ECFieldElement Z1Squared = Z1IsOne ? Z1 : Z1.Square();
M = Three(X1.Add(Z1Squared).Multiply(X1.Subtract(Z1Squared)));
S = Four(Y1Squared.Multiply(X1));
}
else
{
ECFieldElement X1Squared = X1.Square();
M = Three(X1Squared);
if (Z1IsOne)
{
M = M.Add(a4);
}
else if (!a4.IsZero)
{
ECFieldElement Z1Squared = Z1IsOne ? Z1 : Z1.Square();
ECFieldElement Z1Pow4 = Z1Squared.Square();
if (a4Neg.BitLength < a4.BitLength)
{
M = M.Subtract(Z1Pow4.Multiply(a4Neg));
}
else
{
M = M.Add(Z1Pow4.Multiply(a4));
}
}
//S = two(doubleProductFromSquares(X1, Y1Squared, X1Squared, T));
S = Four(X1.Multiply(Y1Squared));
}
ECFieldElement X3 = M.Square().Subtract(Two(S));
ECFieldElement Y3 = S.Subtract(X3).Multiply(M).Subtract(Eight(T));
ECFieldElement Z3 = Two(Y1);
if (!Z1IsOne)
{
Z3 = Z3.Multiply(Z1);
}
// Alternative calculation of Z3 using fast square
//ECFieldElement Z3 = doubleProductFromSquares(Y1, Z1, Y1Squared, Z1Squared);
return new FpPoint(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed);
}
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
return TwiceJacobianModified(true);
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}
public override ECPoint TwicePlus(ECPoint b)
{
if (this == b)
return ThreeTimes();
if (this.IsInfinity)
return b;
if (b.IsInfinity)
return Twice();
ECFieldElement Y1 = this.RawYCoord;
if (Y1.IsZero)
return b;
ECCurve curve = this.Curve;
int coord = curve.CoordinateSystem;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement X1 = this.RawXCoord;
ECFieldElement X2 = b.RawXCoord, Y2 = b.RawYCoord;
ECFieldElement dx = X2.Subtract(X1), dy = Y2.Subtract(Y1);
if (dx.IsZero)
{
if (dy.IsZero)
{
// this == b i.e. the result is 3P
return ThreeTimes();
}
// this == -b, i.e. the result is P
return this;
}
/*
* Optimized calculation of 2P + Q, as described in "Trading Inversions for
* Multiplications in Elliptic Curve Cryptography", by Ciet, Joye, Lauter, Montgomery.
*/
ECFieldElement X = dx.Square(), Y = dy.Square();
ECFieldElement d = X.Multiply(Two(X1).Add(X2)).Subtract(Y);
if (d.IsZero)
{
return Curve.Infinity;
}
ECFieldElement D = d.Multiply(dx);
ECFieldElement I = D.Invert();
ECFieldElement L1 = d.Multiply(I).Multiply(dy);
ECFieldElement L2 = Two(Y1).Multiply(X).Multiply(dx).Multiply(I).Subtract(L1);
ECFieldElement X4 = (L2.Subtract(L1)).Multiply(L1.Add(L2)).Add(X2);
ECFieldElement Y4 = (X1.Subtract(X4)).Multiply(L2).Subtract(Y1);
return new FpPoint(Curve, X4, Y4, IsCompressed);
}
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
return TwiceJacobianModified(false).Add(b);
}
default:
{
return Twice().Add(b);
}
}
}
public override ECPoint ThreeTimes()
{
if (this.IsInfinity)
return this;
ECFieldElement Y1 = this.RawYCoord;
if (Y1.IsZero)
return this;
ECCurve curve = this.Curve;
int coord = curve.CoordinateSystem;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement X1 = this.RawXCoord;
ECFieldElement _2Y1 = Two(Y1);
ECFieldElement X = _2Y1.Square();
ECFieldElement Z = Three(X1.Square()).Add(Curve.A);
ECFieldElement Y = Z.Square();
ECFieldElement d = Three(X1).Multiply(X).Subtract(Y);
if (d.IsZero)
{
return Curve.Infinity;
}
ECFieldElement D = d.Multiply(_2Y1);
ECFieldElement I = D.Invert();
ECFieldElement L1 = d.Multiply(I).Multiply(Z);
ECFieldElement L2 = X.Square().Multiply(I).Subtract(L1);
ECFieldElement X4 = (L2.Subtract(L1)).Multiply(L1.Add(L2)).Add(X1);
ECFieldElement Y4 = (X1.Subtract(X4)).Multiply(L2).Subtract(Y1);
return new FpPoint(Curve, X4, Y4, IsCompressed);
}
case ECCurve.COORD_JACOBIAN_MODIFIED:
{
return TwiceJacobianModified(false).Add(this);
}
default:
{
// NOTE: Be careful about recursions between TwicePlus and ThreeTimes
return Twice().Add(this);
}
}
}
protected virtual ECFieldElement Two(ECFieldElement x)
{
return x.Add(x);
}
protected virtual ECFieldElement Three(ECFieldElement x)
{
return Two(x).Add(x);
}
protected virtual ECFieldElement Four(ECFieldElement x)
{
return Two(Two(x));
}
protected virtual ECFieldElement Eight(ECFieldElement x)
{
return Four(Two(x));
}
protected virtual ECFieldElement DoubleProductFromSquares(ECFieldElement a, ECFieldElement b,
ECFieldElement aSquared, ECFieldElement bSquared)
{
/*
* NOTE: If squaring in the field is faster than multiplication, then this is a quicker
* way to calculate 2.A.B, if A^2 and B^2 are already known.
*/
return a.Add(b).Square().Subtract(aSquared).Subtract(bSquared);
}
// D.3.2 pg 102 (see Note:)
public override ECPoint Subtract(
ECPoint b)
{
if (b.IsInfinity)
return this;
// Add -b
return Add(b.Negate());
}
public override ECPoint Negate()
{
if (IsInfinity)
return this;
ECCurve curve = Curve;
int coord = curve.CoordinateSystem;
if (ECCurve.COORD_AFFINE != coord)
{
return new FpPoint(curve, RawXCoord, RawYCoord.Negate(), RawZCoords, IsCompressed);
}
return new FpPoint(curve, RawXCoord, RawYCoord.Negate(), IsCompressed);
}
protected virtual ECFieldElement CalculateJacobianModifiedW(ECFieldElement Z, ECFieldElement ZSquared)
{
ECFieldElement a4 = this.Curve.A;
if (a4.IsZero)
return a4;
if (ZSquared == null)
{
ZSquared = Z.Square();
}
ECFieldElement W = ZSquared.Square();
ECFieldElement a4Neg = a4.Negate();
if (a4Neg.BitLength < a4.BitLength)
{
W = W.Multiply(a4Neg).Negate();
}
else
{
W = W.Multiply(a4);
}
return W;
}
protected virtual ECFieldElement GetJacobianModifiedW()
{
ECFieldElement[] ZZ = this.RawZCoords;
ECFieldElement W = ZZ[1];
if (W == null)
{
// NOTE: Rarely, twicePlus will result in the need for a lazy W1 calculation here
ZZ[1] = W = CalculateJacobianModifiedW(ZZ[0], null);
}
return W;
}
protected FpPoint TwiceJacobianModified(bool calculateW)
{
ECFieldElement X1 = this.RawXCoord, Y1 = this.RawYCoord, Z1 = this.RawZCoords[0], W1 = GetJacobianModifiedW();
ECFieldElement X1Squared = X1.Square();
ECFieldElement M = Three(X1Squared).Add(W1);
ECFieldElement _2Y1 = Two(Y1);
ECFieldElement _2Y1Squared = _2Y1.Multiply(Y1);
ECFieldElement S = Two(X1.Multiply(_2Y1Squared));
ECFieldElement X3 = M.Square().Subtract(Two(S));
ECFieldElement _4T = _2Y1Squared.Square();
ECFieldElement _8T = Two(_4T);
ECFieldElement Y3 = M.Multiply(S.Subtract(X3)).Subtract(_8T);
ECFieldElement W3 = calculateW ? Two(_8T.Multiply(W1)) : null;
ECFieldElement Z3 = Z1.IsOne ? _2Y1 : _2Y1.Multiply(Z1);
return new FpPoint(this.Curve, X3, Y3, new ECFieldElement[] { Z3, W3 }, IsCompressed);
}
}
/**
* Elliptic curve points over F2m
*/
public class F2mPoint
: ECPointBase
{
/**
* @param curve base curve
* @param x x point
* @param y y point
*/
public F2mPoint(
ECCurve curve,
ECFieldElement x,
ECFieldElement y)
: this(curve, x, y, false)
{
}
/**
* @param curve base curve
* @param x x point
* @param y y point
* @param withCompression true if encode with point compression.
*/
public F2mPoint(
ECCurve curve,
ECFieldElement x,
ECFieldElement y,
bool withCompression)
: base(curve, x, y, withCompression)
{
if ((x == null) != (y == null))
{
throw new ArgumentException("Exactly one of the field elements is null");
}
if (x != null)
{
// Check if x and y are elements of the same field
F2mFieldElement.CheckFieldElements(x, y);
// Check if x and a are elements of the same field
if (curve != null)
{
F2mFieldElement.CheckFieldElements(x, curve.A);
}
}
}
internal F2mPoint(ECCurve curve, ECFieldElement x, ECFieldElement y, ECFieldElement[] zs, bool withCompression)
: base(curve, x, y, zs, withCompression)
{
}
/**
* Constructor for point at infinity
*/
[Obsolete("Use ECCurve.Infinity property")]
public F2mPoint(
ECCurve curve)
: this(curve, null, null)
{
}
protected override ECPoint Detach()
{
return new F2mPoint(null, AffineXCoord, AffineYCoord);
}
public override ECFieldElement YCoord
{
get
{
int coord = this.CurveCoordinateSystem;
switch (coord)
{
case ECCurve.COORD_LAMBDA_AFFINE:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement X = RawXCoord, L = RawYCoord;
if (this.IsInfinity || X.IsZero)
return L;
// Y is actually Lambda (X + Y/X) here; convert to affine value on the fly
ECFieldElement Y = L.Add(X).Multiply(X);
if (ECCurve.COORD_LAMBDA_PROJECTIVE == coord)
{
ECFieldElement Z = RawZCoords[0];
if (!Z.IsOne)
{
Y = Y.Divide(Z);
}
}
return Y;
}
default:
{
return RawYCoord;
}
}
}
}
protected internal override bool CompressionYTilde
{
get
{
ECFieldElement X = this.RawXCoord;
if (X.IsZero)
{
return false;
}
ECFieldElement Y = this.RawYCoord;
switch (this.CurveCoordinateSystem)
{
case ECCurve.COORD_LAMBDA_AFFINE:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// Y is actually Lambda (X + Y/X) here
return Y.TestBitZero() != X.TestBitZero();
}
default:
{
return Y.Divide(X).TestBitZero();
}
}
}
}
/**
* Check, if two ECPoints can be added or subtracted.
* @param a The first ECPoint to check.
* @param b The second ECPoint to check.
* @throws IllegalArgumentException if a and b
* cannot be added.
*/
private static void CheckPoints(
ECPoint a,
ECPoint b)
{
// Check, if points are on the same curve
if (!a.Curve.Equals(b.Curve))
throw new ArgumentException("Only points on the same curve can be added or subtracted");
// F2mFieldElement.CheckFieldElements(a.x, b.x);
}
/* (non-Javadoc)
* @see org.bouncycastle.math.ec.ECPoint#add(org.bouncycastle.math.ec.ECPoint)
*/
public override ECPoint Add(ECPoint b)
{
CheckPoints(this, b);
return AddSimple((F2mPoint) b);
}
/**
* Adds another ECPoints.F2m to this 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 ECPoints.F2m to add to
* this.
* @return this + b
*/
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 Z2IsOne = Z2.IsOne;
ECFieldElement U1 = Z1.Multiply(Y2);
ECFieldElement U2 = Z2IsOne ? Y1 : Y1.Multiply(Z2);
ECFieldElement U = U1.Add(U2);
ECFieldElement V1 = Z1.Multiply(X2);
ECFieldElement V2 = Z2IsOne ? X1 : X1.Multiply(Z2);
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 = Z2IsOne ? Z1 : Z1.Multiply(Z2);
ECFieldElement uv = U.Add(V);
// TODO Delayed modular reduction for sum of products
ECFieldElement A = uv.Multiply(U).Add(VSq.Multiply(curve.A)).Multiply(W).Add(VCu);
ECFieldElement X3 = V.Multiply(A);
ECFieldElement VSqZ2 = Z2IsOne ? VSq : VSq.Multiply(Z2);
// TODO Delayed modular reduction for sum of products
ECFieldElement Y3 = U.Multiply(X1).Add(Y1.Multiply(V)).Multiply(VSqZ2).Add(A.Multiply(uv));
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);
}
// TODO Delayed modular reduction for sum of products
L3 = AU2.Add(B).Square().Add(ABZ2.Multiply(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");
}
}
}
/* (non-Javadoc)
* @see org.bouncycastle.math.ec.ECPoint#subtract(org.bouncycastle.math.ec.ECPoint)
*/
public override ECPoint Subtract(
ECPoint b)
{
CheckPoints(this, b);
return SubtractSimple((F2mPoint) b);
}
/**
* Subtracts another ECPoints.F2m from this
* 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 ECPoints.F2m to subtract from
* this.
* @return this - b
*/
internal F2mPoint SubtractSimple(
F2mPoint b)
{
if (b.IsInfinity)
return this;
// Add -b
return AddSimple((F2mPoint) b.Negate());
}
public virtual F2mPoint Tau()
{
if (this.IsInfinity)
{
return this;
}
ECCurve curve = this.Curve;
int coord = curve.CoordinateSystem;
ECFieldElement X1 = this.RawXCoord;
switch (coord)
{
case ECCurve.COORD_AFFINE:
case ECCurve.COORD_LAMBDA_AFFINE:
{
ECFieldElement Y1 = this.RawYCoord;
return new F2mPoint(curve, X1.Square(), Y1.Square(), IsCompressed);
}
case ECCurve.COORD_HOMOGENEOUS:
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement Y1 = this.RawYCoord, Z1 = this.RawZCoords[0];
return new F2mPoint(curve, X1.Square(), Y1.Square(), new ECFieldElement[] { Z1.Square() }, IsCompressed);
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}
/* (non-Javadoc)
* @see Org.BouncyCastle.Math.EC.ECPoint#twice()
*/
public override ECPoint Twice()
{
if (this.IsInfinity)
return this;
ECCurve curve = this.Curve;
ECFieldElement X1 = this.RawXCoord;
if (X1.IsZero)
{
// A point with X == 0 is it's own additive inverse
return curve.Infinity;
}
int coord = curve.CoordinateSystem;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement Y1 = this.RawYCoord;
ECFieldElement L1 = Y1.Divide(X1).Add(X1);
ECFieldElement X3 = L1.Square().Add(L1).Add(curve.A);
// TODO Delayed modular reduction for sum of products
ECFieldElement Y3 = X1.Square().Add(X3.Multiply(L1.AddOne()));
return new F2mPoint(curve, X3, Y3, IsCompressed);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Y1 = this.RawYCoord, Z1 = this.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement X1Z1 = Z1IsOne ? X1 : X1.Multiply(Z1);
ECFieldElement Y1Z1 = Z1IsOne ? Y1 : Y1.Multiply(Z1);
ECFieldElement X1Sq = X1.Square();
ECFieldElement S = X1Sq.Add(Y1Z1);
ECFieldElement V = X1Z1;
ECFieldElement vSquared = V.Square();
ECFieldElement sv = S.Add(V);
// TODO Delayed modular reduction for sum of products
ECFieldElement h = sv.Multiply(S).Add(curve.A.Multiply(vSquared));
ECFieldElement X3 = V.Multiply(h);
// TODO Delayed modular reduction for sum of products
ECFieldElement Y3 = h.Multiply(sv).Add(X1Sq.Square().Multiply(V));
ECFieldElement Z3 = V.Multiply(vSquared);
return new F2mPoint(curve, X3, Y3, new ECFieldElement[] { Z3 }, IsCompressed);
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
ECFieldElement L1 = this.RawYCoord, Z1 = this.RawZCoords[0];
bool Z1IsOne = Z1.IsOne;
ECFieldElement L1Z1 = Z1IsOne ? L1 : L1.Multiply(Z1);
ECFieldElement Z1Sq = Z1IsOne ? Z1 : Z1.Square();
ECFieldElement a = curve.A;
ECFieldElement aZ1Sq = Z1IsOne ? a : a.Multiply(Z1Sq);
ECFieldElement T = L1.Square().Add(L1Z1).Add(aZ1Sq);
if (T.IsZero)
{
return new F2mPoint(curve, T, curve.B.Sqrt(), IsCompressed);
}
ECFieldElement X3 = T.Square();
ECFieldElement Z3 = Z1IsOne ? T : T.Multiply(Z1Sq);
ECFieldElement b = curve.B;
ECFieldElement L3;
if (b.BitLength < (curve.FieldSize >> 1))
{
ECFieldElement t1 = L1.Add(X1).Square();
ECFieldElement t4;
if (b.IsOne)
{
t4 = aZ1Sq.Add(Z1Sq).Square();
}
else
{
// TODO t2/t3 can be calculated with one square if we pre-compute sqrt(b)
ECFieldElement t2 = aZ1Sq.Square();
ECFieldElement t3 = b.Multiply(Z1Sq.Square());
t4 = t2.Add(t3);
}
L3 = t1.Add(T).Add(Z1Sq).Multiply(t1).Add(t4).Add(X3);
if (a.IsZero)
{
L3 = L3.Add(Z3);
}
else if (!a.IsOne)
{
L3 = L3.Add(a.AddOne().Multiply(Z3));
}
}
else
{
ECFieldElement X1Z1 = Z1IsOne ? X1 : X1.Multiply(Z1);
// TODO Delayed modular reduction for sum of products
L3 = X1Z1.Square().Add(T.Multiply(L1Z1)).Add(X3).Add(Z3);
}
return new F2mPoint(curve, X3, L3, new ECFieldElement[] { Z3 }, IsCompressed);
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}
public override ECPoint TwicePlus(ECPoint b)
{
if (this.IsInfinity)
return b;
if (b.IsInfinity)
return Twice();
ECCurve curve = this.Curve;
ECFieldElement X1 = this.RawXCoord;
if (X1.IsZero)
{
// A point with X == 0 is it's own additive inverse
return b;
}
int coord = curve.CoordinateSystem;
switch (coord)
{
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// NOTE: twicePlus() only optimized for lambda-affine argument
ECFieldElement X2 = b.RawXCoord, Z2 = b.RawZCoords[0];
if (X2.IsZero || !Z2.IsOne)
{
return Twice().Add(b);
}
ECFieldElement L1 = this.RawYCoord, Z1 = this.RawZCoords[0];
ECFieldElement L2 = b.RawYCoord;
ECFieldElement X1Sq = X1.Square();
ECFieldElement L1Sq = L1.Square();
ECFieldElement Z1Sq = Z1.Square();
ECFieldElement L1Z1 = L1.Multiply(Z1);
ECFieldElement T = curve.A.Multiply(Z1Sq).Add(L1Sq).Add(L1Z1);
ECFieldElement L2plus1 = L2.AddOne();
// TODO Delayed modular reduction for sum of products
ECFieldElement A = curve.A.Add(L2plus1).Multiply(Z1Sq).Add(L1Sq).Multiply(T).Add(X1Sq.Multiply(Z1Sq));
ECFieldElement X2Z1Sq = X2.Multiply(Z1Sq);
ECFieldElement B = X2Z1Sq.Add(T).Square();
if (B.IsZero)
{
if (A.IsZero)
{
return b.Twice();
}
return curve.Infinity;
}
if (A.IsZero)
{
return new F2mPoint(curve, A, curve.B.Sqrt(), IsCompressed);
}
ECFieldElement X3 = A.Square().Multiply(X2Z1Sq);
ECFieldElement Z3 = A.Multiply(B).Multiply(Z1Sq);
// TODO Delayed modular reduction for sum of products
ECFieldElement L3 = A.Add(B).Square().Multiply(T).Add(L2plus1.Multiply(Z3));
return new F2mPoint(curve, X3, L3, new ECFieldElement[] { Z3 }, IsCompressed);
}
default:
{
return Twice().Add(b);
}
}
}
public override ECPoint Negate()
{
if (this.IsInfinity)
return this;
ECFieldElement X = this.RawXCoord;
if (X.IsZero)
return this;
ECCurve curve = this.Curve;
int coord = curve.CoordinateSystem;
switch (coord)
{
case ECCurve.COORD_AFFINE:
{
ECFieldElement Y = this.RawYCoord;
return new F2mPoint(curve, X, Y.Add(X), IsCompressed);
}
case ECCurve.COORD_HOMOGENEOUS:
{
ECFieldElement Y = this.RawYCoord, Z = this.RawZCoords[0];
return new F2mPoint(curve, X, Y.Add(X), new ECFieldElement[] { Z }, IsCompressed);
}
case ECCurve.COORD_LAMBDA_AFFINE:
{
ECFieldElement L = this.RawYCoord;
return new F2mPoint(curve, X, L.AddOne(), IsCompressed);
}
case ECCurve.COORD_LAMBDA_PROJECTIVE:
{
// L is actually Lambda (X + Y/X) here
ECFieldElement L = this.RawYCoord, Z = this.RawZCoords[0];
return new F2mPoint(curve, X, L.Add(Z), new ECFieldElement[] { Z }, IsCompressed);
}
default:
{
throw new InvalidOperationException("unsupported coordinate system");
}
}
}
}
}