using System;
using System.Diagnostics;
using Org.BouncyCastle.Utilities;
namespace Org.BouncyCastle.Math.EC{
public abstract class ECFieldElement
{
public abstract BigInteger ToBigInteger();
public abstract string FieldName { get; }
public abstract int FieldSize { get; }
public abstract ECFieldElement Add(ECFieldElement b);
public abstract ECFieldElement Subtract(ECFieldElement b);
public abstract ECFieldElement Multiply(ECFieldElement b);
public abstract ECFieldElement Divide(ECFieldElement b);
public abstract ECFieldElement Negate();
public abstract ECFieldElement Square();
public abstract ECFieldElement Invert();
public abstract ECFieldElement Sqrt();
public override bool Equals(
object obj)
{
if (obj == this)
return true;
ECFieldElement other = obj as ECFieldElement;
if (other == null)
return false;
return Equals(other);
}
protected bool Equals(
ECFieldElement other)
{
return ToBigInteger().Equals(other.ToBigInteger());
}
public override int GetHashCode()
{
return ToBigInteger().GetHashCode();
}
public override string ToString()
{
return this.ToBigInteger().ToString(2);
}
}
public class FpFieldElement
: ECFieldElement
{
private readonly BigInteger q, x;
public FpFieldElement(
BigInteger q,
BigInteger x)
{
if (x.CompareTo(q) >= 0)
throw new ArgumentException("x value too large in field element");
this.q = q;
this.x = x;
}
public override BigInteger ToBigInteger()
{
return x;
}
/**
* return the field name for this field.
*
* @return the string "Fp".
*/
public override string FieldName
{
get { return "Fp"; }
}
public override int FieldSize
{
get { return q.BitLength; }
}
public BigInteger Q
{
get { return q; }
}
public override ECFieldElement Add(
ECFieldElement b)
{
return new FpFieldElement(q, x.Add(b.ToBigInteger()).Mod(q));
}
public override ECFieldElement Subtract(
ECFieldElement b)
{
return new FpFieldElement(q, x.Subtract(b.ToBigInteger()).Mod(q));
}
public override ECFieldElement Multiply(
ECFieldElement b)
{
return new FpFieldElement(q, x.Multiply(b.ToBigInteger()).Mod(q));
}
public override ECFieldElement Divide(
ECFieldElement b)
{
return new FpFieldElement(q, x.Multiply(b.ToBigInteger().ModInverse(q)).Mod(q));
}
public override ECFieldElement Negate()
{
return new FpFieldElement(q, x.Negate().Mod(q));
}
public override ECFieldElement Square()
{
return new FpFieldElement(q, x.Multiply(x).Mod(q));
}
public override ECFieldElement Invert()
{
return new FpFieldElement(q, x.ModInverse(q));
}
// D.1.4 91
/**
* return a sqrt root - the routine verifies that the calculation
* returns the right value - if none exists it returns null.
*/
public override ECFieldElement Sqrt()
{
if (!q.TestBit(0))
throw Platform.CreateNotImplementedException("even value of q");
// p mod 4 == 3
if (q.TestBit(1))
{
// TODO Can this be optimised (inline the Square?)
// z = g^(u+1) + p, p = 4u + 3
ECFieldElement z = new FpFieldElement(q, x.ModPow(q.ShiftRight(2).Add(BigInteger.One), q));
return z.Square().Equals(this) ? z : null;
}
// p mod 4 == 1
BigInteger qMinusOne = q.Subtract(BigInteger.One);
BigInteger legendreExponent = qMinusOne.ShiftRight(1);
if (!(x.ModPow(legendreExponent, q).Equals(BigInteger.One)))
return null;
BigInteger u = qMinusOne.ShiftRight(2);
BigInteger k = u.ShiftLeft(1).Add(BigInteger.One);
BigInteger Q = this.x;
BigInteger fourQ = Q.ShiftLeft(2).Mod(q);
BigInteger U, V;
do
{
Random rand = new Random();
BigInteger P;
do
{
P = new BigInteger(q.BitLength, rand);
}
while (P.CompareTo(q) >= 0
|| !(P.Multiply(P).Subtract(fourQ).ModPow(legendreExponent, q).Equals(qMinusOne)));
BigInteger[] result = fastLucasSequence(q, P, Q, k);
U = result[0];
V = result[1];
if (V.Multiply(V).Mod(q).Equals(fourQ))
{
// Integer division by 2, mod q
if (V.TestBit(0))
{
V = V.Add(q);
}
V = V.ShiftRight(1);
Debug.Assert(V.Multiply(V).Mod(q).Equals(x));
return new FpFieldElement(q, V);
}
}
while (U.Equals(BigInteger.One) || U.Equals(qMinusOne));
return null;
// BigInteger qMinusOne = q.Subtract(BigInteger.One);
//
// BigInteger legendreExponent = qMinusOne.ShiftRight(1);
// if (!(x.ModPow(legendreExponent, q).Equals(BigInteger.One)))
// return null;
//
// Random rand = new Random();
// BigInteger fourX = x.ShiftLeft(2);
//
// BigInteger r;
// do
// {
// r = new BigInteger(q.BitLength, rand);
// }
// while (r.CompareTo(q) >= 0
// || !(r.Multiply(r).Subtract(fourX).ModPow(legendreExponent, q).Equals(qMinusOne)));
//
// BigInteger n1 = qMinusOne.ShiftRight(2);
// BigInteger n2 = n1.Add(BigInteger.One);
//
// BigInteger wOne = WOne(r, x, q);
// BigInteger wSum = W(n1, wOne, q).Add(W(n2, wOne, q)).Mod(q);
// BigInteger twoR = r.ShiftLeft(1);
//
// BigInteger root = twoR.ModPow(q.Subtract(BigInteger.Two), q)
// .Multiply(x).Mod(q)
// .Multiply(wSum).Mod(q);
//
// return new FpFieldElement(q, root);
}
// private static BigInteger W(BigInteger n, BigInteger wOne, BigInteger p)
// {
// if (n.Equals(BigInteger.One))
// return wOne;
//
// bool isEven = !n.TestBit(0);
// n = n.ShiftRight(1);
// if (isEven)
// {
// BigInteger w = W(n, wOne, p);
// return w.Multiply(w).Subtract(BigInteger.Two).Mod(p);
// }
// BigInteger w1 = W(n.Add(BigInteger.One), wOne, p);
// BigInteger w2 = W(n, wOne, p);
// return w1.Multiply(w2).Subtract(wOne).Mod(p);
// }
//
// private BigInteger WOne(BigInteger r, BigInteger x, BigInteger p)
// {
// return r.Multiply(r).Multiply(x.ModPow(q.Subtract(BigInteger.Two), q)).Subtract(BigInteger.Two).Mod(p);
// }
private static BigInteger[] fastLucasSequence(
BigInteger p,
BigInteger P,
BigInteger Q,
BigInteger k)
{
// TODO Research and apply "common-multiplicand multiplication here"
int n = k.BitLength;
int s = k.GetLowestSetBit();
Debug.Assert(k.TestBit(s));
BigInteger Uh = BigInteger.One;
BigInteger Vl = BigInteger.Two;
BigInteger Vh = P;
BigInteger Ql = BigInteger.One;
BigInteger Qh = BigInteger.One;
for (int j = n - 1; j >= s + 1; --j)
{
Ql = Ql.Multiply(Qh).Mod(p);
if (k.TestBit(j))
{
Qh = Ql.Multiply(Q).Mod(p);
Uh = Uh.Multiply(Vh).Mod(p);
Vl = Vh.Multiply(Vl).Subtract(P.Multiply(Ql)).Mod(p);
Vh = Vh.Multiply(Vh).Subtract(Qh.ShiftLeft(1)).Mod(p);
}
else
{
Qh = Ql;
Uh = Uh.Multiply(Vl).Subtract(Ql).Mod(p);
Vh = Vh.Multiply(Vl).Subtract(P.Multiply(Ql)).Mod(p);
Vl = Vl.Multiply(Vl).Subtract(Ql.ShiftLeft(1)).Mod(p);
}
}
Ql = Ql.Multiply(Qh).Mod(p);
Qh = Ql.Multiply(Q).Mod(p);
Uh = Uh.Multiply(Vl).Subtract(Ql).Mod(p);
Vl = Vh.Multiply(Vl).Subtract(P.Multiply(Ql)).Mod(p);
Ql = Ql.Multiply(Qh).Mod(p);
for (int j = 1; j <= s; ++j)
{
Uh = Uh.Multiply(Vl).Mod(p);
Vl = Vl.Multiply(Vl).Subtract(Ql.ShiftLeft(1)).Mod(p);
Ql = Ql.Multiply(Ql).Mod(p);
}
return new BigInteger[]{ Uh, Vl };
}
// private static BigInteger[] verifyLucasSequence(
// BigInteger p,
// BigInteger P,
// BigInteger Q,
// BigInteger k)
// {
// BigInteger[] actual = fastLucasSequence(p, P, Q, k);
// BigInteger[] plus1 = fastLucasSequence(p, P, Q, k.Add(BigInteger.One));
// BigInteger[] plus2 = fastLucasSequence(p, P, Q, k.Add(BigInteger.Two));
//
// BigInteger[] check = stepLucasSequence(p, P, Q, actual, plus1);
//
// Debug.Assert(check[0].Equals(plus2[0]));
// Debug.Assert(check[1].Equals(plus2[1]));
//
// return actual;
// }
//
// private static BigInteger[] stepLucasSequence(
// BigInteger p,
// BigInteger P,
// BigInteger Q,
// BigInteger[] backTwo,
// BigInteger[] backOne)
// {
// return new BigInteger[]
// {
// P.Multiply(backOne[0]).Subtract(Q.Multiply(backTwo[0])).Mod(p),
// P.Multiply(backOne[1]).Subtract(Q.Multiply(backTwo[1])).Mod(p)
// };
// }
public override bool Equals(
object obj)
{
if (obj == this)
return true;
FpFieldElement other = obj as FpFieldElement;
if (other == null)
return false;
return Equals(other);
}
protected bool Equals(
FpFieldElement other)
{
return q.Equals(other.q) && base.Equals(other);
}
public override int GetHashCode()
{
return q.GetHashCode() ^ base.GetHashCode();
}
}
// /**
// * Class representing the Elements of the finite field
// * <code>F<sub>2<sup>m</sup></sub></code> in polynomial basis (PB)
// * representation. Both trinomial (Tpb) and pentanomial (Ppb) polynomial
// * basis representations are supported. Gaussian normal basis (GNB)
// * representation is not supported.
// */
// public class F2mFieldElement
// : ECFieldElement
// {
// /**
// * Indicates gaussian normal basis representation (GNB). Number chosen
// * according to X9.62. GNB is not implemented at present.
// */
// public const int Gnb = 1;
//
// /**
// * Indicates trinomial basis representation (Tpb). Number chosen
// * according to X9.62.
// */
// public const int Tpb = 2;
//
// /**
// * Indicates pentanomial basis representation (Ppb). Number chosen
// * according to X9.62.
// */
// public const int Ppb = 3;
//
// /**
// * Tpb or Ppb.
// */
// private int representation;
//
// /**
// * The exponent <code>m</code> of <code>F<sub>2<sup>m</sup></sub></code>.
// */
// private int m;
//
// /**
// * Tpb: The integer <code>k</code> where <code>x<sup>m</sup> +
// * x<sup>k</sup> + 1</code> represents the reduction polynomial
// * <code>f(z)</code>.<br/>
// * Ppb: 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>.<br/>
// */
// private int k1;
//
// /**
// * Tpb: Always set to <code>0</code><br/>
// * Ppb: 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>.<br/>
// */
// private int k2;
//
// /**
// * Tpb: Always set to <code>0</code><br/>
// * Ppb: 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>.<br/>
// */
// private int k3;
//
// /**
// * Constructor for 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 x The BigInteger representing the value of the field element.
// */
// public F2mFieldElement(
// int m,
// int k1,
// int k2,
// int k3,
// BigInteger x)
// : base(x)
// {
// if ((k2 == 0) && (k3 == 0))
// {
// this.representation = Tpb;
// }
// else
// {
// if (k2 >= k3)
// throw new ArgumentException("k2 must be smaller than k3");
// if (k2 <= 0)
// throw new ArgumentException("k2 must be larger than 0");
//
// this.representation = Ppb;
// }
//
// if (x.SignValue < 0)
// throw new ArgumentException("x value cannot be negative");
//
// this.m = m;
// this.k1 = k1;
// this.k2 = k2;
// this.k3 = k3;
// }
//
// /**
// * Constructor for Tpb.
// * @param m The exponent <code>m</code> of
// * <code>F<sub>2<sup>m</sup></sub></code>.
// * @param k The integer <code>k</code> where <code>x<sup>m</sup> +
// * x<sup>k</sup> + 1</code> represents the reduction
// * polynomial <code>f(z)</code>.
// * @param x The BigInteger representing the value of the field element.
// */
// public F2mFieldElement(
// int m,
// int k,
// BigInteger x)
// : this(m, k, 0, 0, x)
// {
// // Set k1 to k, and set k2 and k3 to 0
// }
//
// public override string FieldName
// {
// get { return "F2m"; }
// }
//
// /**
// * Checks, if the ECFieldElements <code>a</code> and <code>b</code>
// * are elements of the same field <code>F<sub>2<sup>m</sup></sub></code>
// * (having the same representation).
// * @param a field element.
// * @param b field element to be compared.
// * @throws ArgumentException if <code>a</code> and <code>b</code>
// * are not elements of the same field
// * <code>F<sub>2<sup>m</sup></sub></code> (having the same
// * representation).
// */
// public static void CheckFieldElements(
// ECFieldElement a,
// ECFieldElement b)
// {
// if (!(a is F2mFieldElement) || !(b is F2mFieldElement))
// {
// throw new ArgumentException("Field elements are not "
// + "both instances of F2mFieldElement");
// }
//
// if ((a.x.SignValue < 0) || (b.x.SignValue < 0))
// {
// throw new ArgumentException(
// "x value may not be negative");
// }
//
// F2mFieldElement aF2m = (F2mFieldElement)a;
// F2mFieldElement bF2m = (F2mFieldElement)b;
//
// if ((aF2m.m != bF2m.m) || (aF2m.k1 != bF2m.k1)
// || (aF2m.k2 != bF2m.k2) || (aF2m.k3 != bF2m.k3))
// {
// throw new ArgumentException("Field elements are not "
// + "elements of the same field F2m");
// }
//
// if (aF2m.representation != bF2m.representation)
// {
// // Should never occur
// throw new ArgumentException(
// "One of the field "
// + "elements are not elements has incorrect representation");
// }
// }
//
// /**
// * Computes <code>z * a(z) mod f(z)</code>, where <code>f(z)</code> is
// * the reduction polynomial of <code>this</code>.
// * @param a The polynomial <code>a(z)</code> to be multiplied by
// * <code>z mod f(z)</code>.
// * @return <code>z * a(z) mod f(z)</code>
// */
// private BigInteger multZModF(
// BigInteger a)
// {
// // Left-shift of a(z)
// BigInteger az = a.ShiftLeft(1);
// if (az.TestBit(this.m))
// {
// // If the coefficient of z^m in a(z) Equals 1, reduction
// // modulo f(z) is performed: Add f(z) to to a(z):
// // Step 1: Unset mth coeffient of a(z)
// az = az.ClearBit(this.m);
//
// // Step 2: Add r(z) to a(z), where r(z) is defined as
// // f(z) = z^m + r(z), and k1, k2, k3 are the positions of
// // the non-zero coefficients in r(z)
// az = az.FlipBit(0);
// az = az.FlipBit(this.k1);
// if (this.representation == Ppb)
// {
// az = az.FlipBit(this.k2);
// az = az.FlipBit(this.k3);
// }
// }
// return az;
// }
//
// public override ECFieldElement Add(
// ECFieldElement b)
// {
// // No check performed here for performance reasons. Instead the
// // elements involved are checked in ECPoint.F2m
// // checkFieldElements(this, b);
// if (b.x.SignValue == 0)
// return this;
//
// return new F2mFieldElement(this.m, this.k1, this.k2, this.k3, this.x.Xor(b.x));
// }
//
// public override ECFieldElement Subtract(
// ECFieldElement b)
// {
// // Addition and subtraction are the same in F2m
// return Add(b);
// }
//
// public override ECFieldElement Multiply(
// ECFieldElement b)
// {
// // Left-to-right shift-and-add field multiplication in F2m
// // Input: Binary polynomials a(z) and b(z) of degree at most m-1
// // Output: c(z) = a(z) * b(z) mod f(z)
//
// // No check performed here for performance reasons. Instead the
// // elements involved are checked in ECPoint.F2m
// // checkFieldElements(this, b);
// BigInteger az = this.x;
// BigInteger bz = b.x;
// BigInteger cz;
//
// // Compute c(z) = a(z) * b(z) mod f(z)
// if (az.TestBit(0))
// {
// cz = bz;
// }
// else
// {
// cz = BigInteger.Zero;
// }
//
// for (int i = 1; i < this.m; i++)
// {
// // b(z) := z * b(z) mod f(z)
// bz = multZModF(bz);
//
// if (az.TestBit(i))
// {
// // If the coefficient of x^i in a(z) Equals 1, b(z) is added
// // to c(z)
// cz = cz.Xor(bz);
// }
// }
// return new F2mFieldElement(m, this.k1, this.k2, this.k3, cz);
// }
//
//
// public override ECFieldElement Divide(
// ECFieldElement b)
// {
// // There may be more efficient implementations
// ECFieldElement bInv = b.Invert();
// return Multiply(bInv);
// }
//
// public override ECFieldElement Negate()
// {
// // -x == x holds for all x in F2m
// return this;
// }
//
// public override ECFieldElement Square()
// {
// // Naive implementation, can probably be speeded up using modular
// // reduction
// return Multiply(this);
// }
//
// public override ECFieldElement Invert()
// {
// // Inversion in F2m using the extended Euclidean algorithm
// // Input: A nonzero polynomial a(z) of degree at most m-1
// // Output: a(z)^(-1) mod f(z)
//
// // u(z) := a(z)
// BigInteger uz = this.x;
// if (uz.SignValue <= 0)
// {
// throw new ArithmeticException("x is zero or negative, " +
// "inversion is impossible");
// }
//
// // v(z) := f(z)
// BigInteger vz = BigInteger.One.ShiftLeft(m);
// vz = vz.SetBit(0);
// vz = vz.SetBit(this.k1);
// if (this.representation == Ppb)
// {
// vz = vz.SetBit(this.k2);
// vz = vz.SetBit(this.k3);
// }
//
// // g1(z) := 1, g2(z) := 0
// BigInteger g1z = BigInteger.One;
// BigInteger g2z = BigInteger.Zero;
//
// // while u != 1
// while (uz.SignValue != 0)
// {
// // j := deg(u(z)) - deg(v(z))
// int j = uz.BitLength - vz.BitLength;
//
// // If j < 0 then: u(z) <-> v(z), g1(z) <-> g2(z), j := -j
// if (j < 0)
// {
// BigInteger uzCopy = uz;
// uz = vz;
// vz = uzCopy;
//
// BigInteger g1zCopy = g1z;
// g1z = g2z;
// g2z = g1zCopy;
//
// j = -j;
// }
//
// // u(z) := u(z) + z^j * v(z)
// // Note, that no reduction modulo f(z) is required, because
// // deg(u(z) + z^j * v(z)) <= max(deg(u(z)), j + deg(v(z)))
// // = max(deg(u(z)), deg(u(z)) - deg(v(z)) + deg(v(z))
// // = deg(u(z))
// uz = uz.Xor(vz.ShiftLeft(j));
//
// // g1(z) := g1(z) + z^j * g2(z)
// g1z = g1z.Xor(g2z.ShiftLeft(j));
// // if (g1z.BitLength() > this.m) {
// // throw new ArithmeticException(
// // "deg(g1z) >= m, g1z = " + g1z.ToString(2));
// // }
// }
// return new F2mFieldElement(this.m, this.k1, this.k2, this.k3, g2z);
// }
//
// public override ECFieldElement Sqrt()
// {
// throw new ArithmeticException("Not implemented");
// }
//
// /**
// * @return the representation of the field
// * <code>F<sub>2<sup>m</sup></sub></code>, either of
// * {@link F2mFieldElement.Tpb} (trinomial
// * basis representation) or
// * {@link F2mFieldElement.Ppb} (pentanomial
// * basis representation).
// */
// public int Representation
// {
// get { return this.representation; }
// }
//
// /**
// * @return the degree <code>m</code> of the reduction polynomial
// * <code>f(z)</code>.
// */
// public int M
// {
// get { return this.m; }
// }
//
// /**
// * @return Tpb: The integer <code>k</code> where <code>x<sup>m</sup> +
// * x<sup>k</sup> + 1</code> represents the reduction polynomial
// * <code>f(z)</code>.<br/>
// * Ppb: 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>.<br/>
// */
// public int K1
// {
// get { return this.k1; }
// }
//
// /**
// * @return Tpb: Always returns <code>0</code><br/>
// * Ppb: 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>.<br/>
// */
// public int K2
// {
// get { return this.k2; }
// }
//
// /**
// * @return Tpb: Always set to <code>0</code><br/>
// * Ppb: 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>.<br/>
// */
// public int K3
// {
// get { return this.k3; }
// }
//
// public override bool Equals(
// object obj)
// {
// if (obj == this)
// return true;
//
// F2mFieldElement other = obj as F2mFieldElement;
//
// if (other == null)
// return false;
//
// return Equals(other);
// }
//
// protected bool Equals(
// F2mFieldElement other)
// {
// return m == other.m
// && k1 == other.k1
// && k2 == other.k2
// && k3 == other.k3
// && representation == other.representation
// && base.Equals(other);
// }
//
// public override int GetHashCode()
// {
// return base.GetHashCode() ^ m ^ k1 ^ k2 ^ k3;
// }
// }
/**
* Class representing the Elements of the finite field
* <code>F<sub>2<sup>m</sup></sub></code> in polynomial basis (PB)
* representation. Both trinomial (Tpb) and pentanomial (Ppb) polynomial
* basis representations are supported. Gaussian normal basis (GNB)
* representation is not supported.
*/
public class F2mFieldElement
: ECFieldElement
{
/**
* Indicates gaussian normal basis representation (GNB). Number chosen
* according to X9.62. GNB is not implemented at present.
*/
public const int Gnb = 1;
/**
* Indicates trinomial basis representation (Tpb). Number chosen
* according to X9.62.
*/
public const int Tpb = 2;
/**
* Indicates pentanomial basis representation (Ppb). Number chosen
* according to X9.62.
*/
public const int Ppb = 3;
/**
* Tpb or Ppb.
*/
private int representation;
/**
* The exponent <code>m</code> of <code>F<sub>2<sup>m</sup></sub></code>.
*/
private int m;
/**
* Tpb: The integer <code>k</code> where <code>x<sup>m</sup> +
* x<sup>k</sup> + 1</code> represents the reduction polynomial
* <code>f(z)</code>.<br/>
* Ppb: 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>.<br/>
*/
private int k1;
/**
* Tpb: Always set to <code>0</code><br/>
* Ppb: 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>.<br/>
*/
private int k2;
/**
* Tpb: Always set to <code>0</code><br/>
* Ppb: 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>.<br/>
*/
private int k3;
/**
* The <code>IntArray</code> holding the bits.
*/
private IntArray x;
/**
* The number of <code>int</code>s required to hold <code>m</code> bits.
*/
private readonly int t;
/**
* Constructor for 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 x The BigInteger representing the value of the field element.
*/
public F2mFieldElement(
int m,
int k1,
int k2,
int k3,
BigInteger x)
{
// t = m / 32 rounded up to the next integer
this.t = (m + 31) >> 5;
this.x = new IntArray(x, t);
if ((k2 == 0) && (k3 == 0))
{
this.representation = Tpb;
}
else
{
if (k2 >= k3)
throw new ArgumentException("k2 must be smaller than k3");
if (k2 <= 0)
throw new ArgumentException("k2 must be larger than 0");
this.representation = Ppb;
}
if (x.SignValue < 0)
throw new ArgumentException("x value cannot be negative");
this.m = m;
this.k1 = k1;
this.k2 = k2;
this.k3 = k3;
}
/**
* Constructor for Tpb.
* @param m The exponent <code>m</code> of
* <code>F<sub>2<sup>m</sup></sub></code>.
* @param k The integer <code>k</code> where <code>x<sup>m</sup> +
* x<sup>k</sup> + 1</code> represents the reduction
* polynomial <code>f(z)</code>.
* @param x The BigInteger representing the value of the field element.
*/
public F2mFieldElement(
int m,
int k,
BigInteger x)
: this(m, k, 0, 0, x)
{
// Set k1 to k, and set k2 and k3 to 0
}
private F2mFieldElement(int m, int k1, int k2, int k3, IntArray x)
{
t = (m + 31) >> 5;
this.x = x;
this.m = m;
this.k1 = k1;
this.k2 = k2;
this.k3 = k3;
if ((k2 == 0) && (k3 == 0))
{
this.representation = Tpb;
}
else
{
this.representation = Ppb;
}
}
public override BigInteger ToBigInteger()
{
return x.ToBigInteger();
}
public override string FieldName
{
get { return "F2m"; }
}
public override int FieldSize
{
get { return m; }
}
/**
* Checks, if the ECFieldElements <code>a</code> and <code>b</code>
* are elements of the same field <code>F<sub>2<sup>m</sup></sub></code>
* (having the same representation).
* @param a field element.
* @param b field element to be compared.
* @throws ArgumentException if <code>a</code> and <code>b</code>
* are not elements of the same field
* <code>F<sub>2<sup>m</sup></sub></code> (having the same
* representation).
*/
public static void CheckFieldElements(
ECFieldElement a,
ECFieldElement b)
{
if (!(a is F2mFieldElement) || !(b is F2mFieldElement))
{
throw new ArgumentException("Field elements are not "
+ "both instances of F2mFieldElement");
}
F2mFieldElement aF2m = (F2mFieldElement)a;
F2mFieldElement bF2m = (F2mFieldElement)b;
if ((aF2m.m != bF2m.m) || (aF2m.k1 != bF2m.k1)
|| (aF2m.k2 != bF2m.k2) || (aF2m.k3 != bF2m.k3))
{
throw new ArgumentException("Field elements are not "
+ "elements of the same field F2m");
}
if (aF2m.representation != bF2m.representation)
{
// Should never occur
throw new ArgumentException(
"One of the field "
+ "elements are not elements has incorrect representation");
}
}
public override ECFieldElement Add(
ECFieldElement b)
{
// No check performed here for performance reasons. Instead the
// elements involved are checked in ECPoint.F2m
// checkFieldElements(this, b);
IntArray iarrClone = (IntArray) this.x.Clone();
F2mFieldElement bF2m = (F2mFieldElement) b;
iarrClone.AddShifted(bF2m.x, 0);
return new F2mFieldElement(m, k1, k2, k3, iarrClone);
}
public override ECFieldElement Subtract(
ECFieldElement b)
{
// Addition and subtraction are the same in F2m
return Add(b);
}
public override ECFieldElement Multiply(
ECFieldElement b)
{
// Right-to-left comb multiplication in the IntArray
// Input: Binary polynomials a(z) and b(z) of degree at most m-1
// Output: c(z) = a(z) * b(z) mod f(z)
// No check performed here for performance reasons. Instead the
// elements involved are checked in ECPoint.F2m
// checkFieldElements(this, b);
F2mFieldElement bF2m = (F2mFieldElement) b;
IntArray mult = x.Multiply(bF2m.x, m);
mult.Reduce(m, new int[]{k1, k2, k3});
return new F2mFieldElement(m, k1, k2, k3, mult);
}
public override ECFieldElement Divide(
ECFieldElement b)
{
// There may be more efficient implementations
ECFieldElement bInv = b.Invert();
return Multiply(bInv);
}
public override ECFieldElement Negate()
{
// -x == x holds for all x in F2m
return this;
}
public override ECFieldElement Square()
{
IntArray squared = x.Square(m);
squared.Reduce(m, new int[]{k1, k2, k3});
return new F2mFieldElement(m, k1, k2, k3, squared);
}
public override ECFieldElement Invert()
{
// Inversion in F2m using the extended Euclidean algorithm
// Input: A nonzero polynomial a(z) of degree at most m-1
// Output: a(z)^(-1) mod f(z)
// u(z) := a(z)
IntArray uz = (IntArray)this.x.Clone();
// v(z) := f(z)
IntArray vz = new IntArray(t);
vz.SetBit(m);
vz.SetBit(0);
vz.SetBit(this.k1);
if (this.representation == Ppb)
{
vz.SetBit(this.k2);
vz.SetBit(this.k3);
}
// g1(z) := 1, g2(z) := 0
IntArray g1z = new IntArray(t);
g1z.SetBit(0);
IntArray g2z = new IntArray(t);
// while u != 0
while (uz.GetUsedLength() > 0)
// while (uz.bitLength() > 1)
{
// j := deg(u(z)) - deg(v(z))
int j = uz.BitLength - vz.BitLength;
// If j < 0 then: u(z) <-> v(z), g1(z) <-> g2(z), j := -j
if (j < 0)
{
IntArray uzCopy = uz;
uz = vz;
vz = uzCopy;
IntArray g1zCopy = g1z;
g1z = g2z;
g2z = g1zCopy;
j = -j;
}
// u(z) := u(z) + z^j * v(z)
// Note, that no reduction modulo f(z) is required, because
// deg(u(z) + z^j * v(z)) <= max(deg(u(z)), j + deg(v(z)))
// = max(deg(u(z)), deg(u(z)) - deg(v(z)) + deg(v(z))
// = deg(u(z))
// uz = uz.xor(vz.ShiftLeft(j));
// jInt = n / 32
int jInt = j >> 5;
// jInt = n % 32
int jBit = j & 0x1F;
IntArray vzShift = vz.ShiftLeft(jBit);
uz.AddShifted(vzShift, jInt);
// g1(z) := g1(z) + z^j * g2(z)
// g1z = g1z.xor(g2z.ShiftLeft(j));
IntArray g2zShift = g2z.ShiftLeft(jBit);
g1z.AddShifted(g2zShift, jInt);
}
return new F2mFieldElement(this.m, this.k1, this.k2, this.k3, g2z);
}
public override ECFieldElement Sqrt()
{
throw new ArithmeticException("Not implemented");
}
/**
* @return the representation of the field
* <code>F<sub>2<sup>m</sup></sub></code>, either of
* {@link F2mFieldElement.Tpb} (trinomial
* basis representation) or
* {@link F2mFieldElement.Ppb} (pentanomial
* basis representation).
*/
public int Representation
{
get { return this.representation; }
}
/**
* @return the degree <code>m</code> of the reduction polynomial
* <code>f(z)</code>.
*/
public int M
{
get { return this.m; }
}
/**
* @return Tpb: The integer <code>k</code> where <code>x<sup>m</sup> +
* x<sup>k</sup> + 1</code> represents the reduction polynomial
* <code>f(z)</code>.<br/>
* Ppb: 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>.<br/>
*/
public int K1
{
get { return this.k1; }
}
/**
* @return Tpb: Always returns <code>0</code><br/>
* Ppb: 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>.<br/>
*/
public int K2
{
get { return this.k2; }
}
/**
* @return Tpb: Always set to <code>0</code><br/>
* Ppb: 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>.<br/>
*/
public int K3
{
get { return this.k3; }
}
public override bool Equals(
object obj)
{
if (obj == this)
return true;
F2mFieldElement other = obj as F2mFieldElement;
if (other == null)
return false;
return Equals(other);
}
protected bool Equals(
F2mFieldElement other)
{
return m == other.m
&& k1 == other.k1
&& k2 == other.k2
&& k3 == other.k3
&& representation == other.representation
&& base.Equals(other);
}
public override int GetHashCode()
{
return m.GetHashCode()
^ k1.GetHashCode()
^ k2.GetHashCode()
^ k3.GetHashCode()
^ representation.GetHashCode()
^ base.GetHashCode();
}
}
}
|