Bring Fp field element code mostly up-to-date with Java version

1 files changed, 797 insertions, 776 deletions

diff --git a/crypto/src/math/ec/ECFieldElement.cs b/crypto/src/math/ec/ECFieldElement.cs
index 1205cdbcb..180f97fd5 100644
--- a/crypto/src/math/ec/ECFieldElement.cs
+++ b/crypto/src/math/ec/ECFieldElement.cs
@@ -5,49 +5,69 @@ 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 virtual 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 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 virtual int BitLength
+        {
+            get { return ToBigInteger().BitLength; }
+        }
+
+        public virtual bool IsOne
+        {
+            get { return BitLength == 1; }
+        }
+
+        public virtual bool IsZero
+        {
+            get { return 0 == ToBigInteger().SignValue; }
+        }
+
+        public virtual bool TestBitZero()
+        {
+            return ToBigInteger().TestBit(0);
+        }
+
+        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 virtual bool Equals(
+            ECFieldElement other)
+        {
+            return ToBigInteger().Equals(other.ToBigInteger());
+        }
+
+        public override int GetHashCode()
+        {
+            return ToBigInteger().GetHashCode();
+        }
+
+        public override string ToString()
+        {
+            return this.ToBigInteger().ToString(16);
+        }
 
         public virtual byte[] GetEncoded()
         {
@@ -55,320 +75,321 @@ namespace Org.BouncyCastle.Math.EC
         }
     }
 
-	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 this.Equals(z.Square()) ? 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);
-		}
+    public class FpFieldElement
+        : ECFieldElement
+    {
+        private readonly BigInteger q, r, x;
 
-//		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);
-//		}
+        internal static BigInteger CalculateResidue(BigInteger p)
+        {
+            int bitLength = p.BitLength;
+            if (bitLength > 128)
+            {
+                BigInteger firstWord = p.ShiftRight(bitLength - 64);
+                if (firstWord.LongValue == -1L)
+                {
+                    return BigInteger.One.ShiftLeft(bitLength).Subtract(p);
+                }
+            }
+            return null;
+        }
 
-		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)
-//			};
-//		}
+        [Obsolete("Use ECCurve.FromBigInteger to construct field elements")]
+        public FpFieldElement(BigInteger q, BigInteger x)
+            : this(q, CalculateResidue(q), x)
+        {
+        }
+
+        internal FpFieldElement(BigInteger q, BigInteger r, BigInteger x)
+        {
+            if (x == null || x.SignValue < 0 || x.CompareTo(q) >= 0)
+                throw new ArgumentException("value invalid in Fp field element", "x");
+
+            this.q = q;
+            this.r = r;
+            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 bool Equals(
-			object obj)
-		{
-			if (obj == this)
-				return true;
+        public override ECFieldElement Add(
+            ECFieldElement b)
+        {
+            return new FpFieldElement(q, r, ModAdd(x, b.ToBigInteger()));
+        }
+
+        public override ECFieldElement Subtract(
+            ECFieldElement b)
+        {
+            BigInteger x2 = b.ToBigInteger();
+            BigInteger x3 = x.Subtract(x2);
+            if (x3.SignValue < 0)
+            {
+                x3 = x3.Add(q);
+            }
+            return new FpFieldElement(q, r, x3);
+        }
+
+        public override ECFieldElement Multiply(
+            ECFieldElement b)
+        {
+            return new FpFieldElement(q, r, ModMult(x, b.ToBigInteger()));
+        }
+
+        public override ECFieldElement Divide(
+            ECFieldElement b)
+        {
+            return new FpFieldElement(q, r, ModMult(x, b.ToBigInteger().ModInverse(q)));
+        }
+
+        public override ECFieldElement Negate()
+        {
+            return x.SignValue == 0 ? this : new FpFieldElement(q, r, q.Subtract(x));
+        }
 
-			FpFieldElement other = obj as FpFieldElement;
+        public override ECFieldElement Square()
+        {
+            return new FpFieldElement(q, r, ModMult(x, x));
+        }
 
-			if (other == null)
-				return false;
+        public override ECFieldElement Invert()
+        {
+            // TODO Modular inversion can be faster for a (Generalized) Mersenne Prime.
+            return new FpFieldElement(q, r, x.ModInverse(q));
+        }
 
-			return Equals(other);
-		}
+        // 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, r, 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 X = this.x;
+            BigInteger fourX = ModDouble(ModDouble(X)); ;
+
+            BigInteger U, V;
+            Random rand = new Random();
+            do
+            {
+                BigInteger P;
+                do
+                {
+                    P = new BigInteger(q.BitLength, rand);
+                }
+                while (P.CompareTo(q) >= 0
+                    || !(ModMult(P, P).Subtract(fourX).ModPow(legendreExponent, q).Equals(qMinusOne)));
+
+                BigInteger[] result = LucasSequence(P, X, k);
+                U = result[0];
+                V = result[1];
+
+                if (ModMult(V, V).Equals(fourX))
+                {
+                    // Integer division by 2, mod q
+                    if (V.TestBit(0))
+                    {
+                        V = V.Add(q);
+                    }
+
+                    V = V.ShiftRight(1);
+
+                    Debug.Assert(ModMult(V, V).Equals(X));
+
+                    return new FpFieldElement(q, r, V);
+                }
+            }
+            while (U.Equals(BigInteger.One) || U.Equals(qMinusOne));
+
+            return null;
+        }
 
-		protected bool Equals(
-			FpFieldElement other)
-		{
-			return q.Equals(other.q) && base.Equals(other);
-		}
+        private BigInteger[] LucasSequence(
+            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 = ModMult(Ql, Qh);
+
+                if (k.TestBit(j))
+                {
+                    Qh = ModMult(Ql, Q);
+                    Uh = ModMult(Uh, Vh);
+                    Vl = ModReduce(Vh.Multiply(Vl).Subtract(P.Multiply(Ql)));
+                    Vh = ModReduce(Vh.Multiply(Vh).Subtract(Qh.ShiftLeft(1)));
+                }
+                else
+                {
+                    Qh = Ql;
+                    Uh = ModReduce(Uh.Multiply(Vl).Subtract(Ql));
+                    Vh = ModReduce(Vh.Multiply(Vl).Subtract(P.Multiply(Ql)));
+                    Vl = ModReduce(Vl.Multiply(Vl).Subtract(Ql.ShiftLeft(1)));
+                }
+            }
+
+            Ql = ModMult(Ql, Qh);
+            Qh = ModMult(Ql, Q);
+            Uh = ModReduce(Uh.Multiply(Vl).Subtract(Ql));
+            Vl = ModReduce(Vh.Multiply(Vl).Subtract(P.Multiply(Ql)));
+            Ql = ModMult(Ql, Qh);
+
+            for (int j = 1; j <= s; ++j)
+            {
+                Uh = ModMult(Uh, Vl);
+                Vl = ModReduce(Vl.Multiply(Vl).Subtract(Ql.ShiftLeft(1)));
+                Ql = ModMult(Ql, Ql);
+            }
+
+            return new BigInteger[] { Uh, Vl };
+        }
 
-		public override int GetHashCode()
-		{
-			return q.GetHashCode() ^ base.GetHashCode();
-		}
-	}
+        protected virtual BigInteger ModAdd(BigInteger x1, BigInteger x2)
+        {
+            BigInteger x3 = x1.Add(x2);
+            if (x3.CompareTo(q) >= 0)
+            {
+                x3 = x3.Subtract(q);
+            }
+            return x3;
+        }
+
+        protected virtual BigInteger ModDouble(BigInteger x)
+        {
+            BigInteger _2x = x.ShiftLeft(1);
+            if (_2x.CompareTo(q) >= 0)
+            {
+                _2x = _2x.Subtract(q);
+            }
+            return _2x;
+        }
+
+        protected virtual BigInteger ModMult(BigInteger x1, BigInteger x2)
+        {
+            return ModReduce(x1.Multiply(x2));
+        }
+
+        protected virtual BigInteger ModReduce(BigInteger x)
+        {
+            if (r != null)
+            {
+                bool negative = x.SignValue < 0;
+                if (negative)
+                {
+                    x = x.Abs();
+                }
+                int qLen = q.BitLength;
+                while (x.BitLength > (qLen + 1))
+                {
+                    BigInteger u = x.ShiftRight(qLen);
+                    BigInteger v = x.Subtract(u.ShiftLeft(qLen));
+                    if (!r.Equals(BigInteger.One))
+                    {
+                        u = u.Multiply(r);
+                    }
+                    x = u.Add(v);
+                }
+                while (x.CompareTo(q) >= 0)
+                {
+                    x = x.Subtract(q);
+                }
+                if (negative && x.SignValue != 0)
+                {
+                    x = q.Subtract(x);
+                }
+            }
+            else
+            {
+                x = x.Mod(q);
+            }
+            return x;
+        }
+
+        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
@@ -820,439 +841,439 @@ namespace Org.BouncyCastle.Math.EC
 //		}
 //	}
 
-	/**
-	 * 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.Copy();
-			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)
+    /**
+     * 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.Copy();
+            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.Copy();
 
-			// 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)
+            // 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;
+            {
+                // 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)
-				{
+                // If j < 0 then: u(z) <-> v(z), g1(z) <-> g2(z), j := -j
+                if (j < 0)
+                {
                     IntArray uzCopy = uz;
-					uz = vz;
-					vz = uzCopy;
+                    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 = 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();
-		}
-	}
+                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();
+        }
+    }
 }