com.bouncycastle.math.ec.ECFieldElement Maven / Gradle / Ivy
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package com.bouncycastle.math.ec;
import java.math.BigInteger;
import java.util.Random;
public abstract class ECFieldElement
implements ECConstants
{
public abstract BigInteger toBigInteger();
public abstract String getFieldName();
public abstract int getFieldSize();
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 String toString()
{
return this.toBigInteger().toString(2);
}
public static class Fp extends ECFieldElement
{
BigInteger x;
BigInteger q;
public Fp(BigInteger q, BigInteger x)
{
this.x = x;
if (x.compareTo(q) >= 0)
{
throw new IllegalArgumentException("x value too large in field element");
}
this.q = q;
}
public BigInteger toBigInteger()
{
return x;
}
/**
* return the field name for this field.
*
* @return the string "Fp".
*/
public String getFieldName()
{
return "Fp";
}
public int getFieldSize()
{
return q.bitLength();
}
public BigInteger getQ()
{
return q;
}
public ECFieldElement add(ECFieldElement b)
{
return new Fp(q, x.add(b.toBigInteger()).mod(q));
}
public ECFieldElement subtract(ECFieldElement b)
{
return new Fp(q, x.subtract(b.toBigInteger()).mod(q));
}
public ECFieldElement multiply(ECFieldElement b)
{
return new Fp(q, x.multiply(b.toBigInteger()).mod(q));
}
public ECFieldElement divide(ECFieldElement b)
{
return new Fp(q, x.multiply(b.toBigInteger().modInverse(q)).mod(q));
}
public ECFieldElement negate()
{
return new Fp(q, x.negate().mod(q));
}
public ECFieldElement square()
{
return new Fp(q, x.multiply(x).mod(q));
}
public ECFieldElement invert()
{
return new Fp(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 ECFieldElement sqrt()
{
if (!q.testBit(0))
{
throw new RuntimeException("not done yet");
}
// note: even though this class implements ECConstants don't be tempted to
// remove the explicit declaration, some J2ME environments don't cope.
// p mod 4 == 3
if (q.testBit(1))
{
// z = g^(u+1) + p, p = 4u + 3
ECFieldElement z = new Fp(q, x.modPow(q.shiftRight(2).add(ONE), q));
return z.square().equals(this) ? z : null;
}
// p mod 4 == 1
BigInteger qMinusOne = q.subtract(ONE);
BigInteger legendreExponent = qMinusOne.shiftRight(1);
if (!(x.modPow(legendreExponent, q).equals(ONE)))
{
return null;
}
BigInteger u = qMinusOne.shiftRight(2);
BigInteger k = u.shiftLeft(1).add(ONE);
BigInteger Q = this.x;
BigInteger fourQ = Q.shiftLeft(2).mod(q);
BigInteger U, V;
Random rand = new Random();
do
{
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 = lucasSequence(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);
//assert V.multiply(V).mod(q).equals(x);
return new Fp(q, V);
}
}
while (U.equals(ONE) || U.equals(qMinusOne));
return null;
// BigInteger qMinusOne = q.subtract(ECConstants.ONE);
// BigInteger legendreExponent = qMinusOne.shiftRight(1); //divide(ECConstants.TWO);
// if (!(x.modPow(legendreExponent, q).equals(ECConstants.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); //.divide(ECConstants.FOUR);
// BigInteger n2 = n1.add(ECConstants.ONE); //q.add(ECConstants.THREE).divide(ECConstants.FOUR);
//
// BigInteger wOne = WOne(r, x, q);
// BigInteger wSum = W(n1, wOne, q).add(W(n2, wOne, q)).mod(q);
// BigInteger twoR = r.shiftLeft(1); //ECConstants.TWO.multiply(r);
//
// BigInteger root = twoR.modPow(q.subtract(ECConstants.TWO), q)
// .multiply(x).mod(q)
// .multiply(wSum).mod(q);
//
// return new Fp(q, root);
}
// private static BigInteger W(BigInteger n, BigInteger wOne, BigInteger p)
// {
// if (n.equals(ECConstants.ONE))
// {
// return wOne;
// }
// boolean isEven = !n.testBit(0);
// n = n.shiftRight(1);//divide(ECConstants.TWO);
// if (isEven)
// {
// BigInteger w = W(n, wOne, p);
// return w.multiply(w).subtract(ECConstants.TWO).mod(p);
// }
// BigInteger w1 = W(n.add(ECConstants.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(ECConstants.TWO), q)).subtract(ECConstants.TWO).mod(p);
// }
private static BigInteger[] lucasSequence(
BigInteger p,
BigInteger P,
BigInteger Q,
BigInteger k)
{
int n = k.bitLength();
int s = k.getLowestSetBit();
BigInteger Uh = ONE;
BigInteger Vl = TWO;
BigInteger Vh = P;
BigInteger Ql = ONE;
BigInteger Qh = 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 };
}
public boolean equals(Object other)
{
if (other == this)
{
return true;
}
if (!(other instanceof Fp))
{
return false;
}
Fp o = (Fp)other;
return q.equals(o.q) && x.equals(o.x);
}
public int hashCode()
{
return q.hashCode() ^ x.hashCode();
}
}
// /**
// * Class representing the Elements of the finite field
// * F2m 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 static class F2m extends ECFieldElement
// {
// BigInteger x;
//
// /**
// * Indicates gaussian normal basis representation (GNB). Number chosen
// * according to X9.62. GNB is not implemented at present.
// */
// public static final int GNB = 1;
//
// /**
// * Indicates trinomial basis representation (TPB). Number chosen
// * according to X9.62.
// */
// public static final int TPB = 2;
//
// /**
// * Indicates pentanomial basis representation (PPB). Number chosen
// * according to X9.62.
// */
// public static final int PPB = 3;
//
// /**
// * TPB or PPB.
// */
// private int representation;
//
// /**
// * The exponent m of F2m.
// */
// private int m;
//
// /**
// * TPB: The integer k where xm +
// * xk + 1 represents the reduction polynomial
// * f(z).
// * PPB: The integer k1 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// */
// private int k1;
//
// /**
// * TPB: Always set to 0
// * PPB: The integer k2 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// */
// private int k2;
//
// /**
// * TPB: Always set to 0
// * PPB: The integer k3 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// */
// private int k3;
//
// /**
// * Constructor for PPB.
// * @param m The exponent m of
// * F2m.
// * @param k1 The integer k1 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// * @param k2 The integer k2 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// * @param k3 The integer k3 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// * @param x The BigInteger representing the value of the field element.
// */
// public F2m(
// int m,
// int k1,
// int k2,
// int k3,
// BigInteger x)
// {
//// super(x);
// this.x = x;
//
// if ((k2 == 0) && (k3 == 0))
// {
// this.representation = TPB;
// }
// else
// {
// if (k2 >= k3)
// {
// throw new IllegalArgumentException(
// "k2 must be smaller than k3");
// }
// if (k2 <= 0)
// {
// throw new IllegalArgumentException(
// "k2 must be larger than 0");
// }
// this.representation = PPB;
// }
//
// if (x.signum() < 0)
// {
// throw new IllegalArgumentException("x value cannot be negative");
// }
//
// this.m = m;
// this.k1 = k1;
// this.k2 = k2;
// this.k3 = k3;
// }
//
// /**
// * Constructor for TPB.
// * @param m The exponent m of
// * F2m.
// * @param k The integer k where xm +
// * xk + 1 represents the reduction
// * polynomial f(z).
// * @param x The BigInteger representing the value of the field element.
// */
// public F2m(int m, int k, BigInteger x)
// {
// // Set k1 to k, and set k2 and k3 to 0
// this(m, k, 0, 0, x);
// }
//
// public BigInteger toBigInteger()
// {
// return x;
// }
//
// public String getFieldName()
// {
// return "F2m";
// }
//
// public int getFieldSize()
// {
// return m;
// }
//
// /**
// * Checks, if the ECFieldElements a and b
// * are elements of the same field F2m
// * (having the same representation).
// * @param a field element.
// * @param b field element to be compared.
// * @throws IllegalArgumentException if a and b
// * are not elements of the same field
// * F2m (having the same
// * representation).
// */
// public static void checkFieldElements(
// ECFieldElement a,
// ECFieldElement b)
// {
// if ((!(a instanceof F2m)) || (!(b instanceof F2m)))
// {
// throw new IllegalArgumentException("Field elements are not "
// + "both instances of ECFieldElement.F2m");
// }
//
// if ((a.toBigInteger().signum() < 0) || (b.toBigInteger().signum() < 0))
// {
// throw new IllegalArgumentException(
// "x value may not be negative");
// }
//
// ECFieldElement.F2m aF2m = (ECFieldElement.F2m)a;
// ECFieldElement.F2m bF2m = (ECFieldElement.F2m)b;
//
// if ((aF2m.m != bF2m.m) || (aF2m.k1 != bF2m.k1)
// || (aF2m.k2 != bF2m.k2) || (aF2m.k3 != bF2m.k3))
// {
// throw new IllegalArgumentException("Field elements are not "
// + "elements of the same field F2m");
// }
//
// if (aF2m.representation != bF2m.representation)
// {
// // Should never occur
// throw new IllegalArgumentException(
// "One of the field "
// + "elements are not elements has incorrect representation");
// }
// }
//
// /**
// * Computes z * a(z) mod f(z), where f(z) is
// * the reduction polynomial of this.
// * @param a The polynomial a(z) to be multiplied by
// * z mod f(z).
// * @return z * a(z) mod f(z)
// */
// private BigInteger multZModF(final 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 ECFieldElement add(final ECFieldElement b)
// {
// // No check performed here for performance reasons. Instead the
// // elements involved are checked in ECPoint.F2m
// // checkFieldElements(this, b);
// if (b.toBigInteger().signum() == 0)
// {
// return this;
// }
//
// return new F2m(this.m, this.k1, this.k2, this.k3, this.x.xor(b.toBigInteger()));
// }
//
// public ECFieldElement subtract(final ECFieldElement b)
// {
// // Addition and subtraction are the same in F2m
// return add(b);
// }
//
//
// public ECFieldElement multiply(final 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);
// final BigInteger az = this.x;
// BigInteger bz = b.toBigInteger();
// BigInteger cz;
//
// // Compute c(z) = a(z) * b(z) mod f(z)
// if (az.testBit(0))
// {
// cz = bz;
// }
// else
// {
// cz = ECConstants.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 ECFieldElement.F2m(m, this.k1, this.k2, this.k3, cz);
// }
//
//
// public ECFieldElement divide(final ECFieldElement b)
// {
// // There may be more efficient implementations
// ECFieldElement bInv = b.invert();
// return multiply(bInv);
// }
//
// public ECFieldElement negate()
// {
// // -x == x holds for all x in F2m
// return this;
// }
//
// public ECFieldElement square()
// {
// // Naive implementation, can probably be speeded up using modular
// // reduction
// return multiply(this);
// }
//
// public 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.signum() <= 0)
// {
// throw new ArithmeticException("x is zero or negative, " +
// "inversion is impossible");
// }
//
// // v(z) := f(z)
// BigInteger vz = ECConstants.ZERO.setBit(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 = ECConstants.ONE;
// BigInteger g2z = ECConstants.ZERO;
//
// // while u != 1
// while (!(uz.equals(ECConstants.ZERO)))
// {
// // 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)
// {
// final BigInteger uzCopy = uz;
// uz = vz;
// vz = uzCopy;
//
// final 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 ECFieldElement.F2m(
// this.m, this.k1, this.k2, this.k3, g2z);
// }
//
// public ECFieldElement sqrt()
// {
// throw new RuntimeException("Not implemented");
// }
//
// /**
// * @return the representation of the field
// * F2m, either of
// * TPB (trinomial
// * basis representation) or
// * PPB (pentanomial
// * basis representation).
// */
// public int getRepresentation()
// {
// return this.representation;
// }
//
// /**
// * @return the degree m of the reduction polynomial
// * f(z).
// */
// public int getM()
// {
// return this.m;
// }
//
// /**
// * @return TPB: The integer k where xm +
// * xk + 1 represents the reduction polynomial
// * f(z).
// * PPB: The integer k1 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// */
// public int getK1()
// {
// return this.k1;
// }
//
// /**
// * @return TPB: Always returns 0
// * PPB: The integer k2 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// */
// public int getK2()
// {
// return this.k2;
// }
//
// /**
// * @return TPB: Always set to 0
// * PPB: The integer k3 where xm +
// * xk3 + xk2 + xk1 + 1
// * represents the reduction polynomial f(z).
// */
// public int getK3()
// {
// return this.k3;
// }
//
// public boolean equals(Object anObject)
// {
// if (anObject == this)
// {
// return true;
// }
//
// if (!(anObject instanceof ECFieldElement.F2m))
// {
// return false;
// }
//
// ECFieldElement.F2m b = (ECFieldElement.F2m)anObject;
//
// return ((this.m == b.m) && (this.k1 == b.k1) && (this.k2 == b.k2)
// && (this.k3 == b.k3)
// && (this.representation == b.representation)
// && (this.x.equals(b.x)));
// }
//
// public int hashCode()
// {
// return x.hashCode() ^ m ^ k1 ^ k2 ^ k3;
// }
// }
/**
* Class representing the Elements of the finite field
* F2m 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 static class F2m extends ECFieldElement
{
/**
* Indicates gaussian normal basis representation (GNB). Number chosen
* according to X9.62. GNB is not implemented at present.
*/
public static final int GNB = 1;
/**
* Indicates trinomial basis representation (TPB). Number chosen
* according to X9.62.
*/
public static final int TPB = 2;
/**
* Indicates pentanomial basis representation (PPB). Number chosen
* according to X9.62.
*/
public static final int PPB = 3;
/**
* TPB or PPB.
*/
private int representation;
/**
* The exponent m of F2m.
*/
private int m;
/**
* TPB: The integer k where xm +
* xk + 1 represents the reduction polynomial
* f(z).
* PPB: The integer k1 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
*/
private int k1;
/**
* TPB: Always set to 0
* PPB: The integer k2 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
*/
private int k2;
/**
* TPB: Always set to 0
* PPB: The integer k3 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
*/
private int k3;
/**
* The IntArray holding the bits.
*/
private IntArray x;
/**
* The number of ints required to hold m bits.
*/
private int t;
/**
* Constructor for PPB.
* @param m The exponent m of
* F2m.
* @param k1 The integer k1 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
* @param k2 The integer k2 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
* @param k3 The integer k3 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
* @param x The BigInteger representing the value of the field element.
*/
public F2m(
int m,
int k1,
int k2,
int k3,
BigInteger x)
{
// t = m / 32 rounded up to the next integer
t = (m + 31) >> 5;
this.x = new IntArray(x, t);
if ((k2 == 0) && (k3 == 0))
{
this.representation = TPB;
}
else
{
if (k2 >= k3)
{
throw new IllegalArgumentException(
"k2 must be smaller than k3");
}
if (k2 <= 0)
{
throw new IllegalArgumentException(
"k2 must be larger than 0");
}
this.representation = PPB;
}
if (x.signum() < 0)
{
throw new IllegalArgumentException("x value cannot be negative");
}
this.m = m;
this.k1 = k1;
this.k2 = k2;
this.k3 = k3;
}
/**
* Constructor for TPB.
* @param m The exponent m of
* F2m.
* @param k The integer k where xm +
* xk + 1 represents the reduction
* polynomial f(z).
* @param x The BigInteger representing the value of the field element.
*/
public F2m(int m, int k, BigInteger x)
{
// Set k1 to k, and set k2 and k3 to 0
this(m, k, 0, 0, x);
}
private F2m(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 BigInteger toBigInteger()
{
return x.toBigInteger();
}
public String getFieldName()
{
return "F2m";
}
public int getFieldSize()
{
return m;
}
/**
* Checks, if the ECFieldElements a and b
* are elements of the same field F2m
* (having the same representation).
* @param a field element.
* @param b field element to be compared.
* @throws IllegalArgumentException if a and b
* are not elements of the same field
* F2m (having the same
* representation).
*/
public static void checkFieldElements(
ECFieldElement a,
ECFieldElement b)
{
if ((!(a instanceof F2m)) || (!(b instanceof F2m)))
{
throw new IllegalArgumentException("Field elements are not "
+ "both instances of ECFieldElement.F2m");
}
F2m aF2m = (F2m)a;
F2m bF2m = (F2m)b;
if ((aF2m.m != bF2m.m) || (aF2m.k1 != bF2m.k1)
|| (aF2m.k2 != bF2m.k2) || (aF2m.k3 != bF2m.k3))
{
throw new IllegalArgumentException("Field elements are not "
+ "elements of the same field F2m");
}
if (aF2m.representation != bF2m.representation)
{
// Should never occur
throw new IllegalArgumentException(
"One of the field "
+ "elements are not elements has incorrect representation");
}
}
public ECFieldElement add(final 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();
F2m bF2m = (F2m)b;
iarrClone.addShifted(bF2m.x, 0);
return new F2m(m, k1, k2, k3, iarrClone);
}
public ECFieldElement subtract(final ECFieldElement b)
{
// Addition and subtraction are the same in F2m
return add(b);
}
public ECFieldElement multiply(final 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);
F2m bF2m = (F2m)b;
IntArray mult = x.multiply(bF2m.x, m);
mult.reduce(m, new int[]{k1, k2, k3});
return new F2m(m, k1, k2, k3, mult);
}
public ECFieldElement divide(final ECFieldElement b)
{
// There may be more efficient implementations
ECFieldElement bInv = b.invert();
return multiply(bInv);
}
public ECFieldElement negate()
{
// -x == x holds for all x in F2m
return this;
}
public ECFieldElement square()
{
IntArray squared = x.square(m);
squared.reduce(m, new int[]{k1, k2, k3});
return new F2m(m, k1, k2, k3, squared);
}
public 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.isZero())
// 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)
{
final IntArray uzCopy = uz;
uz = vz;
vz = uzCopy;
final 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 F2m(
this.m, this.k1, this.k2, this.k3, g2z);
}
public ECFieldElement sqrt()
{
throw new RuntimeException("Not implemented");
}
/**
* @return the representation of the field
* F2m, either of
* TPB (trinomial
* basis representation) or
* PPB (pentanomial
* basis representation).
*/
public int getRepresentation()
{
return this.representation;
}
/**
* @return the degree m of the reduction polynomial
* f(z).
*/
public int getM()
{
return this.m;
}
/**
* @return TPB: The integer k where xm +
* xk + 1 represents the reduction polynomial
* f(z).
* PPB: The integer k1 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
*/
public int getK1()
{
return this.k1;
}
/**
* @return TPB: Always returns 0
* PPB: The integer k2 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
*/
public int getK2()
{
return this.k2;
}
/**
* @return TPB: Always set to 0
* PPB: The integer k3 where xm +
* xk3 + xk2 + xk1 + 1
* represents the reduction polynomial f(z).
*/
public int getK3()
{
return this.k3;
}
public boolean equals(Object anObject)
{
if (anObject == this)
{
return true;
}
if (!(anObject instanceof F2m))
{
return false;
}
F2m b = (F2m)anObject;
return ((this.m == b.m) && (this.k1 == b.k1) && (this.k2 == b.k2)
&& (this.k3 == b.k3)
&& (this.representation == b.representation)
&& (this.x.equals(b.x)));
}
public int hashCode()
{
return x.hashCode() ^ m ^ k1 ^ k2 ^ k3;
}
}
}
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