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package org.bouncycastle.pqc.math.linearalgebra;
import java.security.SecureRandom;
/**
* This class describes decoding operations of an irreducible binary Goppa code.
* A check matrix H of the Goppa code and an irreducible Goppa polynomial are
* used the operations are worked over a finite field GF(2^m)
*
* @see GF2mField
* @see PolynomialGF2mSmallM
*/
public final class GoppaCode
{
/**
* Default constructor (private).
*/
private GoppaCode()
{
// empty
}
/**
* This class is a container for two instances of {@link GF2Matrix} and one
* instance of {@link Permutation}. It is used to hold the systematic form
* S*H*P = (Id|M) of the check matrix H as returned by
* {@link GoppaCode#computeSystematicForm(GF2Matrix, SecureRandom)}.
*
* @see GF2Matrix
* @see Permutation
*/
public static class MaMaPe
{
private GF2Matrix s, h;
private Permutation p;
/**
* Construct a new {@link MaMaPe} container with the given parameters.
*
* @param s the first matrix
* @param h the second matrix
* @param p the permutation
*/
public MaMaPe(GF2Matrix s, GF2Matrix h, Permutation p)
{
this.s = s;
this.h = h;
this.p = p;
}
/**
* @return the first matrix
*/
public GF2Matrix getFirstMatrix()
{
return s;
}
/**
* @return the second matrix
*/
public GF2Matrix getSecondMatrix()
{
return h;
}
/**
* @return the permutation
*/
public Permutation getPermutation()
{
return p;
}
}
/**
* This class is a container for an instance of {@link GF2Matrix} and one
* int[]. It is used to hold a generator matrix and the set of indices such
* that the submatrix of the generator matrix consisting of the specified
* columns is the identity.
*
* @see GF2Matrix
* @see Permutation
*/
public static class MatrixSet
{
private GF2Matrix g;
private int[] setJ;
/**
* Construct a new {@link MatrixSet} container with the given
* parameters.
*
* @param g the generator matrix
* @param setJ the set of indices such that the submatrix of the
* generator matrix consisting of the specified columns
* is the identity
*/
public MatrixSet(GF2Matrix g, int[] setJ)
{
this.g = g;
this.setJ = setJ;
}
/**
* @return the generator matrix
*/
public GF2Matrix getG()
{
return g;
}
/**
* @return the set of indices such that the submatrix of the generator
* matrix consisting of the specified columns is the identity
*/
public int[] getSetJ()
{
return setJ;
}
}
/**
* Construct the check matrix of a Goppa code in canonical form from the
* irreducible Goppa polynomial over the finite field
* GF(2m).
*
* @param field the finite field
* @param gp the irreducible Goppa polynomial
*/
public static GF2Matrix createCanonicalCheckMatrix(GF2mField field,
PolynomialGF2mSmallM gp)
{
int m = field.getDegree();
int n = 1 << m;
int t = gp.getDegree();
/* create matrix H over GF(2^m) */
int[][] hArray = new int[t][n];
// create matrix YZ
int[][] yz = new int[t][n];
for (int j = 0; j < n; j++)
{
// here j is used as index and as element of field GF(2^m)
yz[0][j] = field.inverse(gp.evaluateAt(j));
}
for (int i = 1; i < t; i++)
{
for (int j = 0; j < n; j++)
{
// here j is used as index and as element of field GF(2^m)
yz[i][j] = field.mult(yz[i - 1][j], j);
}
}
// create matrix H = XYZ
for (int i = 0; i < t; i++)
{
for (int j = 0; j < n; j++)
{
for (int k = 0; k <= i; k++)
{
hArray[i][j] = field.add(hArray[i][j], field.mult(yz[k][j],
gp.getCoefficient(t + k - i)));
}
}
}
/* convert to matrix over GF(2) */
int[][] result = new int[t * m][(n + 31) >>> 5];
for (int j = 0; j < n; j++)
{
int q = j >>> 5;
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
for (int u = 0; u < m; u++)
{
int b = (e >>> u) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
}
}
}
return new GF2Matrix(n, result);
}
/**
* Given a check matrix H, compute matrices S,
* M, and a random permutation P such that
* S*H*P = (Id|M). Return S^-1, M, and
* P as {@link MaMaPe}. The matrix (Id | M) is called
* the systematic form of H.
*
* @param h the check matrix
* @param sr a source of randomness
* @return the tuple (S^-1, M, P)
*/
public static MaMaPe computeSystematicForm(GF2Matrix h, SecureRandom sr)
{
int n = h.getNumColumns();
GF2Matrix hp, sInv;
GF2Matrix s = null;
Permutation p;
boolean found = false;
do
{
p = new Permutation(n, sr);
hp = (GF2Matrix)h.rightMultiply(p);
sInv = hp.getLeftSubMatrix();
try
{
found = true;
s = (GF2Matrix)sInv.computeInverse();
}
catch (ArithmeticException ae)
{
found = false;
}
}
while (!found);
GF2Matrix shp = (GF2Matrix)s.rightMultiply(hp);
GF2Matrix m = shp.getRightSubMatrix();
return new MaMaPe(sInv, m, p);
}
/**
* Find an error vector e over GF(2) from an input
* syndrome s over GF(2m).
*
* @param syndVec the syndrome
* @param field the finite field
* @param gp the irreducible Goppa polynomial
* @param sqRootMatrix the matrix for computing square roots in
* (GF(2m))t
* @return the error vector
*/
public static GF2Vector syndromeDecode(GF2Vector syndVec, GF2mField field,
PolynomialGF2mSmallM gp, PolynomialGF2mSmallM[] sqRootMatrix)
{
int n = 1 << field.getDegree();
// the error vector
GF2Vector errors = new GF2Vector(n);
// if the syndrome vector is zero, the error vector is also zero
if (!syndVec.isZero())
{
// convert syndrome vector to polynomial over GF(2^m)
PolynomialGF2mSmallM syndrome = new PolynomialGF2mSmallM(syndVec
.toExtensionFieldVector(field));
// compute T = syndrome^-1 mod gp
PolynomialGF2mSmallM t = syndrome.modInverse(gp);
// compute tau = sqRoot(T + X) mod gp
PolynomialGF2mSmallM tau = t.addMonomial(1);
tau = tau.modSquareRootMatrix(sqRootMatrix);
// compute polynomials a and b satisfying a + b*tau = 0 mod gp
PolynomialGF2mSmallM[] ab = tau.modPolynomialToFracton(gp);
// compute the polynomial a^2 + X*b^2
PolynomialGF2mSmallM a2 = ab[0].multiply(ab[0]);
PolynomialGF2mSmallM b2 = ab[1].multiply(ab[1]);
PolynomialGF2mSmallM xb2 = b2.multWithMonomial(1);
PolynomialGF2mSmallM a2plusXb2 = a2.add(xb2);
// normalize a^2 + X*b^2 to obtain the error locator polynomial
int headCoeff = a2plusXb2.getHeadCoefficient();
int invHeadCoeff = field.inverse(headCoeff);
PolynomialGF2mSmallM elp = a2plusXb2.multWithElement(invHeadCoeff);
// for all elements i of GF(2^m)
for (int i = 0; i < n; i++)
{
// evaluate the error locator polynomial at i
int z = elp.evaluateAt(i);
// if polynomial evaluates to zero
if (z == 0)
{
// set the i-th coefficient of the error vector
errors.setBit(i);
}
}
}
return errors;
}
}