/// <summary>
/// Compute the result of the division of two polynomials over the field <c>GF(2^m)</c>
/// </summary>
///
/// <param name="A">he first polynomial</param>
/// <param name="F">he second polynomial</param>
///
/// <returns>Returns <c>int[][] {q,r}</c>, where <c>a = q*f+r</c> and <c>deg(r) < deg(f)</c></returns>
private static int[][] Divide(int[] A, int[] F, GF2mField GF2)
{
int df = ComputeDegree(F);
int da = ComputeDegree(A) + 1;
if (df == -1)
{
throw new ArithmeticException("Division by zero.");
}
int[][] result = new int[2][];
result[0] = new int[1];
result[1] = new int[da];
int hc = HeadCoefficient(F);
hc = GF2.Inverse(hc);
result[0][0] = 0;
Array.Copy(A, 0, result[1], 0, result[1].Length);
while (df <= ComputeDegree(result[1]))
{
int[] q;
int[] coeff = new int[1];
coeff[0] = GF2.Multiply(HeadCoefficient(result[1]), hc);
q = MultWithElement(F, coeff[0], GF2);
int n = ComputeDegree(result[1]) - df;
q = MultWithMonomial(q, n);
coeff = MultWithMonomial(coeff, n);
result[0] = Add(coeff, result[0], GF2);
result[1] = Add(q, result[1], GF2);
}
return(result);
}
/// <summary>
/// Reduce a polynomial modulo another polynomial
/// </summary>
///
/// <param name="A">The polynomial</param>
/// <param name="F">The reduction polynomial</param>
///
/// <returns>Returns <c>a mod f</c></returns>
private static int[] Mod(int[] A, int[] F, GF2mField GF2)
{
int df = ComputeDegree(F);
if (df == -1)
{
throw new ArithmeticException("Division by zero");
}
int[] result = new int[A.Length];
int hc = HeadCoefficient(F);
hc = GF2.Inverse(hc);
Array.Copy(A, 0, result, 0, result.Length);
while (df <= ComputeDegree(result))
{
int[] q;
int coeff = GF2.Multiply(HeadCoefficient(result), hc);
q = MultWithMonomial(F, ComputeDegree(result) - df);
q = MultWithElement(q, coeff, GF2);
result = Add(q, result, GF2);
}
return(result);
}
/// <summary>
/// Find an error vector <c>E</c> over <c>GF(2)</c> from an input syndrome <c>S</c> over <c>GF(2^M)</c>
/// </summary>
///
/// <param name="SyndVec">The syndrome</param>
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
/// <param name="SqRootMatrix">The matrix for computing square roots in <c>(GF(2M))<sup>T</sup></c></param>
///
/// <returns>The error vector</returns>
public static GF2Vector SyndromeDecode(GF2Vector SyndVec, GF2mField Field, PolynomialGF2mSmallM Gp, PolynomialGF2mSmallM[] SqRootMatrix)
{
int n = 1 << Field.Degree;
// 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.Head;
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);
}
/// <summary>
/// Return the greatest common divisor of two polynomials over the field <c>GF(2^m)</c>
/// </summary>
///
/// <param name="F">The first polynomial</param>
/// <param name="G">The second polynomial</param>
///
/// <returns>Returns <c>Gcd(f, g)</c></returns>
private static int[] Gcd(int[] F, int[] G, GF2mField GF2)
{
int[] a = F;
int[] b = G;
if (ComputeDegree(a) == -1)
{
return(b);
}
while (ComputeDegree(b) != -1)
{
int[] c = Mod(a, b, GF2);
a = new int[b.Length];
Array.Copy(b, 0, a, 0, a.Length);
b = new int[c.Length];
Array.Copy(c, 0, b, 0, b.Length);
}
int coeff = GF2.Inverse(HeadCoefficient(a));
return(MultWithElement(a, coeff, GF2));
}
/// <summary>
/// Compute the result of the division of two polynomials modulo a third polynomial over the field <c>GF(2^m)</c>
/// </summary>
///
/// <param name="A">The first polynomial</param>
/// <param name="B">The second polynomial</param>
/// <param name="G">The reduction polynomial</param>
///
/// <returns>Returns <c>a * b^(-1) mod g</c></returns>
private static int[] ModDiv(int[] A, int[] B, int[] G, GF2mField GF2)
{
int[] r0 = NormalForm(G);
int[] r1 = Mod(B, G, GF2);
int[] s0 = { 0 };
int[] s1 = Mod(A, G, GF2);
int[] s2;
int[][] q;
while (ComputeDegree(r1) != -1)
{
q = Divide(r0, r1, GF2);
r0 = NormalForm(r1);
r1 = NormalForm(q[1]);
s2 = Add(s0, ModMultiply(q[0], s1, G, GF2), GF2);
s0 = NormalForm(s1);
s1 = NormalForm(s2);
}
int hc = HeadCoefficient(r0);
s0 = MultWithElement(s0, GF2.Inverse(hc), GF2);
return(s0);
}
/// <summary>
/// Compute the matrix for computing square roots in this polynomial ring by inverting the squaring matrix
/// </summary>
private void ComputeSquareRootMatrix()
{
int numColumns = _poly.Degree;
// clone squaring matrix
PolynomialGF2mSmallM[] tmpMatrix = new PolynomialGF2mSmallM[numColumns];
for (int i = numColumns - 1; i >= 0; i--)
{
tmpMatrix[i] = new PolynomialGF2mSmallM(_sqMatrix[i]);
}
// initialize square root matrix as unit matrix
_sqRootMatrix = new PolynomialGF2mSmallM[numColumns];
for (int i = numColumns - 1; i >= 0; i--)
{
_sqRootMatrix[i] = new PolynomialGF2mSmallM(_field, i);
}
// simultaneously compute Gaussian reduction of squaring matrix and unit matrix
for (int i = 0; i < numColumns; i++)
{
// if diagonal element is zero
if (tmpMatrix[i].GetCoefficient(i) == 0)
{
bool foundNonZero = false;
// find a non-zero element in the same row
for (int j = i + 1; j < numColumns; j++)
{
if (tmpMatrix[j].GetCoefficient(i) != 0)
{
// found it, swap columns ...
foundNonZero = true;
SwapColumns(tmpMatrix, i, j);
SwapColumns(_sqRootMatrix, i, j);
// ... and quit searching
j = numColumns;
continue;
}
}
// if no non-zero element was found the matrix is not invertible
if (!foundNonZero)
{
throw new ArithmeticException("Squaring matrix is not invertible.");
}
}
// normalize i-th column
int coef = tmpMatrix[i].GetCoefficient(i);
int invCoef = _field.Inverse(coef);
tmpMatrix[i].MultThisWithElement(invCoef);
_sqRootMatrix[i].MultThisWithElement(invCoef);
if (ParallelUtils.IsParallel)
{
// normalize all other columns
Parallel.For(0, numColumns, j =>
{
if (j != i)
{
int coefp = tmpMatrix[j].GetCoefficient(i);
if (coefp != 0)
{
PolynomialGF2mSmallM tmpSqColumn = tmpMatrix[i].MultWithElement(coefp);
PolynomialGF2mSmallM tmpInvColumn = _sqRootMatrix[i].MultWithElement(coefp);
tmpMatrix[j].AddToThis(tmpSqColumn);
lock (_sqRootMatrix)
_sqRootMatrix[j].AddToThis(tmpInvColumn);
}
}
});
}
else
{
for (int j = 0; j < numColumns; j++)
{
if (j != i)
{
coef = tmpMatrix[j].GetCoefficient(i);
if (coef != 0)
{
PolynomialGF2mSmallM tmpSqColumn = tmpMatrix[i].MultWithElement(coef);
PolynomialGF2mSmallM tmpInvColumn = _sqRootMatrix[i].MultWithElement(coef);
tmpMatrix[j].AddToThis(tmpSqColumn);
lock (_sqRootMatrix)
_sqRootMatrix[j].AddToThis(tmpInvColumn);
}
}
}
}
}
}
/// <summary>
/// Construct the check matrix of a Goppa code in canonical form from the irreducible Goppa polynomial over the finite field <c>GF(2^m)</c>.
/// </summary>
///
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
///
/// <returns>The new GF2Matrix</returns>
public static GF2Matrix CreateCanonicalCheckMatrix(GF2mField Field, PolynomialGF2mSmallM Gp)
{
int m = Field.Degree;
int n = 1 << m;
int t = Gp.Degree;
// create matrix H over GF(2^m)
int[][] hArray = ArrayUtils.CreateJagged <int[][]>(t, n);
// create matrix YZ
int[][] yz = ArrayUtils.CreateJagged <int[][]>(t, n);
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j)));
}
else
{
// here j is used as index and as element of field GF(2^m)
for (int j = 0; j < n; j++)
{
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j));
}
}
for (int i = 1; i < t; i++)
{
// here j is used as index and as element of field GF(2^m)
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
});
}
else
{
for (int j = 0; j < n; j++)
{
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
}
}
}
// create matrix H = XYZ
for (int i = 0; i < t; i++)
{
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
for (int k = 0; k <= i; k++)
{
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
}
});
}
else
{
for (int j = 0; j < n; j++)
{
for (int k = 0; k <= i; k++)
{
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
}
}
}
}
// convert to matrix over GF(2)
int[][] result = ArrayUtils.CreateJagged <int[][]>(t * m, IntUtils.URShift((n + 31), 5));
if (ParallelUtils.IsParallel)
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(j, 5);
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
Parallel.For(0, m, u =>
{
int b = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
});
}
}
}
else
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(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 = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
}
}
}
}
return(new GF2Matrix(n, result));
}
/// <summary>
/// Reduce a polynomial modulo another polynomial
/// </summary>
///
/// <param name="A">The polynomial</param>
/// <param name="F">The reduction polynomial</param>
///
/// <returns>Returns <c>a mod f</c></returns>
private static int[] Mod(int[] A, int[] F, GF2mField GF2)
{
int df = ComputeDegree(F);
if (df == -1)
throw new ArithmeticException("Division by zero");
int[] result = new int[A.Length];
int hc = HeadCoefficient(F);
hc = GF2.Inverse(hc);
Array.Copy(A, 0, result, 0, result.Length);
while (df <= ComputeDegree(result))
{
int[] q;
int coeff = GF2.Multiply(HeadCoefficient(result), hc);
q = MultWithMonomial(F, ComputeDegree(result) - df);
q = MultWithElement(q, coeff, GF2);
result = Add(q, result, GF2);
}
return result;
}
/// <summary>
/// Compute the result of the division of two polynomials modulo a third polynomial over the field <c>GF(2^m)</c>
/// </summary>
///
/// <param name="A">The first polynomial</param>
/// <param name="B">The second polynomial</param>
/// <param name="G">The reduction polynomial</param>
///
/// <returns>Returns <c>a * b^(-1) mod g</c></returns>
private static int[] ModDiv(int[] A, int[] B, int[] G, GF2mField GF2)
{
int[] r0 = NormalForm(G);
int[] r1 = Mod(B, G, GF2);
int[] s0 = { 0 };
int[] s1 = Mod(A, G, GF2);
int[] s2;
int[][] q;
while (ComputeDegree(r1) != -1)
{
q = Divide(r0, r1, GF2);
r0 = NormalForm(r1);
r1 = NormalForm(q[1]);
s2 = Add(s0, ModMultiply(q[0], s1, G, GF2), GF2);
s0 = NormalForm(s1);
s1 = NormalForm(s2);
}
int hc = HeadCoefficient(r0);
s0 = MultWithElement(s0, GF2.Inverse(hc), GF2);
return s0;
}
/// <summary>
/// Return the greatest common divisor of two polynomials over the field <c>GF(2^m)</c>
/// </summary>
///
/// <param name="F">The first polynomial</param>
/// <param name="G">The second polynomial</param>
///
/// <returns>Returns <c>Gcd(f, g)</c></returns>
private static int[] Gcd(int[] F, int[] G, GF2mField GF2)
{
int[] a = F;
int[] b = G;
if (ComputeDegree(a) == -1)
return b;
while (ComputeDegree(b) != -1)
{
int[] c = Mod(a, b, GF2);
a = new int[b.Length];
Array.Copy(b, 0, a, 0, a.Length);
b = new int[c.Length];
Array.Copy(c, 0, b, 0, b.Length);
}
int coeff = GF2.Inverse(HeadCoefficient(a));
return MultWithElement(a, coeff, GF2);
}
/// <summary>
/// Compute the result of the division of two polynomials over the field <c>GF(2^m)</c>
/// </summary>
///
/// <param name="A">he first polynomial</param>
/// <param name="F">he second polynomial</param>
///
/// <returns>Returns <c>int[][] {q,r}</c>, where <c>a = q*f+r</c> and <c>deg(r) < deg(f)</c></returns>
private static int[][] Divide(int[] A, int[] F, GF2mField GF2)
{
int df = ComputeDegree(F);
int da = ComputeDegree(A) + 1;
if (df == -1)
throw new ArithmeticException("Division by zero.");
int[][] result = new int[2][];
result[0] = new int[1];
result[1] = new int[da];
int hc = HeadCoefficient(F);
hc = GF2.Inverse(hc);
result[0][0] = 0;
Array.Copy(A, 0, result[1], 0, result[1].Length);
while (df <= ComputeDegree(result[1]))
{
int[] q;
int[] coeff = new int[1];
coeff[0] = GF2.Multiply(HeadCoefficient(result[1]), hc);
q = MultWithElement(F, coeff[0], GF2);
int n = ComputeDegree(result[1]) - df;
q = MultWithMonomial(q, n);
coeff = MultWithMonomial(coeff, n);
result[0] = Add(coeff, result[0], GF2);
result[1] = Add(q, result[1], GF2);
}
return result;
}
/// <summary>
/// Construct the check matrix of a Goppa code in canonical form from the irreducible Goppa polynomial over the finite field <c>GF(2^m)</c>.
/// </summary>
///
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
///
/// <returns>The new GF2Matrix</returns>
public static GF2Matrix CreateCanonicalCheckMatrix(GF2mField Field, PolynomialGF2mSmallM Gp)
{
int m = Field.Degree;
int n = 1 << m;
int t = Gp.Degree;
// create matrix H over GF(2^m)
int[][] hArray = ArrayUtils.CreateJagged<int[][]>(t, n);
// create matrix YZ
int[][] yz = ArrayUtils.CreateJagged<int[][]>(t, n);
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j)));
}
else
{
// here j is used as index and as element of field GF(2^m)
for (int j = 0; j < n; j++)
yz[0][j] = Field.Inverse(Gp.EvaluateAt(j));
}
for (int i = 1; i < t; i++)
{
// here j is used as index and as element of field GF(2^m)
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
});
}
else
{
for (int j = 0; j < n; j++)
yz[i][j] = Field.Multiply(yz[i - 1][j], j);
}
}
// create matrix H = XYZ
for (int i = 0; i < t; i++)
{
if (ParallelUtils.IsParallel)
{
Parallel.For(0, n, j =>
{
for (int k = 0; k <= i; k++)
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
});
}
else
{
for (int j = 0; j < n; j++)
{
for (int k = 0; k <= i; k++)
hArray[i][j] = Field.Add(hArray[i][j], Field.Multiply(yz[k][j], Gp.GetCoefficient(t + k - i)));
}
}
}
// convert to matrix over GF(2)
int[][] result = ArrayUtils.CreateJagged<int[][]>(t * m, IntUtils.URShift((n + 31), 5));
if (ParallelUtils.IsParallel)
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(j, 5);
int r = 1 << (j & 0x1f);
for (int i = 0; i < t; i++)
{
int e = hArray[i][j];
Parallel.For(0, m, u =>
{
int b = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
});
}
}
}
else
{
for (int j = 0; j < n; j++)
{
int q = IntUtils.URShift(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 = (IntUtils.URShift(e, u)) & 1;
if (b != 0)
{
int ind = (i + 1) * m - u - 1;
result[ind][q] ^= r;
}
}
}
}
}
return new GF2Matrix(n, result);
}
/// <summary>
/// Find an error vector <c>E</c> over <c>GF(2)</c> from an input syndrome <c>S</c> over <c>GF(2^M)</c>
/// </summary>
///
/// <param name="SyndVec">The syndrome</param>
/// <param name="Field">The finite field</param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
/// <param name="SqRootMatrix">The matrix for computing square roots in <c>(GF(2M))<sup>T</sup></c></param>
///
/// <returns>The error vector</returns>
public static GF2Vector SyndromeDecode(GF2Vector SyndVec, GF2mField Field, PolynomialGF2mSmallM Gp, PolynomialGF2mSmallM[] SqRootMatrix)
{
int n = 1 << Field.Degree;
// 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.Head;
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;
}
/// <summary>
/// Compute the inverse of this matrix
/// </summary>
///
/// <returns>Returns the inverse of this matrix (newly created)</returns>
public override Matrix ComputeInverse()
{
if (RowCount != ColumnCount)
{
throw new ArithmeticException("GF2mMatrix: Matrix is not invertible!");
}
// clone this matrix
int[][] tmpMatrix = ArrayUtils.CreateJagged <int[][]>(RowCount, RowCount);
for (int i = RowCount - 1; i >= 0; i--)
{
tmpMatrix[i] = IntUtils.DeepCopy(MatrixN[i]);
}
// initialize inverse matrix as unit matrix
int[][] invMatrix = ArrayUtils.CreateJagged <int[][]>(RowCount, RowCount);
for (int i = RowCount - 1; i >= 0; i--)
{
invMatrix[i][i] = 1;
}
// simultaneously compute Gaussian reduction of tmpMatrix and unit
// matrix
for (int i = 0; i < RowCount; i++)
{
// if diagonal element is zero
if (tmpMatrix[i][i] == 0)
{
bool foundNonZero = false;
// find a non-zero element in the same column
for (int j = i + 1; j < RowCount; j++)
{
if (tmpMatrix[j][i] != 0)
{
// found it, swap rows ...
foundNonZero = true;
SwapColumns(tmpMatrix, i, j);
SwapColumns(invMatrix, i, j);
// ... and quit searching
j = RowCount;
continue;
}
}
// if no non-zero element was found the matrix is not invertible
if (!foundNonZero)
{
throw new ArithmeticException("GF2mMatrix: Matrix is not invertible!");
}
}
// normalize i-th row
int coef = tmpMatrix[i][i];
int invCoef = FieldG.Inverse(coef);
MultRowWithElementThis(tmpMatrix[i], invCoef);
MultRowWithElementThis(invMatrix[i], invCoef);
// normalize all other rows
for (int j = 0; j < RowCount; j++)
{
if (j != i)
{
coef = tmpMatrix[j][i];
if (coef != 0)
{
int[] tmpRow = MultRowWithElement(tmpMatrix[i], coef);
int[] tmpInvRow = MultRowWithElement(invMatrix[i], coef);
AddToRow(tmpRow, tmpMatrix[j]);
AddToRow(tmpInvRow, invMatrix[j]);
}
}
}
}
return(new GF2mMatrix(FieldG, invMatrix));
}