/// <summary>
/// Initialize this class
/// </summary>
///
/// <param name="Field">The finite field</param>
/// <param name="Poly">The reduction polynomial</param>
public PolynomialRingGF2m(GF2mField Field, PolynomialGF2mSmallM Poly)
{
_field = Field;
_poly = Poly;
ComputeSquaringMatrix();
ComputeSquareRootMatrix();
}
/// <summary>
/// Initialize this class
/// </summary>
///
/// <param name="Field">The finite field</param>
/// <param name="Poly">The reduction polynomial</param>
public PolynomialRingGF2m(GF2mField Field, PolynomialGF2mSmallM Poly)
{
_field = Field;
_poly = Poly;
ComputeSquaringMatrix();
ComputeSquareRootMatrix();
}
/// <summary>
/// Swap two columns
/// </summary>
private static void SwapColumns(PolynomialGF2mSmallM[] Matrix, int First, int Second)
{
PolynomialGF2mSmallM tmp = Matrix[First];
Matrix[First] = Matrix[Second];
Matrix[Second] = tmp;
}
/// <summary>
/// Compute the inverse of this polynomial modulo the given polynomial
/// </summary>
///
/// <param name="A">The reduction polynomial</param>
///
/// <returns>Returns <c>this^(-1) mod a</c></returns>
public PolynomialGF2mSmallM ModInverse(PolynomialGF2mSmallM A)
{
int[] unit = { 1 };
int[] resultCoeff = ModDiv(unit, _coefficients, A._coefficients, _field);
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <summary>
/// Copy constructor
/// </summary>
///
/// <param name="Gf">The PolynomialGF2mSmallM to copy</param>
public PolynomialGF2mSmallM(PolynomialGF2mSmallM Gf)
{
// field needs not to be cloned since it is immutable
_field = Gf._field;
_degree = Gf._degree;
_coefficients = IntUtils.DeepCopy(Gf._coefficients);
}
/// <summary>
/// Divide this polynomial by the given polynomial
/// </summary>
/// <param name="F">The polynomial</param>
/// <returns>Returns polynomial pair = {q,r} where this = q*f+r and deg(r) < deg(f)</returns>
public PolynomialGF2mSmallM[] Divide(PolynomialGF2mSmallM F)
{
int[][] resultCoeffs = Divide(_coefficients, F._coefficients, _field);
return(new PolynomialGF2mSmallM[] {
new PolynomialGF2mSmallM(_field, resultCoeffs[0]),
new PolynomialGF2mSmallM(_field, resultCoeffs[1])
});
}
/// <summary>
/// Initialize this class for CCA2 MPKCS
/// </summary>
///
/// <param name="N">Length of the code</param>
/// <param name="K">The dimension of the code</param>
/// <param name="Gf">The finite field <c>GF(2^m)</c></param>
/// <param name="Gp">The irreducible Goppa polynomial</param>
/// <param name="P">The permutation</param>
/// <param name="H">The canonical check matrix</param>
/// <param name="QInv">The matrix used to compute square roots in <c>(GF(2^m))^t</c></param>
internal MPKCPrivateKey(int N, int K, GF2mField Gf, PolynomialGF2mSmallM Gp, Permutation P, GF2Matrix H, PolynomialGF2mSmallM[] QInv)
{
_N = N;
_K = K;
_gField = Gf;
_goppaPoly = Gp;
_P1 = P;
_H = H;
_qInv = QInv;
}
/// <summary>
/// Compute the square root of this polynomial modulo the given polynomial
/// </summary>
///
/// <param name="A">The reduction polynomial</param>
///
/// <returns>Returns <c>this^(1/2) mod A</c></returns>
public PolynomialGF2mSmallM ModSquareRoot(PolynomialGF2mSmallM A)
{
int[] resultCoeff = IntUtils.DeepCopy(_coefficients);
int[] help = ModMultiply(resultCoeff, resultCoeff, A._coefficients, _field);
while (!IsEqual(help, _coefficients))
{
resultCoeff = NormalForm(help);
help = ModMultiply(resultCoeff, resultCoeff, A._coefficients, _field);
}
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <summary>
/// Initialize this class CCA2 MPKCS using encoded byte arrays
/// </summary>
///
/// <param name="N">Length of the code</param>
/// <param name="K">The dimension of the code</param>
/// <param name="Gf">Encoded field polynomial defining the finite field <c>GF(2^m)</c></param>
/// <param name="Gp">Encoded irreducible Goppa polynomial</param>
/// <param name="P">The encoded permutation</param>
/// <param name="H">Encoded canonical check matrix</param>
/// <param name="QInv">The encoded matrix used to compute square roots in <c>(GF(2^m))^t</c></param>
public MPKCPrivateKey(int N, int K, byte[] Gf, byte[] Gp, byte[] P, byte[] H, byte[][] QInv)
{
_N = N;
_K = K;
_gField = new GF2mField(Gf);
_goppaPoly = new PolynomialGF2mSmallM(_gField, Gp);
_P1 = new Permutation(P);
_H = new GF2Matrix(H);
_qInv = new PolynomialGF2mSmallM[QInv.Length];
for (int i = 0; i < QInv.Length; i++)
_qInv[i] = new PolynomialGF2mSmallM(_gField, QInv[i]);
}
/// <summary>
/// Compare this element with another object
/// </summary>
///
/// <param name="Obj">The object for comprison</param>
///
/// <returns>Returns <c>true</c> if the two objects are equal, <c>false</c> otherwise</returns>
public override bool Equals(Object Obj)
{
if (Obj == null || !(Obj is PolynomialGF2mSmallM))
{
return(false);
}
PolynomialGF2mSmallM p = (PolynomialGF2mSmallM)Obj;
if ((_field.Equals(p._field)) && (_degree.Equals(p._degree)) && (Compare.AreEqual(_coefficients, p._coefficients)))
{
return(true);
}
return(false);
}
/// <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 squaring matrix for this polynomial ring, using the base field and the reduction polynomial
/// </summary>
private void ComputeSquaringMatrix()
{
int numColumns = _poly.Degree;
_sqMatrix = new PolynomialGF2mSmallM[numColumns];
PolynomialGF2mSmallM[] _sqMatrix2 = new PolynomialGF2mSmallM[numColumns];
if (ParallelUtils.IsParallel)
{
int nct = numColumns >> 1;
Parallel.For(0, nct, i =>
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
_sqMatrix[i] = new PolynomialGF2mSmallM(_field, monomCoeffs);
});
Parallel.For(nct, numColumns, i =>
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
PolynomialGF2mSmallM monomial = new PolynomialGF2mSmallM(_field, monomCoeffs);
_sqMatrix[i] = monomial.Mod(_poly);
});
}
else
{
for (int i = 0; i < numColumns >> 1; i++)
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
_sqMatrix[i] = new PolynomialGF2mSmallM(_field, monomCoeffs);
}
for (int i = numColumns >> 1; i < numColumns; i++)
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
PolynomialGF2mSmallM monomial = new PolynomialGF2mSmallM(_field, monomCoeffs);
_sqMatrix[i] = monomial.Mod(_poly);
}
}
}
/// <summary>
/// Compute a polynomial pair (a,b) from this polynomial and the given
/// polynomial g with the property b*this = a mod g and deg(a)<=deg(g)/2
/// </summary>
///
/// <param name="G">The reduction polynomial</param>
///
/// <returns>Returns PolynomialGF2mSmallM[] {a,b} with b*this = a mod g and deg(a)<= deg(g)/2</returns>
public PolynomialGF2mSmallM[] ModPolynomialToFracton(PolynomialGF2mSmallM G)
{
int dg = G._degree >> 1;
int[] a0 = NormalForm(G._coefficients);
int[] a1 = Mod(_coefficients, G._coefficients, _field);
int[] b0 = { 0 };
int[] b1 = { 1 };
while (ComputeDegree(a1) > dg)
{
int[][] q = Divide(a0, a1, _field);
a0 = a1;
a1 = q[1];
int[] b2 = Add(b0, ModMultiply(q[0], b1, G._coefficients, _field), _field);
b0 = b1;
b1 = b2;
}
return(new PolynomialGF2mSmallM[] { new PolynomialGF2mSmallM(_field, a1), new PolynomialGF2mSmallM(_field, b1) });
}
/// <summary>
/// Compute the inverse of this polynomial modulo the given polynomial
/// </summary>
///
/// <param name="A">The reduction polynomial</param>
///
/// <returns>Returns <c>this^(-1) mod a</c></returns>
public PolynomialGF2mSmallM ModInverse(PolynomialGF2mSmallM A)
{
int[] unit = { 1 };
int[] resultCoeff = ModDiv(unit, _coefficients, A._coefficients, _field);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Compute the product of this polynomial and the given factor using a Karatzuba like scheme
/// </summary>
///
/// <param name="Factor">the polynomial factor</param>
///
/// <returns>Returns <c>this * factor</c></returns>
public PolynomialGF2mSmallM Multiply(PolynomialGF2mSmallM Factor)
{
int[] resultCoeff = Multiply(_coefficients, Factor._coefficients, _field);
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <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>
/// Reduce this polynomial modulo another polynomial
/// </summary>
///
/// <param name="F">The reduction polynomial</param>
///
/// <returns>Returns <c>this mod f</c></returns>
public PolynomialGF2mSmallM Mod(PolynomialGF2mSmallM F)
{
int[] resultCoeff = Mod(_coefficients, F._coefficients, _field);
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <summary>
/// Add the given polynomial to this polynomial (overwrite this).
/// </summary>
///
/// <param name="Addend">The addend</param>
public void AddToThis(PolynomialGF2mSmallM Addend)
{
_coefficients = Add(_coefficients, Addend._coefficients, _field);
ComputeDegree();
}
/// <summary>
/// Copy constructor
/// </summary>
///
/// <param name="Gf">The PolynomialGF2mSmallM to copy</param>
public PolynomialGF2mSmallM(PolynomialGF2mSmallM Gf)
{
// field needs not to be cloned since it is immutable
_field = Gf._field;
_degree = Gf._degree;
_coefficients = IntUtils.DeepCopy(Gf._coefficients);
}
/// <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>
/// Compute the product of this polynomial and the given factor using a Karatzuba like scheme
/// </summary>
///
/// <param name="Factor">the polynomial factor</param>
///
/// <returns>Returns <c>this * factor</c></returns>
public PolynomialGF2mSmallM Multiply(PolynomialGF2mSmallM Factor)
{
int[] resultCoeff = Multiply(_coefficients, Factor._coefficients, _field);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Compute the square root of this polynomial using a square root matrix
/// </summary>
///
/// <param name="M">The matrix for computing square roots in <c>(GF(2^m))^t</c> the polynomial ring defining the square root matrix</param>
///
/// <returns>Returns <c>this^(1/2)</c> modulo the reduction polynomial implicitly given via the square root matrix</returns>
public PolynomialGF2mSmallM ModSquareRootMatrix(PolynomialGF2mSmallM[] M)
{
int length = M.Length;
int[] resultCoeff = new int[length];
// do matrix multiplication
for (int i = 0; i < length; i++)
{
// compute scalar product of i-th row and j-th column
for (int j = 0; j < length; j++)
{
if (i >= M[j]._coefficients.Length)
continue;
if (j < _coefficients.Length)
{
int scalarTerm = _field.Multiply(M[j]._coefficients[i], _coefficients[j]);
resultCoeff[i] = _field.Add(resultCoeff[i], scalarTerm);
}
}
}
// compute the square root of each entry of the result coefficients
for (int i = 0; i < length; i++)
resultCoeff[i] = _field.Sqrt(resultCoeff[i]);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Compute the square root of this polynomial modulo the given polynomial
/// </summary>
///
/// <param name="A">The reduction polynomial</param>
///
/// <returns>Returns <c>this^(1/2) mod A</c></returns>
public PolynomialGF2mSmallM ModSquareRoot(PolynomialGF2mSmallM A)
{
int[] resultCoeff = IntUtils.DeepCopy(_coefficients);
int[] help = ModMultiply(resultCoeff, resultCoeff, A._coefficients, _field);
while (!IsEqual(help, _coefficients))
{
resultCoeff = NormalForm(help);
help = ModMultiply(resultCoeff, resultCoeff, A._coefficients, _field);
}
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Square this polynomial using a squaring matrix
/// </summary>
///
/// <param name="M">The squaring matrix</param>
///
/// <returns>Returns <c>this^2</c> modulo the reduction polynomial implicitly given via the squaring matrix</returns>
public PolynomialGF2mSmallM ModSquareMatrix(PolynomialGF2mSmallM[] M)
{
int length = M.Length;
int[] resultCoeff = new int[length];
int[] thisSquare = new int[length];
// square each entry of this polynomial
for (int i = 0; i < _coefficients.Length; i++)
thisSquare[i] = _field.Multiply(_coefficients[i], _coefficients[i]);
// do matrix-vector multiplication
for (int i = 0; i < length; i++)
{
// compute scalar product of i-th row and coefficient vector
for (int j = 0; j < length; j++)
{
if (i >= M[j]._coefficients.Length)
continue;
int scalarTerm = _field.Multiply(M[j]._coefficients[i], thisSquare[j]);
resultCoeff[i] = _field.Add(resultCoeff[i], scalarTerm);
}
}
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Compute a polynomial pair (a,b) from this polynomial and the given
/// polynomial g with the property b*this = a mod g and deg(a)<=deg(g)/2
/// </summary>
///
/// <param name="G">The reduction polynomial</param>
///
/// <returns>Returns PolynomialGF2mSmallM[] {a,b} with b*this = a mod g and deg(a)<= deg(g)/2</returns>
public PolynomialGF2mSmallM[] ModPolynomialToFracton(PolynomialGF2mSmallM G)
{
int dg = G._degree >> 1;
int[] a0 = NormalForm(G._coefficients);
int[] a1 = Mod(_coefficients, G._coefficients, _field);
int[] b0 = { 0 };
int[] b1 = { 1 };
while (ComputeDegree(a1) > dg)
{
int[][] q = Divide(a0, a1, _field);
a0 = a1;
a1 = q[1];
int[] b2 = Add(b0, ModMultiply(q[0], b1, G._coefficients, _field), _field);
b0 = b1;
b1 = b2;
}
return new PolynomialGF2mSmallM[] { new PolynomialGF2mSmallM(_field, a1), new PolynomialGF2mSmallM(_field, b1) };
}
/// <summary>
/// Compute the product of this polynomial and another polynomial modulo a third polynomial
/// </summary>
///
/// <param name="A">The polynomial</param>
/// <param name="B">The reduction polynomial</param>
///
/// <returns>Returns <c>this * a mod b</c></returns>
public PolynomialGF2mSmallM ModMultiply(PolynomialGF2mSmallM A, PolynomialGF2mSmallM B)
{
int[] resultCoeff = ModMultiply(_coefficients, A._coefficients, B._coefficients, _field);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Compute the sum of this polynomial and the given polynomial
/// </summary>
///
/// <param name="Addend">he addend</param>
///
/// <returns>Return <c>this + a</c> (newly created)</returns>
public PolynomialGF2mSmallM Add(PolynomialGF2mSmallM Addend)
{
int[] resultCoeff = Add(_coefficients, Addend._coefficients, _field);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Reduce this polynomial modulo another polynomial
/// </summary>
///
/// <param name="F">The reduction polynomial</param>
///
/// <returns>Returns <c>this mod f</c></returns>
public PolynomialGF2mSmallM Mod(PolynomialGF2mSmallM F)
{
int[] resultCoeff = Mod(_coefficients, F._coefficients, _field);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}
/// <summary>
/// Compute the squaring matrix for this polynomial ring, using the base field and the reduction polynomial
/// </summary>
private void ComputeSquaringMatrix()
{
int numColumns = _poly.Degree;
_sqMatrix = new PolynomialGF2mSmallM[numColumns];
PolynomialGF2mSmallM[] _sqMatrix2 = new PolynomialGF2mSmallM[numColumns];
if (ParallelUtils.IsParallel)
{
int nct = numColumns >> 1;
Parallel.For(0, nct, i =>
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
_sqMatrix[i] = new PolynomialGF2mSmallM(_field, monomCoeffs);
});
Parallel.For(nct, numColumns, i =>
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
PolynomialGF2mSmallM monomial = new PolynomialGF2mSmallM(_field, monomCoeffs);
_sqMatrix[i] = monomial.Mod(_poly);
});
}
else
{
for (int i = 0; i < numColumns >> 1; i++)
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
_sqMatrix[i] = new PolynomialGF2mSmallM(_field, monomCoeffs);
}
for (int i = numColumns >> 1; i < numColumns; i++)
{
int[] monomCoeffs = new int[(i << 1) + 1];
monomCoeffs[i << 1] = 1;
PolynomialGF2mSmallM monomial = new PolynomialGF2mSmallM(_field, monomCoeffs);
_sqMatrix[i] = monomial.Mod(_poly);
}
}
}
/// <summary>
/// Compute the sum of this polynomial and the given polynomial
/// </summary>
///
/// <param name="Addend">he addend</param>
///
/// <returns>Return <c>this + a</c> (newly created)</returns>
public PolynomialGF2mSmallM Add(PolynomialGF2mSmallM Addend)
{
int[] resultCoeff = Add(_coefficients, Addend._coefficients, _field);
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <summary>
/// Divide this polynomial by the given polynomial
/// </summary>
/// <param name="F">The polynomial</param>
/// <returns>Returns polynomial pair = {q,r} where this = q*f+r and deg(r) < deg(f)</returns>
public PolynomialGF2mSmallM[] Divide(PolynomialGF2mSmallM F)
{
int[][] resultCoeffs = Divide(_coefficients, F._coefficients, _field);
return new PolynomialGF2mSmallM[]{
new PolynomialGF2mSmallM(_field, resultCoeffs[0]),
new PolynomialGF2mSmallM(_field, resultCoeffs[1])};
}
/// <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>
/// Swap two columns
/// </summary>
private static void SwapColumns(PolynomialGF2mSmallM[] Matrix, int First, int Second)
{
PolynomialGF2mSmallM tmp = Matrix[First];
Matrix[First] = Matrix[Second];
Matrix[Second] = tmp;
}
/// <summary>
/// Compute the result of the division of this polynomial by another polynomial modulo a third polynomial
/// </summary>
///
/// <param name="Divisor">The divisor</param>
/// <param name="Modulus">The reduction polynomial</param>
///
/// <returns>Returns <c>this * divisor^(-1) mod modulus</c></returns>
public PolynomialGF2mSmallM ModDivide(PolynomialGF2mSmallM Divisor, PolynomialGF2mSmallM Modulus)
{
int[] resultCoeff = ModDiv(_coefficients, Divisor._coefficients, Modulus._coefficients, _field);
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <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>
/// Compute the product of this polynomial and another polynomial modulo a third polynomial
/// </summary>
///
/// <param name="A">The polynomial</param>
/// <param name="B">The reduction polynomial</param>
///
/// <returns>Returns <c>this * a mod b</c></returns>
public PolynomialGF2mSmallM ModMultiply(PolynomialGF2mSmallM A, PolynomialGF2mSmallM B)
{
int[] resultCoeff = ModMultiply(_coefficients, A._coefficients, B._coefficients, _field);
return(new PolynomialGF2mSmallM(_field, resultCoeff));
}
/// <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>
/// Generate an encryption Key pair
/// </summary>
///
/// <returns>A McElieceKeyPair containing public and private keys</returns>
public IAsymmetricKeyPair GenerateKeyPair()
{
// finite field GF(2^m)
GF2mField field = new GF2mField(_M, _fieldPoly);
// irreducible Goppa polynomial
PolynomialGF2mSmallM gp = new PolynomialGF2mSmallM(field, _T, PolynomialGF2mSmallM.RANDOM_IRREDUCIBLE_POLYNOMIAL, _rngEngine);
PolynomialRingGF2m ring = new PolynomialRingGF2m(field, gp);
// matrix for computing square roots in (GF(2^m))^t
PolynomialGF2mSmallM[] qInv = ring.SquareRootMatrix;
// generate canonical check matrix
GF2Matrix h = GoppaCode.CreateCanonicalCheckMatrix(field, gp);
// compute short systematic form of check matrix
GoppaCode.MaMaPe mmp = GoppaCode.ComputeSystematicForm(h, _rngEngine);
GF2Matrix shortH = mmp.SecondMatrix;
Permutation p = mmp.Permutation;
// compute short systematic form of generator matrix
GF2Matrix shortG = (GF2Matrix)shortH.ComputeTranspose();
// obtain number of rows of G (= dimension of the code)
int k = shortG.RowCount;
// generate keys
IAsymmetricKey pubKey = new MPKCPublicKey(_N, _T, shortG);
IAsymmetricKey privKey = new MPKCPrivateKey(_N, k, field, gp, p, h, qInv);
// return key pair
return new MPKCKeyPair(pubKey, privKey);
}
/// <summary>
/// Add the given polynomial to this polynomial (overwrite this).
/// </summary>
///
/// <param name="Addend">The addend</param>
public void AddToThis(PolynomialGF2mSmallM Addend)
{
_coefficients = Add(_coefficients, Addend._coefficients, _field);
ComputeDegree();
}
private void Dispose(bool Disposing)
{
if (!_isDisposed && Disposing)
{
try
{
if (_gField != null)
{
_gField.Clear();
_gField = null;
}
if (_goppaPoly != null)
{
_goppaPoly.Clear();
_goppaPoly = null;
}
if (_H != null)
{
_H.Clear();
_H = null;
}
if (_P1 != null)
{
_P1.Clear();
_P1 = null;
}
if (_qInv != null)
{
for (int i = 0; i < _qInv.Length; i++)
{
_qInv[i].Clear();
_qInv[i] = null;
}
_qInv = null;
}
_K = 0;
_N = 0;
}
catch { }
_isDisposed = true;
}
}
/// <summary>
/// Compute the result of the division of this polynomial by another polynomial modulo a third polynomial
/// </summary>
///
/// <param name="Divisor">The divisor</param>
/// <param name="Modulus">The reduction polynomial</param>
///
/// <returns>Returns <c>this * divisor^(-1) mod modulus</c></returns>
public PolynomialGF2mSmallM ModDivide(PolynomialGF2mSmallM Divisor, PolynomialGF2mSmallM Modulus)
{
int[] resultCoeff = ModDiv(_coefficients, Divisor._coefficients, Modulus._coefficients, _field);
return new PolynomialGF2mSmallM(_field, resultCoeff);
}