/// <summary>
/// Converts the given element in representation according to this field to a new element in
/// representation according to B1 using the change-of-basis matrix calculated by computeCOBMatrix.
/// </summary>
///
/// <param name="Elem">The GF2nElement to convert</param>
/// <param name="Basis">The basis to convert <c>Elem</c> to</param>
///
/// <returns>Returns <c>Elem</c> converted to a new element representation according to <c>basis</c></returns>
public GF2nElement Convert(GF2nElement Elem, GF2nField Basis)
{
if (Basis == this)
return (GF2nElement)Elem.Clone();
if (FieldPoly.Equals(Basis.FieldPoly))
return (GF2nElement)Elem.Clone();
if (DegreeN != Basis.DegreeN)
throw new Exception("GF2nField.Convert: B1 has a different degree and thus cannot be coverted to!");
int i;
GF2Polynomial[] COBMatrix;
i = Fields.IndexOf(Basis);
if (i == -1)
{
ComputeCOBMatrix(Basis);
i = Fields.IndexOf(Basis);
}
COBMatrix = (GF2Polynomial[])Matrices[i];
GF2nElement elemCopy = (GF2nElement)Elem.Clone();
if (elemCopy is GF2nONBElement)
((GF2nONBElement)elemCopy).ReverseOrder();
GF2Polynomial bs = new GF2Polynomial(DegreeN, elemCopy.ToFlexiBigInt());
bs.ExpandN(DegreeN);
GF2Polynomial result = new GF2Polynomial(DegreeN);
for (i = 0; i < DegreeN; i++)
{
if (bs.VectorMult(COBMatrix[i]))
result.SetBit(DegreeN - 1 - i);
}
if (Basis is GF2nPolynomialField)
{
return new GF2nPolynomialElement((GF2nPolynomialField)Basis, result);
}
else if (Basis is GF2nONBField)
{
GF2nONBElement res = new GF2nONBElement((GF2nONBField)Basis, result.ToFlexiBigInt());
res.ReverseOrder();
return res;
}
else
{
throw new Exception("GF2nField.convert: B1 must be an instance of GF2nPolynomialField or GF2nONBField!");
}
}
/// <summary>
/// Computes the change-of-basis matrix for basis conversion according to 1363.
/// The result is stored in the lists fields and matrices.
/// </summary>
///
/// <param name="B1">The GF2nField to convert to</param>
public override void ComputeCOBMatrix(GF2nField B1)
{
// we are in B0 here!
if (DegreeN != B1.Degree)
throw new ArgumentException("GF2nPolynomialField.computeCOBMatrix: B1 has a different degree and thus cannot be coverted to!");
if (B1 is GF2nONBField)
{
// speedup (calculation is done in PolynomialElements instead of ONB)
B1.ComputeCOBMatrix(this);
return;
}
int i, j;
GF2nElement[] gamma;
GF2nElement u;
GF2Polynomial[] COBMatrix = new GF2Polynomial[DegreeN];
for (i = 0; i < DegreeN; i++)
COBMatrix[i] = new GF2Polynomial(DegreeN);
// find Random Root
do
{
// u is in representation according to B1
u = B1.RandomRoot(FieldPoly);
}
while (u.IsZero());
// build gamma matrix by multiplying by u
if (u is GF2nONBElement)
{
gamma = new GF2nONBElement[DegreeN];
gamma[DegreeN - 1] = GF2nONBElement.One((GF2nONBField)B1);
}
else
{
gamma = new GF2nPolynomialElement[DegreeN];
gamma[DegreeN - 1] = GF2nPolynomialElement.One((GF2nPolynomialField)B1);
}
gamma[DegreeN - 2] = u;
for (i = DegreeN - 3; i >= 0; i--)
gamma[i] = (GF2nElement)gamma[i + 1].Multiply(u);
if (B1 is GF2nONBField)
{
// convert horizontal gamma matrix by vertical Bitstrings
for (i = 0; i < DegreeN; i++)
{
for (j = 0; j < DegreeN; j++)
{
// TODO remember: ONB treats its Bits in reverse order !!!
if (gamma[i].TestBit(DegreeN - j - 1))
COBMatrix[DegreeN - j - 1].SetBit(DegreeN - i - 1);
}
}
}
else
{
// convert horizontal gamma matrix by vertical Bitstrings
for (i = 0; i < DegreeN; i++)
{
for (j = 0; j < DegreeN; j++)
{
if (gamma[i].TestBit(j))
COBMatrix[DegreeN - j - 1].SetBit(DegreeN - i - 1);
}
}
}
// store field and matrix for further use
Fields.Add(B1);
Matrices.Add(COBMatrix);
// store field and inverse matrix for further use in B1
B1.Fields.Add(this);
B1.Matrices.Add(InvertMatrix(COBMatrix));
}
/// <summary>
/// Computes a random root of the given polynomial
/// </summary>
///
/// <param name="P">A polynomial</param>
///
/// <returns>A random root of the polynomial</returns>
public override GF2nElement RandomRoot(GF2Polynomial P)
{
// We are in B1!!!
GF2nPolynomial c;
GF2nPolynomial ut;
GF2nElement u;
GF2nPolynomial h;
int hDegree;
// 1. Set g(t) <- f(t)
GF2nPolynomial g = new GF2nPolynomial(P, this);
int gDegree = g.Degree;
int i;
// 2. while deg(g) > 1
while (gDegree > 1)
{
do
{
// 2.1 choose random u (element of) GF(2^m)
u = new GF2nONBElement(this, _secRand);
ut = new GF2nPolynomial(2, GF2nONBElement.Zero(this));
// 2.2 Set c(t) <- ut
ut.Set(1, u);
c = new GF2nPolynomial(ut);
// 2.3 For i from 1 to m-1 do
for (i = 1; i <= DegreeN - 1; i++)
{
// 2.3.1 c(t) <- (c(t)^2 + ut) mod g(t)
c = c.MultiplyAndReduce(c, g);
c = c.Add(ut);
}
// 2.4 set h(t) <- GCD(c(t), g(t))
h = c.Gcd(g);
// 2.5 if h(t) is constant or deg(g) = deg(h) then go to
// step 2.1
hDegree = h.Degree;
gDegree = g.Degree;
}
while ((hDegree == 0) || (hDegree == gDegree));
// 2.6 If 2deg(h) > deg(g) then set g(t) <- g(t)/h(t) ...
if ((hDegree << 1) > gDegree)
g = g.Quotient(h);
else // ... else g(t) <- h(t)
g = new GF2nPolynomial(h);
gDegree = g.Degree;
}
// 3. Output g(0)
return g.At(0);
}