/// <summary>
/// Shifts left <c>this</c> by <c>N</c> and stores the result in <c>this</c> PolynomialGF2n
/// </summary>
///
/// <param name="N">The amount the amount to shift the coefficients</param>
public void ShiftThisLeft(int N)
{
if (N > 0)
{
int i;
int oldSize = _size;
GF2nField f = _coeff[0].GetField();
Enlarge(_size + N);
for (i = oldSize - 1; i >= 0; i--)
{
_coeff[i + N] = _coeff[i];
}
if (_coeff[0] is GF2nPolynomialElement)
{
for (i = N - 1; i >= 0; i--)
{
_coeff[i] = GF2nPolynomialElement.Zero((GF2nPolynomialField)f);
}
}
else if (_coeff[0] is GF2nONBElement)
{
for (i = N - 1; i >= 0; i--)
{
_coeff[i] = GF2nONBElement.Zero((GF2nONBField)f);
}
}
}
}
/// <summary>
/// Enlarges the size of this PolynomialGF2n to <c>k</c> + 1
/// </summary>
///
/// <param name="K">The new maximum degree</param>
public void Enlarge(int K)
{
if (K <= _size)
{
return;
}
int i;
GF2nElement[] res = new GF2nElement[K];
Array.Copy(_coeff, 0, res, 0, _size);
GF2nField f = _coeff[0].GetField();
if (_coeff[0] is GF2nPolynomialElement)
{
for (i = _size; i < K; i++)
{
res[i] = GF2nPolynomialElement.Zero((GF2nPolynomialField)f);
}
}
else if (_coeff[0] is GF2nONBElement)
{
for (i = _size; i < K; i++)
{
res[i] = GF2nONBElement.Zero((GF2nONBField)f);
}
}
_size = K;
_coeff = res;
}
/// <summary>
/// Calculates <c>this</c> to the power of <c>K</c> and returns the result in a new GF2nPolynomialElement
/// </summary>
///
/// <param name="K">The power</param>
///
/// <returns>Returns <c>this</c>^<c>K</c> in a new GF2nPolynomialElement</returns>
public GF2nPolynomialElement Power(int K)
{
if (K == 1)
{
return(new GF2nPolynomialElement(this));
}
GF2nPolynomialElement result = GF2nPolynomialElement.One((GF2nPolynomialField)mField);
if (K == 0)
{
return(result);
}
GF2nPolynomialElement x = new GF2nPolynomialElement(this);
x.polynomial.ExpandN((x.mDegree << 1) + 32); // increase performance
x.polynomial.ReduceN();
for (int i = 0; i < mDegree; i++)
{
if ((K & (1 << i)) != 0)
{
result.MultiplyThisBy(x);
}
x.Square();
}
return(result);
}
/// <summary>
/// Squares this GF2nPolynomialElement by shifting left its Bitstring and reducing.
/// <para>This is supposed to be the slowest method. Use SquarePreCalc or SquareMatrix instead.</para>
/// </summary>
///
/// <returns></returns>
public GF2nPolynomialElement SquareBitwise()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareThisBitwise();
result.ReduceThis();
return(result);
}
/// <summary>
/// Squares this GF2nPolynomialElement using GF2nField's squaring matrix.
/// <para>This is supposed to be fast when using a polynomial (no tri- or pentanomial) as fieldpolynomial.
/// Use SquarePreCalc when using a tri- or pentanomial as fieldpolynomial instead.</para>
/// </summary>
///
/// <returns>Returns <c>this^2</c> (newly created)</returns>
public GF2nPolynomialElement SquareMatrix()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareThisMatrix();
result.ReduceThis();
return(result);
}
/// <summary>
/// Compute the sum of this element and <c>Addend</c>.
/// </summary>
///
/// <param name="Addend">The addend</param>
///
/// <returns>Returns <c>this + other</c></returns>
public override IGFElement Add(IGFElement Addend)
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.AddToThis(Addend);
return(result);
}
/// <summary>
/// Increase the element by one
/// </summary>
///
/// <returns>Returns <c>this</c> + 'one'</returns>
public override GF2nElement Increase()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.IncreaseThis();
return(result);
}
/// <summary>
/// Compute the product of this element and <c>factor</c>
/// </summary>
///
/// <param name="Factor">he factor</param>
///
/// <returns>Returns <c>this * factor</c> </returns>
public override IGFElement Multiply(IGFElement Factor)
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.MultiplyThisBy(Factor);
return(result);
}
/// <summary>
/// Compute the square root of this element and return the result in a new GF2nPolynomialElement
/// </summary>
///
/// <returns>Returns <c>this^1/2</c> (newly created)</returns>
public override GF2nElement SquareRoot()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareRootThis();
return(result);
}
/// <summary>
/// Squares this GF2nPolynomialElement by using precalculated values and reducing.
/// <para>This is supposed to de fastest when using a trinomial or pentanomial as field polynomial.
/// Use SquareMatrix when using a ordinary polynomial as field polynomial.</para>
/// </summary>
///
/// <returns></returns>
public GF2nPolynomialElement SquarePreCalc()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareThisPreCalc();
result.ReduceThis();
return(result);
}
/// <summary>
/// Compute a random root of the given GF2Polynomial
/// </summary>
///
/// <param name="G">The polynomial</param>
///
/// <returns>Returns a random root of <c>G</c></returns>
public override GF2nElement RandomRoot(GF2Polynomial G)
{
// We are in B1!!!
GF2nPolynomial c;
GF2nPolynomial ut;
GF2nElement u;
GF2nPolynomial h;
int hDegree;
// 1. Set g(t) <- f(t)
GF2nPolynomial g = new GF2nPolynomial(G, 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 GF2nPolynomialElement(this, new Random());
ut = new GF2nPolynomial(2, GF2nPolynomialElement.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
{
g = new GF2nPolynomial(h); // ... else g(t) <- h(t)
}
gDegree = g.Degree;
}
// 3. Output g(0)
return(g.At(0));
}
/// <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("GF2nField.computeCOBMatrix: B1 has a different degree and thus cannot be coverted to!");
}
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());
gamma = new GF2nPolynomialElement[DegreeN];
// build gamma matrix by squaring
gamma[0] = (GF2nElement)u.Clone();
for (i = 1; i < DegreeN; i++)
{
gamma[i] = gamma[i - 1].Square();
}
// 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);
}
}
}
Fields.Add(B1);
Matrices.Add(COBMatrix);
B1.Fields.Add(this);
B1.Matrices.Add(InvertMatrix(COBMatrix));
}
/// <summary>
/// Calculates the multiplicative inverse of <c>this</c> and returns the result in a new GF2nPolynomialElement
/// </summary>
///
/// <returns>Returns <c>this</c>^(-1)</returns>
public GF2nPolynomialElement InvertSquare()
{
GF2nPolynomialElement n;
GF2nPolynomialElement u;
int i, j, k, b;
if (IsZero())
{
throw new ArithmeticException();
}
// b = (n-1)
b = mField.Degree - 1;
// n = a
n = new GF2nPolynomialElement(this);
n.polynomial.ExpandN((mDegree << 1) + 32); // increase performance
n.polynomial.ReduceN();
// k = 1
k = 1;
// for i = (r-1) downto 0 do, r=bitlength(b)
for (i = BigMath.FloorLog(b) - 1; i >= 0; i--)
{
// u = n
u = new GF2nPolynomialElement(n);
// for j = 1 to k do
for (j = 1; j <= k; j++)
{
u.SquareThisPreCalc(); // u = u^2
}
// n = nu
n.MultiplyThisBy(u);
// k = 2k
k <<= 1;
// if b(i)==1
if ((b & _bitMask[i]) != 0)
{
// n = n^2 * b
n.SquareThisPreCalc();
n.MultiplyThisBy(this);
// k = k+1
k += 1;
}
}
// outpur n^2
n.SquareThisPreCalc();
return(n);
}
/// <summary>
/// Returns the trace of this GF2nPolynomialElement
/// </summary>
///
/// <returns>The trace of this GF2nPolynomialElement</returns>
public override int Trace()
{
GF2nPolynomialElement t = new GF2nPolynomialElement(this);
int i;
for (i = 1; i < mDegree; i++)
{
t.SquareThis();
t.AddToThis(this);
}
if (t.IsOne())
{
return(1);
}
return(0);
}
/// <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 GF2nPolynomialElement))
{
return(false);
}
GF2nPolynomialElement otherElem = (GF2nPolynomialElement)Obj;
if (mField != otherElem.mField)
{
if (!mField.FieldPolynomial.Equals(otherElem.mField.FieldPolynomial))
{
return(false);
}
}
return(polynomial.Equals(otherElem.polynomial));
}
/// <summary>
/// Solves the quadratic equation <c>z^2 + z = this</c> if such a solution exists.
/// <para>This method returns one of the two possible solutions.
/// The other solution is <c>z + 1</c>. Use z.Increase() to compute this solution.</para>
/// </summary>
///
/// <returns>Returns a GF2nPolynomialElement representing one z satisfying the equation <c>z^2 + z = this</c></returns>
public override GF2nElement SolveQuadraticEquation()
{
if (IsZero())
{
return(Zero((GF2nPolynomialField)mField));
}
if ((mDegree & 1) == 1)
{
return(HalfTrace());
}
// TODO this can be sped-up by precomputation of p and w's
GF2nPolynomialElement z, w;
do
{
// step 1.
GF2nPolynomialElement p = new GF2nPolynomialElement(
(GF2nPolynomialField)mField, new Random());
// step 2.
z = Zero((GF2nPolynomialField)mField);
w = (GF2nPolynomialElement)p.Clone();
// step 3.
for (int i = 1; i < mDegree; i++)
{
// compute z = z^2 + w^2 * this
// and w = w^2 + p
z.SquareThis();
w.SquareThis();
z.AddToThis(w.Multiply(this));
w.AddToThis(p);
}
}while (w.IsZero()); // step 4.
if (!Equals(z.Square().Add(z)))
{
throw new Exception();
}
// step 5.
return(z);
}
/// <summary>
/// Returns the half-trace of this GF2nPolynomialElement
/// </summary>
///
/// <returns>Returns a GF2nPolynomialElement representing the half-trace of this GF2nPolynomialElement</returns>
private GF2nPolynomialElement HalfTrace()
{
if ((mDegree & 0x01) == 0)
{
throw new Exception();
}
int i;
GF2nPolynomialElement h = new GF2nPolynomialElement(this);
for (i = 1; i <= ((mDegree - 1) >> 1); i++)
{
h.SquareThis();
h.SquareThis();
h.AddToThis(this);
}
return(h);
}
/// <summary>
/// Creates a new PolynomialGF2n from the given Bitstring <c>G</c> over the GF2nField <c>B1</c>
/// </summary>
///
/// <param name="G">The Bitstring to use</param>
/// <param name="B1">The field</param>
public GF2nPolynomial(GF2Polynomial G, GF2nField B1)
{
_size = B1.Degree + 1;
_coeff = new GF2nElement[_size];
int i;
if (B1 is GF2nONBField)
{
for (i = 0; i < _size; i++)
{
if (G.TestBit(i))
{
_coeff[i] = GF2nONBElement.One((GF2nONBField)B1);
}
else
{
_coeff[i] = GF2nONBElement.Zero((GF2nONBField)B1);
}
}
}
else if (B1 is GF2nPolynomialField)
{
for (i = 0; i < _size; i++)
{
if (G.TestBit(i))
{
_coeff[i] = GF2nPolynomialElement.One((GF2nPolynomialField)B1);
}
else
{
_coeff[i] = GF2nPolynomialElement.Zero((GF2nPolynomialField)B1);
}
}
}
else
{
throw new ArgumentException("GF2nPolynomial: PolynomialGF2n(Bitstring, GF2nField): B1 must be an instance of GF2nONBField or GF2nPolynomialField!");
}
}
/**
* .
*
* @param other t
*/
/// <summary>
/// Creates a new GF2nPolynomialElement by cloning the given GF2nPolynomialElement <c>Ge</c>
/// </summary>
///
/// <param name="Ge">The GF2nPolynomialElement to clone</param>
public GF2nPolynomialElement(GF2nPolynomialElement Ge)
{
mField = Ge.mField;
mDegree = Ge.mDegree;
polynomial = new GF2Polynomial(Ge.polynomial);
}
/// <summary>
/// Increase the element by one
/// </summary>
///
/// <returns>Returns <c>this</c> + 'one'</returns>
public override GF2nElement Increase()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.IncreaseThis();
return result;
}
/// <summary>
/// Squares this GF2nPolynomialElement by using precalculated values and reducing.
/// <para>This is supposed to de fastest when using a trinomial or pentanomial as field polynomial.
/// Use SquareMatrix when using a ordinary polynomial as field polynomial.</para>
/// </summary>
///
/// <returns></returns>
public GF2nPolynomialElement SquarePreCalc()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareThisPreCalc();
result.ReduceThis();
return result;
}
/// <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 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>
/// Calculates <c>this</c> to the power of <c>K</c> and returns the result in a new GF2nPolynomialElement
/// </summary>
///
/// <param name="K">The power</param>
///
/// <returns>Returns <c>this</c>^<c>K</c> in a new GF2nPolynomialElement</returns>
public GF2nPolynomialElement Power(int K)
{
if (K == 1)
return new GF2nPolynomialElement(this);
GF2nPolynomialElement result = GF2nPolynomialElement.One((GF2nPolynomialField)mField);
if (K == 0)
return result;
GF2nPolynomialElement x = new GF2nPolynomialElement(this);
x.polynomial.ExpandN((x.mDegree << 1) + 32); // increase performance
x.polynomial.ReduceN();
for (int i = 0; i < mDegree; i++)
{
if ((K & (1 << i)) != 0)
result.MultiplyThisBy(x);
x.Square();
}
return result;
}
/// <summary>
/// Solves the quadratic equation <c>z^2 + z = this</c> if such a solution exists.
/// <para>This method returns one of the two possible solutions.
/// The other solution is <c>z + 1</c>. Use z.Increase() to compute this solution.</para>
/// </summary>
///
/// <returns>Returns a GF2nPolynomialElement representing one z satisfying the equation <c>z^2 + z = this</c></returns>
public override GF2nElement SolveQuadraticEquation()
{
if (IsZero())
return Zero((GF2nPolynomialField)mField);
if ((mDegree & 1) == 1)
return HalfTrace();
// TODO this can be sped-up by precomputation of p and w's
GF2nPolynomialElement z, w;
do
{
// step 1.
GF2nPolynomialElement p = new GF2nPolynomialElement(
(GF2nPolynomialField)mField, new Random());
// step 2.
z = Zero((GF2nPolynomialField)mField);
w = (GF2nPolynomialElement)p.Clone();
// step 3.
for (int i = 1; i < mDegree; i++)
{
// compute z = z^2 + w^2 * this
// and w = w^2 + p
z.SquareThis();
w.SquareThis();
z.AddToThis(w.Multiply(this));
w.AddToThis(p);
}
}
while (w.IsZero()); // step 4.
if (!Equals(z.Square().Add(z)))
throw new Exception();
// step 5.
return z;
}
/// <summary>
/// Squares this GF2nPolynomialElement by shifting left its Bitstring and reducing.
/// <para>This is supposed to be the slowest method. Use SquarePreCalc or SquareMatrix instead.</para>
/// </summary>
///
/// <returns></returns>
public GF2nPolynomialElement SquareBitwise()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareThisBitwise();
result.ReduceThis();
return result;
}
/// <summary>
/// Squares this GF2nPolynomialElement using GF2nField's squaring matrix.
/// <para>This is supposed to be fast when using a polynomial (no tri- or pentanomial) as fieldpolynomial.
/// Use SquarePreCalc when using a tri- or pentanomial as fieldpolynomial instead.</para>
/// </summary>
///
/// <returns>Returns <c>this^2</c> (newly created)</returns>
public GF2nPolynomialElement SquareMatrix()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareThisMatrix();
result.ReduceThis();
return result;
}
/// <summary>
/// Compute the sum of this element and <c>Addend</c>.
/// </summary>
///
/// <param name="Addend">The addend</param>
///
/// <returns>Returns <c>this + other</c></returns>
public override IGFElement Add(IGFElement Addend)
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.AddToThis(Addend);
return result;
}
/**
* .
*
* @param other t
*/
/// <summary>
/// Creates a new GF2nPolynomialElement by cloning the given GF2nPolynomialElement <c>Ge</c>
/// </summary>
///
/// <param name="Ge">The GF2nPolynomialElement to clone</param>
public GF2nPolynomialElement(GF2nPolynomialElement Ge)
{
mField = Ge.mField;
mDegree = Ge.mDegree;
polynomial = new GF2Polynomial(Ge.polynomial);
}
/// <summary>
/// Returns the half-trace of this GF2nPolynomialElement
/// </summary>
///
/// <returns>Returns a GF2nPolynomialElement representing the half-trace of this GF2nPolynomialElement</returns>
private GF2nPolynomialElement HalfTrace()
{
if ((mDegree & 0x01) == 0)
throw new Exception();
int i;
GF2nPolynomialElement h = new GF2nPolynomialElement(this);
for (i = 1; i <= ((mDegree - 1) >> 1); i++)
{
h.SquareThis();
h.SquareThis();
h.AddToThis(this);
}
return h;
}
/// <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("GF2nField.computeCOBMatrix: B1 has a different degree and thus cannot be coverted to!");
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());
gamma = new GF2nPolynomialElement[DegreeN];
// build gamma matrix by squaring
gamma[0] = (GF2nElement)u.Clone();
for (i = 1; i < DegreeN; i++)
gamma[i] = gamma[i - 1].Square();
// 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);
}
}
Fields.Add(B1);
Matrices.Add(COBMatrix);
B1.Fields.Add(this);
B1.Matrices.Add(InvertMatrix(COBMatrix));
}
/// <summary>
/// Compute the product of this element and <c>factor</c>
/// </summary>
///
/// <param name="Factor">he factor</param>
///
/// <returns>Returns <c>this * factor</c> </returns>
public override IGFElement Multiply(IGFElement Factor)
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.MultiplyThisBy(Factor);
return result;
}
/// <summary>
/// Compute a random root of the given GF2Polynomial
/// </summary>
///
/// <param name="G">The polynomial</param>
///
/// <returns>Returns a random root of <c>G</c></returns>
public override GF2nElement RandomRoot(GF2Polynomial G)
{
// We are in B1!!!
GF2nPolynomial c;
GF2nPolynomial ut;
GF2nElement u;
GF2nPolynomial h;
int hDegree;
// 1. Set g(t) <- f(t)
GF2nPolynomial g = new GF2nPolynomial(G, 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 GF2nPolynomialElement(this, new Random());
ut = new GF2nPolynomial(2, GF2nPolynomialElement.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
g = new GF2nPolynomial(h); // ... else g(t) <- h(t)
gDegree = g.Degree;
}
// 3. Output g(0)
return g.At(0);
}
/// <summary>
/// Calculates the multiplicative inverse of <c>this</c> and returns the result in a new GF2nPolynomialElement
/// </summary>
///
/// <returns>Returns <c>this</c>^(-1)</returns>
public GF2nPolynomialElement InvertSquare()
{
GF2nPolynomialElement n;
GF2nPolynomialElement u;
int i, j, k, b;
if (IsZero())
throw new ArithmeticException();
// b = (n-1)
b = mField.Degree - 1;
// n = a
n = new GF2nPolynomialElement(this);
n.polynomial.ExpandN((mDegree << 1) + 32); // increase performance
n.polynomial.ReduceN();
// k = 1
k = 1;
// for i = (r-1) downto 0 do, r=bitlength(b)
for (i = BigMath.FloorLog(b) - 1; i >= 0; i--)
{
// u = n
u = new GF2nPolynomialElement(n);
// for j = 1 to k do
for (j = 1; j <= k; j++)
u.SquareThisPreCalc(); // u = u^2
// n = nu
n.MultiplyThisBy(u);
// k = 2k
k <<= 1;
// if b(i)==1
if ((b & _bitMask[i]) != 0)
{
// n = n^2 * b
n.SquareThisPreCalc();
n.MultiplyThisBy(this);
// k = k+1
k += 1;
}
}
// outpur n^2
n.SquareThisPreCalc();
return n;
}
/// <summary>
/// Returns the trace of this GF2nPolynomialElement
/// </summary>
///
/// <returns>The trace of this GF2nPolynomialElement</returns>
public override int Trace()
{
GF2nPolynomialElement t = new GF2nPolynomialElement(this);
int i;
for (i = 1; i < mDegree; i++)
{
t.SquareThis();
t.AddToThis(this);
}
if (t.IsOne())
return 1;
return 0;
}
/// <summary>
/// Compute the square root of this element and return the result in a new GF2nPolynomialElement
/// </summary>
///
/// <returns>Returns <c>this^1/2</c> (newly created)</returns>
public override GF2nElement SquareRoot()
{
GF2nPolynomialElement result = new GF2nPolynomialElement(this);
result.SquareRootThis();
return result;
}
