/// <summary>
/// Compute the square root of this element and return the result in a new GF2nElement
/// </summary>
///
/// <returns>Returns <c>this^1/2</c> (newly created)</returns>
public override GF2nElement SquareRoot()
{
GF2nONBElement result = new GF2nONBElement(this);
result.SquareRootThis();
return(result);
}
/// <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>
/// 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)
{
GF2nONBElement result = new GF2nONBElement(this);
result.AddToThis(Addend);
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)
{
GF2nONBElement result = new GF2nONBElement(this);
result.MultiplyThisBy(Factor);
return(result);
}
/// <summary>
/// Compute <c>this</c> element + 1
/// </summary>
///
/// <returns>Returns <c>this</c> + 1</returns>
public override GF2nElement Increase()
{
GF2nONBElement result = new GF2nONBElement(this);
result.IncreaseThis();
return(result);
}
/// <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>
/// Compute the multiplicative inverse of this element
/// </summary>
///
/// <returns>Returns <c>this^-1</c> (newly created)</returns>
public override IGFElement Invert()
{
GF2nONBElement result = new GF2nONBElement(this);
result.InvertThis();
return(result);
}
/// <summary>
/// Copy the field values from another GF2nONBElement instance
/// </summary>
///
/// <param name="Gf2n">The GF2nONBElement to copy</param>
public GF2nONBElement(GF2nONBElement Gf2n)
{
m_Field = Gf2n.m_Field;
m_Degree = m_Field.Degree;
m_Length = ((GF2nONBField)m_Field).GetONBLength();
m_Bit = ((GF2nONBField)m_Field).GetONBBit();
m_Pol = new long[m_Length];
Assign(Gf2n.GetElement());
}
/// <summary>
/// Copy the field values from another GF2nONBElement instance
/// </summary>
///
/// <param name="Gf2n">The GF2nONBElement to copy</param>
public GF2nONBElement(GF2nONBElement Gf2n)
{
mField = Gf2n.mField;
mDegree = mField.Degree;
_mLength = ((GF2nONBField)mField).GetONBLength();
_mBit = ((GF2nONBField)mField).GetONBBit();
_mPol = new long[_mLength];
Assign(Gf2n.GetElement());
}
/// <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, m_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));
}
/// <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 GF2nONBElement))
{
return(false);
}
GF2nONBElement otherElem = (GF2nONBElement)Obj;
for (int i = 0; i < m_Length; i++)
{
if (m_Pol[i] != otherElem.m_Pol[i])
{
return(false);
}
}
return(true);
}
/// <summary>
/// Multiplicatively invert of this element (overwrite <c>this</c>)
/// </summary>
public void InvertThis()
{
if (IsZero())
{
throw new ArithmeticException();
}
int r = 31; // m_Degree kann nur 31 Bits lang sein!!!
// Bitlaenge von m_Degree:
for (bool found = false; !found && r >= 0; r--)
{
if (((m_Degree - 1) & _mBitmask[r]) != 0)
{
found = true;
}
}
r++;
GF2nElement m = Zero((GF2nONBField)m_Field);
GF2nElement n = new GF2nONBElement(this);
int k = 1;
for (int i = r - 1; i >= 0; i--)
{
m = (GF2nElement)n.Clone();
for (int j = 1; j <= k; j++)
{
m.SquareThis();
}
n.MultiplyThisBy(m);
k <<= 1;
if (((m_Degree - 1) & _mBitmask[i]) != 0)
{
n.SquareThis();
n.MultiplyThisBy(this);
k++;
}
}
n.SquareThis();
}
/// <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!");
}
}
/// <summary>
/// Copy the field values from another GF2nONBElement instance
/// </summary>
///
/// <param name="Gf2n">The GF2nONBElement to copy</param>
public GF2nONBElement(GF2nONBElement Gf2n)
{
mField = Gf2n.mField;
mDegree = mField.Degree;
_mLength = ((GF2nONBField)mField).GetONBLength();
_mBit = ((GF2nONBField)mField).GetONBBit();
_mPol = new long[_mLength];
Assign(Gf2n.GetElement());
}
/// <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>
/// 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)
{
GF2nONBElement result = new GF2nONBElement(this);
result.AddToThis(Addend);
return result;
}
/// <summary>
/// Multiplicatively invert of this element (overwrite <c>this</c>)
/// </summary>
public void InvertThis()
{
if (IsZero())
throw new ArithmeticException();
int r = 31; // mDegree kann nur 31 Bits lang sein!!!
// Bitlaenge von mDegree:
for (bool found = false; !found && r >= 0; r--)
{
if (((mDegree - 1) & _mBitmask[r]) != 0)
found = true;
}
r++;
GF2nElement m = Zero((GF2nONBField)mField);
GF2nElement n = new GF2nONBElement(this);
int k = 1;
for (int i = r - 1; i >= 0; i--)
{
m = (GF2nElement)n.Clone();
for (int j = 1; j <= k; j++)
m.SquareThis();
n.MultiplyThisBy(m);
k <<= 1;
if (((mDegree - 1) & _mBitmask[i]) != 0)
{
n.SquareThis();
n.MultiplyThisBy(this);
k++;
}
}
n.SquareThis();
}
/// <summary>
/// Compute the multiplicative inverse of this element
/// </summary>
///
/// <returns>Returns <c>this^-1</c> (newly created)</returns>
public override IGFElement Invert()
{
GF2nONBElement result = new GF2nONBElement(this);
result.InvertThis();
return result;
}
/// <summary>
/// Compute <c>this</c> element + 1
/// </summary>
///
/// <returns>Returns <c>this</c> + 1</returns>
public override GF2nElement Increase()
{
GF2nONBElement result = new GF2nONBElement(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)
{
GF2nONBElement result = new GF2nONBElement(this);
result.MultiplyThisBy(Factor);
return result;
}
/// <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>
/// Compute the square root of this element and return the result in a new GF2nElement
/// </summary>
///
/// <returns>Returns <c>this^1/2</c> (newly created)</returns>
public override GF2nElement SquareRoot()
{
GF2nONBElement result = new GF2nONBElement(this);
result.SquareRootThis();
return result;
}
/// <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);
}