/// <summary>
/// Multiplies <c>this</c> by <c>P</c> and returns the result in a new PolynomialGF2n
/// </summary>
///
/// <param name="P">The PolynomialGF2n to multiply</param>
///
/// <returns>Returns <c>this</c> * <c>P</c></returns>
public GF2nPolynomial Multiply(GF2nPolynomial P)
{
int i, j;
int aDegree = Size;
int bDegree = P.Size;
if (aDegree != bDegree)
{
throw new ArgumentException("PolynomialGF2n.Multiply: this and b must have the same size!");
}
GF2nPolynomial result = new GF2nPolynomial((aDegree << 1) - 1);
for (i = 0; i < Size; i++)
{
for (j = 0; j < P.Size; j++)
{
if (result._coeff[i + j] == null)
{
result._coeff[i + j] = (GF2nElement)_coeff[i].Multiply(P._coeff[j]);
}
else
{
result._coeff[i + j] = (GF2nElement)result._coeff[i + j].Add(_coeff[i].Multiply(P._coeff[j]));
}
}
}
return(result);
}
/// <summary>
/// Divides <c>this</c> by <c>B</c> and stores the quotient in a new PolynomialGF2n
/// </summary>
///
/// <param name="B">The divisor</param>
///
/// <returns>Returns the quotient <c>this</c> / <c>B</c></returns>
public GF2nPolynomial Quotient(GF2nPolynomial B)
{
GF2nPolynomial[] result = new GF2nPolynomial[2];
result = Divide(B);
return(result[0]);
}
/// <summary>
/// Divides <c>this</c> by <c>b</c> and stores the remainder in a new PolynomialGF2n
/// </summary>
///
/// <param name="B">The divisor</param>
///
/// <returns>Returns The remainder <c>this</c> % <c>b</c></returns>
public GF2nPolynomial Remainder(GF2nPolynomial B)
{
GF2nPolynomial[] result = new GF2nPolynomial[2];
result = Divide(B);
return(result[1]);
}
/// <summary>
/// Adds the PolynomialGF2n <c>G</c> to <c>this</c> and returns the result in a new <c>PolynomialGF2n</c>
/// </summary>
///
/// <param name="P">The <c>PolynomialGF2n</c> to add</param>
///
/// <returns>Returns <c>this + b</c></returns>
public GF2nPolynomial Add(GF2nPolynomial P)
{
GF2nPolynomial result;
if (Size >= P.Size)
{
result = new GF2nPolynomial(Size);
int i;
for (i = 0; i < P.Size; i++)
{
result._coeff[i] = (GF2nElement)_coeff[i].Add(P._coeff[i]);
}
for (; i < Size; i++)
{
result._coeff[i] = _coeff[i];
}
}
else
{
result = new GF2nPolynomial(P.Size);
int i;
for (i = 0; i < Size; i++)
{
result._coeff[i] = (GF2nElement)_coeff[i].Add(P._coeff[i]);
}
for (; i < P.Size; i++)
{
result._coeff[i] = P._coeff[i];
}
}
return(result);
}
/// <summary>
/// Creates a new PolynomialGF2n by cloning the given PolynomialGF2n <c>G</c>
/// </summary>
///
/// <param name="G">The PolynomialGF2n to clone</param>
public GF2nPolynomial(GF2nPolynomial G)
{
int i;
_coeff = new GF2nElement[G._size];
_size = G._size;
for (i = 0; i < _size; i++)
_coeff[i] = (GF2nElement)G._coeff[i].Clone();
}
/// <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>
/// Multiplies the scalar <c>E</c> to each coefficient of this PolynomialGF2n and returns the result in a new PolynomialGF2n
/// </summary>
///
/// <param name="E">The scalar to multiply</param>
///
/// <returns>Returns <c>this</c> x <c>E</c></returns>
public GF2nPolynomial ScalarMultiply(GF2nElement E)
{
GF2nPolynomial result = new GF2nPolynomial(Size);
for (int i = 0; i < Size; i++)
{
result._coeff[i] = (GF2nElement)_coeff[i].Multiply(E);
}
return(result);
}
/// <summary>
/// Creates a new PolynomialGF2n by cloning the given PolynomialGF2n <c>G</c>
/// </summary>
///
/// <param name="G">The PolynomialGF2n to clone</param>
public GF2nPolynomial(GF2nPolynomial G)
{
int i;
_coeff = new GF2nElement[G._size];
_size = G._size;
for (i = 0; i < _size; i++)
{
_coeff[i] = (GF2nElement)G._coeff[i].Clone();
}
}
/// <summary>
/// Creates a new PolynomialGF2n by cloning the given PolynomialGF2n <c>G</c>
/// </summary>
///
/// <param name="G">The PolynomialGF2n to clone</param>
public GF2nPolynomial(GF2nPolynomial G)
{
int i;
m_coeff = new GF2nElement[G.m_Size];
m_Size = G.m_Size;
for (i = 0; i < m_Size; i++)
{
m_coeff[i] = (GF2nElement)G.m_coeff[i].Clone();
}
}
/// <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>
///
/// <returns>The shifted polynomial</returns>
public GF2nPolynomial ShiftLeft(int N)
{
if (N <= 0)
{
return(new GF2nPolynomial(this));
}
GF2nPolynomial result = new GF2nPolynomial(_size + N, _coeff[0]);
result.AssignZeroToElements();
for (int i = 0; i < _size; i++)
{
result._coeff[i + N] = _coeff[i];
}
return(result);
}
/// <summary>
/// Divides <c>this</c> by <c>b</c> and stores the result in a new PolynomialGF2n[2], quotient in result[0] and remainder in result[1]
/// </summary>
///
/// <param name="B">The divisor</param>
///
/// <returns>Retuens the quotient and remainder of <c>this</c> / <c>b</c></returns>
public GF2nPolynomial[] Divide(GF2nPolynomial B)
{
GF2nPolynomial[] result = new GF2nPolynomial[2];
GF2nPolynomial a = new GF2nPolynomial(this);
a.Shrink();
GF2nPolynomial shift;
GF2nElement factor;
int bDegree = B.Degree;
GF2nElement inv = (GF2nElement)B._coeff[bDegree].Invert();
if (a.Degree < bDegree)
{
result[0] = new GF2nPolynomial(this);
result[0].AssignZeroToElements();
result[0].Shrink();
result[1] = new GF2nPolynomial(this);
result[1].Shrink();
return(result);
}
result[0] = new GF2nPolynomial(this);
result[0].AssignZeroToElements();
int i = a.Degree - bDegree;
while (i >= 0)
{
factor = (GF2nElement)a._coeff[a.Degree].Multiply(inv);
shift = B.ScalarMultiply(factor);
shift.ShiftThisLeft(i);
a = a.Add(shift);
a.Shrink();
result[0]._coeff[i] = (GF2nElement)factor.Clone();
i = a.Degree - bDegree;
}
result[1] = a;
result[0].Shrink();
return(result);
}
/// <summary>
/// Computes the greatest common divisor of <c>this</c> and <c>g</c> and returns the result in a new PolynomialGF2n
/// </summary>
///
/// <param name="G">The GF2nPolynomial</param>
///
/// <returns>Returns gcd(<c>this</c>, <c>g</c>)</returns>
public GF2nPolynomial Gcd(GF2nPolynomial G)
{
GF2nPolynomial a = new GF2nPolynomial(this);
GF2nPolynomial b = new GF2nPolynomial(G);
a.Shrink();
b.Shrink();
GF2nPolynomial c;
GF2nPolynomial result;
GF2nElement alpha;
while (!b.IsZero())
{
c = a.Remainder(b);
a = b;
b = c;
}
alpha = a._coeff[a.Degree];
result = a.ScalarMultiply((GF2nElement)alpha.Invert());
return(result);
}
/// <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 GF2nPolynomial))
{
return(false);
}
GF2nPolynomial otherPol = (GF2nPolynomial)Obj;
if (Degree != otherPol.Degree)
{
return(false);
}
for (int i = 0; i < _size; i++)
{
if (!_coeff[i].Equals(otherPol._coeff[i]))
{
return(false);
}
}
return(true);
}
/// <summary>
/// Reduces <c>this</c> by <c>G</c> and returns the result in a newPolynomialGF2n
/// </summary>
///
/// <param name="G">The modulus</param>
///
/// <returns>Returns <c>this</c> % <c>G</c></returns>
public GF2nPolynomial Reduce(GF2nPolynomial G)
{
return Remainder(G);
}
/// <summary>
/// Multiplies <c>this</c> by <c>P</c> and returns the result in a new PolynomialGF2n
/// </summary>
///
/// <param name="P">The PolynomialGF2n to multiply</param>
///
/// <returns>Returns <c>this</c> * <c>P</c></returns>
public GF2nPolynomial Multiply(GF2nPolynomial P)
{
int i, j;
int aDegree = Size;
int bDegree = P.Size;
if (aDegree != bDegree)
throw new ArgumentException("PolynomialGF2n.Multiply: this and b must have the same size!");
GF2nPolynomial result = new GF2nPolynomial((aDegree << 1) - 1);
for (i = 0; i < Size; i++)
{
for (j = 0; j < P.Size; j++)
{
if (result._coeff[i + j] == null)
result._coeff[i + j] = (GF2nElement)_coeff[i].Multiply(P._coeff[j]);
else
result._coeff[i + j] = (GF2nElement)result._coeff[i + j].Add(_coeff[i].Multiply(P._coeff[j]));
}
}
return result;
}
/// <summary>
/// Divides <c>this</c> by <c>b</c> and stores the result in a new PolynomialGF2n[2], quotient in result[0] and remainder in result[1]
/// </summary>
///
/// <param name="B">The divisor</param>
///
/// <returns>Retuens the quotient and remainder of <c>this</c> / <c>b</c></returns>
public GF2nPolynomial[] Divide(GF2nPolynomial B)
{
GF2nPolynomial[] result = new GF2nPolynomial[2];
GF2nPolynomial a = new GF2nPolynomial(this);
a.Shrink();
GF2nPolynomial shift;
GF2nElement factor;
int bDegree = B.Degree;
GF2nElement inv = (GF2nElement)B._coeff[bDegree].Invert();
if (a.Degree < bDegree)
{
result[0] = new GF2nPolynomial(this);
result[0].AssignZeroToElements();
result[0].Shrink();
result[1] = new GF2nPolynomial(this);
result[1].Shrink();
return result;
}
result[0] = new GF2nPolynomial(this);
result[0].AssignZeroToElements();
int i = a.Degree - bDegree;
while (i >= 0)
{
factor = (GF2nElement)a._coeff[a.Degree].Multiply(inv);
shift = B.ScalarMultiply(factor);
shift.ShiftThisLeft(i);
a = a.Add(shift);
a.Shrink();
result[0]._coeff[i] = (GF2nElement)factor.Clone();
i = a.Degree - bDegree;
}
result[1] = a;
result[0].Shrink();
return result;
}
/// <summary>
/// Divides <c>this</c> by <c>b</c> and stores the remainder in a new PolynomialGF2n
/// </summary>
///
/// <param name="B">The divisor</param>
///
/// <returns>Returns The remainder <c>this</c> % <c>b</c></returns>
public GF2nPolynomial Remainder(GF2nPolynomial B)
{
GF2nPolynomial[] result = new GF2nPolynomial[2];
result = Divide(B);
return result[1];
}
/// <summary>
/// Computes the greatest common divisor of <c>this</c> and <c>g</c> and returns the result in a new PolynomialGF2n
/// </summary>
///
/// <param name="G">The GF2nPolynomial</param>
///
/// <returns>Returns gcd(<c>this</c>, <c>g</c>)</returns>
public GF2nPolynomial Gcd(GF2nPolynomial G)
{
GF2nPolynomial a = new GF2nPolynomial(this);
GF2nPolynomial b = new GF2nPolynomial(G);
a.Shrink();
b.Shrink();
GF2nPolynomial c;
GF2nPolynomial result;
GF2nElement alpha;
while (!b.IsZero())
{
c = a.Remainder(b);
a = b;
b = c;
}
alpha = a._coeff[a.Degree];
result = a.ScalarMultiply((GF2nElement)alpha.Invert());
return result;
}
/// <summary>
/// Reduces <c>this</c> by <c>G</c> and returns the result in a newPolynomialGF2n
/// </summary>
///
/// <param name="G">The modulus</param>
///
/// <returns>Returns <c>this</c> % <c>G</c></returns>
public GF2nPolynomial Reduce(GF2nPolynomial G)
{
return(Remainder(G));
}
/// <summary>
/// Multiplies <c>this</c> by <c>B</c>, reduces the result by <c>G</c> and returns it in a new PolynomialGF2n
/// </summary>
///
/// <param name="B">The PolynomialGF2n to multiply</param>
/// <param name="G">The modulus</param>
///
/// <returns>Returns <c>this</c> * <c>B</c> mod <c>G</c></returns>
public GF2nPolynomial MultiplyAndReduce(GF2nPolynomial B, GF2nPolynomial G)
{
return(Multiply(B).Reduce(G));
}
/// <summary>
/// Multiplies <c>this</c> by <c>B</c>, reduces the result by <c>G</c> and returns it in a new PolynomialGF2n
/// </summary>
///
/// <param name="B">The PolynomialGF2n to multiply</param>
/// <param name="G">The modulus</param>
///
/// <returns>Returns <c>this</c> * <c>B</c> mod <c>G</c></returns>
public GF2nPolynomial MultiplyAndReduce(GF2nPolynomial B, GF2nPolynomial G)
{
return Multiply(B).Reduce(G);
}
/// <summary>
/// Multiplies the scalar <c>E</c> to each coefficient of this PolynomialGF2n and returns the result in a new PolynomialGF2n
/// </summary>
///
/// <param name="E">The scalar to multiply</param>
///
/// <returns>Returns <c>this</c> x <c>E</c></returns>
public GF2nPolynomial ScalarMultiply(GF2nElement E)
{
GF2nPolynomial result = new GF2nPolynomial(Size);
for (int i = 0; i < Size; i++)
result._coeff[i] = (GF2nElement)_coeff[i].Multiply(E);
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>
///
/// <returns>The shifted polynomial</returns>
public GF2nPolynomial ShiftLeft(int N)
{
if (N <= 0)
return new GF2nPolynomial(this);
GF2nPolynomial result = new GF2nPolynomial(_size + N, _coeff[0]);
result.AssignZeroToElements();
for (int i = 0; i < _size; i++)
result._coeff[i + N] = _coeff[i];
return result;
}
/// <summary>
/// Adds the PolynomialGF2n <c>G</c> to <c>this</c> and returns the result in a new <c>PolynomialGF2n</c>
/// </summary>
///
/// <param name="P">The <c>PolynomialGF2n</c> to add</param>
///
/// <returns>Returns <c>this + b</c></returns>
public GF2nPolynomial Add(GF2nPolynomial P)
{
GF2nPolynomial result;
if (Size >= P.Size)
{
result = new GF2nPolynomial(Size);
int i;
for (i = 0; i < P.Size; i++)
result._coeff[i] = (GF2nElement)_coeff[i].Add(P._coeff[i]);
for (; i < Size; i++)
result._coeff[i] = _coeff[i];
}
else
{
result = new GF2nPolynomial(P.Size);
int i;
for (i = 0; i < Size; i++)
result._coeff[i] = (GF2nElement)_coeff[i].Add(P._coeff[i]);
for (; i < P.Size; i++)
result._coeff[i] = P._coeff[i];
}
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>
/// Divides <c>this</c> by <c>B</c> and stores the quotient in a new PolynomialGF2n
/// </summary>
///
/// <param name="B">The divisor</param>
///
/// <returns>Returns the quotient <c>this</c> / <c>B</c></returns>
public GF2nPolynomial Quotient(GF2nPolynomial B)
{
GF2nPolynomial[] result = new GF2nPolynomial[2];
result = Divide(B);
return result[0];
}
