/// <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 InvertEEA()
{
if (IsZero())
throw new ArithmeticException();
GF2Polynomial b = new GF2Polynomial(mDegree + 32, "ONE");
b.ReduceN();
GF2Polynomial c = new GF2Polynomial(mDegree + 32);
c.ReduceN();
GF2Polynomial u = GetGF2Polynomial();
GF2Polynomial v = mField.FieldPolynomial;
GF2Polynomial h;
int j;
u.ReduceN();
while (!u.IsOne())
{
u.ReduceN();
v.ReduceN();
j = u.Length - v.Length;
if (j < 0)
{
h = u;
u = v;
v = h;
h = b;
b = c;
c = h;
j = -j;
c.ReduceN(); // this increases the performance
}
u.ShiftLeftAddThis(v, j);
b.ShiftLeftAddThis(c, j);
}
b.ReduceN();
return new GF2nPolynomialElement((GF2nPolynomialField)mField, b);
}
/// <summary>
/// Returns the remainder of <c>this</c> divided by <c>G</c> in a new GF2Polynomial
/// </summary>
///
/// <param name="G">A GF2Polynomial != 0</param>
///
/// <returns>Returns a new GF2Polynomial (<c>this</c> % <c>G</c>)</returns>
public GF2Polynomial Remainder(GF2Polynomial G)
{
// a div b = q / r
GF2Polynomial a = new GF2Polynomial(this);
GF2Polynomial b = new GF2Polynomial(G);
GF2Polynomial j;
int i;
if (b.IsZero())
throw new Exception();
a.ReduceN();
b.ReduceN();
if (a._length < b._length)
return a;
i = a._length - b._length;
while (i >= 0)
{
j = b.ShiftLeft(i);
a.SubtractFromThis(j);
a.ReduceN();
i = a._length - b._length;
}
return a;
}
/// <summary>
/// Checks if <c>this</c> is irreducible, according to IEEE P1363, A.5.5, p103.
/// <para>Note: The algorithm from IEEE P1363, A5.5 can be used to check a polynomial with coefficients in GF(2^r) for irreducibility.
/// As this class only represents polynomials with coefficients in GF(2), the algorithm is adapted to the case r=1.</para>
/// </summary>
///
/// <returns>Returns true if <c>this</c> is irreducible</returns>
public bool IsIrreducible()
{
if (IsZero())
return false;
GF2Polynomial f = new GF2Polynomial(this);
int d, i;
GF2Polynomial u, g;
GF2Polynomial dummy;
f.ReduceN();
d = f._length - 1;
u = new GF2Polynomial(f._length, "X");
for (i = 1; i <= (d >> 1); i++)
{
u.SquareThisPreCalc();
u = u.Remainder(f);
dummy = u.Add(new GF2Polynomial(32, "X"));
if (!dummy.IsZero())
{
g = f.Gcd(dummy);
if (!g.IsOne())
return false;
}
else
{
return false;
}
}
return true;
}
/// <summary>
/// Returns the absolute quotient of <c>this</c> divided by <c>G</c> in a new GF2Polynomial
/// </summary>
///
/// <param name="G">A GF2Polynomial != 0</param>
///
/// <returns>Returns a new GF2Polynomial |_ <c>this</c> / <c>G</c></returns>
public GF2Polynomial Quotient(GF2Polynomial G)
{
// a div b = q / r
GF2Polynomial q = new GF2Polynomial(_length);
GF2Polynomial a = new GF2Polynomial(this);
GF2Polynomial b = new GF2Polynomial(G);
GF2Polynomial j;
int i;
if (b.IsZero())
throw new Exception();
a.ReduceN();
b.ReduceN();
if (a._length < b._length)
return new GF2Polynomial(0);
i = a._length - b._length;
q.ExpandN(i + 1);
while (i >= 0)
{
j = b.ShiftLeft(i);
a.SubtractFromThis(j);
a.ReduceN();
q.XorBit(i);
i = a._length - b._length;
}
return q;
}
/// <summary>
/// Divides <c>this</c> by <c>G</c> and returns the quotient and remainder in a new GF2Polynomial[2], quotient in [0], remainder in [1]
/// </summary>
///
/// <param name="G">A GF2Polynomial != 0</param>
///
/// <returns>Returns a new GF2Polynomial[2] containing quotient and remainder</returns>
public GF2Polynomial[] Divide(GF2Polynomial G)
{
// a div b = q / r
GF2Polynomial[] result = new GF2Polynomial[2];
GF2Polynomial q = new GF2Polynomial(_length);
GF2Polynomial a = new GF2Polynomial(this);
GF2Polynomial b = new GF2Polynomial(G);
GF2Polynomial j;
int i;
if (b.IsZero())
throw new Exception();
a.ReduceN();
b.ReduceN();
if (a._length < b._length)
{
result[0] = new GF2Polynomial(0);
result[1] = a;
return result;
}
i = a._length - b._length;
q.ExpandN(i + 1);
while (i >= 0)
{
j = b.ShiftLeft(i);
a.SubtractFromThis(j);
a.ReduceN();
q.XorBit(i);
i = a._length - b._length;
}
result[0] = q;
result[1] = a;
return result;
}