/// <param name="s">Output sequence as polynomial</param>
/// <param name="C">Fills the given polynomial with LFSR Connection Polynomial</param>
/// <returns>Returns the Linear Complexity of given sequence</returns>
public static int BerlekampMassey(Polynomial s, out Polynomial Conn)
{
Polynomial C = new Polynomial(s.Field, 1);
Polynomial B = new Polynomial(s.Field, 1);
int N = s.Degree + 1;
int L = 0;
int m = 1;
FiniteFieldElement b = s.Field.Elements[1];
int n;
for (n = 0; n < N; n++)
{
// calculate discrepency
FiniteFieldElement d = new FiniteFieldElement();
d = s.Coefficients[n];
for (int i = 1; i <= L; i++)
{
try
{
d += C.Coefficients[i] * s.Coefficients[n - i];
}
catch { }
}
if (d.Value == 0)
{
m++;
}
else if (2 * L <= n)
{
Polynomial T = new Polynomial();
T = C;
Polynomial c1 = Polynomial.GenerateFromTerm(s.Field, d * b.Inverse, m);
C = C - (c1 * B);
L = n + 1 - L;
B = T;
b = d;
m = 1;
}
else
{
Polynomial c1 = Polynomial.GenerateFromTerm(s.Field, d * b.Inverse, m);
C = C - (c1 * B);
m++;
}
}
Conn = C;
return L;
}
private void Form1_Load(object sender, EventArgs e)
{
GF5 = new FiniteField(5);
GF56Primitive = new Polynomial(GF5, 2, 0, 1, 4, 1, 0, 1);
ExtF = new ExtensionField(GF56Primitive);
lblInfo.Text = "Current Field Info: " + ExtF.ToString();
}
public LFSR(Polynomial feedbackPolynomial, FiniteFieldElement[] initialState)
{
if (feedbackPolynomial.Field.Characteristic != initialState[0].Field.Characteristic)
throw new Exception("Both values should be defined in the same field.");
if ((initialState.Length - 1) < feedbackPolynomial.Degree)
throw new ArgumentOutOfRangeException("Initial state size must be greater then feedback polynomial degree.");
this.currentState = initialState;
this._feedbackPolynomial = feedbackPolynomial;
periodOncedenHesaplandiysa = false;
}
/// <returns>Exact division q of polynomials a and b s.t. a = b*q + r</returns>
public static Polynomial operator /(Polynomial a, Polynomial b)
{
if (a.Field.Characteristic != b.Field.Characteristic)
{
throw new Exception("Both polynomials should be defined in the same field.");
}
//if (a.Degree < b.Degree)
//{
// throw new Exception("First polynomial should have higher degree.");
//}
Polynomial result = new Polynomial(a.Field, 0);
Polynomial r = a, q = new Polynomial(a.Field, 0);
while (r.Degree >= b.Degree && b.Degree != 0)
{
q = Polynomial.GenerateFromTerm(r.Field, r._coefficients[r._coefficients.Length - 1] / b._coefficients[b._coefficients.Length - 1], r.Degree - b.Degree);
result += q;
r = r - (b * q);
}
return result;
}
public static Polynomial ModPow(Polynomial poly, int exponent, Polynomial modulus)
{
Polynomial result = new Polynomial(poly.Field, 1);
for (int i = 0; i < exponent; i++)
{
result = (result * poly) % modulus;
//if (i % 1000 == 0)
// Debug.WriteLine(i + "/" + exponent);
}
return result;
}
public static Polynomial operator *(Polynomial poly1, Polynomial poly2)
{
if (poly1.Field.Characteristic != poly2.Field.Characteristic)
{
throw new Exception("Both polynomials should be defined in the same field.");
}
Polynomial p = new Polynomial(poly1.Field, new int[poly1.Degree + poly2.Degree + 1]);
for (int i = 0; i < poly1._coefficients.Length; i++)
{
for (int j = 0; j < poly2._coefficients.Length; j++)
p._coefficients[i + j] += poly1._coefficients[i] * poly2._coefficients[j];
}
int plen = p._coefficients.Length;
for (int i = 0; i < plen - 1; i++)
{
if (p._coefficients[plen - i - 1].Value == 0)
Array.Resize(ref p._coefficients, p._coefficients.Length - 1);
else
break;
}
return p;
}
public static Polynomial operator -(Polynomial poly1, Polynomial poly2)
{
if (poly1.Field.Characteristic != poly2.Field.Characteristic)
{
throw new Exception("Both polynomials should be defined in the same field.");
}
FiniteFieldElement[] elts = poly1.Degree > poly2.Degree ? (FiniteFieldElement[])poly1.Coefficients.Clone() : (FiniteFieldElement[])poly2.Coefficients.Clone();
Polynomial p = new Polynomial(elts);
int len = Math.Min(poly1.Degree, poly2.Degree);
for (int i = 0; i <= len; i++)
p._coefficients[i] = poly1._coefficients[i] - poly2._coefficients[i];
FiniteFieldElement sifir = new FiniteFieldElement();
sifir.Field = p.Field;
sifir.Value = 0;
for (int i = len + 1; i < p._coefficients.Length; i++)
p._coefficients[i] = sifir - p._coefficients[i];
int plen = p._coefficients.Length;
for (int i = 0; i < plen - 1; i++)
{
if (p._coefficients[plen - i - 1].Value == 0)
Array.Resize(ref p._coefficients, p._coefficients.Length - 1);
else
break;
}
return p;
}
public ExtensionField(Polynomial primitivePolynomial)
{
this._baseField = primitivePolynomial.Field;
this._primitivePolynomial = primitivePolynomial;
this.rnd = new Random();
}
/// <summary>
/// Decides if the element generates the extension field with over base field with given primitive polynomial.
/// </summary>
/// <param name="primitivePolynomial">Should be in the same field with the element selected.</param>
public bool IsPrimitiveElement(Polynomial primitivePolynomial)
{
if (primitivePolynomial.Field.Characteristic != this.Field.Characteristic)
{
throw new Exception("Given primitive polynomial should be in the same field with the polynomial.");
}
int[] dv = MathTools.Divisors(Math.Pow(this.Field.Characteristic, primitivePolynomial.Degree) - 1);
for (int i = 0; i < dv.Length - 1; i++)
{
Polynomial test = Polynomial.ModPow(this, dv[i], primitivePolynomial);
if (test.IsIdentity)
return false;
}
return true;
}
/// <summary>
/// Generates a monic polynomial of given degree over given field F.
/// </summary>
public static Polynomial RandomMonic(FiniteField F, int degree)
{
Random rnd = new Random();
int[] coefs = new int[degree + 1];
for (int i = 0; i < coefs.Length - 1; i++)
coefs[i] = rnd.Next(F.Elements[0].Value, F.Elements[F.Elements.Length - 1].Value + 1);
coefs[coefs.Length - 1] = 1;
Polynomial p = new Polynomial(F, coefs);
return p;
}
public ExtensionFieldElement(ExtensionField ExtField, Polynomial ElementValue)
{
this.Value = ElementValue;
this.Field = ExtField;
}
public static Polynomial Pow(Polynomial p, int exp)
{
Polynomial result = new Polynomial(p.Field, 1);
for (int i = 0; i < exp; i++)
{
result *= p;
//if (i % 1000 == 0)
// Debug.WriteLine(i + "/" + exp);
}
return result;
}