public void FactorizeRR(Polynomial Q, int i, Polynomial f, List <Polynomial> fList)
{
Q = NormalizeX(Q);
#if VERBOSE
Console.WriteLine("i: " + i + " - k: " + k);
#elif !MUTE
Console.Write(".");
#endif
for (int j = 0; j < Fq.q; j++)
{
FFE eval = Q.Evaluate(Fq[0], Fq[j]);
if (eval == Fq[0])
{
#if VERBOSE
Console.WriteLine(Q);
Console.WriteLine("Q(0," + Fq[j] + ") = " + Q.Evaluate(Fq[0], Fq[j]));
#endif
Polynomial otherF = f + Fq[j] * (Polynomial.GetX() ^ i);
if (i == k - 1)
{
fList.Add(otherF);
}
else
{
Polynomial nextQ = Y_To_XY_Plus_Gamma(Q, Fq[j]);
FactorizeRR(nextQ, i + 1, otherF, fList);
}
}
}
}
public FFE[] Encode(FFE[] message)
{
Polynomial f = new Polynomial(message);
FFE[] x = new FFE[n];
for (int i = 0; i < n; i++)
{
x[i] = f.Evaluate(alphas[i], FFE.DEFAULT_FIELD[0]);
}
return(x);
}
protected FiniteField(int p, int n)
{
this.p = p;
this.n = n;
this.q = (int)Math.Pow(p, n);
this.elements = new FFE[q];
for (int i = 0; i < q; i++)
{
elements[i] = new FFE(i);
}
FFE.DEFAULT_FIELD = this;
}
public FFE Evaluate(FFE xVal, FFE yVal)
{
FFE res = FFE.DEFAULT_FIELD[0];
for (int i = 0; i <= GetMaxDegX(); i++)
{
for (int j = 0; j <= GetMaxDegY(); j++)
{
res += this.values[i, j] * (xVal ^ i) * (yVal ^ j);
}
}
return(res);
}
public FFE[] Decode(double[,] RM)
{
if (RM == null)
{
throw new ArgumentNullException("RM cannot be null");
}
if (RM.GetLength(0) != n | RM.GetLength(1) != Fq.q)
{
throw new ArgumentException("RM matrix should have size n x q (" + n + " x " + Fq.q + ").");
}
int[,] M = GreedyMAA(RM);
int omega = ComputeOmega(Cost(M));
List <Polynomial> polyList = ListDecode(M, omega);
if (polyList.Count == 0)
{
//throw new Exception("Decoding failed");
return(new FFE[n]);
}
double maxProba = 0;
Polynomial best = polyList[0];
foreach (Polynomial poly in polyList)
{
double proba = 1;
for (int i = 0; i < n; i++)
{
for (int j = 0; j < Fq.q; j++)
{
if (poly.Evaluate(alphas[i], Fq[0]) == Fq[j])
{
proba *= RM[i, j];
}
}
}
if (proba > maxProba)
{
maxProba = proba;
best = poly;
}
}
#if !MUTE
Console.WriteLine("\n--== RESULT ==--\n" + best);
#endif
FFE[] x = new FFE[n];
for (int i = 0; i < n; i++)
{
x[i] = best.Evaluate(alphas[i], Fq[0]);
}
return(x);
}
public Polynomial Y_To_XY_Plus_Gamma(Polynomial Q, FFE gamma)
{
Polynomial result = new Polynomial(Q.GetMaxDegX() + Q.GetMaxDegY(), Q.GetMaxDegY());
for (int i = 0; i <= Q.GetMaxDegX(); i++)
{
for (int j = 0; j <= Q.GetMaxDegY(); j++)
{
for (int d = 0; d <= j; d++)
{
result[i + d, d] += Fq[(int)C(d, j) % Fq.p] * Q[i, j] * (gamma ^ (j - d));
}
}
}
return(result);
}
FFE[] Decoder.Decode(double[,] RM)
{
FFE[] y = new FFE[n];
for (int i = 0; i < n; i++)
{
int maxJ = 0;
double max = 0;
for (int j = 0; j < Fq.q; j++)
{
if (RM[i, j] > max)
{
max = RM[i, j];
maxJ = j;
}
}
y[i] = Fq[maxJ];
}
return(this.Decode(y));
}
public static FFE[] GetDefautLocation(FiniteField Fq, int n)
{
if (n > Fq.q)
{
throw new ArgumentException("More locations than field elements is not allowed. You must ensure n <= q.");
}
FFE[] alphas = new FFE[n];
if (n < Fq.q)
{
for (int i = 0; i < n; i++)
{
alphas[i] = Fq[i + 1];
}
}
else // n == q
{
for (int i = 0; i < n; i++)
{
alphas[i] = Fq[i];
}
}
return(alphas);
}
private Polynomial[] GetQuotientRemainder(Polynomial dividend, Polynomial divisor)
{
if (divisor == Fq[0])
{
throw new DivideByZeroException();
}
if (dividend.GetDegreeX() < divisor.GetDegreeX())
{
throw new ArgumentException("The degree of the dividend cannot be less than the degree of the divisor.");
}
Polynomial remainder = Fq[1] * dividend; // to make a "by value copy" of the dividend
Polynomial quotient = new Polynomial(0, 0);
Polynomial x = Polynomial.GetX();
int degDiv = divisor.GetDegreeX();
FFE leadingCoeficient = divisor[degDiv, 0];
for (int deg = dividend.GetDegreeX(); deg >= degDiv; deg--)
{
FFE coef = remainder[deg, 0] / leadingCoeficient;
quotient += coef * (x ^ (deg - degDiv));
remainder -= coef * divisor * (x ^ (deg - degDiv));
}
return(new Polynomial[] { quotient, remainder });
}
public override string ToString()
{
String ans = "";
FFE zero = FFE.DEFAULT_FIELD[0];
for (int j = 0; j <= GetMaxDegY(); j++)
{
for (int i = 0; i <= GetMaxDegX(); i++)
{
if (this[i, j] != zero)
{
ans += this[i, j] + (i == 0 ? "" : "x^" + i) + (j == 0 ? "" : "y^" + j) + " + ";
}
}
}
if (ans.Length > 0)
{
return(ans.Substring(0, ans.Length - 2));
}
else
{
return("0");
}
}
public Polynomial FindQ(int[,] M, int omega)
{
int L = omega / (k - 1);
Polynomial[] Q = new Polynomial[L + 1];
int[] wdeg = new int[L + 1];
for (int l = 0; l <= L; l++)
{
Q[l] = new Polynomial(omega, L);
Q[l][0, l] = Fq[1];
wdeg[l] = l * (k - 1);
}
FFE[] lambdas = new FFE[L + 1];
int lowestL;
for (int i = 0; i < alphas.Length; i++)
{
for (int j = 0; j < Fq.q; j++)
{
if (M[i, j] != 0)
{
#if VERBOSE
Console.WriteLine("i,j: " + alphas[i] + "," + Fq[j]);
#endif
for (int u = 0; u < M[i, j]; u++)
{
for (int v = 0; v < M[i, j] - u; v++)
{
#if !MUTE
#if !VERBOSE
Console.Write(".");
#endif
#endif
lowestL = -1;
for (int l = 0; l <= L; l++)
{
Polynomial Duv = HasseDerivative(Q[l], u, v);
lambdas[l] = Duv.Evaluate(alphas[i], Fq[j]);
#if VERBOSE
Console.WriteLine("Q[" + l + "](x,y) = " + Q[l] + "\n => D_" + u + "," + v + " = " + Duv + "\n => lambda (" + alphas[i] + "," + Fq[j] + ") = " + lambdas[l]);
#endif
if (lambdas[l] != Fq[0])
{
if (lowestL == -1 || wdeg[l] < wdeg[lowestL])
{
lowestL = l;
}
}
}
#if VERBOSE
Console.WriteLine("\n" + "Lowest l is: " + lowestL + "\n");
#endif
if (lowestL != -1)
{
for (int l = 0; l <= L; l++)
{
if (lambdas[l] != Fq[0] & l != lowestL)
{
Q[l] = Q[l] - (lambdas[l] / lambdas[lowestL]) * Q[lowestL];
}
}
Q[lowestL] = (Polynomial.GetX() - alphas[i]) * Q[lowestL];
wdeg[lowestL] += 1;
}
}
}
}
}
}
return(Q[ArgMin(wdeg)]);
}
private FFE[] Decode(FFE[] y)
{
Polynomial[] p_i = new Polynomial[3];
Polynomial[] v_i = new Polynomial[3] {
Fq[0], Fq[0], Fq[1]
};
Polynomial x = Polynomial.GetX();
// p_1 = prod_j (x - alpha_j)
p_i[1] = Fq[1];
for (int i = 0; i < n; i++)
{
p_i[1] *= (x - alphas[i]);
}
// p_2 = sum y_i prod (x - alpha_j)/(alpha_i - alpha_j)
p_i[2] = Fq[0];
for (int i = 0; i < n; i++)
{
Polynomial fact = y[i];
for (int j = 0; j < n; j++)
{
if (j != i)
{
fact *= Fq[1] / (alphas[i] - alphas[j]) * (x - alphas[j]);
}
}
p_i[2] += fact;
}
if (p_i[2].GetDegreeX() < k)
{
FFE[] hatx = new FFE[n];
for (int i = 0; i < n; i++)
{
hatx[i] = p_i[2].Evaluate(alphas[i], Fq[0]);
}
return(hatx);
}
// the Euclidean division begins...
Polynomial[] QR;
do
{
p_i[0] = p_i[1];
p_i[1] = p_i[2];
v_i[0] = v_i[1];
v_i[1] = v_i[2];
QR = GetQuotientRemainder(p_i[0], p_i[1]);
p_i[2] = QR[1];
v_i[2] = -Fq[1] * QR[0] * v_i[1] + v_i[0];
}while(2 * p_i[2].GetDegreeX() >= n + k);
//p_i[2] = f * v[2] + r
QR = GetQuotientRemainder(p_i[2], v_i[2]);
if (QR[1] == Fq[0])
{
// Success
FFE[] hatx = new FFE[n];
for (int i = 0; i < n; i++)
{
hatx[i] = QR[0].Evaluate(alphas[i], Fq[0]);
}
return(hatx);
}
else
{
// Failure
return(new FFE[n]);
}
}
public abstract FFE Add(FFE a, FFE b);
public abstract FFE Substract(FFE a, FFE b);
public int[] AsCoefs(FFE a)
{
return(AsCoefs(a.value));
}
public abstract FFE Power(FFE a, int exp);
public abstract FFE Opposite(FFE a);
public abstract FFE Divide(FFE a, FFE b);
public abstract FFE Multiply(FFE a, FFE b);