public IntegerPolynomial ToIntegerPolynomial()
{
IntegerPolynomial i = F1.Multiply(F2.ToIntegerPolynomial());
i.Add(F3.ToIntegerPolynomial());
return(i);
}
public IntegerPolynomial Multiply(IntegerPolynomial b)
{
IntegerPolynomial c = F1.Multiply(b);
c = F2.Multiply(c);
c.Add(F3.Multiply(b));
return(c);
}
/**
* Computes the inverse mod <code>q; q</code> must be a power of 2.<br>
* Returns <code>null</code> if the polynomial is not invertible.
*
* @param q the modulus
* @return a new polynomial
*/
public IntegerPolynomial InvertFq(int q)
{
int N = coeffs.Length;
int k = 0;
IntegerPolynomial b = new IntegerPolynomial(N + 1);
b.coeffs[0] = 1;
IntegerPolynomial c = new IntegerPolynomial(N + 1);
IntegerPolynomial f = new IntegerPolynomial(N + 1);
Array.Copy(coeffs, f.coeffs, N);
f.ModPositive(2);
// set g(x) = x^N − 1
IntegerPolynomial g = new IntegerPolynomial(N + 1);
g.coeffs[0] = 1;
g.coeffs[N] = 1;
while (true)
{
while (f.coeffs[0] == 0)
{
for (int i = 1; i <= N; i++)
{
f.coeffs[i - 1] = f.coeffs[i]; // f(x) = f(x) / x
c.coeffs[N + 1 - i] = c.coeffs[N - i]; // c(x) = c(x) * x
}
f.coeffs[N] = 0;
c.coeffs[0] = 0;
k++;
if (f.EqualsZero())
{
return(null); // not invertible
}
}
if (f.EqualsOne())
{
break;
}
if (f.Degree() < g.Degree())
{
// exchange f and g
IntegerPolynomial temp = f;
f = g;
g = temp;
// exchange b and c
temp = b;
b = c;
c = temp;
}
f.Add(g, 2);
b.Add(c, 2);
}
if (b.coeffs[N] != 0)
{
return(null);
}
// Fq(x) = x^(N-k) * b(x)
IntegerPolynomial Fq = new IntegerPolynomial(N);
int j = 0;
k %= N;
for (int i = N - 1; i >= 0; i--)
{
j = i - k;
if (j < 0)
{
j += N;
}
Fq.coeffs[j] = b.coeffs[i];
}
return(Mod2ToModq(Fq, q));
}
/**
* Computes the inverse mod 3.
* Returns <code>null</code> if the polynomial is not invertible.
*
* @return a new polynomial
*/
public IntegerPolynomial InvertF3()
{
int N = coeffs.Length;
int k = 0;
IntegerPolynomial b = new IntegerPolynomial(N + 1);
b.coeffs[0] = 1;
IntegerPolynomial c = new IntegerPolynomial(N + 1);
IntegerPolynomial f = new IntegerPolynomial(N + 1);
Array.Copy(coeffs, f.coeffs, N);
f.ModPositive(3);
// set g(x) = x^N − 1
IntegerPolynomial g = new IntegerPolynomial(N + 1);
g.coeffs[0] = -1;
g.coeffs[N] = 1;
while (true)
{
while (f.coeffs[0] == 0)
{
for (int i = 1; i <= N; i++)
{
f.coeffs[i - 1] = f.coeffs[i]; // f(x) = f(x) / x
c.coeffs[N + 1 - i] = c.coeffs[N - i]; // c(x) = c(x) * x
}
f.coeffs[N] = 0;
c.coeffs[0] = 0;
k++;
if (f.EqualsZero())
{
return(null); // not invertible
}
}
if (f.EqualsAbsOne())
{
break;
}
if (f.Degree() < g.Degree())
{
// exchange f and g
IntegerPolynomial temp = f;
f = g;
g = temp;
// exchange b and c
temp = b;
b = c;
c = temp;
}
if (f.coeffs[0] == g.coeffs[0])
{
f.Sub(g, 3);
b.Sub(c, 3);
}
else
{
f.Add(g, 3);
b.Add(c, 3);
}
}
if (b.coeffs[N] != 0)
{
return(null);
}
// Fp(x) = [+-] x^(N-k) * b(x)
IntegerPolynomial Fp = new IntegerPolynomial(N);
int j = 0;
k %= N;
for (int i = N - 1; i >= 0; i--)
{
j = i - k;
if (j < 0)
{
j += N;
}
Fp.coeffs[j] = f.coeffs[0] * b.coeffs[i];
}
Fp.EnsurePositive(3);
return(Fp);
}
/**
* Karazuba multiplication
*/
private IntegerPolynomial MultiplyRecursive(IntegerPolynomial poly2)
{
int[] a = coeffs;
int[] b = poly2.coeffs;
int n = poly2.coeffs.Length;
if (n <= 32)
{
int cn = 2 * n - 1;
IntegerPolynomial c = new IntegerPolynomial(new int[cn]);
for (int k = 0; k < cn; k++)
{
for (int i = System.Math.Max(0, k - n + 1); i <= System.Math.Min(k, n - 1); i++)
{
c.coeffs[k] += b[i] * a[k - i];
}
}
return(c);
}
else
{
int n1 = n / 2;
int[] a1temp = new int[n1];
int[] a2temp = new int[n - n1];
int[] b1temp = new int[n1];
int[] b2temp = new int[n - n1];
Array.Copy(a, a1temp, n1);
Array.Copy(a, n1, a2temp, 0, n - n1);
Array.Copy(b, b1temp, n1);
Array.Copy(b, n1, b2temp, 0, n - n1);
IntegerPolynomial a1 = new IntegerPolynomial(a1temp);
IntegerPolynomial a2 = new IntegerPolynomial(a2temp);
IntegerPolynomial b1 = new IntegerPolynomial(b1temp);
IntegerPolynomial b2 = new IntegerPolynomial(b2temp);
IntegerPolynomial A = (IntegerPolynomial)a1.Clone();
A.Add(a2);
IntegerPolynomial B = (IntegerPolynomial)b1.Clone();
B.Add(b2);
IntegerPolynomial c1 = a1.MultiplyRecursive(b1);
IntegerPolynomial c2 = a2.MultiplyRecursive(b2);
IntegerPolynomial c3 = A.MultiplyRecursive(B);
c3.Sub(c1);
c3.Sub(c2);
IntegerPolynomial c = new IntegerPolynomial(2 * n - 1);
for (int i = 0; i < c1.coeffs.Length; i++)
{
c.coeffs[i] = c1.coeffs[i];
}
for (int i = 0; i < c3.coeffs.Length; i++)
{
c.coeffs[n1 + i] += c3.coeffs[i];
}
for (int i = 0; i < c2.coeffs.Length; i++)
{
c.coeffs[2 * n1 + i] += c2.coeffs[i];
}
return(c);
}
}