public Tuple <Polynomial, Polynomial, Polynomial> FindBezoutCoefficients(Polynomial firstArg, Polynomial secondArg)
{
var polyMath = new PolynomialMath(-1);
Polynomial u0 = new Polynomial(new BigInteger[] { 1 });
Polynomial u1 = new Polynomial(new BigInteger[] { 0 });
Polynomial v0 = new Polynomial(new BigInteger[] { 0 });
Polynomial v1 = new Polynomial(new BigInteger[] { 1 });
Polynomial u2, v2, q0, r2;
Polynomial r0 = firstArg;
Polynomial r1 = secondArg;
while (r1.Deg != -1)
{
q0 = polyMath.Div(r0, r1);
u2 = polyMath.Sub(u0, polyMath.Mul(q0, u1));
v2 = polyMath.Sub(v0, polyMath.Mul(q0, v1));
r2 = polyMath.Add(polyMath.Mul(u2, firstArg), polyMath.Mul(v2, secondArg));
r0 = r1;
r1 = r2;
u0 = u1;
u1 = u2;
v0 = v1;
v1 = v2;
}
return(new Tuple <Polynomial, Polynomial, Polynomial>(u0, v0, r0));
}
public List <long> FindRoots(Polynomial polynomial, long mod)
{
if (mod < 100)
{
var bruteForceMethod = new BruteforceRootFinder();
return(bruteForceMethod.FindRoots(polynomial, mod));
}
var result = new List <long>();
var gcd = new PolynomialGcd();
var polyMath = new PolynomialMath(mod);
var x = new Polynomial(new BigInteger[] { 0, 1 });
var powX = polyMath.ModPow(x, mod, polynomial);
var h = polyMath.Sub(powX, x);
var g = gcd.Calculate(polynomial, h, mod);
if (g.Value(0) % mod == 0)
{
result.Add(0);
g = polyMath.Div(g, new Polynomial(new BigInteger[] { 0, 1 }));
}
Roots(g, mod, result);
return(result);
}
public bool IsIrreducible(Polynomial poly, BigInteger mod)
{
var factors = new List <int>();
var deg = poly.Deg;
if (deg % 2 == 0)
{
factors.Add(2);
}
for (int i = 3; i <= deg; i += 2)
{
if (deg % i == 0)
{
factors.Add(i);
}
}
var polyMath = new PolynomialMath(mod);
var x = new Polynomial(new BigInteger[] { 0, 1 });
var xPow = polyMath.ModPow(x, BigInteger.Pow(mod, deg), poly);
var bigPoly = polyMath.Sub(xPow, x);
if (bigPoly.Deg != -1)
{
return(false);
}
var polynomialGcd = new PolynomialGcd();
foreach (var factor in factors)
{
xPow = polyMath.ModPow(x, BigInteger.Pow(mod, deg / factor), poly);
bigPoly = polyMath.Sub(xPow, x);
var gcd = polynomialGcd.Calculate(bigPoly, poly, mod);
if (gcd.Deg != 0 || (gcd[0] + mod) % mod != 1)
{
return(false);
}
}
return(true);
}
public PolynomialOverFiniteField Div(PolynomialOverFiniteField firstArg, PolynomialOverFiniteField secondArg)
{
if (secondArg.Deg == -1)
{
throw new DivideByZeroException();
}
if (firstArg.Deg < secondArg.Deg)
{
return(new PolynomialOverFiniteField(new Polynomial[] { new Polynomial(new BigInteger[] { 0 }) }));
}
var firstDeg = firstArg.Deg;
var secondDeg = secondArg.Deg;
var newDeg = firstDeg - secondDeg;
var result = new Polynomial[newDeg + 1];
var reminder = (Polynomial[])firstArg.Coefficients.Clone();
Polynomial firstLc;
Polynomial secondLc = secondArg[secondDeg];
var inverse = new PolynomialInverse();
secondLc = inverse.Inverse(secondLc, Mod);
for (int i = 0; i <= newDeg; i++)
{
firstLc = reminder[firstDeg - i];
result[newDeg - i] = _polynomialMath.Mul(firstLc, secondLc);
for (int j = 0; j <= secondDeg; j++)
{
reminder[firstDeg - secondDeg + j - i] =
_polynomialMath.Sub(reminder[firstDeg - secondDeg + j - i], (_polynomialMath.Mul(result[newDeg - i], secondArg[j])));
}
}
return(new PolynomialOverFiniteField(result, Mod, FieldChar));
}