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);
}
void Roots(Polynomial polynomial, long mod, List <long> roots)
{
if (polynomial.Deg < 1)
{
return;
}
if (polynomial.Deg == 2)
{
if (mod % 2 == 0)
{
for (long i = 0; i < mod; i++)
{
if (polynomial.Value(i) % mod == 0)
{
roots.Add(i);
}
}
}
else
{
var a = polynomial[2];
var b = polynomial[1];
var c = polynomial[0];
var sqrt = new ModularSqrt();
var inverse = new ModularInverse();
var inv2A = inverse.Inverse(2 * a, mod);
var tmp = inv2A * b;
var root = sqrt.Sqrt(tmp * tmp - inverse.Inverse(a, mod) * c, mod);
roots.Add((long)((-tmp + root) % mod));
roots.Add((long)((-tmp - root) % mod));
}
return;
}
if (polynomial.Deg == 1)
{
var lc = (polynomial[1] + mod) % mod;
if (lc == 1)
{
roots.Add((long)(-polynomial[0]));
}
else if (lc == mod - 1)
{
roots.Add((long)(polynomial[0]));
}
else
{
var inverse = new ModularInverse();
roots.Add((long)(-polynomial[0] * inverse.Inverse(polynomial[1], mod)));
}
return;
}
var polyMath = new PolynomialMath(mod);
var arr = new BigInteger[2];
arr[0] = _random.Next() % mod;
arr[1] = 1;
var h = new Polynomial(arr);
h = polyMath.ModPow(h, (mod - 1) / 2, polynomial);
h[0] = h[0] - 1;
var gcd = new PolynomialGcd();
var g = gcd.Calculate(polynomial, h, mod);
while (g.Deg == 0 || g == h)
{
arr[0] = _random.Next() % mod;
h = new Polynomial(arr);
h = polyMath.ModPow(h, (mod - 1) / 2, polynomial);
h[0] -= 1;
g = gcd.Calculate(polynomial, h, mod);
}
Roots(g, mod, roots);
Roots(polyMath.Div(polynomial, g), mod, roots);
}