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 PolynomialOverFiniteField(Polynomial[] arr, Polynomial mod, BigInteger fieldChar)
{
var polyMath = new PolynomialMath(fieldChar);
_deg = arr.Length - 1;
for (int i = _deg; i >= 0; i--)
{
if (polyMath.Rem(arr[i], mod) == new Polynomial(new BigInteger[] { 0 }))
{
_deg--;
}
else
{
break;
}
}
var length = _deg + 1;
if (length == 0)
{
length++;
}
_coefficients = new Polynomial[length];
for (int i = 0; i <= _deg; i++)
{
_coefficients[i] = polyMath.Rem(arr[i], mod);
}
}
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 Polynomial Calculate(Polynomial firstArg, Polynomial secondArg, BigInteger mod)
{
if (firstArg.Deg == secondArg.Deg && firstArg.Deg == -1)
{
throw new InvalidOperationException();
}
if (firstArg.Deg == 0 || secondArg.Deg == 0)
{
return(new Polynomial(new BigInteger[] { 1 }));
}
if (firstArg.Deg == -1)
{
return(secondArg);
}
if (secondArg.Deg == -1)
{
return(firstArg);
}
if (firstArg.Deg < secondArg.Deg || secondArg.Deg == -1)
{
var tmp = secondArg;
secondArg = firstArg;
firstArg = tmp;
}
var polyMath = new PolynomialMath(mod);
firstArg = firstArg.Reduce(mod);
secondArg = secondArg.Reduce(mod);
while (secondArg.Deg != -1)
{
var tmp = firstArg;
firstArg = secondArg;
secondArg = polyMath.Rem(tmp, secondArg);
}
return(polyMath.Normalize(firstArg));
}
public FiniteFieldMath(Polynomial mod, BigInteger fieldChar)
{
Mod = mod;
FieldChar = fieldChar;
_polynomialMath = new PolynomialMath(FieldChar);
}
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);
}