public Polynomial FindMinimalPolynomial(Polynomial polynomial)
{
var cyclicSet = new List <PolynomialOverFiniteField>();
cyclicSet.Add(new PolynomialOverFiniteField(new Polynomial[] { (_char - 1) * polynomial, new Polynomial(new ulong[] { 1 }) }, this));
var curr = Pow(polynomial, _char);
while (curr != polynomial)
{
cyclicSet.Add(new PolynomialOverFiniteField(new Polynomial[] { (_char - 1) * curr, new Polynomial(new ulong[] { 1 }) }, this));
curr = Pow(curr, _char);
}
PolynomialOverFiniteField result = cyclicSet[0];
var deg = 1;
for (int k = 1; k < cyclicSet.Count; k++)
{
result = result * cyclicSet[k];
}
List <ulong> coefficients = new List <ulong>();
for (int i = 0; i < result.Deg + 1; i++)
{
coefficients.Add(result[i][0]);
}
return(new Polynomial(coefficients.ToArray()));
}
private List <Polynomial> FindRoots(List <Polynomial> coefficients)
{
var poly = new PolynomialOverFiniteField(coefficients.ToArray(), _field);
var points = _field.GetAllElements();
var result = new List <Polynomial>();
var i = 0;
while (result.Count != poly.Deg)
{
if (poly.Value(points[i]) == Polynomial.Zero)
{
result.Add(points[i]);
}
i++;
}
return(result);
}
public Polynomial Sqrt(Polynomial sqr, Polynomial mod, Polynomial modDerivative, BigInteger sqrtNorm)
{
BigInteger max = 0;
for (int i = 0; i <= sqr.Deg; i++)
{
if (BigInteger.Abs(sqr[i]) > max)
{
max = BigInteger.Abs(sqr[i]);
}
}
var nextPrime = new NextPrime();
var bound = max * 10;
var primes = new List <BigInteger>();
var sieve = new EratosthenesSieve();
var primesToCheck = sieve.GetPrimes(100000000, 110000000);
BigInteger bigMod = 1;
foreach (var primeToCheck in primesToCheck)
{
var irreducibilityTest = new IrreducibilityTest();
if (!irreducibilityTest.IsIrreducible(mod, primeToCheck))
{
continue;
}
bigMod *= primeToCheck;
primes.Add(primeToCheck);
if (bigMod > bound)
{
break;
}
}
var prime = 2 * primes[primes.Count - 1];
while (bigMod < bound)
{
prime = nextPrime.GetNext(prime);
var irreducibilityTest = new IrreducibilityTest();
while (!irreducibilityTest.IsIrreducible(mod, prime))
{
prime = nextPrime.GetNext(prime + 1);
}
bigMod *= prime;
primes.Add(prime);
}
var sqrts = new List <Polynomial>();
foreach (var fieldChar in primes)
{
var polyMath = new PolynomialMath(fieldChar);
var four = new Polynomial(new BigInteger[] { 4 });
Polynomial b;
var isSqr = false;
for (int i = 0; true; i++)
{
b = new Polynomial(new BigInteger[] { i });
var tmp = polyMath.ModPow(polyMath.Sub(polyMath.Mul(b, b), polyMath.Mul(four, sqr)),
(BigInteger.Pow(fieldChar, mod.Deg) - 1) / 2, mod);
isSqr = (tmp[0] == 1 || tmp[0] == -(fieldChar - 1)) && (tmp.Deg == 0);
if (!isSqr)
{
break;
}
}
var gfMath = new FiniteFieldMath(mod, fieldChar);
var modPoly = new PolynomialOverFiniteField(new Polynomial[] { sqr, b, new Polynomial(new BigInteger[] { 1 }) });
var y = new PolynomialOverFiniteField(new Polynomial[] { new Polynomial(new BigInteger[] { 0 }), new Polynomial(new BigInteger[] { 1 }) });
var sqrt = gfMath.ModPow(y, (BigInteger.Pow(fieldChar, mod.Deg) + 1) / 2, modPoly)[0];
var sqrtNormMod = sqrtNorm * polyMath.ModPow(modDerivative, (BigInteger.Pow(fieldChar, mod.Deg) - 1) / (fieldChar - 1), mod)[0];
var norm = polyMath.ModPow(sqrt, (BigInteger.Pow(fieldChar, mod.Deg) - 1) / (fieldChar - 1), mod)[0];
sqrtNormMod %= fieldChar;
if (sqrtNormMod < 0)
{
sqrtNormMod = sqrtNormMod + fieldChar;
}
if (norm < 0)
{
norm = norm + fieldChar;
}
if (norm != sqrtNormMod)
{
sqrt = polyMath.ConstMul(-1, sqrt);
}
sqrts.Add(sqrt);
}
BigInteger[] result = new BigInteger[mod.Deg];
var crt = new GarnerCrt();
for (int i = 0; i < mod.Deg; i++)
{
var coefficients = new List <BigInteger>();
for (int j = 0; j < primes.Count; j++)
{
coefficients.Add(sqrts[j][i]);
}
result[i] = crt.Calculate(coefficients, primes);
}
var resultPoly = new Polynomial(result);
for (int i = 0; i <= resultPoly.Deg; i++)
{
var coeff = BigInteger.Abs(resultPoly[i]);
if (coeff > max)
{
if (resultPoly[i] < 0)
{
resultPoly[i] += bigMod;
}
else
{
resultPoly[i] -= bigMod;
}
}
}
return(resultPoly);
}
public Polynomial Syndrome(Polynomial message, Polynomial point)
{
var word = new PolynomialOverFiniteField(message, _field);
return(word.Value(point));
}
