public static RingDecomposeList HenselLifting(IntPolynomial f, RingDecomposeList fFactorization, List <RingPolynomial> GCDCoeffs, int liftingDegree)
{
IntPolynomial hasSquares;
IntPolynomial.GCD(f, f.Derivative(), out hasSquares);
IntPolynomial SquareFreef = (f / hasSquares).Quotient;
BigInteger squareCoeff = hasSquares[hasSquares.size - 1];
SquareFreef *= squareCoeff;
BigInteger p = RingBint.mod;
BigInteger OriginalCoeff = SquareFreef[SquareFreef.size - 1];
List <IntPolynomial> LiftingList = new List <IntPolynomial>();
for (int i = 0; i < fFactorization.CountUniq; i++)
{
LiftingList.Add(new IntPolynomial(fFactorization[i]));
}
LiftingList[0] *= OriginalCoeff;
for (int t = 1; t < liftingDegree; t++)
{
IntPolynomial multiplyFactor = new IntPolynomial {
1
};
for (int i = 0; i < LiftingList.Count; i++)
{
multiplyFactor *= new IntPolynomial(LiftingList[i]);
}
var temp = SquareFreef - multiplyFactor;
RingPolynomial d = new RingPolynomial(temp / BigInteger.Pow(p, t));
for (int i = 0; i < fFactorization.CountUniq; i++)
{
RingPolynomial CurrUniqPoly = fFactorization[i];
if (i == 0)
{
CurrUniqPoly *= fFactorization.polyCoef;
}
RingPolynomial Gc = (d * GCDCoeffs[i] / CurrUniqPoly).Reminder;
SetModContext(p, t + 1);
LiftingList[i] = LiftingList[i] + new IntPolynomial(Gc) * BigInteger.Pow(p, t);
SetModContext(p);
}
Program.Log("Разложение поднято до " + BigInteger.Pow(p, t + 1));
}
SetModContext(p, liftingDegree);
RingDecomposeList liftigRes = new RingDecomposeList();
for (int i = 0; i < LiftingList.Count; i++)
{
liftigRes.Add(new RingPolynomial(LiftingList[i]));
}
liftigRes.polyCoef = OriginalCoeff;
return(liftigRes);
}
public virtual void AddRange(RingDecomposeList polys)
{
foreach (var divisor in polys.divisors)
{
for (int i = 0; i < divisor.count; i++)
{
Add(divisor.poly);
}
}
polyCoef *= polys.polyCoef;
}
public RingDecomposeList FactorIntPolynomialOverBigModule(out RingDecomposeList fFactorization)
{
IntPolynomial f = this;
IntPolynomial hasSquares;
IntPolynomial.GCD(f, f.Derivative(), out hasSquares);
IntPolynomial SquareFreef = (f / hasSquares).Quotient;
RingDecomposeList LiftedFactorisation;
if (SquareFreef.degree > 1)
{
BigInteger mod = SelectAppropriateMod(f, SquareFreef);
RingPolynomial.SetModContext(mod);
RingPolynomial fRing = new RingPolynomial(f);
fFactorization = fRing.BerlekampFactor();
List <RingPolynomial> GCDCoeffs = RingPolynomial.GetGCDCoefficientForHensel(fFactorization);
LiftedFactorisation = RingPolynomial.HenselLiftingUntilTheEnd(f, fFactorization, GCDCoeffs);
LiftedFactorisation.polyCoef = f[f.size - 1];
}
else
{
RingPolynomial.SetModContext(5);
// f состоит из кратных множителей первой степени
LiftedFactorisation = new RingDecomposeList();
BigInteger coeff = SquareFreef.CoeffGCD();
IntPolynomial Multiplier = SquareFreef / coeff;
// находим наибольшее число в f, чтобы подобрать модуль
BigInteger biggestCoeff = f[0];
for (int i = 0; i < f.size; i++)
{
if (f[i] > biggestCoeff)
{
biggestCoeff = f[i];
}
}
BigInteger mod = getNextPrime(biggestCoeff);
RingPolynomial.SetModContext(mod);
for (int i = 0; i < f.size - 1; i++)
{
LiftedFactorisation.Add(new RingPolynomial(Multiplier));
}
LiftedFactorisation.polyCoef *= (RingBint)coeff;
fFactorization = LiftedFactorisation;
}
return(LiftedFactorisation);
}
static RingDecomposeList BerlekampDecompose(RingPolynomial poly)
{
RingDecomposeList res = new RingDecomposeList {
poly
};
Program.Log("Поиск " + (_BerlekampDecomposingPoly.Length + 1) + " множителей с помощью разлагающих многочленов для:");
poly.Print('c');
Program.Log("Найденые множители:");
for (int m = 0; m < res.CountUniq; m++)
{
for (int i = 0; i < _BerlekampDecomposingPoly.Length; i++)
{
RingPolynomial oneDecomposePoly = new RingPolynomial(_BerlekampDecomposingPoly[i]);
for (int j = 0; j < RingBint.mod; j++)
{
RingPolynomial gcd_result;
GCD(res[m], oneDecomposePoly, out gcd_result);
gcd_result *= (new RingBint(1) / gcd_result[gcd_result.degree]);
if (gcd_result.degree > 0)
{
if (gcd_result == res[m])
{
break;
}
else
{
res.Add(gcd_result);
res[m] = (res[m] / gcd_result).Quotient;
if (res.CountUniq == _BerlekampDecomposingPoly.Length + 1)
{
for (; m < res.CountUniq; m++)
{
res[m].Print('g');
}
Program.Log("На данный момент найдены все множители (" + res.CountUniq + "). Выход.");
return(res);
}
}
}
oneDecomposePoly[0] += 1;
}
}
res[m].Print('g');
}
return(res);
}
static void PrintDecompositionResult(RingDecomposeList factorRes)
{
if (Program.LogEnabled)
{
for (int i = 0; i < factorRes.CountUniq; i++)
{
factorRes[i].Print('g');
if (factorRes.divisors[i].count > 1)
{
Program.Log("\tКратность множителя: " + factorRes.divisors[i].count);
}
}
}
}
public RingDecomposeList(RingDecomposeList list)
{
polyCoef = list.polyCoef;
divisors = new List <Divisor>(list.divisors);
}
public static RingDecomposeList HenselLiftingUntilTheEnd(IntPolynomial f, RingDecomposeList fFactorization, List <RingPolynomial> GCDCoeffs)
{
IntPolynomial hasSquares;
IntPolynomial.GCD(f, f.Derivative(), out hasSquares);
IntPolynomial SquareFreef = (f / hasSquares).Quotient;
BigInteger squareCoeff = hasSquares[hasSquares.size - 1];
SquareFreef *= squareCoeff;
BigInteger p = RingBint.mod;
BigInteger OriginalCoeff = f[f.size - 1];
BigInteger biggestCoeff = SquareFreef[0];
for (int i = 1; i < SquareFreef.size; i++)
{
if (SquareFreef[i] > biggestCoeff)
{
biggestCoeff = SquareFreef[i];
}
}
List <IntPolynomial> LiftingList = new List <IntPolynomial>();
for (int i = 0; i < fFactorization.CountUniq; i++)
{
LiftingList.Add(new IntPolynomial(fFactorization[i]));
}
LiftingList[0] *= OriginalCoeff;
bool Decomposed = false;
int t = 1;
BigInteger currMod = p;
while (!Decomposed)
{
currMod = BigInteger.Pow(p, t + 1);
IntPolynomial multiplyFactor = new IntPolynomial {
1
};
for (int i = 0; i < LiftingList.Count; i++)
{
multiplyFactor *= LiftingList[i];
}
var temp = SquareFreef - multiplyFactor;
RingPolynomial d = new RingPolynomial(temp / BigInteger.Pow(p, t));
for (int i = 0; i < fFactorization.CountUniq; i++)
{
RingPolynomial CurrUniqPoly = new RingPolynomial(fFactorization[i]);
if (i == 0)
{
CurrUniqPoly *= fFactorization.polyCoef;
}
RingPolynomial Gc = ((d * GCDCoeffs[i]) / CurrUniqPoly).Reminder;
SetModContext(p, t + 1);
LiftingList[i] = LiftingList[i] + new IntPolynomial(Gc) * BigInteger.Pow(p, t);
SetModContext(p);
}
t++;
Program.Log("Разложение поднято до " + currMod);
if (currMod > biggestCoeff)
{
SetModContext(p, t);
RingPolynomial res = new RingPolynomial {
1
};
RingPolynomial FirstWithoutCoeff = new RingPolynomial(LiftingList[0]) / LiftingList[0][LiftingList[0].degree];
for (int u = 0; u < fFactorization.CountUniq; u++)
{
for (int j = 0; j < fFactorization.divisors[u].count; j++)
{
if (j != 0 && u == 0)
{
res *= FirstWithoutCoeff;
}
else
{
res *= new RingPolynomial(LiftingList[u]);
}
}
}
IntPolynomial resInt = new IntPolynomial(res);
if (f == resInt)
{
Decomposed = true;
}
}
}
SetModContext(p, t);
RingDecomposeList liftigRes = new RingDecomposeList();
for (int i = 0; i < LiftingList.Count; i++)
{
liftigRes.Add(new RingPolynomial(LiftingList[i]));
}
liftigRes.polyCoef = OriginalCoeff;
return(liftigRes);
}
public static List <RingPolynomial> GetGCDCoefficientForHensel(RingDecomposeList fDecompose)
{
List <RingPolynomial> factorsOfCoeff = new List <RingPolynomial>();
List <RingPolynomial> resultCoeff = new List <RingPolynomial>();
// изначально задаем все значения коэффициентов единичными
// и множителей разложения тоже
for (int i = 0; i < fDecompose.CountUniq; i++)
{
resultCoeff.Add(new RingPolynomial {
1
});
factorsOfCoeff.Add(new RingPolynomial {
1
});
}
// запоминаем все сомножители искомого разложения
for (int i = 0; i < fDecompose.CountUniq; i++)
{
for (int j = 0; j < fDecompose.CountUniq; j++)
{
if (i != j)
{
factorsOfCoeff[i] *= fDecompose[j];
}
}
if (i != 0)
{
// считаем, что первый множитель разложения содержит степень исходного многочлена
factorsOfCoeff[i] *= fDecompose.polyCoef;
}
}
RingPolynomial currentGCD = new RingPolynomial();
for (int i = 0; i < fDecompose.CountUniq - 1; i++)
{
GCDResult currCoefficient = GCD(factorsOfCoeff[i], factorsOfCoeff[i + 1], out currentGCD);
// раскрываем последовательно GCD, ища GCD соседних множителей [gcd(a, b, c) = gcd(gcd(a, b), c) = gcd(currentGCD, c)]
factorsOfCoeff[i + 1] = currentGCD;
for (int j = 0; j < i + 1; j++)
{
resultCoeff[j] *= currCoefficient.Coef1;
}
resultCoeff[i + 1] *= currCoefficient.Coef2;
}
// Заменяем коэффициенты их остатками от деления на i элемент факторизации (deg(resultCoeff[i]) < deg(fDecompose[i]))
for (int i = 0; i < fDecompose.CountUniq; i++)
{
RingPolynomial currFact = new RingPolynomial(fDecompose[i]);
if (i == 0)
{
currFact *= fDecompose.polyCoef;
}
resultCoeff[i] = (resultCoeff[i] / currFact).Reminder;
for (int j = 0; j < resultCoeff[i].size; j++)
{
resultCoeff[i][j] /= currentGCD[currentGCD.degree]; // нормализуем
}
}
return(resultCoeff);
}
public RingDecomposeList BerlekampFactor()
{
Program.recDepth++;
RingPolynomial poly = this.Normalize();
if (poly.degree < 2) // если многочлен меньше второй степени, то и раскладывать смысла нет
{
Program.Log("Разложение для " + poly + " не требуется.");
//end
Program.recDepth--;
return(new RingDecomposeList {
poly
});
}
Program.Log("Поиск разложения для многочлена:");
if (Program.LogEnabled)
{
poly.Print('a');
}
RingDecomposeList factorRes = new RingDecomposeList(); // результат - список делителей
factorRes.polyCoef = poly[poly.degree];
poly *= 1 / factorRes.polyCoef;
// 1) убираем кратные множители
bool recheckPower = false;
RingPolynomial multipleFators, newFactors;
GCD(poly, poly.Derivative(), out multipleFators);
if (multipleFators.degree > 0)
{
//1.1) Обнаружены кратные множители. Разделение исходного многочлена a(x) на b(x) * c(x)
// - c(x) гарантированно не содержит кратных корней
// - b(x) - это оставшаяся часть обрабатывается пробным делением в конце алгоритма
multipleFators *= (new RingBint(1) / multipleFators[multipleFators.degree]);
Program.Log("Обнаружены кратные множители. b(x) = " + multipleFators);
poly = (poly / multipleFators).Quotient; //текущий многочлен теперь c(x)
recheckPower = true;
}
newFactors = multipleFators;
multipleFators = new RingPolynomial();
while (multipleFators != newFactors && newFactors.degree > 0)
{
multipleFators = newFactors;
GCD(multipleFators, multipleFators.Derivative(), out newFactors);
}
if (multipleFators == newFactors)
{
// 1.2) особый случай, когда производная равна 0. Тогда f(x) = g(x)^p, но тогда f(x) = g(x^p) - воспользуемся этим
Program.Log("Обнаружена общая степень кратная модулю поля (" + RingBint.mod + ")");
RingPolynomial g = new RingPolynomial((newFactors.size - 1) / (int)RingBint.mod + 1);
for (int i = 0; i < g.size; i++)
{
g[i] = newFactors[i * (int)RingBint.mod];
}
g *= 1 / g[g.degree];
RingDecomposeList gFactorRes = g.BerlekampFactor();
factorRes.AddRange(gFactorRes);
recheckPower = true;
}
// 2) Обработка c(x)
Program.Log("Разложение многочлена без кратных корней c(x) = " + poly);
if (poly.degree < 2) // если многочлен меньше второй степени, то и раскладывать смысла нет
{
Program.Log("Разложение для " + poly + " не требуется.");
if (poly.degree == 1)
{
factorRes.Add(poly);
}
else
{
factorRes.polyCoef *= poly[0];
}
//end
if (recheckPower)
{
for (int i = 0; i < factorRes.CountUniq; i++)
{
RingPolynomial divisor = factorRes.divisors[i].poly;
int count = 1;
RingPolynomial temp = divisor;
while (true)
{
temp *= divisor;
if ((this / temp).Reminder.IsNull())
{
count++;
}
else
{
break;
}
}
while (count > factorRes.divisors[i].count)
{
factorRes.Add(factorRes.divisors[i].poly);
}
}
}
Program.Log("Общее разложение на данном этапе: ");
PrintDecompositionResult(factorRes);
Program.recDepth--;
return(factorRes);
}
// 2.1) Построение матрицы и нахождение ФСР ОСЛУ
RingBint[,] matrix = new RingBint[poly.degree, poly.degree - 1]; // транспонированное заполнение матрицы
RingBint[][] decomposingPoly;
for (int i = 1; i < poly.degree; i++)
{
RingPolynomial edinica = new RingPolynomial()
{
[i * (int)RingBint.mod] = 1
};
var reminder = (edinica / poly).Reminder;
for (int j = 0; j < poly.degree; j++)
{
matrix[j, i - 1] = reminder[j];
}
matrix[i, i - 1] -= 1;
}
decomposingPoly = RingMatrix.SolveHSLE(matrix);
// 2.2) рекурсивное разложение с помощью разлазающих многочленов (в статической переменной)
if (decomposingPoly.Length > 0)
{
_BerlekampDecomposingPoly = decomposingPoly;
RingBint leadingCoeff = poly[poly.degree];
poly *= 1 / (RingBint)leadingCoeff; //normalize senior coeff
var decomposeResult = BerlekampDecompose(poly);
factorRes.AddRange(decomposeResult);
factorRes.polyCoef *= leadingCoeff;
}
else
{
//end
Program.Log("Разложение для " + poly + " не требуется. Многочлен неприводимый.");
factorRes.Add(poly);
if (recheckPower)
{
for (int i = 0; i < factorRes.CountUniq; i++)
{
RingPolynomial divisor = factorRes.divisors[i].poly;
int count = 1;
RingPolynomial temp = divisor;
while (true)
{
temp *= divisor;
if ((this / temp).Reminder.IsNull())
{
count++;
}
else
{
break;
}
}
while (count > factorRes.divisors[i].count)
{
factorRes.Add(factorRes.divisors[i].poly);
}
}
}
Program.Log("Общее разложение на данном этапе: ");
PrintDecompositionResult(factorRes);
Program.recDepth--;
return(factorRes);
}
//end
if (recheckPower)
{
for (int i = 0; i < factorRes.CountUniq; i++)
{
RingPolynomial divisor = factorRes.divisors[i].poly;
int count = 1;
RingPolynomial temp = divisor;
while (true)
{
temp *= divisor;
if ((this / temp).Reminder.IsNull())
{
count++;
}
else
{
break;
}
}
while (count > factorRes.divisors[i].count)
{
factorRes.Add(factorRes.divisors[i].poly);
}
}
}
Program.Log("Общее разложение на данном этапе: ");
PrintDecompositionResult(factorRes);
Program.recDepth--;
return(factorRes);
}
