public static BigInteger Run(BigInteger N, int B, int M)
{
// Find basis for sieving
var primes = nt.Primes(B).Where(p => nt.IsQuadResidue(p, N)).ToList();
var sqN = N.Sqrt();
Func<int, BigInteger> Q = x => (x + sqN) * (x + sqN) - N;
// use quadsieve on V
BigInteger[] V = Enumerable.Range(0, M).Select(x => Q(x)).ToArray();
int[,] coeff; // coeff[i,j] : Q(i) = Prod p_j ^e_ij
Sieve(N, primes, ref V, out coeff);
int K = primes.Count;
int L = V.Count(w => w.IsOne);
int[] indices = new int[L]; // indices[i] = x_i, Q(x_i) : B-smooth
int[,] vectors = new int[K, L];
for(int i=0, cnt=0; i<V.Length && cnt < L; i++)
{
if (!V[i].IsOne) continue;
for(int j=0; j<K; j++)
{
indices[cnt] = i;
vectors[j, cnt] = coeff[i, j] % 2; // transposed
}
cnt++;
}
var exponent = GaussianElimination(vectors);
BigInteger ans1, ans2;
ans1 = ans2 = BigInteger.One;
for (int i = 0; i < exponent.Length; i++)
{
if (exponent[i] == 0) continue;
ans1 *= (indices[i] + sqN);
ans2 *= Q(indices[i]);
}
ans2 = ans2.Sqrt();
var a = BigInteger.GreatestCommonDivisor(ans1 - ans2, N);
if (a.IsOne)
a = BigInteger.GreatestCommonDivisor(ans1 + ans2, N);
return a;
}
/// <summary> v cQ1
/// Initializes a SievingRequest based on given parameters, and finds the quadratic residues for primes specified
/// </summary>
/// <param name="N">The number to factor</param>
/// <param name="B">The limit for smooth numbers</param>
/// <param name="f">The polynomial to sieve</param>
/// <param name="sievereq">The SieveRequest that will be initialized</param>
public static void InitSievingRequest(BigInteger N, int B, PolynomialFunction f, SieveRequest sievereq)
{
sievereq.AStart = (int)N.Sqrt() + 1;
sievereq.StartIdx = 0;
sievereq.polyFunction = f;
sievereq.L = primeSupply[B];
SievingData sievedat = new SievingData();
EvaluatePoly(sievereq, sievedat);
List<List<int>> tmpPrimeStarts = new List<List<int>>();
List<int> tmpPrimeIntervals = new List<int>();
for (int pI = 0; pI < B; pI++)
{
int p = primeSupply[pI];
List<int> tmp = new List<int>();
for (int a = 0; a < p; a++)
{
if (sievedat.V[a] % p == 0)
{
tmp.Add(a);
}
}
if (tmp.Count > 0)
{
tmpPrimeIntervals.Add(p);
tmpPrimeStarts.Add(tmp);
}
}
sievereq.PrimeIntervals = tmpPrimeIntervals.ToArray();
sievereq.PrimeStarts = tmpPrimeStarts.ToArray();
sievereq.B = sievereq.PrimeIntervals.Length;
}
private static void Sieve(BigInteger N, List<int> primes, ref BigInteger[] V, out int[,] coeff)
{
var sqN = N.Sqrt();
int M = V.Length;
int K = primes.Count;
// exponent vectors
coeff = new int[M, K];
for (int k = 0; k < K; k++)
{
int p = primes[k];
int a;
if (p == 2)
{
a = (sqN.IsEven ^ N.IsEven) ? 1 : 0;
for (int i = a; i < M; i += 2)
{
if (!V[i].IsEven) throw new ArithmeticException();
V[i] /= 2;
coeff[i, k]++;
}
continue;
}
int sr = nt.ModSqrt(N, p);
a = (p+sr - sqN.Mod(p)) % p;
int b = (2*p - sr - sqN.Mod(p)) % p;
for (int i = a; i < M; i += p)
{
if (V[i] % p != 0) throw new ArithmeticException();
V[i] /= p;
coeff[i, k]++;
}
for (int i = b; i < M; i += p)
{
if (V[i] % p != 0) throw new ArithmeticException();
V[i] /= p;
coeff[i, k]++;
}
}
}
/// <summary>
/// Checks whether a number is a prime number or not.
/// </summary>
/// <param name="n">The number to test.</param>
/// <param name="checkCache">Whether to check cache for primality.</param>
/// <returns>Whether the number is prime.</returns>
public static bool IsPrime(BigInteger n, bool checkCache = true)
{
if (n <= 1) return false;
if (checkCache && n <= BagMath.LargestSmallPrime) return Array.BinarySearch<int>(BagMath.SmallPrimes, (int)n) >= 0;
if (n < 4) return true;
if (n % 2 == 0) return false;
if (n < 9) return true;
if (n % 3 == 0) return false;
BigInteger r = n.Sqrt(),
f = 5;
while (f <= r)
{
if (n % f == 0) return false;
if (n % (f + 2) == 0) return false;
f += 6;
}
return true;
}
public static void Test_Sqrt_Neg1throwsException()
{
var neg1 = new BigInteger(-1);
var sqrt = neg1.Sqrt();
}