private ulong Rho(ulong n, ulong xInit, ulong c, IReducer <ulong> reducer)
{
if ((n & 1) == 0)
{
return(2);
}
var x = reducer.ToResidue(xInit);
var y = x.Copy();
var ys = x.Copy();
var r = 1;
var m = batchSize;
var cPrime = reducer.ToResidue(c);
var one = reducer.ToResidue(1);
var diff = one.Copy();
var q = one.Copy();
var g = (ulong)1;
do
{
x.Set(y);
for (int i = 0; i < r; i++)
{
AdvanceF(y, cPrime);
}
var k = 0;
while (k < r && g == 1)
{
ys.Set(y);
var limit = Math.Min(m, r - k);
q.Set(one);
for (int i = 0; i < limit; i++)
{
AdvanceF(y, cPrime);
q.Multiply(diff.Set(x).Subtract(y));
}
g = IntegerMath.GreatestCommonDivisor(q.Value, n);
k += limit;
}
r <<= 1;
}while (g == 1);
if (g.CompareTo(n) == 0)
{
// Backtrack.
do
{
AdvanceF(ys, cPrime);
g = IntegerMath.GreatestCommonDivisor(diff.Set(x).Subtract(ys).Value, n);
}while (g == 1);
}
if (g.CompareTo(n) == 0)
{
return(0);
}
return(g);
}
private long Rho(long n, long xInit, long c)
{
if ((n & 1) == 0)
{
return(2);
}
var x = xInit;
var y = xInit;
var ys = y;
var r = 1;
var m = batchSize;
var g = (long)1;
do
{
x = y;
for (int i = 0; i < r; i++)
{
y = F(y, c, n);
}
var k = 0;
while (k < r && g == 1)
{
ys = y;
var limit = Math.Min(m, r - k);
var q = (long)1;
for (int i = 0; i < limit; i++)
{
y = F(y, c, n);
q = IntegerMath.ModularProduct(q, x - y, n);
}
g = IntegerMath.GreatestCommonDivisor(q, n);
k += limit;
}
r <<= 1;
}while (g == 1);
if (g == n)
{
do
{
ys = F(ys, c, n);
g = IntegerMath.GreatestCommonDivisor(x - ys, n);
}while (g == 1);
}
return(g);
}
public MertensFunctionWheel64(int threads)
{
this.threads = threads;
wheelSubtotal = new int[wheelSize];
wheelInclude = new bool[wheelSize >> 1];
wheelNext = new int[wheelSize >> 1];
var total = 0;
var first = -1;
for (var i = 0; i < wheelSize; i++)
{
var include = IntegerMath.GreatestCommonDivisor(i, wheelSize) == 1;
if (include)
{
if (first == -1)
{
first = i;
}
++total;
}
wheelSubtotal[i] = total;
wheelInclude[i >> 1] = include;
}
Debug.Assert(total == wheelCount);
var next = first - 1;
for (var i = (wheelSize >> 1) - 1; i >= 0; i--)
{
next += 2;
wheelNext[i] = next;
if (wheelInclude[i])
{
next = 0;
}
}
}
public override Rational GreatestCommonDivisor(Rational a, Rational b)
{
return(IntegerMath.GreatestCommonDivisor((BigInteger)a, (BigInteger)b));
}
public override uint GreatestCommonDivisor(uint a, uint b)
{
return(IntegerMath.GreatestCommonDivisor(a, b));
}
public MertensFunctionWheel(int threads)
{
this.threads = threads;
wheelSubtotal = new int[wheelSize];
wheelInclude = new bool[wheelSize >> 1];
wheelNext = new byte[wheelSize >> 1];
wheelPrev = new byte[wheelSize >> 1];
var total = 0;
for (var i = 0; i < wheelSize; i++)
{
var include = IntegerMath.GreatestCommonDivisor(i, wheelSize) == 1;
if (include)
{
++total;
}
wheelSubtotal[i] = total;
wheelInclude[i >> 1] = include;
}
Debug.Assert(total == wheelCount);
var next = 0;
for (var i = (wheelSize >> 1) - 1; i >= 0; i--)
{
next += 2;
wheelNext[i] = (byte)next;
if (wheelInclude[i])
{
next = 0;
}
}
var prev = 0;
for (var i = 0; i < (wheelSize >> 1); i++)
{
prev += 2;
wheelPrev[i] = (byte)prev;
if (wheelInclude[i])
{
prev = 0;
}
}
#if DEBUG
for (var i = 1; i < wheelSize; i += 2)
{
var skip = wheelNext[i >> 1];
for (var j = 2; j < skip; j += 2)
{
if (wheelInclude[((i + j) % wheelSize) >> 1])
{
Debugger.Break();
}
}
if (!wheelInclude[((i + skip) % wheelSize) >> 1])
{
Debugger.Break();
}
}
for (var i = 1; i < wheelSize; i += 2)
{
var skip = wheelPrev[i >> 1];
for (var j = 2; j < skip; j += 2)
{
if (wheelInclude[((i - j + wheelSize) % wheelSize) >> 1])
{
Debugger.Break();
}
}
if (!wheelInclude[((i - skip + wheelSize) % wheelSize) >> 1])
{
Debugger.Break();
}
}
#endif
wheelFactors = IntegerMath.Factors(wheelSize).ToArray();
wheelMobius = wheelFactors.Select(factor => IntegerMath.Mobius(factor)).ToArray();
}
/*-----------------------------------------------------------------------*/
uint squfof_one_cycle(squfof_data_t data, uint mult_idx,
uint num_iter, out uint factor)
{
/* Perform one unit of work for SQUFOF with one multiplier */
uint sqrtn = data.sqrtn[mult_idx];
uint cutoff = data.cutoff[mult_idx];
uint num_saved = data.num_saved[mult_idx];
uint multiplier = 2 * multipliers[mult_idx];
uint coarse_cutoff = cutoff * multiplier;
ushort[] saved = data.saved[mult_idx];
uint q0 = data.q0[mult_idx];
uint p1 = data.p1[mult_idx];
uint q1 = data.q1[mult_idx];
uint i, j;
uint p0 = 0;
uint sqrtq = 0;
ulong scaledn;
factor = 0;
for (i = 0; i < num_iter; i++)
{
uint q, bits, tmp;
/* compute (sqrtn+p1)/q1; since this is unity
* more than half the time, special case the
* division to save some effort */
tmp = sqrtn + p1 - q1;
q = 1;
if (tmp >= q1)
{
q += tmp / q1;
}
/* compute the next numerator and denominator */
p0 = q * q1 - p1;
q0 = q0 + (p1 - p0) * q;
/* In order to avoid trivial factorizations,
* save q1 values that are small in a list. Only
* values that do not contain factors of the
* multiplier must be saved; however, the GCD
* is 5x more expensive than all the rest of the
* work performed in the loop. To avoid most GCDs,
* only attempt one if q1 is less than the cutoff
* times the full multiplier
*
* If the queue overflows (very rare), signal that
* this multiplier failed and move on to another one */
if (q1 < coarse_cutoff)
{
tmp = q1 / IntegerMath.GreatestCommonDivisor(q1, multiplier);
if (tmp < cutoff)
{
if (num_saved >= QSIZE)
{
data.failed[mult_idx] = true;
return(i);
}
saved[num_saved++] = (ushort)tmp;
}
}
/* if q0 is a perfect square, then the factorization
* has probably succeeded. Most of the squareness
* tests out there require multiple divisions and
* complicated loops. We can approximate these tests
* by doing two things: testing that the number of
* trailing zeros in q0 is even, and then testing
* if q0 shifted right this many places is 1 mod 8. */
bits = 0;
tmp = q0;
while ((tmp & 1) == 0)
{
bits++;
tmp >>= 1;
}
if ((bits & 1) == 0 && ((tmp & 7) == 1))
{
/* q0 is probably a perfect square. Take the
* square root by cheating */
sqrtq = (uint)(Math.Sqrt((double)q0));
if (sqrtq * sqrtq == q0)
{
/* it *is* a perfect square. If it has
* not appeared previously in the list
* for this multiplier, then we're almost
* finished */
for (j = 0; j < num_saved; j++)
{
if (saved[j] == sqrtq)
{
break;
}
}
if (j == num_saved)
{
break;
}
}
}
/* perform the odd half of the SQUFOF cycle */
tmp = sqrtn + p0 - q0;
q = 1;
if (tmp >= q0)
{
q += tmp / q0;
}
p1 = q * q0 - p0;
q1 = q1 + (p0 - p1) * q;
if (q0 < coarse_cutoff)
{
tmp = q0 / IntegerMath.GreatestCommonDivisor(q0, multiplier);
if (tmp < cutoff)
{
if (num_saved >= QSIZE)
{
data.failed[mult_idx] = true;
return(i);
}
saved[num_saved++] = (ushort)tmp;
}
}
}
if (sqrtq == 1)
{
/* the above found a trivial factor, so this
* multiplier has failed */
data.failed[mult_idx] = true;
return(i);
}
else if (i == num_iter)
{
/* no square root found; save the parameters
* and go on to the next multiplier */
data.q0[mult_idx] = q0;
data.p1[mult_idx] = p1;
data.q1[mult_idx] = q1;
data.num_saved[mult_idx] = num_saved;
return(i);
}
/* square root found; the algorithm cannot fail now.
* Compute the inverse quadratic form and iterate */
q0 = sqrtq;
p1 = p0 + sqrtq * ((sqrtn - p0) / sqrtq);
scaledn = data.n * (ulong)multipliers[mult_idx];
q1 = (uint)((scaledn - (ulong)p1 * (ulong)p1) / (ulong)q0);
while (true)
{
uint q, tmp;
tmp = sqrtn + p1 - q1;
q = 1;
if (tmp >= q1)
{
q += tmp / q1;
}
p0 = q * q1 - p1;
q0 = q0 + (p1 - p0) * q;
if (p0 == p1)
{
q0 = q1;
break;
}
tmp = sqrtn + p0 - q0;
q = 1;
if (tmp >= q0)
{
q += tmp / q0;
}
p1 = q * q0 - p0;
q1 = q1 + (p0 - p1) * q;
if (p0 == p1)
{
break;
}
}
/* q0 is the factor of n. Remove factors that exist
* in the multiplier and save whatever's left */
q0 = q0 / IntegerMath.GreatestCommonDivisor(q0, multiplier);
factor = q0;
return(i);
}
public override ulong GreatestCommonDivisor(ulong a, ulong b)
{
return(IntegerMath.GreatestCommonDivisor(a, b));
}
public override double GreatestCommonDivisor(double a, double b)
{
return((double)IntegerMath.GreatestCommonDivisor(ToBigInteger(a), ToBigInteger(b)));
}
public override Complex GreatestCommonDivisor(Complex a, Complex b)
{
return((Complex)IntegerMath.GreatestCommonDivisor(ToBigInteger(a), ToBigInteger(b)));
}
public override BigInteger GreatestCommonDivisor(BigInteger a, BigInteger b)
{
return(IntegerMath.GreatestCommonDivisor(a, b));
}