Brent's Method

Brent's improvement (in 1980) to Pollard's rho method is to extend the core idea by generalizing the cycle by powers of two.
Full details here, with due reference to Floyd -
http://web.comlab.ox.ac.uk/oucl/work/richard.brent/pd/rpb051i.pdf

The following from Wikipedia:
Input: n, the integer to be factored; x0, such that 0 ≤ x0 ≤ n; m such that m > 0; and f(x), a pseudo-random function modulo n.
Output: a non-trivial factor of n, or failure.
1. y ← x0, r ← 1, q ← 1.
2. Do:
1. x ← y
2. For i = 1 To r:
1. y ← f(y)
3. k ← 0
4. Do:
1. ys ← y
2. For i = 1 To min(m, r − k):
1. y ← f(y)
2. q ← (q × |x − y|) mod n
3. g ← GCD(q, n)
4. k ← k + m
5. Until (k ≥ r or g > 1)
6. r ← 2r
3. Until g > 1
4. If g = n then
1. Do:
1. ys ← f(ys)
2. g ← GCD(|x − ys|, n)
2. Until g > 1
5. If g = n then return failure, else return g

Also, the pseudocode immediately below is taken from a PD document by Connelly Barnes of Oregon State University
http://oregonstate.edu/~barnesc/documents/factoring.pdf

function brentFactor(N)
# Initial values x(i) and x(m) for i = 0.
xi := 2
xm := 2
for i from 1 to infinity
# Find x(i) from x(i-1).
xi := (xi ^ 2 + 1) % N
s := gcd(xi - xm, N)
if s <> 1 and s <> N then
return s, N/s
end if
if integralPowerOf2(i) then
xm := xi
end if
end do
end function

Here is its Java implementation from superfac9:

BigInteger factorizebrent(BigInteger n) {
BigInteger k = new BigInteger("1");
BigInteger r = new BigInteger("1");
BigInteger i = new BigInteger("1");
BigInteger m = new BigInteger("1");
BigInteger iter = new BigInteger("1");
BigInteger z = new BigInteger("1");
BigInteger x = new BigInteger("1");
BigInteger y = new BigInteger("1");
BigInteger q = new BigInteger("1");
BigInteger ys = new BigInteger("1");

m=TEN;
r=ONE;
iter=ZERO;
z=ZERO;
y=z;
q=ONE;

do {
x=y;
for (i=ONE;i.compareTo(r)<=0;i=i.add(ONE)) y=((y.multiply(y)).add(THREE)).mod(n);
k=ZERO;
do {
iter=iter.add(ONE);
// System.out.print("iter=" + iter.toString() + '\r');
ys=y;
for (i=ONE;i.compareTo(mr_min(m,r.subtract(k)))<=0;i=i.add(ONE)) {
y=((y.multiply(y)).add(THREE)).mod(n);
q=((y.subtract(x)).multiply(q)).mod(n);
}
z=n.gcd(q);
k=k.add(m);
} while (k.compareTo(r)<0 && z.compareTo(ONE)==0);
r=r.multiply(TWO);
} while (z.compareTo(ONE)==0 && iter.compareTo(TENTHOUSAND)<0);

if (z.compareTo(n)==0) do {
ys=((ys.multiply(ys)).add(THREE)).mod(n);
z=n.gcd(ys.subtract(x));
} while (z.compareTo(ONE)==0);

return z;
}

Achievements of this method include the factorization, in 1980, of the eighth Fermat number:
http://wwwmaths.anu.edu.au/~brent/pub/pub061.html