The second part of Pollard rho factorization method concerns detection of the fact that a sequence has become periodic. Pollard's
original suggestion was to use the idea attributed to Floyd of comparing 
to 
for all 
. Brent's improvement to Pollard's method concerns how to detect
periodicity, and replaces Floyd's method with the following algorithm. Keep only
one running copy of 
. If 
is a power of a base 
, let 
, and at each step, compare the current value 
with the saved value 
. In the factorization case, instead of comparing 
with 
, compute
 |
More generally, Brent (1980) considered using any base 
for saving values instead of 
. However, he found 
to be very close to optimal.
