TITLE: Are there any fast algorithms for factoring integers that don't work by searching for smooth numbers?
QUESTION [7 upvotes]: All of the fast algorithms that I have seen which factor integers work by searching for smooth numbers. Are there any fast algorithms for factoring integers that don't work by searching for smooth numbers? Is there any reason to believe that it is possible to construct fast algorithms for factoring integers that don't work by searching for smooth numbers?

REPLY [2 votes]: Depends a bit on what you call a "fast algorithm", but Pollard rho doesn't search for smooth numbers.