A deterministic version of Pollard’s $p-1$ algorithm

Abstract:

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algo- rithms.

We begin with Pollard’s $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit.

We also investigate reductions of factoring to computing Euler’s totient function $\varphi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\varphi (n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $\varphi$ are sufficient to completely factor any integer $n$ in less than $\exp \Bigl ((1+o(1))(\ln n)^{\frac {1}{3}} (\ln \ln n)^{\frac {2}{3}}\Bigr )$ deterministic time.