A deterministic version of Pollard's p-1 algorithm

TL;DR

This paper demonstrates how smooth numbers can be used to derandomize Pollard's p-1 algorithm and related factoring methods, enabling deterministic polynomial-time factorization under certain conditions.

Contribution

It provides a deterministic polynomial-time version of Pollard's p-1 algorithm and extends derandomization techniques to cyclotomic methods and factoring via Euler's totient function.

Findings

Prime factors with smooth p-1 can be found deterministically in polynomial time.

Partial derandomization of cyclotomic factoring methods achieved.

Complete factorization from Euler's totient with O(log n) queries within subexponential time.

Abstract

In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. 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 $\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\phi(n)$. 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 $\phi$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr)$ deterministic time.