A deterministic version of Pollard's p-1 algorithm

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 ≥ 2) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler’s totient function φ. We point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given φ(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(lnn) oracle queries for values of φ are sufficient to completely factor any integer n in less than exp ( (1 + o(1))(lnn) 1 3 (ln lnn) 2 3 )