  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.
