Deterministic factoring with oracles

We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer $N$ unconditionally, in deterministic polynomial time, given the value of the Euler totient $(\varphi(N)$? We show that this can be done, under certain size conditions on the prime factors of $N$. The key technique is lattice basis reduction using the LLL algorithm. Among our results, we show for example that if $N$ is a squarefree integer with a prime factor $p>\sqrt N$ , then we can recover $p$ in deterministic polynomial time given $\varphi(N))$. We also shed some light on the analogous problems for Carmichael's function, and the order oracle that is used in Shor's quantum factoring algorithm.