Deterministic factoring with oracles

Abstract

Can we factor an integer \(N\) unconditionally, in deterministic polynomial time, given the value of its 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 that if \(N\) has 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 factorization problems given oracles for the sum-of-divisors function, Carmichael’s function, and the order oracle that is used in Shor’s quantum factoring algorithm.