Deterministic factoring with oracles

TL;DR

This paper demonstrates that integer factorization can be achieved deterministically in polynomial time using number-theoretic oracles like Euler's totient, under specific conditions on the prime factors, employing lattice basis reduction techniques.

Contribution

It establishes the first unconditional deterministic polynomial-time factoring method given the totient, under certain prime size conditions, using lattice reduction.

Findings

Factorization possible in polynomial time with totient for certain N

Lattice basis reduction (LLL) is effective in this context

Results extend to Carmichael's function and order oracles

Abstract

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 $$\Phi$(N)$? We show that this can be done, under certain size conditions on the prime factors of N. The key technique is lattice basis reduction using the LLL algorithm. Among our results, we show for example that if $N$ is a squarefree integer with a prime factor $p > $\sqrt$ N$ , then we can recover p in deterministic polynomial time given $$\Phi$(N)$. 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.