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.

给定欧拉整数\ ( \ varphi (N) \)的值，我们能否在确定性多项式时间内无条件分解整数\ (N \)？我们表明，这可以在\ (N \)的主要因子的特定大小条件下完成。关键技术是使用 LLL 算法的格基约简。在我们的结果中，我们表明，如果\ (N \)有一个质因子\ (p> \ sqrt {N} \)，那么我们可以在给定\ (\ varphi (N) 的确定性多项式时间内恢复\ (p \ ) \)。我们还阐明了给定除数和函数、Carmichael 函数和 Shor 量子因式分解算法中使用的阶数预言机的类似因式分解问题。