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.

翻译：我们用数字理论或手腕重新审视整数因子化的问题,包括一个众所周知的问题:鉴于Euler Tent $\Phi$(N)的价值,我们能否在确定性多元时间中无条件地将整数纳美元乘以整数因子化的问题?我们显示,在N的质因子的一定规模条件下,可以做到这一点。关键技术是使用LLL算法降低拉特基数。例如,我们显示,如果美元是一个无正方形的整数,其质因子值大于$/sqrt$(N),那么我们可以在确定性多元时恢复p,给$\Phi$(N)。我们还就Carmichael功能的类似问题以及Shor量量因子算法中使用的顺序问题做了一些说明。