We show that given the order of a single element selected uniformly at random from \({\mathbb {Z}}_N^*\), we can with very high probability, and for any integer N, efficiently find the complete factorization of N in polynomial time. This implies that a single run of the quantum part of Shor’s factoring algorithm is usually sufficient. All prime factors of N can then be recovered with negligible computational cost in a classical post-processing step. The classical algorithm required for this step is essentially due to Miller.

中文翻译：

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在一次寻序算法中有效地完全分解任何整数

我们表明，给定的一单个元件的从以随机均匀地选择的顺序\（{\ mathbb {Z}} _ N R个* \） ，我们可以以非常高的概率，并且对于任何整数Ñ，有效地找到的完整因式分解Ñ在多项式时间内。这意味着 Shor 因子分解算法的量子部分的单次运行通常就足够了。然后可以在经典的后处理步骤中以可忽略的计算成本恢复N 的所有主要因子。这一步所需的经典算法主要归功于米勒。