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We show that given the order of a single element selected uniformly at random from    $${\mathbb {Z}}_N^*$$  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.

On completely factoring any integer efficiently in a single run of an order-finding algorithm