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

TL;DR

This paper demonstrates that knowing the order of a single random element from _N^* allows for efficient, complete factorization of any integer N in polynomial time, often with just one quantum order-finding run.

Contribution

It introduces a method to fully factorize any integer N using a single quantum order-finding operation combined with classical post-processing.

Findings

Complete factorization achieved in polynomial time

Single quantum run often sufficient for full factorization

Classical post-processing is efficient and negligible in cost

Abstract

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.