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

Martin Ekerå

doi:10.1007/s11128-021-03069-1

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arXiv · 2007.10044

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

Martin Ekerå

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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.

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Martin Ekerå. 2020-07-20. On completely factoring any integer efficiently in a single run of an order finding algorithm. https://doi.org/10.1007/s11128-021-03069-1

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