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

Martin Ekerå

TL;DR

It is shown that with very high probability, and for any integer N, the complete factorization of N in polynomial time can be efficiently found, which implies that a single run of the quantum part of Shor’s factoring algorithm is usually sufficient.

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.

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

TL;DR

It is shown that with very high probability, and for any integer N, the complete factorization of N in polynomial time can be efficiently found, which implies that a single run of the quantum part of Shor’s factoring algorithm is usually sufficient.

Abstract

We show that given the order of a single element selected uniformly at random from , we can with very high probability, and for any integer , efficiently find the complete factorization of 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 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.

Paper Structure

This paper contains 23 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Earlier works
  3. On the success probability of quantum order finding
  4. Tradeoffs in quantum order finding
  5. Improvements for odd orders
  6. Improvements for special form integers
  7. Other related works on factoring via order finding
  8. On the relation to our contribution
  9. Our contribution
  10. Notes on our original intuition for this work
  11. The algorithm
  12. Notes on efficient implementation
  13. Notes on performing order finding for a multiple of $N$
  14. Notes on analogies with Miller's and Rabin's works
  15. Notes on analogies with Pollard's works
  16. ...and 8 more sections

Key Result

Lemma 1

The probability that $p_i$ is unlucky is at most $\log p_i / (m' \log m')$.

Theorems & Definitions (11)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Claim 1
  • proof
  • Theorem 1
  • ...and 1 more