A quantum primality test with order finding

TL;DR

This paper introduces a quantum primality test leveraging order finding and Fermat's theorem, capable of efficiently determining if a number is prime with high certainty and providing a quantum advantage over classical methods.

Contribution

The paper presents a novel quantum primality testing algorithm based on order finding, improving efficiency and certainty in primality verification compared to classical approaches.

Findings

The quantum algorithm can identify primes with high probability.

It can also certify compositeness with certainty in many cases.

The computational complexity is significantly reduced using fast multiplication techniques.

Abstract

Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an element of the multiplicative group of integers modulo $N$ with order $N-1$. If one is found, the number is known to be prime. During the test, we can also show most of the times $N$ is composite with certainty (and a witness) or, after $\log\log N$ unsuccessful attempts to find an element of order $N-1$, declare it composite with high probability. The algorithm requires $O((\log n)^2 n^3)$ operations for a number $N$ with $n$ bits, which can be reduced to $O(\log\log n (\log n)^3 n^2)$ operations in the asymptotic limit if we use fast multiplication.