An exact quantum order finding algorithm and its applications

TL;DR

This paper introduces an efficient exact quantum algorithm for order finding when a multiple of the order is known, improving quantum algorithms' precision and applicability in number theory and cryptography.

Contribution

It presents a novel exact quantum order finding algorithm combining Fourier transform and amplitude amplification, with applications to primality testing and primitive element finding.

Findings

Derandomizes quantum primality testing.

Enables efficient exact quantum primitive element search.

Improves quantum order finding precision.

Abstract

We present an efficient exact quantum algorithm for order finding problem when a multiple $m$ of the order $r$ is known. The algorithm consists of two main ingredients. The first ingredient is the exact quantum Fourier transform proposed by Mosca and Zalka in [MZ03]. The second ingredient is an amplitude amplification version of Brassard and Hoyer in [BH97] combined with some ideas from the exact discrete logarithm procedure by Mosca and Zalka in [MZ03]. As applications, we show how the algorithm derandomizes the quantum algorithm for primality testing proposed by Donis-Vela and Garcia-Escartin in [DVGE18], and serves as a subroutine of an efficient exact quantum algorithm for finding primitive elements in arbitrary finite fields. .