Deterministically finding an element of large order in $\mathbb{Z}_N^*$

TL;DR

This paper introduces a deterministic algorithm to find elements of large order modulo N, improving efficiency and applicability in factoring algorithms, with results comparable to concurrent research.

Contribution

It presents a faster deterministic algorithm for finding large order elements, advancing the state-of-the-art in factoring-related computations.

Findings

Algorithm runs in O(D^{1/2 + o(1)}) time for large D

Returns a large order element, a non-trivial factor, or reports primality

Similar results obtained independently by other researchers

Abstract

In this paper, we present an improvement for the problem of deterministically finding an element of large multiplicative order modulo some integer $N$. This problem arises as a key subroutine in current deterministic factoring algorithms, such as those proposed by Harvey and Hittmeir [Mathematics of Computation, 2021]. Specifically, let $D<N$ be positive integers with \begin{equation}\label{eq:abs} D > \exp\left(\sqrt{2\log N \log \log N}\right). \end{equation} We give a deterministic algorithm that does one of the following: Returns an element $a \in \mathbb{Z}_N^*$ with $\operatorname{ord}_N(a) > D$; Returns a non-trivial factor of $N$; Or reports that $N$ is prime. The running time of our algorithm is $O(D^{1/2 + o(1)})$. Similar results were independently and concurrently obtained by Harvey and Hittmeir [arXiv:2601.11131, 2026] in work that appeared while this manuscript was in preparation. Prior to these works, the best known algorithm for finding an element with order larger than $D$ was given by Oznovich and Volk [SODA 2026], requiring $D > N^{\frac{1}{6}}$. We also present a simpler algorithm that applies for any $D < N$ and runs in $O(D^{2.5+o(1)}\operatorname{polylog}(N))$.