On Deterministically Finding an Element of High Order Modulo a Composite

TL;DR

This paper presents a deterministic algorithm that efficiently finds elements of high order modulo a composite number or factors the number itself, improving previous bounds and aiding in integer factorization.

Contribution

The authors introduce a new deterministic algorithm that finds high-order elements or factors with improved bounds, extending prior work and impacting integer factorization methods.

Findings

Runs in time $D^{1/2+o(1)}$ for target order $D \\ge N^{1/6}$

Improves upon Hittmeir's algorithm with weaker assumptions on $D$

Supports cases where $N$ has an $r$-power divisor with $r \\ge 2$

Abstract

We give a deterministic algorithm that, given a composite number $N$ and a target order $D \ge N^{1/6}$, runs in time $D^{1/2+o(1)}$ and finds either an element $a \in \mathbb{Z}_N^*$ of multiplicative order at least $D$, or a nontrivial factor of $N$. Our algorithm improves upon an algorithm of Hittmeir (arXiv:1608.08766), who designed a similar algorithm under the stronger assumption $D \ge N^{2/5}$. Hittmeir's algorithm played a crucial role in the recent breakthrough deterministic integer factorization algorithms of Hittmeir and Harvey (arXiv:2006.16729, arXiv:2010.05450, arXiv:2105.11105). When $N$ is assumed to have an $r$-power divisor with $r\ge 2$, our algorithm provides the same guarantees assuming $D \ge N^{1/6r}$.