On Deterministically Finding an Element of High Order Modulo a Composite
Ziv Oznovich, Ben Lee Volk
TL;DR
This work advances deterministic factorization by giving an algorithm that, for composite $N$ and target order $D\ge N^{1/6}$, outputs either a nontrivial factor of $N$ or an element of $\oldsymbol{Z}_N^*$ with order at least $D$ in time $D^{1/2+o(1)}$. Building on Hittmeir’s framework, the authors tighten the analysis of consecutive-root occurrences to allow larger intermediate orders and replace the final factoring step with a residue-class factoring approach that leverages knowledge of prime divisors’ residues modulo a chosen $s$. They also extend the method to the case where $N$ has an $r$-power divisor, achieving the same running time under the weaker bound $D\ge N^{1/(6r)}$. The results integrate with deterministic factoring algorithms by supplying a robust subroutine that deterministically yields high-order elements or factors, potentially improving the overall dependence on $N$ in contemporary derandomized factorization frameworks. A concurrent line of work has produced related results with stronger assumptions, underscoring the growing viability of deterministic high-order element approaches in factoring.
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 $D \ge N^{1/6r}$.