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 $O(D^{2.5+o(1)}\operatorname{polylog}(N))$.

翻译：在本文中，我们改进了在模整数 $N$ 下确定性地寻找大乘法阶元的问题。该问题是当前确定性分解算法（如 Harvey 和 Hittmeir [Mathematics of Computation, 2021] 提出的算法）中的关键子程序。具体地，设 $D<N$ 为正整数，且满足 \begin{equation}\label{eq:abs} D > \exp\left(\sqrt{2\log N \log \log N}\right). \end{equation} 我们给出一个确定性算法，该算法执行以下操作之一：返回一个满足 $\operatorname{ord}_N(a) > D$ 的元素 $a \in \mathbb{Z}_N^*$；返回 $N$ 的一个非平凡因子；或报告 $N$ 为素数。该算法的时间复杂度为 $O(D^{1/2 + o(1)})$。在本文手稿准备期间，Harvey 和 Hittmeir 独立且同时地获得了类似结果 [arXiv:2601.11131, 2026]。在这些工作之前，寻找阶大于 $D$ 的元素的最佳已知算法由 Oznovich 和 Volk [SODA 2026] 给出，该算法要求 $D > N^{\frac{1}{6}}$。我们还提出一个更简单的算法，适用于任意 $D < N$，其运行时间为 $O(D^{2.5+o(1)}\operatorname{polylog}(N))$。