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 $D \ge N^{1/6r}$.

翻译：我们提出一种确定性算法，给定合数$N$与目标阶数$D \ge N^{1/6}$，该算法在$D^{1/2+o(1)}$时间内运行，并找到乘法阶至少为$D$的元素$a \in \mathbb{Z}_N^*$，或$N$的一个非平凡因子。我们的算法改进了Hittmeir（arXiv:1608.08766）的算法，后者在更强的假设$D \ge N^{2/5}$下设计了类似算法。Hittmeir的算法在近期Hittmeir与Harvey（arXiv:2006.16729、arXiv:2010.05450、arXiv:2105.11105）突破性的确定性整数分解算法中发挥了关键作用。当假设$N$具有$r$次幂因子且$r\ge 2$时，我们的算法在假设$D \ge N^{1/6r}$的条件下提供相同的保证。