On Deterministically Finding an Element of High Order Modulo a Composite

We give a deterministic algorithm that, given a composite number $N$ and a target order $D\ge N^{1\mathrm{/}6}$, runs in time $D^{1\mathrm{/}2+o(1)}$ and finds either an element $a\in {\mathbb{Z}}_N^{\ast }$ 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 $D\ge N^{2\mathrm{/}5}$. 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 $D\ge N^{1\mathrm{/}6r}$.