Paper detail

Deterministic methods for finding elements of large multiplicative order

We revisit the problem of rigorously and deterministically finding elements of large order in the multiplicative group of integers modulo a natural number $N$. Solving this problem is an essential step in several recent deterministic algorithms for factoring $N$, including the currently fastest ones. In 2018, the second author gave an algorithm that for a given target order $D \geq N^{2/5}$, finds either an element of order exceeding $D$, or a nontrivial divisor of $N$, or proves that $N$ is prime. The running time was \[ O\left(\frac{D^{1/2}}{(\log \log D)^{1/2}} \log^2 N \right) \] bit operations, asymptotically the same as the cost of computing the order of a single element using Sutherland's optimisation of the classical babystep-giantstep method. Subsequent work by several authors weakened the hypothesis $D \geq N^{2/5}$ to $D \geq N^{1/6}$. In this paper, we show that the hypothesis may be dropped altogether. Moreover, if $N$ is prime, we can guarantee returning an element of order exceeding $D$, rather than a proof that $N$ is prime.