Deterministic methods for finding elements of large multiplicative order

TL;DR

This paper presents a deterministic algorithm that finds elements of large multiplicative order modulo N without restrictions on D, improving previous methods and ensuring success even when N is prime.

Contribution

It removes the previous lower bound requirement on D and guarantees finding an element of large order if N is prime, advancing deterministic factorization techniques.

Findings

Eliminates the D ≥ N^{2/5} hypothesis from earlier algorithms.

Guarantees finding an element of large order when N is prime.

Maintains similar asymptotic complexity as previous methods.

Abstract

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.