Finding Primitive Elements in Finite Fields of Small Characteristic

TL;DR

This paper introduces a deterministic polynomial-time algorithm for finding primitive elements in finite fields of small characteristic, leveraging relation generation techniques and heuristic assumptions for success.

Contribution

It presents a novel deterministic algorithm for primitive element discovery in finite fields, improving reliability under certain conditions.

Findings

Algorithm runs in polynomial time in p and n.

Succeeds under heuristic assumptions for general cases.

Enhanced success guarantees for special cases with small order of p in (Z/nZ)^×.

Abstract

We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field $\mathbb{F}_{p^n}$ where $p$ is a prime. In time polynomial in $p$ and $n$, the algorithm either outputs an element that is provably a generator or declares that it has failed in finding one. The algorithm relies on a relation generation technique in Joux's heuristically $L(1/4)$-method for discrete logarithm computation. Based on a heuristic assumption, the algorithm does succeed in finding a generator. For the special case when the order of $p$ in $(\mathbb{Z}/n\mathbb{Z})^\times$ is small (that is $(\log_p(n))^{\mathcal{O}(1)}$), we present a modification with greater guarantee of success while making weaker heuristic assumptions.