# Quadratic Probabilistic Algorithms for Normal Bases

## TL;DR

This paper presents a probabilistic algorithm for efficiently finding normal elements in finite Galois extensions with abelian or metacyclic Galois groups, improving computational methods for constructing normal bases.

## Contribution

It introduces a novel probabilistic algorithm that reduces the problem of finding normal elements to invertibility checks, with quadratic or near-quadratic complexity depending on the group type.

## Key findings

- Algorithm is quadratic in the size of G for metacyclic groups.
- Algorithm is slightly subquadratic for abelian groups.
- Deciding normality reduces to invertibility in the group algebra.

## Abstract

It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $\alpha \in K$ whose orbit $G\cdot\alpha$ forms an $F$-basis of $K$. Such an element $\alpha$ is called \emph{normal} and $G\cdot\alpha$ is called a normal basis. In this paper we introduce a probabilistic algorithm for finding a normal element when $G$ is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether a random element $\alpha \in K$ is normal can be reduced to deciding whether $\sum_{\sigma \in G} \sigma(\alpha)\sigma \in K[G]$ is invertible. In an algebraic model, the cost of our algorithm is quadratic in the size of $G$ for metacyclic $G$ and slightly subquadratic for abelian $G$.