Explicit Euclidean division algorithms for some degree 8 number rings

TL;DR

This paper develops explicit Euclidean division algorithms for certain degree 8 number rings, including cyclotomic rings, based on Lenstra's proof and closest vector computations, with implementation and complexity analysis.

Contribution

It introduces explicit Euclidean division algorithms for specific degree 8 number rings, previously lacking such algorithms, and provides their implementation and complexity study.

Findings

Algorithm successfully performs Euclidean division in targeted rings.

Implementation in SageMath demonstrates practical applicability.

Analysis of remainder sizes offers insights into algorithm efficiency.

Abstract

This article focuses on some rings of integers of number fields which are known to be norm-Euclidean domains, but for which no explicit algorithm computing the Euclidean division has yet been studied or implemented. The rings of integers we are interested in were proven to be Euclidean by H.W. Lenstra, Jr in 1978; they include the $n$-th cyclotomic rings for $n=15,20,24$. We present an algorithm performing Euclidean division in these rings based on Lenstra's proof and a closest vector computation by Conway and Sloane, and study its complexity. We give a complete implementation of the algorithm in SageMath. We also estimate the size of the remainders obtained when computing Euclidean divisions with this algorithm.