Explicit Euclidean division algorithms for some degree 8 number rings

2026-02-13

Technology Category

• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Matrix & Tensor Methods

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• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Euclidean division
• norm-Euclidean domains
• cyclotomic rings
• number fields
• integer rings

Innovation

• Euclidean division
• norm-Euclidean domains
• cyclotomic rings
• closest vector problem
• algorithm implementation