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Explicit Euclidean division algorithms for some degree 8 number rings

Christophe Levrat

TL;DR

This work addresses explicit Euclidean division in norm-Euclidean rings of integers for four number fields $\mathbb{Q}(\zeta_{15})$, $\mathbb{Q}(\zeta_{20})$, $\mathbb{Q}(\zeta_{24})$, and $\mathbb{Q}(\sqrt{3},\sqrt{5},\sqrt{-1})$ by reframing Lenstra's geometric proof as a closest-vector problem on the $E_8$ lattice. It leverages the Conway–Sloane CVP algorithm, analyzes bit-complexity to be $O(M(n))$ with $M(n)=O(n\log n)$, and provides a complete SageMath implementation alongside estimates of remainder sizes via Euclidean minima, showing near-optimal behavior in most cases. The results establish a practical, efficient path to explicit division in these rings and illuminate how far current methods are from optimality, while outlining open questions on extending to other cyclotomic rings and constructing explicit norm-Euclidean functions for unsettled cases. The work thus bridges abstract norm-Euclidean proofs with concrete computational techniques, enabling gcd computations and potential cryptographic applications in these arithmetic settings.

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.

Explicit Euclidean division algorithms for some degree 8 number rings

TL;DR

This work addresses explicit Euclidean division in norm-Euclidean rings of integers for four number fields , , , and by reframing Lenstra's geometric proof as a closest-vector problem on the lattice. It leverages the Conway–Sloane CVP algorithm, analyzes bit-complexity to be with , and provides a complete SageMath implementation alongside estimates of remainder sizes via Euclidean minima, showing near-optimal behavior in most cases. The results establish a practical, efficient path to explicit division in these rings and illuminate how far current methods are from optimality, while outlining open questions on extending to other cyclotomic rings and constructing explicit norm-Euclidean functions for unsettled cases. The work thus bridges abstract norm-Euclidean proofs with concrete computational techniques, enabling gcd computations and potential cryptographic applications in these arithmetic settings.

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 -th cyclotomic rings for . 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.
Paper Structure (18 sections, 4 theorems, 19 equations, 1 figure, 2 algorithms)

This paper contains 18 sections, 4 theorems, 19 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Euclidean and Norm-Euclidean rings of integers
  3. Cyclotomic rings
  4. Euclidean cyclotomic rings
  5. Effectively Euclidean number rings
  6. Lenstra's proof
  7. Our contribution
  8. Root systems and lattices
  9. Euclidean division in $F$: Lenstra's proof
  10. Link with a closest vector problem
  11. Generic algorithms
  12. The case of $E_8$: an algorithm due to Conway and Sloane
  13. Algorithm and Complexity
  14. Euclidean minima and size of the computed remainders
  15. Estimating $A_F$
  16. ...and 3 more sections

Key Result

Theorem 1

Under these hypotheses, the ring of integers $\mathcal{O}_F$ of $F$ is norm-Euclidean.

Figures (1)

  • Figure 1: Average time needed to compute a division of two elements in $\mathbb{Z}[\zeta_{15}]$ whose coordinates in the base $B$ are fractions of integers of bit length $n$

Theorems & Definitions (12)

  • Remark 1.1
  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Remark 5.1
  • Definition 3
  • Lemma 1
  • ...and 2 more