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The complexity of elliptic normal bases

Daniel Panario, Mohamadou Sall, Qiang Wang

TL;DR

This work addresses the complexity of elliptic normal bases arising from elliptic-period constructions, focusing on the weight of the multiplication tensor and its dependence on elliptic-curve data. It introduces a precise framework tying $C({\mathcal N})$ to vector weights derived from two points on an elliptic curve and proves upper and lower bounds that depend on these weights. The study finds that many contributing vectors tend to be dense, complicating efforts to achieve low-complexity bases, and demonstrates how the curve choice can markedly affect actual complexity through concrete Magma-based examples. The results guide future research toward selecting elliptic-curve data that yield low-weight special vectors to realize more efficient elliptic normal bases in practice.

Abstract

We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight of some special vectors related to the multiplication table of those bases. This analysis leads us to some perspectives on the search for low complexity normal bases from elliptic periods.

The complexity of elliptic normal bases

TL;DR

This work addresses the complexity of elliptic normal bases arising from elliptic-period constructions, focusing on the weight of the multiplication tensor and its dependence on elliptic-curve data. It introduces a precise framework tying to vector weights derived from two points on an elliptic curve and proves upper and lower bounds that depend on these weights. The study finds that many contributing vectors tend to be dense, complicating efforts to achieve low-complexity bases, and demonstrates how the curve choice can markedly affect actual complexity through concrete Magma-based examples. The results guide future research toward selecting elliptic-curve data that yield low-weight special vectors to realize more efficient elliptic normal bases in practice.

Abstract

We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight of some special vectors related to the multiplication table of those bases. This analysis leads us to some perspectives on the search for low complexity normal bases from elliptic periods.
Paper Structure (15 sections, 6 theorems, 47 equations, 3 tables)

This paper contains 15 sections, 6 theorems, 47 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background material
  3. Elliptic curves
  4. Normal bases over finite fields
  5. Elliptic normal bases
  6. The complexity of elliptic normal bases
  7. Notation and previous results
  8. Bounds on the complexity of elliptic normal bases
  9. Weight of special vectors
  10. Examples of elliptic normal basis
  11. First example
  12. Second example
  13. Analysis of the elliptic normal basis from $E_2$
  14. Analysis of the elliptic normal basis from $E_3$
  15. Conclusions and further work

Key Result

Proposition 1

The complexity $C_{{\mathcal{N}}}$ of a normal basis ${\mathcal{N}}$ of ${\mathbb F}_{q^n}$ over ${\mathbb F}_q$ is bounded by

Theorems & Definitions (9)

  • Proposition 1: MOVW89
  • Theorem 1: CL09
  • Remark 1
  • Lemma 1
  • Remark 2
  • Theorem 2
  • proof
  • Proposition 2: Silv09, Proposition II.3.1
  • Proposition 3