On the number of zero trace elements in polynomial bases for \(\mathbb F_{2^n}\)
type: Item
instance of: scholarly article
title: On the number of zero trace elements in polynomial bases for \(\mathbb F_{2^n}\)
zbMATH Open document ID: 1069.11051
author: Igor E. Shparlinski
published in: Revista Matemática Complutense
publication date: 2005-07-13T00:00:00Z
full work available at URL: https://eudml.org/doc/44549
review text: Let \(\mathbb F_{2^n}\) denote the finite field of \(2^n\) elements, \(\alpha\) a defining element of \(\mathbb F_{2^n}\), i.e., \(\mathbb F_2(\alpha)=\mathbb F_{2^n}\), and denote by \(N(\alpha)\) the number of zero trace elements in the polynomial basis \(\{1,\alpha,\ldots,\alpha^{n-1}\}\). The author proves that there exists a defining element \(\alpha\in \mathbb F_{2^n}\) such that \[ N(\alpha)\geq n-\log n-2,\quad n\geq 21. \] The proof is based on the Weil bound. The question about the spectrum of possible values of \(N(\alpha)\), when \(\alpha\) runs through all defining elements of \(F_{2^n}\), has been introduced in \textit{O. Ahmadi} and \textit{A. Menezes} [On the number of trace-one elements in polynomial bases for \(\mathbb F_{2^n}\), Des. Codes Cryptography 37, No. 3, 493--507 (2005; Zbl 1142.11377)].
reviewed by: Arne Winterhof
Mathematics Subject Classification ID: 11T30
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 94B60
zbMATH DE Number: 2187074
zbMATH Keywords: bases of finite fields
zbMATH Keywords: Weil bound
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2187074