Existence and properties of k-normal elements over finite fields

Sophie Huczynska, Gary L. Mullen, Daniel Panario, David Thomson

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Abstract

An element αFqn is normal over Fq if {α,αq,.,αqn-1} is a basis for Fqn over Fq. It is well known that αFqn is normal over F q if and only if the polynomials (x)=αxn- 1+αqxn-2+â̄+αqn- 2x+αqn-1 and xn-1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. An element αFqn is k-normal over Fq if the greatest common divisor of the polynomials (x) and xn-1 in Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal. In this paper, we introduce and characterize k-normal elements, establish a formula and numerical bounds for the number of k-normal elements and prove an existence result for primitive 1-normal elements.

Original languageEnglish (US)
Pages (from-to)170-183
Number of pages14
JournalFinite Fields and their Applications
Volume24
DOIs
Publication statusPublished - Aug 8 2013

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Algebra and Number Theory
  • Engineering(all)
  • Applied Mathematics

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