TY - JOUR
T1 - Existence and properties of k-normal elements over finite fields
AU - Huczynska, Sophie
AU - Mullen, Gary L.
AU - Panario, Daniel
AU - Thomson, David
N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.
PY - 2013
Y1 - 2013
N2 - 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 gα(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 gα(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.
AB - 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 gα(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 gα(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.
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U2 - 10.1016/j.ffa.2013.07.004
DO - 10.1016/j.ffa.2013.07.004
M3 - Article
AN - SCOPUS:84881020926
VL - 24
SP - 170
EP - 183
JO - Finite Fields and Their Applications
JF - Finite Fields and Their Applications
SN - 1071-5797
ER -