On the Density of Normal Bases in Finite Fields

Authors:
Gudmund Skovbjerg Frandsen
Affiliations:
BRICSBasic Research in Computer Science, Centre of the Danish National Research Foundation., Department of Computer Science, University of Aarhus, Ny Munkegade, Building 540, DK-8000, Aarhus C, De ...
Venue:
Finite Fields and Their Applications
Year:
2000

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Abstract

Let F"q"^"n denote the finite field with q^n elements, for q a prime power. F"q"^"n may be regarded as an n-dimensional vector space over F"q. @a@?F"q"^"n generates a normal basis for this vector space (F"q"^"n:F"q), if {@a, @a^q, @a^q^^^2 , ... , @a^q^^^n^^^-^^^1} are linearly independent over F"q. Let N"q"("n") denote the number of elements in F"q"^"n that generate a normal basis for F"q"^"n:F"q, and let @n"q(n)=N"q(n)/q^n denote the frequency of such elements. We show that there exists a constant c0 such that and this is optimal up to a constant factor in that we show We also obtain an explicit lower bound: