On the Density of Normal Bases in Finite Fields

Let Fqn denote the finite field with q elements, for q being a prime power. Fqn may be regarded as an n-dimensional vector space over Fq. α ∈ Fqn generates a normal basis for this vector space (Fqn : Fq), if {α,α , αq2 , . . . , αq} are linearly independent over Fq. Let N(q, n) denote the number of elements in Fqn that generate a normal basis for Fqn : Fq, and let ν(q, n) = N(q,n) qn denote the frequency of such elements. We show that there exists a constant c > 0 such that ν(q, n) ≥ c 1 √ dlogq ne , for all n, q ≥ 2 and this is optimal up to a constant factor in that we show 0.28477 ≤ lim n→∞ inf ν(q, n) √ logq n ≤ 0.62521, for all q ≥ 2 We also obtain an explicit lower bound: ν(q, n) ≥ 1 edlogq ne , for all n, q ≥ 2 ∗Supported by the ESPRIT Long Term Research Programme of the EU, under project number 20244 (ALCOM-IT) †Basic Research in Computer Science, Centre of the Danish National Research Foundation.