On the existence of primitive completely normal bases of finite fields

Abstract

Let F q be the finite field of characteristic p with q elements and F q n its extension of degree n. We prove that there exists a primitive element of F q n that produces a completely normal basis of F q n over F q , provided that n = p ℓ m with (m, p) = 1 and q > m.

The existence of primitive normal elements of quadratic forms over finite fields

“…(9,4) Similarly, proceeding in similar manner, we have the following pair as the solitary possible exceptional pair from the remaining cases in which m ′ ≤ 22. (9,8) For each of the individual pairs (q, m) listed above that do not satisfy the sufficient condition based on Lemma 5.3, we can test them more precisely by means of the sufficient condition (4.2) after factorising completely x m − 1 and q m − 1 and making a choice of polynomial divisor g of x m − 1 and factor d of q m − 1. In practice, the best choice is to choose p 1 , .…”

Further Results on the Morgan–Mullen Conjecture

“…The aim of this work is to establish the existence of primitive and completely normal elements for a larger range for the parameters q, n. We prove new sharper bounds for the number of completely normal elements of a given extension and use it to establish the existence of primitive and completely normal elements, using the method laid out in [7]. Our results hold asymptotically for n up to roughly q 2 with the additional assumption that q − 1 ∤ n when n is even in Theorem 1.3.…”

“…Recently, Hachenberger [11], using elementary methods, proved the validity of Conjecture 1.2 for q ≥ n 3 and n ≥ 37. In [7], the range was improved to n ≤ q.…”

Computational Results on the Existence of Primitive Complete Normal Basis Generators

“…This is used to settle the asymptotic result mentioned in Subsection 2.7 and in order to show that P CN (q, n) > 0 whenever q ≥ n 7/2 and n ≥ 7, or when q ≥ n 3 and n ≥ 37 (see [13]). Recently, the latter could be improved considerably by Garefalakis and Kapatenakis [5]:…”

“…We shall derive an alternative sufficient criterion, next. In fact, it is adopted from, and improves the correponding result in [5]; the improvement rests on the fact that we work with the non-trivial (q, n)-essential sets which are based on the CN-graphs rather than the trivial one, {d ∈ N : d | n, d = n}, which in fact is never optimal. Throughout, let ω = ω(q n − 1) denote the number of all distinct prime divisors of q n − 1.…”