Gauss periods as constructions of low complexity normal bases

Optimal normal bases are special cases of the so-called Gauss periods (Disquisitiones Arithmeticae, Articles 343–366); in particular, optimal normal bases are Gauss periods of type ( n , 1) for any characteristic and of type ( n , 2) for characteristic 2. We present the multiplication tables and complexities of Gauss periods of type ( n , t ) for all n and t = 3, 4, 5 over any finite field and give a slightly weaker result for Gauss periods of type ( n , 6). In addition, we give some general results on the so-called cyclotomic numbers, which are intimately related to the structure of Gauss periods. We also present the general form of a normal basis obtained by the trace of any normal basis in a finite extension field. Then, as an application of the trace construction, we give upper bounds on the complexity of the trace of a Gauss period of type ( n , 3).