Abstract. Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for F q n over Fq, where r is a prime and nk = r − 1 for some integer k. We generalize this construction by allowing arbitrary integers r with nk = ϕ(r), and find in many cases smaller values of k than is possible with the previously known approach.

Abstract.A probabilistic algorithm is exhibited that calculates the gcd of many integers using gcds of pairs of integers; the expected number of pairwise gcds required is less than two.