Let q be a power of a prime p, and let r=nk+1\documentclass[12pt]{minimal}
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\begin{document}$$r=nk+1$$\end{document} be a prime such that r∤q\documentclass[12pt]{minimal}
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\begin{document}$$r\not \mid q$$\end{document}, where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of Fqn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_{q}^{n}$$\end{document} over Fq\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_q$$\end{document}; the complexity of the resulting normal basis of Fqn\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_{q}^{n}$$\end{document} over Fq\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_q$$\end{document} is denoted by C(n, k; p). Recent works determined C(n, k; p) for k≤7\documentclass[12pt]{minimal}
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\begin{document}$$k\le 7$$\end{document} and all qualified n and q. In this paper, we show that for any given k>0\documentclass[12pt]{minimal}
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\begin{document}$$k>0$$\end{document}, C(n, k; p) is given by an explicit formula except for finitely many primes r=nk+1\documentclass[12pt]{minimal}
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\begin{document}$$r=nk+1$$\end{document} and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes r=nk+1\documentclass[12pt]{minimal}
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\begin{document}$$r=nk+1$$\end{document}. Our numerical results cover C(n, k; p) for k≤20\documentclass[12pt]{minimal}
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\begin{document}$$k\le 20$$\end{document} and all qualified n and q.
