Circulant matrices, gauss sums and mutually unbiased I. The prime number case

In this paper, we consider the problem of Mutually Unbiased Bases in prime dimension $d$. It is known to provide exactly $d+1$ mutually unbiased bases. We revisit this problem using a class of circulant $d\times d$ matrices. The constructive proof of a set of $d+1$ mutually unbiased bases follows, together with a set of properties of Gauss sums, and of bi-unimodular sequences.