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 × 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.