Lisoněk recently reformulated the characterization of Charpin and Gong of a large class of hyper-bent functions in terms of cardinalities of hyperelliptic curves following previous ideas of Lachaud and Wolfmann, and Katz and Livné. In this paper, we present a generic approach of such ideas and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a pol...