Characterizations and constructions of n-to-1 mappings over finite fields

n-to-1 mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of n-to-1 mappings over finite fields are studied. First, we provide a characterization of general n-to-1 mappings over Fpm by means of the Walsh transform. Then, we completely determine 3-to-1 polynomials with degree no more than 4 over Fpm. Furthermore, we obtain an AGW-like criterion for characterizing some close relationship between the n-to-1 property of a mapping over finite set A and that of another mapping over a subset of A. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit n-to-1 mappings. Especially, three explicit constructions of the form xrh(xs) from the cyclotomic perspective, and several classes of n-to-1 mappings of the form g(xqk−x+δ)+cx are provided.