On generalized bent functions

Abstract

Bent functions were first introduced by Rothaus in 1976 as an interesting combinatorial object with the important property of having the maximum distance to all affine functions. Bent functions have many applications to coding theory, cryptography and sequence designs. For many years the focus was on the construction of binary bent functions. There are several known examples of binary monomial and binomial bent functions. In 1985, Kumar, Scholtz and Welch generalized bent functions to the case of an arbitrary finite field. In the recent years, new results on nonbinary bent functions have appeared. This paper gives an updated overview of some of the recent results and open problems on generalized bent functions. This includes some recent constructions of weakly regular monomial and binomial bent functions and examples of non-weakly regular bent functions.