Chapter 15 - Algebraic Generalizations of Bent Functions

Abstract

Generalizations of bent functions with respect to their algebraic, combinatorial, and cryptographic properties are becoming more numerous and more widely studied from year to year. It is quite difficult not only to determine connections between generalizations, but also to collect information about all of them and briefly review the progress in this area. In this chapter and the next two chapters we provide a systematic survey of the existing generalizations of bent functions and try whenever possible to establish relations between various generalizations. In this chapter the following algebraic generalizations are considered: q-valued bent functions, p-ary bent functions, bent functions over a finite field, generalized Boolean bent functions of Schmidt, bent functions from a finite Abelian group into the set of complex numbers on the unit circle, bent functions from a finite Abelian group into a finite Abelian group, non-Abelian bent functions, vectorial G-bent functions, and multidimensional bent functions on a finite Abelian group.