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There are many generalizations of the classical Boolean bent functions. Let ,  be finite groups and let  be a finite -set. -perfect nonlinear functions from  to  have been studied in several papers. They are generalizations of perfect nonlinear functions from  itself to . By introducing the concept of a (, )-related difference family of , we obtain a characterization of -perfect nonlinear functions on  in terms of a (, )-related difference family. When  is abelian, we prove that there is a normalized -dual set  of , and characterize a -difference set of  by the Fourier transform on a normalized -dual set . We will also investigate the existence and constructions of -perfect nonlinear functions and -bent functions. Several known results (IEEE Trans Inf Theory 47(7):2934–2943, ; Des Codes Cryptogr 46:83–96, ; GESTS Int Trans Comput Sci Eng 12:1–14, ; Linear Algebra Appl 452:89–105, ) are direct consequences of our results.

Nonlinear functions and difference sets on group actions