Bent and generalized bent Boolean functions

摘要

In this paper, we investigate the properties of generalized bent functions defined on Z_2~n with values in Z_q, where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana-McFarland type bent functions and Dillon's functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana-McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on Z_2~n with values in Z_4 and Z_8. Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.