Further Results on Generalized Bent Functions and Their Complete Characterization

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon's $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh-Hadamard transform of their component functions, and we generalize these results to the case of odd prime.