Complete Characterization of Generalized Bent and 2k-Bent Boolean Functions

In this paper, we investigate properties of generalized bent Boolean functions and <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent (i.e., negabent, octabent, hexadecabent, <italic>et al.</italic>) Boolean functions in a uniform framework. From the Hadamard matrices, Hodžić and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodžić and Pasalic’s conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semi-bent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent functions, demonstrate that <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent functions are actually a special class of generalized bent functions, and completely characterize <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent functions.

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