Decomposing Generalized Bent and Hyperbent Functions

In this paper, we introduce generalized hyperbent functions from <inline-formula> <tex-math notation="LaTeX">$ {\mathbb F}_{2^{n}}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$ {\mathbb Z}_{2^{k}}$ </tex-math></inline-formula>, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">$ {\mathbb F}_{2^{n}}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$ {\mathbb Z}_{2^{k}}$ </tex-math></inline-formula> consist of components which are generalized (hyper)bent functions from <inline-formula> <tex-math notation="LaTeX">$ {\mathbb F}_{2^{n}}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$ {\mathbb Z}_{2^{k^\prime }}$ </tex-math></inline-formula> for some <inline-formula> <tex-math notation="LaTeX">$k^\prime < k$ </tex-math></inline-formula>. For even <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, most notably we show that the g-hyperbentness of <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> is equivalent to the hyperbentness of the components of <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> with some conditions on the Walsh–Hadamard coefficients. For odd <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, where the associated Boolean functions are bent.

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