Abstract
In this paper we investigate generalized Boolean functions whose spectrum is flat with respect to a set of Walsh-Hadamard transforms defined using various complex primitive roots of 1. We also study some differential properties of the generalized Boolean functions in even dimension defined in terms of these different characters. We show that those functions have similar properties to the vectorial bent functions. We next clarify the case of gbent functions in odd dimension. As a by-product of our proofs, more generally, we also provide several results about plateaued functions. Furthermore, we find characterizations of plateaued functions with respect to different characters in terms of second derivatives and fourth moments.
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