A Note on Generalized Bent Criteria for Boolean Functions
Sugata Gangopadhyay, Enes Pašalić, Pantelimon Stănică · IEEE Transactions on Information Theory · 2012
In this paper, we consider the spectra of Boolean functions with respect to the action of unitary transforms obtained by taking tensor products of the Hadamard kernel, denoted byH, and the nega-Hadamard kernel, denoted byN. The set of all such transforms is denoted by {H,N}n. A Boolean function is said to be bent4if its spectrum with respect to at least one unitary transform in {H,N}nis flat. We obtain a relationship between bent, semibent, and bent4functions, which is a generalization of the relationship between bent and negabent Boolean functions proved by Parker and Pott [cf., LNCS 4893 (2007), 9-23]. As a corollary to this result, we prove that the maximum possible algebraic degree of a bent4function onnvariables is [n/2] and, hence, solve an open problem posed by Riera and Parker [cf., IEEE-TIT 52:9 (2006), 4142-4159].