Decomposing generalized bent and hyperbent functions

In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. For odd $n$, 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 $n$, where the associated Boolean functions are bent.Comment: 24 page

On Walsh Spectrum of Cryptographic Boolean Function

Walsh transformation of a Boolean function ascertains a number of cryptographic properties of the Boolean function viz, non-linearity, bentness, regularity, correlation immunity and many more. The functions, for which the numerical value of Walsh spectrum is fixed, constitute a class of Boolean functions known as bent functions. Bent functions possess maximum possible non-linearity and therefore have a significant role in design of cryptographic systems. A number of generalisations of bent function in different domains have been proposed in the literature. General expression for Walsh transformation of generalised bent function (GBF) is derived. Using this condition, a set of Diophantine equations whose solvability is a necessary condition for the existence of GBF is also derived. Examples to demonstrate how these equations can be utilised to establish non-existence and regularity of GBFs is presented

A note on generalized bent criteria for Boolean functions

The article of record as published may be found at http://dx.doi.org/10.1109/TIT.2012.2235908.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 by H, and the nega–Hadamard kernel, denoted by N. The set of all such transforms is denoted by {H,N}n. A Boolean function is said to be bent4 if its spectrum with respect to at least one unitary transform in {H,N}n is flat. We obtain a relationship between bent, semi–bent and bent4 functions, 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 bent4 function on n variables is [n/2], and hence solve an open problem posed by Riera and Parker [cf. IEEE-TIT 52:9 (2006), 4142–4159]