**A note on generalized bent criteria for Boolean functions**

*Sugata Gangopadhyay, Enes Pasalic and Pantelimon Stanica*

**Abstract: **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, denoted by $H$, and the
nega--Hadamard, denoted by $N$,
kernels. The set of all such transforms is denoted by $\{H, N\}^n$.
A Boolean function is said to be bent$_4$ if its spectrum
with respect to at least one unitary transform in $\{H, N\}^n$ is flat.
We prove that the maximum possible algebraic degree of a bent$_4$
function on $n$ variables is $\lceil \frac{n}{2} \rceil$, and hence
solve an open problem posed by Riera and Parker [cf. IEEE-IT: 52(2)(2006) 4142--4159].
We obtain a relationship between bent and bent$_4$ 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].

**Category / Keywords: **secret-key cryptography /

**Date: **received 8 Jun 2012, last revised 2 Jul 2012

**Contact author: **gsugata at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20120702:124756 (All versions of this report)

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