International Association
for Cryptologic Research

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].