Characterization of Basic 5-Value Spectrum Functions Through Walsh-Hadamard Transform

The first and the third authors recently introduced a spectral construction of plateaued and of 5-value spectrum functions. In particular, the design of the latter class requires a specification of integers
$\{W(u):u\in \mathbb {F}^{n}_{2}\}$
, where
$W(u)\in \left\{{0, \pm 2^{\frac {n+s_{1}}{2}}, \pm 2^{\frac {n+s_{2}}{2}}}\right\}$
, so that the sequence
$\{W(u):u\in \mathbb {F}^{n}_{2}\}$
is a valid spectrum of a Boolean function (recovered using the inverse Walsh transform). Technically, this is done by allocating a suitable Walsh support
$S=S^{[{1}]}\cup S^{[{2}]}\subset \mathbb {F}^{n}_{2}$
, where
$S^{[i]}$
corresponds to those
$u \in \mathbb {F} _{2}^{n}$
for which
$W(u)=\pm 2^{\frac {n+s_{i}}{2}}$
. In addition, two
dual
functions
$g_{[i]}:S^{[i]}\rightarrow \mathbb {F}_{2}$
(with
$\#S^{[i]}=2^{\lambda _{i}}$
) are employed to specify the signs through
$W(u)=2^{\frac {n+s_{i}}{2}}(-1)^{g_{[i]}(u)}$
for
$u\in S^{[i]}$
whereas
$W(u)=0$
for
$u\not \in S$
. In this work, two closely related problems are considered. Firstly, the specification of plateaued functions (duals)
$g_{[i]}$
, which additionally satisfy the so-called totally disjoint spectra property, is fully characterized (so that
$W(u)$
is a spectrum of a Boolean function) when the Walsh support
$S$
is given as a union of two disjoint affine subspaces
$S^{[i]}$
. Especially, when plateaued dual functions
$g_{[i]}$
themselves have affine Walsh supports, an efficient spectral design that utilizes arbitrary bent functions (as duals of
$g_{[i]}$
) on the corresponding ambient spaces is given. The problem of specifying affine inequivalent 5-value spectra functions is also addressed and an efficient construction method that ensures the inequivalence property is derived (sufficient condition being a selection of affine inequivalent duals). In the second part of this work, we investigate duals of plateaued functions with affine Walsh supports. For a given such plateaued function, we show that different orderings of its Walsh support which are employing the Sylvester-Hadamard recursion actually induce bent duals which are affine equivalent.