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

### Abstract

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 $\{W(u)\colon u \in {F}_2^n\}$, where $W(u) \in \{0, \pm 2n+s_1/2, \pm 2n+s_2/2\}$, so that the sequence $\{W(u)\colon u \in \mathbb{F}^n_2\}$ 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 $u \in \mathbb{F}^n_2$ for which $W(u) = \pm 2n+s_i/2$. In addition, two dual functions $g[i]\colon S[i] \to \mathbb{F}^n_2$ (with $\# S[i] = 2^{\lambda_i}$) are employed to specify the signs through $W(u) = 2n+ s_i /2 (-1)^{g[i]} (u)$ for $u \in S[i]$ whereas $W(u) = 0$ for $u \notin S$. 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.

Publication
IEEE Transactions on Information Theory