Characterization of Basic 5-Value Spectrum Functions Through Walsh-Hadamard Transform
Samir Hodžić, Peter Horák, Enes Pašalić · IEEE Transactions on Information Theory · 2020
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, twodualfunctions$g_{[i]}:S^{[i]}\rightarrow \mathbb {F}_{2}$(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$u ot \in S$. 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.