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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{W(u):u\in \mathbb {F}^{n}_{2}\}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W(u)\in \left\{{0, \pm 2^{\frac {n+s_{1}}{2}}, \pm 2^{\frac {n+s_{2}}{2}}}\right\}$ </tex-math></inline-formula> , so that the sequence <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{W(u):u\in \mathbb {F}^{n}_{2}\}$ </tex-math></inline-formula> is a valid spectrum of a Boolean function (recovered using the inverse Walsh transform). Technically, this is done by allocating a suitable Walsh support <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S=S^{[{1}]}\cup S^{[{2}]}\subset \mathbb {F}^{n}_{2}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S^{[i]}$ </tex-math></inline-formula> corresponds to those <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u \in \mathbb {F} _{2}^{n}$ </tex-math></inline-formula> for which <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W(u)=\pm 2^{\frac {n+s_{i}}{2}}$ </tex-math></inline-formula> . In addition, two <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dual</i> functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g_{[i]}:S^{[i]}\rightarrow \mathbb {F}_{2}$ </tex-math></inline-formula> (with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\#S^{[i]}=2^{\lambda _{i}}$ </tex-math></inline-formula> ) are employed to specify the signs through <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W(u)=2^{\frac {n+s_{i}}{2}}(-1)^{g_{[i]}(u)}$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u\in S^{[i]}$ </tex-math></inline-formula> whereas <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W(u)=0$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u\not \in S$ </tex-math></inline-formula> . In this work, two closely related problems are considered. Firstly, the specification of plateaued functions (duals) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g_{[i]}$ </tex-math></inline-formula> , which additionally satisfy the so-called totally disjoint spectra property, is fully characterized (so that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$W(u)$ </tex-math></inline-formula> is a spectrum of a Boolean function) when the Walsh support <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> is given as a union of two disjoint affine subspaces <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S^{[i]}$ </tex-math></inline-formula> . Especially, when plateaued dual functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g_{[i]}$ </tex-math></inline-formula> themselves have affine Walsh supports, an efficient spectral design that utilizes arbitrary bent functions (as duals of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$g_{[i]}$ </tex-math></inline-formula> ) 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.