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 ∈ Fn2}, where W(u) ∈ {0, ±2n+s1/2, ±2n+s2/2}, so that the sequence {W(u): u ∈ Fn2} 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] ∪ S[2] ⊂ Fn2, where S[i] corresponds to those u ∈ Fn2 for which W(u) = ±2n+si/2. In addition, two dual functions g[i]: S[i] → F2 (with #S[i] = 2λ i) are employed to specify the signs through W(u) = 2n+si/2(-1)g[i](u) for u ∈ S[i] whereas W(u) = 0 for u ∉ 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.