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

摘要

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 ${W(u):uin mathbb {F}^{n}_{2}}$ , where $W(u)in left{{0, pm 2^{rac {n+s_{1}}{2}}, pm 2^{rac {n+s_{2}}{2}}}ight}$ , so that the sequence ${W(u):uin mathbb {F}^{n}_{2}}$ 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 $W(u)=pm 2^{rac {n+s_{i}}{2}}$ . In addition, two dual functions $g_{[i]}:S^{[i]}ightarrow mathbb {F}_{2}$ (with $#S^{[i]}=2^{lambda _{i}}$ ) are employed to specify the signs through $W(u)=2^{rac {n+s_{i}}{2}}(-1)^{g_{[i]}(u)}$ for $uin S^{[i]}$ whereas $W(u)=0$ for $uot in S$ . 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 (suffi