Generalized Bent Criteria for Boolean Functions (I)

Abstract

In the first part of this paper [16], some results on how to compute the flat spectra of Boolean constructions w.r.t. the transforms {I, H} n , {H, N } n and {I, H, N } n were presented, and the relevance of Local Complementation to the quadratic case was indicated. In this second part, the results are applied to develop recursive formulae for the numbers of flat spectra of some structural quadratics. Observations are made as to the generalised Bent properties of boolean functions of algebraic degree greater than two, and the number of flat spectra w.r.t. {I, H, N } n are computed for some of them., where i 2 = −1, the Negahadamard kernel, and I the 2 × 2 identity matrix.We say that a Boolean function p(x) : GF(2) n → GF (2) is Bent [17] if P = 2 −n/2 ( n−1 i=0 H)(−1) p(x) has a flat spectrum, or, in other words, if P = (P k ) ∈ C 2 n is such that |P k | = 1 ∀ k ∈ GF(2) n . Bent boolean functions are desirable cryptographic primitives as they optimise resistance to linear cryptanalysis. If the function is quadratic, we can associate to it a simple non-directed graph, and in this case a flat spectrum is obtained iff Γ, the adjacency matrix of the graph, has maximum rank mod 2. In Part I, we generalised C. Riera is with the

One and Two-Variable Interlace Polynomials: A Spectral Interpretation

“…As the interlace polynomial, Q, is LC-invariant [1], it is also an invariant of the corresponding QECC. This result implies that Q is invariant under the application of certain LU transforms to the multipartite quantum state associated with the QECC [18], for it turns out that LC-equivalence for graph states can be characterised by LU-transformation via the set of transforms {I, H, N } n [20]. Therefore Q can be used to summarise some important properties of an associated quantum graph state.…”

“…. , x n−1 ), where p(x) = i<j Γ ij x i x j [20]. This identification allows us to interpret q(G, 1) as the number of flat spectra of p(x) w.r.t.…”

“…{I, H} n and {I, H, N } n , respectively, as defined in [20,21], and use them to compute the interlace polynomial of the clique (complete graph), and clique-line-clique.…”

“…Therefore Q can be used to summarise some important properties of an associated quantum graph state. More specifically, an analysis of the spectra of a Boolean function provides measures of entanglement of the associated quantum multipartite state, as defined by the QECC and/or its associated quadratic Boolean function [20,18,13].…”

“…the Walsh-Hadamard transform (WHT), and such spectra are conveniently summarised by their respective interlace polynomials. As seen in [20], just an enumeration of the flat spectra of a function w.r.t. {I, H, N } n or its subsets provides a good measure of the 'strength' of the function in various contexts.…”