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
Abstract. We relate the one-and two-variable interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms. By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement. We establish the form of the interlace polynomial for certain functions, provide new one and two-variable interlace polynomials, and propose a generalisation of the interlace polynomial to hypergraphs. We also prove conjectures from [15] and equate certain spectral metrics with various evaluations of the interlace polynomial.
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