Close Encounters with Boolean Functions of Three Different Kinds

Abstract

Abstract. Complex arrays with good aperiodic properties are characterised and it is shown how the joining of dimensions can generate sequences which retain the aperiodic properties of the parent array. For the case of 2 × 2 × . . . × 2 arrays we define two new notions of aperiodicity by exploiting a unitary matrix represention. In particular, we apply unitary rotations by members of a size-3 cyclic subgroup of the local Clifford group to the aperiodic description. It is shown how the three notions of aperiodicity relate naturally to the autocorrelations described by the action of the Heisenberg-Weyl group. Finally, after providing some cryptographic motivation for two of the three aperiodic descriptions, we devise new constructions for complementary pairs of Boolean functions of three different kinds, and give explicit examples for each.

Reed-Muller Codes for Peak Power Control in Multicarrier CDMA

“…r , where W = 2 r−1 (m − r + 2). The recursion of Boolean functions in (29) equivalently derives the recursion of generator matrices of…”

“…A recursive construction of Boolean functions is then presented for the Reed-Muller subcodes, where the PAPR of the MC-CDMA signal encoded by the subcode is proven to be theoretically bounded. The author of [29] pointed out that the construction is equivalent to Type-III sequences in [29] where he made a general and mathematical study for Boolean functions with bounded PAPR, not considering the application to MC-CDMA. We also discuss a connection between the code rate of the subcode and the maximum PAPR.…”

Boolean functions whose restrictions are highly nonlinear

“…, ∀θ ∈ R}, be infinite sets of 2 × 2 unitaries, called type-II, and type-III unitaries, respectively [2], [8], [9]. We can show that…”

“…Typical cryptanalysis considers WalshHadamard spectra of Boolean functions, and we examine spectra with respect to more general transform sets. This is one in a series of papers that tackles such ideas [2], [4], [8]- [10]. We also propose two new types of multivariate complementary set to add to the one proposed in [7].…”