Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic boolean functions are related to simple graphs and it is shown that the orbit generated by successive Local Complementations on a graph can be found within the transform spectra under investigation. The flat spectra of ...

It is presently shown that the Deutsch-Jozsa algorithm connected to concept of bent function. Particularly, it noticeable quantum circuit used denote well-known by itself computer performs Walsh transform a Boolean Consequently, output from when hidden function corresponds flat spectrum states.

Hromkovif, J., S.A. Loikin, A.I. Rybko, A.A. Sapoienko and N.A. Skalikova, Lower bounds on the area complexity of Boolean circuits, Theoretical Computer Science 97 (1992) 2855300. The layout area of Boolean circuits is considered as a complexity measure of Boolean functions. Introducing the communication complexity of Boolean circuits and proving that this communication complexity squared provi...

Algebraic immunity of Boolean function f is defined as the minimal degree of a nonzero g such that fg = 0 or (f + 1)g = 0. Given a positive even integer n, it is found that the weight distribution of any n-variable symmetric Boolean function with maximum algebraic immunity n 2 is determined by the binary expansion of n. Based on the foregoing, all n-variable symmetric Boolean functions with max...

Boolean networks are special types of finite state timediscrete dynamical systems. A Boolean network can be described by a function from an n-dimensional vector space over the field of two elements to itself. A fundamental problem in studying these dynamical systems is to link their long term behaviors to the structures of the functions that define them. In this paper, a method for deriving a B...

Bent functions (Dillon 1974; Rothaus 1976) are extremal objects in combinatorics and Boolean function theory. They have been studied for about 40 years; even more, under the name of difference sets in elementary Abelian 2-groups. The motivation for the study of these particular difference sets is mainly cryptographic (but bent functions play also a role in coding theory and sequences; and as di...

We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.

Given a positive Boolean function f and a subset ∆ of its variables, we give a combinatorial condition characterizing the existence of a prime implicant D̂ of the Boolean dual fd of f , having the property that every variable in ∆ appears in D̂. We show that the recognition of this property is an NP-complete problem, suggesting an inherent computational difficulty of Boolean dualization, independ...

This article introduces various Boolean operators which are used in discussing the properties and stability of a 2's complement circuit. We present the deenitions and related theorems for the following logical operators which include negative input/output: 'and2a', 'or2a', 'xor2a' and 'nand2a', 'nor2a', etc. We formalize the concept of a 2's complement circuit, deene the structures of comple-me...

This paper explores derivative operations of the Boolean differential calculus for lattices of Boolean functions. Such operations are needed to design circuits with short delay and low power consumption [3] as well as to calculate minimal complete sets of fitting test patterns [4]. It will be shown that each derivative operation of a lattice of Boolean functions creates again a lattice of Boole...