Generalized bent functions and their properties

Given a bent function $$f(\varvec{x})$$ of n variables, its max-weight and min-weight functions are introduced as the Boolean functions $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ whose supports are the sets $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} | w(f \oplus l_{\varvec{a}}) = 2^{n-1}+2^{\frac{n}{2}-1}\}$$ and $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} | w(f \oplus l_{\varvec{a}}) = 2^{n-1}-2^{\frac{n}{2}-1}\}$$ respectively, where $$w(f \oplus l_{\varvec{a}})$$ denotes the Hamming weight of the Boolean function $$f(\varvec{x}) \oplus l_{\varvec{a}}(\varvec{x})$$ and $$l_{\varvec{a}}(\varvec{x})$$ is the linear function defined by $$\varvec{a} \in {\mathbb {F}}_{2}^{n}$$ . $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple $$(f_{0}(\varvec{x}), f_{1}(\varvec{x}), f_{2}(\varvec{x}), f_{3}(\varvec{x}))$$ of bent functions of n variables such that $$f_{0}(\varvec{x}) \oplus f_{1}(\varvec{x}) \oplus f_{2}(\varvec{x}) \oplus f_{3}(\varvec{x}) = 1$$ , a bent function of $$n+2$$ variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for four decades. Since the complete classification of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this paper, we show that linear involutions (which are an important class of permutations) over finite fields give rise to bent functions in bivariate representations. In particular, we exhibit new constructions of bent functions involving binomial linear involutions whose dual functions are directly obtained without computation. The existence of bent functions from involutions heavily relies on solving systems of equations over finite fields.

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis $${\widehat{G}}$$ , consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on $${\widehat{G}}$$ , and obtain characterizations of a bent function by its Fourier transforms (as functions on $${\widehat{G}}$$ ). For a function f from G to another finite group, we define a dual function $${\widetilde{f}}$$ on $${\widehat{G}}$$ , and characterize the nonlinearity of f by its dual function $${\widetilde{f}}$$ . Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.

Partial spreads and spreads play an important role in some constructions of bent functions.

In this article we study two party Communication Complexity of Boolean bent functions from Maiorana–McFarland class. In particular, we describe connections between Maiorana–McFarland construction of bent functions and operations on matrix form of Boolean functions and show that bent functions of 2n variables from Maiorana–McFarland class have deterministic communication complexity equal n + 1. Finally, we show that not all bent functions have high communication complexity lower bound by giving the example of such function.

$$\mathbb {Z}$$-bent functions, mappings from $$\mathbb {F}_2^n$$ to a subset of $$\mathbb {Z}$$, were introduced by Dobbertin and Leander (Des Codes Cryptogr 49:3–22, 2008) as an attempt to capture the origin of standard bent functions and in particular to understand better a recursive construction framework of bent functions. Nevertheless, many questions have been left open in Dobbertin and Leander (2008) such as efficient construction methods of $$\mathbb {Z}$$-bent functions of different levels, where these levels specify precisely a subset of $$\mathbb {Z}$$ containing both the image values of f and its normalized Fourier coefficients. In this article, using different design techniques, we provide several generic construction methods of $$\mathbb {Z}$$-bent functions of arbitrary levels, thereby solving an open problem posed in Dobbertin and Leander (2008). On the other hand, apart from an independent theoretical interest in these objects, our rigor treatment of the so-called gluing technique reveals that this approach is equivalent to a classical concept of concatenation. More precisely, gluing four suitable n-variables $$\mathbb {Z}$$-bent functions of level one to obtain an $$(n+2)$$-variable bent function directly corresponds to a concatenation of four suitable n-variable Boolean functions. Nevertheless, the recursive framework based on $$\mathbb {Z}$$-bent functions remains to be investigated further in this context.

The solution to an open problem on the bentness of Mesnager's functions

A system of transparent layers on top of an arbitrary underlying subsystem is considered. It is shown that the layer refractive index and the subsystem generalized Fresnel coefficients can be expressed in a simple and explicit way by the Fourier coefficients of the ellipsometric function ρ with the layer thickness as an expansion parameter. Thus the inverse ellipsometric problem is reduced to a much simpler and well-defined problem of finding these Fourier coefficients. Analysis shows that the ellipsometric inverse task must be considered separately, depending on whether the modulus of the s-polarization Fresnel coefficient for the ambient–layer interface is smaller than, equal to, or greater than that for the layer–substrate system.

In two preceding papers by Jagodzinski (Phys. Stat. Sol.(a)146(1994) 477; Acta Crystallogr. A52(1996) 675), a new method of phase determination has been described. Using latent lattices of point scatterers for a single kind of atoms, the structure is generated by shift vectors, displacing the atoms from the origins of the latent lattice into their correct positions. The mathematical treatment of this procedure resulted in very complex formulae for structure factorsE′(h), normalised such thatE′(0) = 1. The three components of the displacement vectors define a new real and reciprocal shift space. The method is briefly described in the first three sections of this paper. There is an important difference between one and higher dimensions: the number of latent lattices with different shift functions is infinite, but only very few of them are optimal. Phases and amplitudes of the Fourier coefficients of the shift function are strongly correlated. This property results from the transformation of diffraction data and the prohibition of trespassing of points, caused by large Fourier coefficients of the shift function. In addition to the well known influence of strong reflections, the approximate [unk]-symmetries of the diffraction data and the areas of minimum intensity play an outstanding role for the selection of the optimal latent lattice. Its corresponding shift function has small displacements and small amplitudes of the Fourier coefficients. In the case of complicated structures, these properties demand a careful computer evaluation of the intensity distribution in reciprocal space, in order to determine the optimal latent lattice of a given structure. – The unit cell of the structure containsM(= number of atoms) cells of the latent lattice. As long asMis a product of three integersMxMyMz, there is no difficulty in finding the optimal latent lattice in diffraction. Each set of directions has to be tested in reciprocal space. In all other cases difficulties arise which may only be overcome with the aid of latent sublattices (smaller cell, more atoms in the structure), or latent superlattices (larger cell, less atoms in the structure). Both cases are discussed, and it is shown that the existence of the various types of latent lattices may be found in the normal diffraction pattern, applying the criteria derived in this paper. The meaning of latent sub- and superlattices is discussed from the structural point of view, and phase relations, caused by the prohibition of trespassing of points, are derived. The great power of the new method will be discussed in a later paper, where differences of shift functions are introduced. Their important consequences for phase determination will be demonstrated with the aid of certain systems of linear equations.

The Fourier coefficients are investigated for such a periodic time function whose magnitude keeps constant during the time of every half-period. In this case the relationship between the mean value and the Fourier coefficients is achieved using appropriate mathematical theory. It was applied successfully to the study of torsional harmonic excitations due to gas pressure in a four-cycle IC engine. A formula relating the truncated Fourier coefficient series to MIP has been established with very little error. The series simulation of Fourier coefficients is utilized in the determination and analysis of excitations.

The studies about the analytic representations of some wavelength functions have been made by many investigators.The purpose of this paper is to investigate the method of expressing the fundamental wavelength functions of C. I. E. system with Fourier series.The numerical data of Fourier's coefficients of the important wavelength functions and some examples calculating the chromaticities by means of Fourier's coefficients are explained.

A simple growth condition is found for the Fourier coefficients of continuous 1-periodic functions of bounded fc-entropy norm.This growth condition is equivalent to the condition given by Dabrowski.The entropy norms of certain random Fourier series are considered. The growth conditionLet k : [0, 1] - [0, 1] be nondecreasing, concave, with k(0+) = 0, k(l) = 1, and k'(0) = lim^Q k(x)/x = oo .We define the entropy norm based on k as in Korenblum [7], and let Ck be the space of continuous 1-periodic functions of finite entropy norm.Let p G PM(0, 1] be the probability measure associated with k as in Dabrowski [4], so that k(x) = -dp(u)dt. Jo J(t,l] uThen the entropy norm of / G Ck may be computed by \\f\\k= f f osc[i+;c t]fdp(x)dt JtJ(o,i] 'where T = R mod 1 is identified with [0, 1], andfor any interval /.Let h(x) = (cosx -I + x /2)/x for x > 0. Let n = f(0 xx2nnh(2nnx)dp(x).In Dabrowski [2] (and [3]), the theorem is given: If f G Ck then Sn^o \anJn\ < wnere / ~ Y^anen ls tne Fourier series of / (and en(t) = e nm for n G Z). Our goal is to simplify this result.We will estimate n .

We compare rates of decay of the cosine Fourier coefficients of a 2φ-periodic function f with that of Riemann sum of f over [0, 2φ] minus .

In this paper we study the Fourier coefficients of theta functions attached to Dirichlet characters at cusps other than infinity. The method is based on expressing them in terms of explicit elements of the adelic Schwartz space and studying the action of the adelic metaplectic group on these elements. We derive explicit formulae for the Fourier coefficients at all cusps. For the sake of simplicity, some restrictions are placed on the Dirichlet characters considered.

The implication is a transformation of a Harker synthesis that has the property of indicating the possible positions of atoms in the crystal structure. The location of the atoms is subject to a certain ambiguity which can be defined by an ambiguity factor, m = M/q. M is a basic ambiguity characteristic of the symmetry axis and arising from group products, while q is a degeneracy factor expressing a particular kind of equivalence of these products. The ambiguity factor can also be determined more easily with the aid of missing spectra. It is shown that M is the modulus of the transformation from the implication cell to the crystal cell, and 1/q is the fraction of spectra present. The ambiguity factor of any implication synthesis can, therefore, be easily determined from the symmetry and characteristic missing spectra of the space group. The ambiguity of an implication can be resolved with the aid of m Patterson line syntheses. The implication synthesis, therefore, provides a theoretical device for untangling the Patterson synthesis. This suggests that there should exist a simple relation between the Fourier coefficients in the implication function and the corresponding coefficients in the Fourier expression of the projected electron density. The details of this relation are dealt with in another paper.