PROPERTIES AND CALCULATION OF PAIRED HAAR TRANSFORM

Different properties of recently introduced Paired Haar transform have been shown. Nonpolynomial Haar Pxpansion of incompletely specified Boolean functions has been presented. Based on the above properties and expansion some applications of Paired Haar spectrum have been proposed. Algorithm for the calculation of Haar Pair spectrum from disjoint cubes for systems of incompletely specified Boolean functions has also been developed.     

代数免疫度为1的布尔函数

布尔函数的代数免疫度是在流密码的代数攻击中所产生的重要概念.研究了代数免疫度为1的布尔函数,得到的主要结果有:对代数免疫度为1的布尔函数给出了一个谱刻画,给出了其个数的精确计数公式,最后给出了此类函数的非线性度的紧的上界.     

Bent函数的一般构造法

本文用概率方法给出小项表示的布尔函数谱的性质,据此得到了Ｂｅｎｔ函数的特征矩阵的等价刻画,原则上给出了Ｂｅｎｔ函数的一般构造法,并为Ｂｅｎｔ函数的计数问题提供了一个模型。文中还提出了Ｂｅｎｔ矩阵的概念,考察了Ｂｅｎｔ矩阵的性质,并借助Ｂｅｎｔ矩阵得到由已知Ｂｅｎｔ函数构造新的Ｂｅｎｔ函数构造新的Ｂｅｎｔ函数的方法。     

Directional Haar wavelet frames on triangles

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C²-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.     

Local limit theorems for multiplicative free convolutions

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici–Pata bijection.     

On stationary classes of three-value logic functions

This work is devoted to the properties of transformations of the vector of values of three-value logic functions to the vector of coefficients of their polynomials. A similar transformation of Boolean functions is used in cryptology, and its properties have been thoroughly studied. Stationary classes of three-value logic functions are introduced, and their hierarchy and the exact number of functions in them are obtained.     

New construction of wavelets base on floor function

In this paper, the properties of the floor function has been used to find a function which is one on the interval [0, 1) and is zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a,b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine–Cosine wavelet, Block-Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions.     

Structure of stationary classes of three-value logic functions

This work deals with the properties of transformations of the vectors of values of three-value logic functions to the vectors of the coefficients of their polynomials. A similar transformation of Boolean functions is used in cryptology, and its properties have been thoroughly studied. Stationary classes of three-value logic functions are introduced, and their hierarchy and the exact number of functions in them are obtained. The complete structure of these classes is described.     

State Analysis and Optimal Control of Time-Varying Discrete Systems via Haar Wavelets

The state analysis and optimal control of time-varying discrete systems via Haar wavelets are the main tasks of this paper. First, we introduce the definition of discrete Haar wavelets. Then, a comparison between Haar wavelets and other orthogonal functions is given. Based upon some useful properties of the Haar wavelets, a special product matrix and a related coefficient matrix are proposed; also, a shift matrix and a summation matrix are derived. These matrices are very effective in solving our problems. The local property of the Haar wavelets is applied to shorten the calculation procedures.     

广义部分Bent函数和广义Bent函数的关系

Bent函数是一类特殊的布尔函数,因其非线性性和稳定性在密码学和通信等领域有很重要的应用,但它们数量少,不平衡且无相关免疫性,为了弥补Bent函数的不足,Claud Carlet提出了部分Bent函数的概念,部分Bent函数是包含Bent函数的更大的函数类,后来,人们又将这两种函数概念先后都拓广到了环zm^n(m为正整数）上,分别被称为zm^n上的广义Bent函数和广义部分Bent函数,本文利用zp^n(p为素数）上广义部分Bent函数的Chrestenson循环谱特征讨论了zp^n上的广义部分Bent函数和广义Bent函数之间的关系,给出了这两种函数之间的函数关系式和谱值关系式。     

Boolean Functions: Degree and Support

In this paper we establish some properties about Boolean functions that allow us to relate their degree and their support. These properties allow us to compute the degree of a Boolean function without having to calculate its algebraic normal form. Furthermore, we introduce some linear algebra properties that allow us to obtain the degree of a Boolean function from the dimension of a linear or affine subspace. Finally we derive some algorithms and compute the average time to obtain the degree of some Boolean functions from its support.     

The expressive power of binary submodular functions

We investigate whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This question has been considered in several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively.Our results have several corollaries. First, we characterise precisely which submodular polynomials of arity 4 can be expressed by binary submodular polynomials. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish, for the first time, that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.     

A unified framework for testing linear‐invariant properties

In the study of property testing, a particularly important role has been played by linear invariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed‐Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear‐invariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following:

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Cross products of complete orthonormal systems of functions

The orthonormal kernel is a continuous analog for an orthonormal system of functions. The cross product of any two orthonormal systems, complete in L₂, is an example of a complete orthonormal kernel with respect to Lebesgue measure. In this note we continue our study of the properties of the cross product of a Haar system with an arbitrary orthonormal system of functions, complete in L₂, and totally bounded. We investigate certain properties of the cross product of a Haar system with another Haar system.Translated from Matematicheskie Zametki, Vol. 15, No. 2, pp. 331–340, February, 1974.The author thanks Professor N. Ya. Vilenkin for helpful discussions during the course of this work.     

Level structure of Zhegalkin polynomials,properties of test sets,and an annihilator search algorithm

Properties of the Boolean functions specified by the Zhegalkin polynomials in n variables of degree not greater than k are investigated from the viewpoint of placing their unit (zero) points on a unit cube. Properties of test sets for the Zhegalkin polynomials are considered, where the key role is played by the irredundant test sets. A deterministic algorithm for finding all the annihilators for a given polynomial is described including minimal-degree annihilators that have applications in cryptology. In the available algorithms for finding annihilators, the problem is reduced to solving systems of linear Boolean equations. Reducing the dimension of these systems decreases the algorithmic complexity of solving the problem. The proposed algorithm makes it possible to decrease the complexity of finding annihilators by reducing the dimension of such systems but it does not reduce the asymptotic complexity of solving systems of linear Boolean equations.     

Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via rationalized Haar functions

Rationalized Haar functions are developed to approximate of the nonlinear Volterra–Fredholm–Hammerstein integral equations. The properties of rationalized Haar functions are first presented, and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.     

Concertina-like movements of the error curve in the alternation theorem

If a continuous function f is approximated by elements of a Haar space in the maximum norm on an interval, the error curve of the best approximation has well known alternation properties. It is shown that if f is adjoined to the Haar space all zeros of the error function are monotonously increasing functions of the endpoints, and that under an additional hypothesis, the entire graph of the error curve is shifted to the left or right when the endpoints are moved accordingly.     

具有特定非零Walsh谱值个数的布尔函数的研究及构造

布尔函数与其变元的相关性与流密码的相关攻击有紧密联系，Walsh变换则是研究布尔函数相关特性的主要工具，本文研究了非零Walsh谱值个数k=9，10的布尔函数，证明了k=9的函数的不存在性，并构造了所有k=10的函数。     

Construction of balanced Boolean functions with high nonlinearity,good local and global avalanche characteristics

Boolean functions possessing multiple cryptographic criteria play an important role in the design of symmetric cryptosystems. The following criteria for cryptographic Boolean functions are often considered: high nonlinearity, balancedness, strict avalanche criterion, and global avalanche characteristics. The trade-off among these criteria is a difficult problem and has attracted many researchers. In this paper, two construction methods are provided to obtain balanced Boolean functions with high nonlinearity. Besides, the constructed functions satisfy strict avalanche criterion and have good global avalanche characteristics property. The algebraic immunity of the constructed functions is also considered.