On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.     

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。     

Normal correlation coefficient of non-normal variables using piece-wise linear approximation

The correlation coefficient of non-normal variables is expressed as a function of the correlation coefficient of normal variables using piece-wise linear approximation of each univariate transform of normal to anything, and the second order moments of a multiply truncated bivariate normal distribution. For the inverse problem, an algorithm iterates this analytic function in order to assign a normal correlation coefficient to two non-normal variables. The algorithm is applied for the generation of randomized bivariate samples with given correlation coefficient and marginal distributions and used in a randomization test for bivariate nonlinearity. The test correctly does not reject the null hypothesis of linear correlation if the nonlinearity is plausible and due to the sample transform alone.     

On CCZ-equivalence of addition mod 2 n

We show that addition mod 2 ⁿ is CCZ-equivalent to a quadratic vectorial Boolean function. We use this to reduce the solution of systems of differential equations of addition to the solution of an equivalent system of linear equations and to derive a fully explicit formula for the correlation coefficients, which leads to enhanced results about the Walsh transform of addition mod 2 ⁿ . The results have direct applications in the cryptanalysis of cryptographic primitives which use addition mod 2 ⁿ .     

Modified Cross-Validation for Penalized High-Dimensional Linear Regression Models

In this article, for Lasso penalized linear regression models in high-dimensional settings, we propose a modified cross-validation (CV) method for selecting the penalty parameter. The methodology is extended to other penalties, such as Elastic Net. We conduct extensive simulation studies and real data analysis to compare the performance of the modified CV method with other methods. It is shown that the popular K-fold CV method includes many noise variables in the selected model, while the modified CV works well in a wide range of coefficient and correlation settings. Supplementary materials containing the computer code are available online.     

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

So far there is no systematic attempt to construct Boolean functions with maximum annihilator immunity. In this paper we present a construction keeping in mind the basic theory of annihilator immunity. This construction provides functions with the maximum possible annihilator immunity and the weight, nonlinearity and algebraic degree of the functions can be properly calculated under certain cases. The basic construction is that of symmetric Boolean functions and applying linear transformation on the input variables of these functions, one can get a large class of non-symmetric functions too. Moreover, we also study several other modifications on the basic symmetric functions to identify interesting non-symmetric functions with maximum annihilator immunity. In the process we also present an algorithm to compute the Walsh spectra of a symmetric Boolean function with O(n²) time and O(n) space complexity. We use the term “Annihilator Immunity” instead of “Algebraic Immunity” referred in the recent papers [3–5, 9, 18, 19]. Please see Remark 1 for the details of this notational change     

Spectral Decomposition of Cyclic Operators in Discrete Harmonic Analysis

For the operators of the discrete Fourier transform, the discrete Vilenkin–Christenson transform, and all linear transpositions of the discrete Walsh transform, we obtain their spectral decompositions and calculate the dimensions of eigenspaces. For complex operators, namely, the discrete Fourier transform and the Vilenkin–Christenson transform, we obtain real projectors on eigenspaces. For the discrete Walsh transform, we consider in detail the Paley and Walsh orderings and a new ordering in which the matrices of operators are symmetric. For operators of linear transpositions of the discrete Walsh transforms with nonsymmetric matrices, we obtain a spectral decomposition with complex projectors on eigenspaces. We also present the Parseval frame for eigenspaces of the discrete Walsh transform.     

A practical non-parametric copula algorithm for system reliability with correlations

System reliability analysis involving correlated random variables is challenging because the failure probability cannot be uniquely determined under the given probability information. This paper proposes a system reliability evaluation method based on non-parametric copulas. The approximated joint probability distribution satisfying the constraints specified by correlations has the maximal relative entropy with respect to the joint probability distribution of independent random variables. Thus the reliability evaluation is unbiased from the perspective of information theory. The estimation of the non-parametric copula parameters from Pearson linear correlation, Spearman rank correlation, and Kendall rank correlation are provided, respectively. The approximated maximum entropy distribution is then integrated with the first and second order system reliability method. Four examples are adopted to illustrate the accuracy and efficiency of the proposed method. It is found that traditional system reliability method encodes excessive dependence information for correlated random variables and the estimated failure probability can be significantly biased.     

Results on rotation symmetric bent functions

In this paper we analyze the combinatorial properties related to the Walsh spectra of rotation symmetric Boolean functions on even number of variables. These results are then applied in studying rotation symmetric bent functions. For the first time we could present an enumeration strategy for all the 10-variable rotation symmetric bent functions.     

广义Walsh变式与一极值问题

<正> 设p为大于1的整数,t为非负实数,t的p进表示为     

Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros

In this paper we study the neighbourhood of 15-variable Patterson-Wiedemann (PW) functions, i.e., the functions that differ by a small Hamming distance from the PW functions in terms of truth table representation. We exploit the idempotent structure of the PW functions and interpret them as Rotation Symmetric Boolean Functions (RSBFs). We present techniques to modify these RSBFs to introduce zeros in the Walsh spectra of the modified functions with minimum reduction in nonlinearity. Our technique demonstrates 15-variable balanced and 1-resilient functions with currently best known nonlinearities 16272 and 16264 respectively. In the process, we find functions for which the autocorrelation spectra and algebraic immunity parameters are best known till date.     

中国股市行业板块的动态相关性建模——基于DCC—MVGARCH模型

相关性的讨论是现代金融分析的重要内容,行业板块的相关分析是组合投资的关键步骤。基于能刻画动态相关的DCC-MVGARCH模型对我国波动剧烈的六个股市行业板块进行了相关性研究,结果表明：十个板块相关性序列可看成常量,Pearson相关系数仍能刻画其相对大小,这为机构投资者按Pearson相关系数进行组合构建提供了实证依据;五个动态相关性序列是宽平稳而非严平稳的,适合采用随机过程建模以实现预测,另外动态相关性与时变波动率存在一定的关系,当波动率增强时,相关性有随之增大的趋势。     

A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order “rogue wave” solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.     

Binary linear codes with few weights from Boolean functions

Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de, and the duals of some of them are distance-optimal with respect to the sphere packing bound.

     

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.     

斜变换ST的演化生成与快速算法

１．引言 含有“斜”基向量的正交变换（斜变换 ＳＴ）概念是由 Ｅｎｏｍｏｔｏ ＆ Ｓｈｉｂａｔａ（１９７１）提出的［１］．斜向量是一个在其范围内呈均匀阶梯下降的离散锯齿波形．对于亮度逐渐改变的图象,用斜向量来表示是适合的． Ｅｎｏｍｏｔｏ 　＆ 　Ｓｈｉｂａｔａ仅考虑了斜向量长度为 ４和 ８的情况．Ｐｒａｔｔ等人利用递推性将 ＳＴ推广到 Ｎ＝ ２ｍ阶的情形,给出了 ＳＴ的一般定义［２］,并与其它变换进行了比较［３］．ＳＴ已成功地用在图象编码上,而且在非正弦类交换编码的应用中,斜变换的效果最好［２,３］． Ａｈ…     

Refined estimates of the number of repetition-free boolean functions in the full binary basis {&, ∨, ⊕, −

We consider repetition-free Boolean functions in the basis {&, ∨, ⊕, ?}, and prove a formula expressing the number of such functions of n variables as a product of Fibonacci numbers. These products are estimated; as a result, we obtain asymptotic estimates for the number of repetition-free Boolean functions. These estimates involve Euler numbers of second order and can be reduced by well-known methods to the form of an exponential-power series. These estimates can be used to construct the final asymptotics of the number of repetition-free Boolean functions in the full binary basis.     

基于类Pearson综合相关系数的概率语言TOPSIS多属性决策方法

概率语言术语集(PLTS)包含了语言术语信息及其相应的概率信息,使得原始决策信息得到充分利用,大大提高了语言术语多属性决策的科学性.文章研究了一种基于概率语言术语集类Pearson综合相关系数的TOPSIS多属性决策方法.首先考虑了PLTSs的3个特征因素:均值、方差和长度,然后基于传统Pearson相关系数的思想,提...     

An exact penalty approach for solving a class of minimization problems with boolean variables

An exact penalty approach for solving minimization problems with a concave objective function, linear constraints and Boolean variables is proposed. The penalty problems have continuous variables. An estimation of the penalty parameter which guarantees the exactness can be calculated on the base of an auxiliary problem. The results are applied to problems with an arbitrary quadratic objective function, linear constraints and Boolean variables. This leads to a modified Lagrangean approach for the latter problems. In the general case, the penalty approach is compared with a direct application of results of global optimization to a modification of the initial problem.