A note on two classes of Boolean functions with optimal algebraic immunity

Tu and Deng proposed a class of bent functions which are of optimal algebraic immunity under the assumption of a combinatorial conjecture.In this paper,the authors compute the dual of the Tu-Deng functions and then show that they are still of optimal algebraic immunity under the assumption of the same conjecture.For another class of Boolean functions constructed by Tang,et al.which are of optimal algebraic immunity with similar forms to Tu-Deng functions,the authors show that they are not bent functions by using some basic properties of binary complete Kloosterman sums.     

A Construction of 1-Resilient Boolean Functions with Good Cryptographic Properties

This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic immunity are also considered. A sufficient condition of the modified functions with optimal algebraic degree in terms of the Siegenthaler bound is proposed. The authors obtain a lower bound on the nonlinearity of the Tang-Carlet-Tang functions, which is slightly better than the known result. If the authors do not break the “continuity” of the support and zero sets, the functions constructed in this paper have suboptimal algebraic immunity. Finally, four specific classes of 1-resilient Boolean functions constructed from this construction and with the mentioned good cryptographic properties are proposed. Experimental results show that there are many 1-resilient Boolean functions have higher nonlinearities than known 1-resilient functions modified by Tu-Deng and Tang-Carlet-Tang functions.     

A Combinatorial Condition and Boolean Functions with Optimal Algebraic Immunity

This paper first proposes an infinite class of 2k-variable Boolean functions with high nonlinearity and high algebraic degree. Then an infinite class of balanced Boolean functions are proposed by modifying the above Boolean functions. This class of balanced Boolean functions have optimal algebraic degree and high nonlinearity. Both classes have optimal algebraic immunity based on a general combinatorial conjecture.     

On algebraic immunity of trace inverse functions on finite fields of characteristic two

The trace inverse functions Tr(λx~(-1)) over the finite field F_(2~n) are a class of very important Boolean functions and are used in many stream ciphers such as SFINKS,RAKAPOSHI,the simple counter stream cipher(SCSC) presented by Si W and Ding C(2012),etc.In order to evaluate the security of those ciphers in resistance to(fast) algebraic attacks,the authors need to characterize algebraic properties of Tr(λx~(-1)).However,currently only some bounds on algebraic immunity of Tr(λx~(-1)) are given in the public literature,for example,the NGG upper bound and the Bayev lower bound,etc.This paper gives the exact value of the algebraic immunity of Tr(λx~(-1)) over F_(2~n),that is,AI(Tr(λx~(-1))) =[2n~(1/2)]- 2,where n ≥ 2,A ∈ F_(2~n) and λ≠ 0,which shows that Dalai's conjecture on the algebraic immunity of Tr(λx~(-1)) is correct.What is more,the authors demonstrate some weak properties of Tr(λx~(-1)) against fast algebraic attacks.     

The Log-Concavity of Kazhdan-Lusztig Polynomials of Uniform Matroids

Elias, et al.(2016) conjectured that the Kazhdan-Lusztig polynomial of any matroid is logconcave. Inspired by a computer proof of Moll’s log-concavity conjecture given by Kauers and Paule,the authors use a computer algebra system to prove the conjecture for arbitrary uniform matroids.     

Constructions of unitary space-time codes with full diversity

The problem of constructing unitary space-time codes with high diversity product has been studied in many prior works. Recently, constructions of parametric fully diverse unitary space-time codes for prime number antennas system have been introduced. In this paper, the authors propose new construction methods based on these constructions. And fully diverse codes of any number antennas are obtained from these constructions. Unitary space-time codes from present constructions are found to have better error performance than many best known ones.     

Application of Sturm Theorem in the Global Controllability of a Class of High Dimensional Polynomial Systems

In this paper, the global controllability for a class of high dimensional polynomial systems has been investigated and a constructive algebraic criterion algorithm for their global controllability has been obtained. By the criterion algorithm, the global controllability can be determined in finite steps of arithmetic operations. The algorithm is imposed on the coefficients of the polynomials only and the analysis technique is based on Sturm Theorem in real algebraic geometry and its modern progress.Finally, the authors will give some examples to show the application of our results.     

Exponentially adaptive synchronization of an uncertain delayed dynamical network

Over the past decades, complex networks have been prosperous greatly in various fields of sciences and engineering. Much attention has been given to investigate the synchronization of complex networks in recent years. However, few work has done for the networks with uncertain parameters and unknown topology. In this paper, to further reveal the dynamical mechanism in complex networks with time delays, an uncertain general complex dynamical network with delayed nodes is studied. By constructing a drive network and a suitable slave network, several novel criteria for the networks consisting of the identical nodes and different nodes have been obtained based on the adaptive feedback method. Particularly, the hypotheses and the proposed adaptive laws for network synchronization are simple and can be readily applied in practical applications. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed synchronization criteria.     

B-spline patches and transfinite interpolation method for PDE controlled simulation

This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method developed by Klaus Hollig to derive this new idea.First of all,the authors replace the R-function method with transfinite interpolation to build a function which vanishes on boundaries. Secondly,the authors simulate the partial differential equation by directly applying differential operators to basis functions,which is similar to the RBF method rather than Hollig’s method.These new strategies then make the constructing of bases and the linear system much more straightforward.And as the interpolation is brought in,the design of schemes for solving practical PDEs can be more flexible. This new method is easy to carry out and suitable for simulations in the fields such as graphics to achieve rapid rendering.Especially when the specified precision is not very high,this method performs much faster than WEB-spline method.     

Results on permutation symmetric Boolean functions

This paper provides a systematic method on the enumeration of various permutation symmetric Boolean functions. The results play a crucial role on the search of permutation symmetric Boolean functions with good cryptographic properties. The proposed method is algebraic in nature. As a by-product, the authors correct and generalize the corresponding results of St?nic? and Maitra (2008). Further, the authors give a complete classification of block-symmetric bent functions based on the results of Zhao and Li (2006), and the result is the only one classification of a certain class of permutation symmetric bent functions after the classification of symmetric bent functions proposed by Savicky (1994).     

Dembowski-Ostrom polynomials from reversed Dickson polynomials

This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form x~(2+2~α) is almost perfect nonlinear.It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.     

Parametric equation solving and quantifier elimination in finite fields with the characteristic set method

For a parametric algebraic system in finite fields,this paper presents a method for computing the cover and the refined cover based on the characteristic set method.From the cover,the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains.By the refined cover,the author gives a complete classification of the number of solutions for this system,that is,the author divides the parameter space into several disjoint components,and on every component the system has a fix number of solutions.Moreover,the author develops a method of quantifier elimination for first order formulas in finite fields.     

基于贝叶斯优化构建DBN结构优化算法

针对动态贝叶斯网络(DBN)结构学习问题,提出了一种基于贝叶斯优化(BOA)的DBN结构寻优算法。首先,从传统进化优化机制的基本理论和基本操作入手,刻划了基于概率模型进化算法的基本思想。其次,通过描述基于概率模型进化算法的构图基础,引出了DBN结构学习机制,即基于BOA的DBN结构寻优算法。BOA算法的关键是根据优良解集学习得到DBN,以及根据DBN推理生成新个体,前者更为重要,依据基于贪婪机理的遗传算法解决动态网络学习,再应用DBN前向模拟完成后一步。仿真结果表明了该算法的可行性。     

可重入柔性调度问题研究:模型、算法与应用

近二十年来,在企业提高生产效率和增加生产灵活性方面的内在需求推动下,一类广泛存在于半导体制造、钢铁生产、汽车制造、武器生产、化工、微生物检测、金融财务等制造和服务领域,且具有多阶段、并行机拓扑结构的柔性生产调度问题的研究和应用正得到学术界和工业界越来越多的重视.考虑到在实际生产过程中,工件在部分阶段的加工工艺路径为循环的技术要求,与实际生产过程更加贴近的可重入系统的特性被引入到柔性制造系统.鉴于可重入柔性调度问题(reentran flexible scheduling problem,REFSP)在理论上的重要性及其对于实际调度的指导性,本文综述了可重入柔性调度问题的研究进展.首先介绍了可重入柔性调度问题的基本数学模型,并基于α|β|γ表示法对已有的复杂环境下的可重入柔性调度问题进行了归类,接着对其优化求解方法进行了归纳,并梳理了其主要应用领域,最后提出了有待进一步研究的若干方向和内容,旨在推进国内在可重入柔性调度问题研究领域的理论研究和工程应用.     

Multi-facility ordered median problems in directed networks

This paper uses a finite dominating set (FDS) to investigate the multi-facility ordered median problem (OMP) in a strongly connected directed network. The authors first prove that the multi-facility OMP has an FDS in the node set, which not only generalizes the FDS result provided by Kalcsics, et al. (2002), but also extends the FDS result from the single-facility case to the multiple case, filling an important gap. Then, based on this FDS result, the authors develop an exact algorithm to solve the problem. However, if the number of facilities is large, it is not practical to find the optimal solution, because the multi-facility OMP in directed networks is NP-hard. Hence, we present a constant-approximation algorithm for the p-median problem in directed networks. Finally, we pose an open problem for future research.     

A Hybrid Procedure for Finding Real Points on a Real Algebraic Set

Journal of Systems Science and Complexity - Motivated by the idea of Shen, et al.’s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy...     

A BSDE Approach to Stochastic Differential Games Involving Impulse Controls and HJBI Equation

This paper focuses on zero-sum stochastic differential games in the framework of forward-backward stochastic differential equations on a finite time horizon with both players adopting impulse controls. By means of BSDE methods, in particular that of the notion from Peng’s stochastic backward semigroups, the authors prove a dynamic programming principle for both the upper and the lower value functions of the game. The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle. As a consequence, the uniqueness implies that the upper and lower value functions coincide and the game admits a value.

     

On Basis and Pure Nash Equilibrium of Finite Pure Harmonic Games

This paper investigates the basis and pure Nash equilibrium of finite pure harmonic games (FPHGs) based on the vector space structure. First, a new criterion is proposed for the construction of pure harmonic subspace, based on which, a more concise basis is constructed for the pure harmonic subspace. Second, based on the new basis of FPHGs and auxiliary harmonic vector, a more easily verifiable criterion is presented for the existence of pure Nash equilibrium in basis FPHGs. Third, by constructing a pure Nash equilibrium cubic matrix, the verification of pure Nash equilibrium in three-player FPHGs is given.

     

基于A-H股溢价率的价值投资策略研究

寻找有效定价因子并构建可获得超额收益的投资策略一直是金融理论界和实务界关注的核心问题.本文运用组合价差法分别研究了基于A-H股溢价率的3种价值投资策略1998年至2007年的超额收益,发现空间策略和时间策略都可获得超额期望收益,时空交叉策略又可进一步提高超额收益.基于Fama等(1992)构造合溢价率的4因素模型,证明了在控制市场风险、B/M比值和公司规模3个因素后,溢价率效应依然存在.     

Exponential Sums over Finite Fields

This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open problems are presented along the way, with particular emphasis on Gauss periods, Kloosterman sums and one variable exponential sums. One main tool is the applications of various p-adic methods. For this reason, the author has also included a brief exposition of certain p-adic estimates of exponential sums. The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University. Notes were taken by Shaoshi Chen and Ruichen Xu.