轻量级迭代MDS矩阵的构造

随着具有最大分支数的扩散层在分组密码和hash函数中的应用,轻量级MDS矩阵的构造受到广泛关注.基于迭代构造是实现低成本MDS矩阵的一种有效方法.将低成本的矩阵通过迭代一定次数后成为MDS矩阵,通过实现该低成本的矩阵来实现MDS性质.但是这种方法需要以增加迭代次数个时钟周期等待时间为代价.本文通过减少矩阵迭代次数,从而降低矩阵实现的延迟,来构造更加轻量的迭代MDS矩阵.在迭代次数小于等于阶数时,本文给出了有限域F_{2^m}上4阶矩阵在不同迭代次数下,能够成为迭代MDS矩阵所含非零元个数的下界;进一步地,通过矩阵置换相似分类及MDS条件,不断减小可行空间,实现了非零元个数达到下界的迭代MDS矩阵的穷搜,从而找到在该迭代次数下异或数达到下界的4阶迭代MDS矩阵.     

轻量级迭代MDS矩阵的构造

随着具有最大分支数的扩散层在分组密码和hash函数中的应用,轻量级MDS矩阵的构造受到广泛关注.基于迭代构造是实现低成本MDS矩阵的一种有效方法.将低成本的矩阵通过迭代一定次数后成为MDS矩阵,通过实现该低成本的矩阵来实现MDS性质.但是这种方法需要以增加迭代次数个时钟周期等待时间为代价.本文通过减少矩阵迭代次数,从而降低矩阵实现的延迟,来构造更加轻量的迭代MDS矩阵.在迭代次数小于等于阶数时,本文给出了有限域F_{2^m}上4阶矩阵在不同迭代次数下,能够成为迭代MDS矩阵所含非零元个数的下界;进一步地,通过矩阵置换相似分类及MDS条件,不断减小可行空间,实现了非零元个数达到下界的迭代MDS矩阵的穷搜,从而找到在该迭代次数下异或数达到下界的4阶迭代MDS矩阵.     

基于Cauchy矩阵的线性变换的研究

P-置换是实现分组密码扩散原则的关键组件.一般来说,分支数越大,扩散效果越明显.人们利用MDS矩阵设计最优线性变换作为分组密码组件的扩散层.在达到最优线性变换的同时,针对扩散矩阵还应满足矩阵中元素尽量少的要求,对Cauchy型MDS矩阵分别与Hadmard矩阵和循环移位矩阵的相互结合方式构造最优线性层的方法进行了研究.对Cauchy-Hadmard矩阵(同时是Cauchy矩阵和Hadmard矩阵)构造线性变换的一种方法进行了分析,给出了算法的C语言的关键程序,根据算法给出了一个最优线性变换的示例;对循环移位矩阵构造Cauchy矩阵进行了尝试和证明.结果显示Cauchy-Hadmard矩阵满足矩阵元素最少和运算复杂度低的要求,利用循环移位矩阵无法构造出Cauchy矩阵.这些结论为设计分组密码组件的扩散层提供了重要的方法参考.     

MDS矩阵的比特变换性质及计数研究

本文对某些特定形式MDS矩阵的比特变换性质进行了研究,得到了输入输出具有相同比特奇偶性的等价条件,并给出了理论证明。另外,还给出了MDS矩阵的计数公式并给出了相应证明。最后,在理论分析的基础上进行了计算机统计实验,实验结果与理论推导结果一致。     

基于二元关系的矩阵变换及应用

介绍了关系矩阵变换和关系同构的概念,证明了关系同构的两个充要条件.讨论了同构关系之逆关系、关系的传递闭包的同构性质等.给出了用关系矩阵变换判定关系等价性的几个充要条件.     

实际意义下n-D单边矩阵方程有解的充要条件

本文在讨论n-D矩阵方程求解条件的基础上, 针对由Agathoklis等人提出的具有实际意义的n维（n-D）离散系统,给出了判定n-D矩阵 方程是否有解的充要条件,并给出了判定系统是否具有实际意义稳定性的方法,以算例加以 说明．     

实际意义下n—D单边矩阵方程有解的充要条件

本文在讨论了ｎ－Ｄ矩阵方程求解条件的基础上，针对由Ａｇａｔｈｏｋｌｉｓ等人提出的具有实际意义的ｎ维离散系统，给出了判定ｎ－Ｄ矩阵方程是否有解的充要条件，并给出了判定系统是否具有实际意义稳定性的方法，以算例加以说明。     

两个非对称图量子MDS码的构造

量子纠错编码技术在量子信息理论中一直以来有着重要的地位,在量子纠错编码方案中,Schingemann和Werner两人提出了通过构造具有某些性质的图（矩阵）来构造非二元量子码的方法,他们利用这种图论方法构造出很多好的量子码,特别给出量子码[[[5,1,3]]p][(p≥3)]存在性的一个新证明。此方法可从对称量子码推广至非对称量子码的构造,利用推广方法证明了非对称图量子MDS码[[[5,1,4/2]]p],[(p>5)]和[[[7,1,6/2]]p][(p>7)]的存在性。     

基于图学习正则判别非负矩阵分解的人脸识别

基于图正则非负矩阵分解（NMF）算法充分利用了高维数据通常位于一个低维流形空间的假设从而构造拉普拉斯矩阵,但该算法的缺点是构造出的拉普拉斯矩阵是提前计算得到的,并没有在乘性更新过程中对它进行迭代。为了解决这个问题,结合子空间学习中的自表示方法生成表示系数,并进一步计算相似性矩阵从而得到拉普拉斯矩阵,而且在更新过程中对拉普拉斯矩阵进行迭代。另外,利用训练集的标签信息构造类别指示矩阵,并引入两个不同的正则项分别对该类别指示矩阵进行重构。该算法被称为图学习正则判别非负矩阵分解（GLDNMF）,并给出了相应的乘性更新规则和目标函数的收敛性证明。在两个标准数据集上的人脸识别实验结果显示,和现有典型算法相比,所提算法的人脸识别的准确率提升了1% ~ 5%,验证了其有效性。     

基于Q矩阵的LDPC码编码设计方法

给出了Q矩阵的定义,在此基础上提出了一种基于Q矩阵的LDPC码编码器设计方法.此设计方案将奇偶校验矩阵H分解成两个子矩阵Hp和Hd,通过对这两个子矩阵分别设计,构造出H矩阵.其中Hp是上三角双对角矩阵,Hd矩阵由Q矩阵按照一定的行重列重,根据某种排列规则构造而成.着重研究了Q排列规则,并找到了一种较好方法,使得这种编码方案得到了很好的性能.采用的这种编码方案的编码器算法具有较低的实现复杂度,为LDPC码编码器设计提供了很好的选择方案.     

Inversion of confluent Vandermonde matrices

This paper is concerned with the inversion of confluent Vandermonde matrices. A novel and simple recursive algorithm for inverting confluent Vandermonde matrices is presented. The algorithm is suitable for classroom use in both numerical as well as symbolic computation. Examples are included to illustrate the proposed algorithm.     

On the quadratic stability of switched linear systems associated with symmetric transfer function matrices

In this paper we give necessary and sufficient conditions for weak and strong quadratic stability of a class of switched linear systems consisting of two subsystems, associated with symmetric transfer function matrices. These conditions can simply be tested by checking the eigenvalues of the product of two subsystem matrices. This result is an extension of the result by Shorten and Narendra for strong quadratic stability, and the result by Shorten et al. on weak quadratic stability for switched linear systems. Examples are given to illustrate the usefulness of our results.     

An improved method for determining the stability of interval matrices

In this paper, the problem of the stability of interval matrices has been tackled using the properties of real stability radius. Based on these, a necessary and sufficient condition has been developed for the Hurwitz (Schur) stability of an interval matrix. An algorithm has been suggested on the basis of the above results to determine the stability of such a system. This work provides an alternative tool to that proposed by Wang et al. and it has been claimed, on the basis of comparative results, that it is a more efficient method in terms of both the computation time and the number of matrices to be checked.     

分组密码二元扩散结构的几点注记

0-1矩阵常用于设计分组密码的扩散结构,首先证明,当GF(2n)上的矩阵重新定义在扩域GF(2mn)上时其分支教保持不变,据此补充了Choy等人关于GF(2n)上二元矩阵分支数上界的证明.构造了一批分支数达到最优的8阶二元可逆矩阵,给出了一类差分分支数和线性分支数相等的二元可逆矩阵,并从中搜索出了大量16阶分支数达到最优的二元矩阵和对合二元矩阵.     

On the calculus of complex matrices

In this note, elements of matrix calculus are extended to complex matrices. With this extension, the Cauchy-Riemann conditions in matrix notation are introduced. It is shown that for many cases where the complex matrix derivative does not exist, a complex gradient and Hessian matrix—which are defined by partial derivatives with respect to real and imaginary matrix parts—do exist. Further, these first and second-order derivative matrices are used to provide necessary and sufficient conditions for an extremum of a real scalar function of a complex matrix. Some illustrative examples are provided.     

Hankel/Toeplitz matrices and the static output feedback stabilization problem

The static output feedback (SOF) stabilization problem for general linear, continuous-time and discrete-time systems is discussed. A few novel necessary and sufficient conditions are proposed, and a modified SOF stabilization problem with performance is studied. For multiple-input single-output (or single-input multiple-output) systems the relation with a class of Hankel matrices, and their inverses, in the continuous-time case and with a class of Toeplitz matrices, in the discrete-time case, is established. These relationships are used to construct conceptual numerical algorithms. Finally, it is shown that, in the continuous-time case, the problem can be recast as a concave–convex programming problem. A few worked out examples illustrate the underlying theory.     

On the static output feedback stabilisation of discrete event dynamic systems based upon the approach of semi-tensor product of matrices

In this paper, the static output feedback stabilisation of discrete event dynamic systems (DEDSs) is investigated via the semi-tensor product (STP) of matrices. Firstly, the dynamics of DEDSs modelled by deterministic Moore-type finite automata are converted into an matrix expression in the STP frame. Secondly, necessary and sufficient conditions for the existence of a static output feedback control pattern, stabilising the controlled discrete event dynamic systems to some equilibrium point, are given, and constructive algorithms to seek the static output feedback control pattern including an effective specific solution algorithm and an analytic solution algorithm are proposed. Thirdly, the equilibrium-based static output feedback stabilisation of DEDSs is extended to the set-based static output feedback stabilisation and necessary and sufficient conditions and constructive algorithms to seek the corresponding control pattern are provided. Finally, some examples are presented to illustrate the effectiveness of the proposed approach.     

区间矩阵二次稳定的充分必要条件

提出了区间矩阵二次稳定的充分必要条件,以及相应的稳定裕度的计算方法.结论以线性矩阵不等式(LMI)的形式给出.利用功能强大的LMI工具,求解非常方便.所给实例表明,该方法用于确定区间矩阵的鲁棒稳定性及其稳定裕度,非常有效.