几类基于0-线性转换的置换多项式

有限域上的置换多项式在密码学,编码理论和序列设计等领域中有着广泛的应用.至今,对于置换多项式的研究已取得一系列的进展,研究者提出利用AGW准则、分段函数等来构造和证明置换,而对于置换多项式的分类仅有少数几种被提出,因此构造不同类型的置换多项式是一个值得研究的问题.本文利用迹函数和线性化多项式构造了一类有限域上具有特殊形式的无限类置换多项式.首先,我们对一类线性化多项式和一类二次齐次多项式进行讨论,当其存在线性转换时,给出其需要满足的条件.进一步地,当这两类多项式存在0-线性转换时,我们利用这两类函数构造出了有限域上具有特殊形式的两类置换多项式.     

求解本原多项式的快速算法

本原元和本原多项式是有限域理论中的2个重要的概念.本原元的求解问题是解决实际密码序列问题的前提条件,而本原元的求解问题又可以归结为本原多项式的求解问题.该文结合求解最小多项式的方法给出一个在二元有限域上本原多项式的求解算法,在求解过程中同时给出了相应的最小多项式,并给出了算法相应的效能分析.     

三元域上三次和四次剩余码的幂等生成元

有限域上高次剩余码的生成多项式都是多项式[xn-1]的因式。针对多项式[xn-1]在有限域上分解的困难性,给出了三元域[F3]上三次和四次剩余码的幂等生成元表达式。利用计算机软件求解这些幂等生成元与[xn-1]最大公因式就可得到三次和四次剩余码生成多项式而不用分解[xn-1]。     

二元域上三次和四次剩余码的幂等生成元

有限域上高次剩余码的生成多项式都是多项式[xn-1]的因式。针对多项式[xn-1]在有限域上分解的困难性,给出了二元域[F2]上三次和四次剩余码的幂等生成元表达式。利用计算机软件求解该幂等生成元与[xn-1]最大公因式就可得到三次和四次剩余码生成多项式而不用分解[xn-1]。     

有限域F2n上的2类正形置换多项式研究

研究有限域 上的正形置换多项式,针对有限域 上2d-1次和2d次正形置换多项式存在性的问题,利用同余类知识和有限域上乘积多项式的次数分布规律,分析其原因并给出有限域 上2d-1次正形置换多项式不存在和2d次正形置换多项式存在的判定结果。     

有限域上置换多项式的研究进展

AGW准则和分段方法是构造有限域上置换多项式的两种主要方法。介绍有限域上置换多项式在密码学和编码理论中的应用,总结利用AGW准则和分段方法构造有限域上置换多项式和逆置换的研究进展,阐述置换多项式存在的问题,并对下一步研究工作进行展望。     

一种基于多项式的代理保护代理签名体制研究

文章首先分析了张青坡等人中提出的多项式形式的E1Gamal签名体制的安全缺陷,然后基于有限域上多项式的性质,提出了有限域上多项式形式代理保护代理签名方案;新的签名方案中,利用多项式进行签名权利的委托,并由改进的有限域上的多项式形式的E1Gamal签名体制生成代理签名。新方案的安全性基于离散对数的难解性。     

基于近似最大公因多项式问题的公钥密码方案

研究了有限域F2上有随机噪声的一组多项式的近似最大公因式问题,提出了基于近似最大公因多项式问题的公钥密码方案。证明了方案的正确性并归约证明了方案的安全性等价于求解近似最大公因式问题,同时讨论了对于该方案可能的攻击方式。通过与现有公钥系统比较,该方案的安全性和可靠性较高,运算速度较快。     

混合安全双向认证密钥协商协议*

结合ECC密码体制优点和有限域上离散对数问题,提出了一种基于混沌映射的混合安全双向认证密钥协商协议。协议基于有限域上切比雪夫多项式的半群特性,运用ECC密码算法隐藏通信双方产生的有限域上切比雪夫多项式值,实现了通信双方双向认证,避免了Bergamo攻击,抵抗了中间人攻击、重放攻击,保证了密钥协商的安全性。理论上分析表明,该协议不仅具有强安全性,而且具有高效性特点。     

基于有限域上Chebyshev多项式的密钥协商方案

利用传统RSA算法和有限域上离散对数问题,提出一种新的基于混沌映射的密钥协商方案。该方案基于有限域上Chebyshev多项式良好的半群特性,运用RSA算法巧妙地隐藏通信双方产生的有限域上的Cheby-shev多项式值,从而避免了以往的种种主动攻击,保证了密钥协商的安全;同时,该密钥协商方案还实现了身份认证功能。理论分析和软件实现证明了该方案的可行性、正确性和安全性。     

A Method for Fast Computation of the Fourier Transform over a Finite Field

We consider the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of algorithms for the Fourier transform with complexity less than that of the best known analogs are given.     

A linearized polynomial mixed integer programming model for the integration of process planning and scheduling

This paper presents a linearized polynomial mixed-integer programming model (PMIPM) for the integration of process planning and scheduling problem. First, the integration problem is modeled as a PMIPM in which some of the terms are of products of up to three variables, of both binary and continuous in nature. Then, an equivalent linearized model is derived from the polynomial model by applying certain linearization techniques. Although the linearized models have more variables and constraints than their polynomial counterparts, they are potentially solvable to the optimum in comparison to their equivalent polynomial models. Experiments show that the linearized model possesses certain characteristics that are absent from other models in the literature, and provides a fundamental framework for further research in this area.     

On the Deterministic Complexity of Factoring Polynomials

The paper focuses on the deterministic complexity of factoring polynomials over finite fields assuming the extended Riemann hypothesis (ERH). By the works of Berlekamp (1967, 1970) and Zassenbaus (1969), the general problem reduces deterministically in polynomial time to finding a proper factor of any squarefree and completely splitting polynomial over a prime field F_p. Algorithms are designed to split such polynomials. It is proved that a proper factor of a polynomial can be found deterministically in polynomial time, under ERH, if its roots do not satisfy some stringent condition, called super square balanced. It is conjectured that super square balanced polynomials do not exist.     

Robust stability of polynomials with multilinear parameter dependence

The problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way. We show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials (obtained by setting parameters at their extreme values), while checking for unstable complex roots normally involves examining the real solutions of up to m + 1 simultaneous polynomial equations, where m is the number of parameters. When m = 2, this is an especially simple task.     

Exact Computation of Polynomial Zeros Expressible by Square Roots

In this paper we give an efficient algorithm to find symbolically correct zeros of a polynomial f ∈ R[X] which can be represented by square roots. R can be any domain if a factorization algorithm over R[X] is given, including finite rings or fields, integers, rational numbers, and finite algebraic or transcendental extensions of those. Asymptotically, the algorithm needs O(T_f(d²)) operations in R, where T_f(d) are the operations for the factorization algorithm over R[X] for a polynomial of degree d. Thus, the algorithm has polynomial running time for instance for polynomials over finite fields or the rationals. We also present a quick test for deciding whether a given polynomial has zeros expressible by square roots and describe some additional methods for special cases.     

Zeros of equivariant vector fields: Algorithms for an invariant approach

We present a symbolic algorithm to solve for the zeros of a polynomial vector field equivariant with respect to a finite subgroup of O (n). We prove that the module of equivariant. polynomial maps for a finite matrix group is Cohen-Macaulay and give an algorithm to compute a fundamental basis. Equivariant normal forms are easily computed from this basis. We use this basis to transform the problem of finding the zeros of an equivariant map to the problem of finding zeros of a set of invariant polynomials. Solving for the values of fundamental polynomial invariants at the zeros effectively reduces each group orbit of solutions to a single point. Our emphasis is on a computationally effective algorithm and we present our techniques applied to two examples.     

Polynomial complete problems in automata theory

Kozen (1977) proved that the emptiness problem for regular languages intersection is polynomial complete. In this paper we show that many other problems concerning deterministic finite state automata are polynomial complete and therefore intractable for solution. On the other hand, simplified versions of these problems can be solved in polynomial time by deterministic algorithms. This work is a part of the research on automata theory carried out at the Institute of Cybernetics headed by academician V.M. Glushkov.     

Idempotent Computation over Finite Fields

In this paper, we provide an account of several new techniques for computing the primitive idempotents of a commutative artinian algebra over a finite field. Examples of such algebras include the center of a finite group algebra or any finite dimensional quotient of a polynomial ring. The computational methods described are applicable in fairly general situations and the algorithms presented are easily programmed. Both pseudocode and operation counts are provided. As an application, the problem of factoring polynomials over finite fields is discussed.