摘要
在交通信息工程与控制中,交通信息安全是一个重要的研究领域,其中保密性与认证性是交通信息安全的两个基本属性。基于格论的公钥密码体制和数字签名方案由于具有安全性好、带宽占有少等突出性能,近年来已受到学术界的极大关注。本文主要研究格论中既有序结构也有代数结构的空间—格序代数。具有特殊结构的格序代数可用于设计密码算法或数字签名方案。研究成果具有重要的理论意义与潜在的应用价值。
本文属于格论的理论研究部分,主要讨论了格序代数的序共轭的各类乘积空间的格序性质,f-代数的一次序共轭上的特殊算子,格序代数上f-模与d-模,以及格序代数的商空间与代数性质。主要研究成果分为以下四个部分。
第一部分研究了格序代数,主要是.f-代数、几乎f-代数和d-代数的序共轭的乘积空间的格序性质。得到了在一定的条件下这三类代数的二次序共轭与一次序共轭,一次序共轭与原代数在Arens乘积下的乘积空间为一次序共轭的序理想的结论。并利用这些结果推导了f-代数的二次序连续共轭为半素的一个新的充分必要条件,即：二次序共轭与一次序共轭的乘积空间是一次序共轭的一个序稠密理想。
第二部分主要讨论了f-代数的一次序共轭上的三类特殊算子：正交射,f-正交射和f-线性算子。引入并定义了新的算子：f-正交射和f-线性算子,讨论了这两类算子空间的格序性质。重点研究了这三类算子之间的互推关系,证明了正交射一定是f-线性算子,而f-线性算子也是f-正交射的结论。特别的,当f-代数为平方根闭时,得到了这三类算子为同一类算子的结论。从而对已知的f-代数的一次序共轭上的正交射给出了新的刻画方法。
第三部分研究了格序代数上的两类特殊的模结构—f-模与d-模。考察了保不交算子和f-模上的线性算子之间的关系。证明了在一定条件下,保不交算子和f-模上的线性算子等价的结论,并得到了保不交算子的逆算子的线性性质。其次,引入了一类新的模结构—格序代数上的d-模,研究了d-模的序共轭,证明了d-模的二次序共轭仍为d-模的结论。同时研究了由格同态或保区间算子所生成的主理想上的特殊d-模,得到了d-模的实例。
最后一部分主要讨论了格序代数的商空间及其主要的代数性质。首先给出了格序代数的商的概念。特别的,引入了商f-代数、商几乎f-代数和商d-代数的概念,并给出其等价的刻画方法。其次,讨论了这三类格序代数的商的主要代数性质,包括：半素性、交换性和逆元存在性。举例比较了格序代数与其商在这些性质方面的异同。更深入的,给出了格序代数的商的半素性的等价刻画,得到了交换性成立的若干条件,并证明了商f-代数单位元和逆元存在的条件。
In the traffic information engineering and control, traffic information security is an important research field in which confidentiality and authentication are two basic attributes. The lattice-based public key cryptosystem and digital signature scheme with good safety and low bandwidth have received great attention in recent years. In this paper our main research object is lattice algebra which is a vector space with ordered and algebraic structure. The special lattice algebra can be used to design cryptographic algorithms or digital signature scheme. The research has important theoretical significance and potential applications.
This paper is devoted to the theoretical study on order dual of a lattice algebra, the f-module and d-module on a lattice algebra and the quotient of lattice algebra. Actually, the present work consists of four main parts.
In the first part, we investigate the ordered properties of multiplicative spaces of order dual of the lattice algebra, especially the f-algebra, the almost f-algebra and the d-algebra. It's proved that under some conditions these multiplicative spaces equipped with Arens multiplications are order ideals in the order dual of lattice algebra. And as an application, a necessary and sufficient condition is derived under which the continuous order bidual of an f-algebra to be semiprime.
In the second part, the orthomorphisms, the f-orthomorphisms and the f-linear operators on the order dual of an f-algebra are considered. We introduce the new concepts of f-orthomorphisms and the f-linear operators. Then we discuss the ordered properties of these operators'spaces, and study the relationships between these operators. It is proved that the orthomorphisms must be f-linear operators, and f-linear operators must be f-orthomorphisms. Especially, when the f-algebra is square root closed, they are the same class of operators. As a result, we give a new characterization for the orthomorphisms.
In the third part, the f-module and d-module on a lattice algebra have been studied. Firstly, the relationship between the disjointness preserving operators and linear operators on an f-module is discussed. It is obtained that these operators are equvalent under some conditions. Then the linear proerty of the inverse of a disjointness preserving operator is given. Moreover we introduce the concept of the d-module, and obtain that the order bidual of a d-module is likewise a d-module. We study the d-modules over the ideals generated by a lattice homomorphism or an interval preserving operator with some examples.
Finally, the quotient space of a lattice algebra and their algebraic properties have been fully studied. The concepts of the quotient lattice algebra, especially the quotient f-algebra, quotient almost f-algebra and quotient d-algebra are given, as well as the equivalent characterizations. We also investigate the algebraic properties of a quotient lattice algebra. The examples are given to compare these properties. The equivalent condition under which a quotient lattice algebra is semiprime is obtained. At last we discuss the commutative property and show the conditions under which the unit and the inverse in a quotient f-algebra exist.
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