代数学

Using the theories of probability, algebra and spectral theory comprehensively, we investigate some related characteristics of logic functions in cryptography: Firstly, we introduce m order generalized s - correlation immunity of Boolean vector functions and prove that the higher order generalized ε- correlation immunity can guarantee the lower order generalized ε- correlation immunity. Then by applying decomposition formula of joint distribution of Boolean random vectors, we give a spectrum criterion of m order generalized e - correlation immunity of Boolean vector functions. Furthermore, we show that the algebraic degree of m order generalized e - correlation immune Boolean vector functions is not restricted by the correlation immune orders.

本文主要运用概率论的思想和方法，并结合代数学和频谱理论的相关知识，对密码学中逻辑函数的有关性质进行了研究，主要包括以下三个方面的内容：首先，对布尔向量函数的相关免疫性进行了拓展，给出了k维布尔向量函数m阶广义ε-相关免疫的概念，证明了布尔向量函数的高阶广义ε-相关免疫性蕴含低阶广义ε-相关免疫性，并根据布尔随机向量联合分布分解式得到了布尔向量函数m阶广义ε-相关免疫的一个谱判别条件，还说明了m阶广义ε-相关免疫布尔向量函数的代数次数不受相关免疫阶数的制约。

We describe the algebraic characteristic of the public-key cryptosystems based on algebraic structures such as elliptic curves,bilinear pairings,as well as braid group s,with the emphasis on the difficulties of constructing public-key cryptosystems based on non-commutative algebraic structures.

通过分析基于椭圆曲线、双线性对以及基于辫子群的公钥密码体制的代数学特征，着重讨论了构建基于非交换代数的公钥密码体制所面临的困难。

Rational polynomial coefficient in the Rational domain can not be the judge about substituting mathematical theorem is the most important issue.

有理系数多项式在有理数域上不可约的判定定理是代数学中的重要问题。

At completeness in theory, we give the base of mathematical morphology-complete lattice, and some nature of complete lattice.

作为理论上的完备性，我们给出了数学形态的代数学基础——完备格理论，并简单的介绍了完备格及其性质。

As the foundation of algebra, this course also offers indispensable preparation for subsequent core curriculum and sample curriculum such as differential equations, numerical calculation, functional analysis, probability theory and multivariate analysis, etc.

它不仅是代数学的基础，而且为微分方程，数值计算，泛函分析，概率论，多元分析等后继的基础课及选修课程提供了必要的

Matrix is an important mathematical concept,which is among the major research fields of Algebra study . Diagonalization of the matrix is the key content of Matrix Theory.

矩阵是数学中的一个重要的基本概念，是代数学的一个主要研究对象，而矩阵的对角化是矩阵论中的一个重点内容。

Our studies mainly focus on the free groups with finite ranks . Free groups with finite ranks are the fundamental subjects of combinatorial group theory, and they have central importance for the studies of both topology and algebra.

我们研究比较多的是有限秩的自由群，有限秩自由群是组合群论的中心课题，是一个在拓扑学及代数学上都具有重要意义的研究对象，获取有关这些自由群的结构的信息的一个主要方法是研究其上的各种自同构作用。

The proof of fundamental theorem of algebra involves much algebra knowledge which is difficult to understand.

代数学基本定理的经典证明用到较多的代数知识，且难以理解，文章探讨用数学分析的方法予以证明。

This paper explores ways to prove fundamental theorem of algebra with mathematical analysis.

该证明从复变多项式无非零最小模引入，并在此基础上简单证明了代数学的基本定理。

Nor are the rules of algebra governed by social conventions: a gawky 14-year-old who clams up in interviews can still do well.

社会习俗也绝然不能支配代数学的规律：一名在面试中沉默笨拙的14岁小孩仍然能够在这方面做的很好。

In the negative and interrogative forms, of course, this is identical to the non-emphatic forms.

。但是，在否定句或疑问句里，这种带有"do"的方法表达的效果却没有什么强调的意思。

Go down on one's knees;kneel down

屈膝跪下。。。下跪祈祷