双线性函数及其应用

3双线性函数在不同基下的矩阵……………………………………………………………4
3.1 双线性函数在不同基下的矩阵之间的关系………………………………………4
3.2 相同基下，不同的双线性函数所对应的矩阵……………………………………5
4 双线性函数与辛空间及对偶空间…………………………………………………………6
1 常用的欧式空间……………………………………………………………………………1
2双线性函数…………………………………………………………………………………2
2.1线性函数的简单性质………………………………………………………………2
2.1.1 线性函数的定义……………………………………………………………2
4.1双线性函数与辛空间………………………………………………………………7
4.2双线性函数与对偶空间…………………………………………………………10
5双线性函数的应用领域…………………………………………………………………13
6结束语……………………………………………………………………………………14
Key words：Doublelinearfunction,andthematrixofthecontract,thematrixofthesimilar
前言
双线性函数是线性代数理论的一个重要内容．它涉及很多内容，如对称阵、反对称阵、二次型、正交阵、辛阵等，特别地双线性函数与线性函数有密切关系．由于研究关联着多个因素的量所引起的问题，则需要考察多元函数。如果所研究的关联性是线性的，那么称这个问题为线性问题。历史上线性代数的第一个问题是关于解线性方程组的问题，而线性方程组理论的发展又促成了作为工具的矩阵论和行列式理论的创立与发展，这些内容已成为我们线性代数教材的主要部分。最初的线性方程组问题大都是来源于生活实践，正是实际问题刺激了线性代数这一学科的诞生与发展。另外，近现代数学分析与几何学等数学分支的要求也促使了线性代数的进一步发展。
关键词：双线性函数；矩阵的合同；矩阵的相似
Abstract：In the past the cryptography studies, bilinear pairing function (Weil pairing Tate and matching) are usually used in analysis in learning, password: through the use of matching function, can will some of the elliptic curve discrete logarithm problem about reduced to a limited domain of discrete logarithm problem. In recent years, cryptography, home found that, if properly to visual function changes, and application in some appropriate elliptic curve, it can be constructed out of the low bandwidth, can prove safe (provable secure), based on bilinear pairings function of encryption, signatures and key agreement protocol, etc. These breakthrough for the construction of the password agreement opened up a new train: because bilinear pairings is the features of a function, can be used to design some has certain types of password agreement, these agreements with other method very hard commonly, or even can realize, its efficiency and no based on bilinear pairings function of high. For example, three square round short signature of key agreement protocol, identity based encryption scheme. This paper makes a study of the bilinear pairings function in the construction of new password agreement applications. The main research contents include: (1) summarized the bilinear pairings function concept, has the characteristics, and introduced the diffie-hellman problem and bilinear pairings function in the application of cryptography; (2) put forward a using bilinear pairings of function to safety before digital signature scheme: in a based on bilinear pairing the signature scheme based on the structure of a prior to the safety of the signature scheme. In this paper the safety of the scheme are analyzed, and some have to safety before the signature schemes are compared, andthe results show that the scheme in efficiency and signature length have a certain advantages; (3) in this paper put forward such a solution: multiple users will be encrypted data (use Alice public key) sent to not completely reliable data storage server (such as mail servers and file servers, etc.). If Alice wants to let the server can inquires documentation is contain certain words encryption and feedback result, but at the same time and don't want to give the server decrypt data ability. In this case, the need for special technology to deal with. This paper constructs a can inquire, based on public key and and flow of the combination of the password, using bilinear pairings function encryption system, it can make the query server, and do not break data confidentiality. In this scheme, the server and can't understand the results more than inquires about expressly information; And when only a given ciphertext, not trusted server can't get about expressly information. (4) put forward a blind signature scheme polymerization, it combines blind signature and polymerization signature advantage of the two, to generate the blind signature polymerization as a signature polymerization, saving time and storage space, also reduced of transmission bandwidth requirements.
2.1.2 线性空间的性质……………………………………………………………3
2.1.3 对偶基………………………………………………………………………3
2.2双线性函数的内容及性质…………………………………………………………3
2.2.1 双线性函数的性质…………………函数的内容…………………………………………………………3
本科生毕业论文(设计)
题目：双线性函数及其应用
专业：数学与应用数学
学号:
学生姓名：
摘要（关键词）…………………………………………………………………………………1
Abstract（Key words）…………………………………………………………………………1
前言……………………………………………………………………………………………2
参考文献……………………………………………………………………………………14
致谢…………………………………………………………………………………………1
双线性函数及其应用
摘要：在以往的密码学研究当中,双线性配对函数(Weil配对和Tate配对)通常被用在密码分析学中:通过使用配对函数,可以将某些椭圆曲线上的离散对数问题约减到有限域上的离散对数问题。近些年来,密码学家发现,如果对配对函数进行适当的改动,并应用在某些合适的椭圆曲线上,就可以构造出低带宽的、可证明安全的(provable secure)、基于双线性配对函数的加密、签名和密钥协商等协议。这些突破性的工作为密码协议的构造开辟了新的思路:由于双线性配对函数所具有的特性,可以用来设计一些具有特殊性质的密码协议,这些协议一般很难用其他方法实现,或者即使可以实现,其效率也没有基于双线性配对函数的高。例如短签名、三方一轮的密钥协商协议、基于身份的加密方案等。本文主要研究双线性配对函数在构造新的密码协议方面的应用。主要研究内容包括:(1)总结了双线性配对函数的概念、所具有的特性,并介绍了Diffie-Hellman难题以及双线性配对函数在密码学中的应用;(2)提出了一个使用双线性配对函数的前向安全的数字签名方案:在一个基于双线性配对函数的签名方案的基础上构造了一个前向安全的签名方案。文中对方案的安全性进行了分析,并与已有的一些前向安全的签名方案进行了比较,结果表明该方案在效率和签名长度上有一定的优势;(3)本文对这样一种情况提出了解决方案:多个用户将加密数据(使用Alice的公钥)发送到不完全可信的数据存储服务器上(例如邮件服务器和文件服务器等)。如果Alice想让服务器能够查询加密文档是否含有某些单词并反馈结果,但同时又不希望给予服务器解密数据的能力。在这种情况下,需要特殊的技术来处理。本文构造了一个可查询的、基于公钥并与流密码结合的、使用双线性配对函数的加密系统,它能让服务器进行查询,而又不失数据的机密性。在该方案中,服务器并不能了解比查询结果更多的关于明文的信息;且当只给定密文时,不被信任的服务器不能得到关于明文的信息。(4)提出了一个盲聚合签名方案,它结合了盲签名和聚合签名两者的优点,使生成的盲签名聚合为一个聚合签名,节省了时间和存储空间,也降低了对传输带宽的要求。