Algorithms and cryptography in global fields

在过去的几十年里，计算机和通信的使用出现了前所未有的增长 技术.除此之外，还迫切需要保护信息：必须保护机密数据，防止未经授权的当事方获取或修改，网上银行用户需要经过认证，以确保他们是他们声称的人，电子邮件的来源必须可核实，以确定它们是否合法。可悲的是，公众已经习惯了关于病毒感染的计算机，黑客攻击，污损网站，身份盗窃案件，甚至互联网恐怖主义的潜在威胁的新闻。 密码学是保护数字信息的机密性、完整性和真实性的有效工具。许多现代密码系统的安全性依赖于某些数学问题，专家们认为这些问题非常难以解决。这种系统设计的基本思想是，对手必须解决这些非常困难的问题之一的一个实例，才能破解这个方案。因此，深入研究这些类型的问题并设计出利用它们的新密码系统，在理论和实践上都具有很大的意义。为此，我建议研究一种称为全局场的数学结构。某些全球领域为高效和非常安全的密码系统提供了一个很好的框架;这包括著名的椭圆曲线密码系统，它已被部署在一系列消费电子产品中，如蓝光技术和黑莓。本文对全局域上快速算法的研究将使相应的密码体制更加高效。此外，我的探索，在全球领域的某些困难的数学问题，将导致更好地理解这种计划的安全性。拟议的工作的成果将是一个集合的数论计算以及一套安全和有效的密码协议，可用于保护电子通信的理解算法。

The last few decades have seen an unprecedented increase in the use of computing and communication technology. Coupled with this development is a crucial need for information protection: confidential data must be guarded against access or modification by unauthorized parties, users of online banking need to be authenticated to ensure that they are who they claim they are, and the origin of e-mails must be verifiable to ascertain if they are legitimate. Sadly, the public has become accustomed to news flashes about virus-infected computers, hacking, defaced web sites, cases of identity theft, and even potential threats of internet terrorism. Cryptography is an effective vehicle for preserving the confidentiality, integrity and authenticity of digital information. The security of many modern cryptosystems relies on certain mathematical problems that experts believe to be extremely difficult to solve. The idea underlying the design of such a system is that an adversary would have to solve an instance of one of these very hard problems in order to crack the scheme. It is therefore of great interest, both theoretical and practical, to thoroughly study these types of problems and devise new cryptosystems that make use of them. To this end, I propose to investigate a type of mathematical structure called a global field. Certain global fields provide an excellent framework for efficient and very secure cryptosystems; this includes the celebrated elliptic curve cryptosystems which have been deployed in a range of consumer electronics products such as BluRay technology and the BlackBerry. My research into fast arithmetic in global fields will make the corresponding cryptographic schemes more efficient. Moreover, my exploration into the difficulty of certain hard mathematical problems in global fields will lead to a better understanding of the security of such schemes. The outcomes of the proposed work will be a collection of well understood algorithms for number theoretic computations as well as a suite of secure and efficient cryptographic protocols that can be used to protect electronic communications.