Algorithms and cryptography in global fields

在过去的几十年里，计算机和通信技术的使用出现了前所未有的增长。与这一发展相结合的是对信息保护的关键需求：必须防止未经授权的各方访问或修改机密数据，需要对网上银行用户进行身份验证，以确保他们是他们所声称的人，并且必须验证电子邮件的来源，以确定它们是否合法。可悲的是，公众已经习惯了关于电脑被病毒感染、黑客攻击、网站被污损、身份被盗、甚至网络恐怖主义潜在威胁的新闻。密码学是保护数字信息的保密性、完整性和真实性的有效手段。许多现代密码系统的安全性依赖于某些专家认为极难解决的数学问题。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中的一个实例才能破解该方案。因此，深入研究这些类型的问题并设计利用它们的新密码系统在理论上和实践上都是非常有趣的。为此，我建议研究一种称为全局场的数学结构。某些全局字段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，它已部署在一系列消费电子产品中，如蓝光技术和黑莓。我对全局领域快速算法的研究将使相应的密码方案更加高效。此外，我对全球领域中某些数学难题的难度的探索将有助于更好地理解这些方案的安全性。提出的工作成果将是一套易于理解的数论计算算法，以及一套安全有效的加密协议，可用于保护电子通信。

The last few decades have seen an unprecedented increase in the use of computing and communicationtechnology. 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.