Algorithms and cryptography in algebraic function 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。在全球范围内，通过互联网和移动电话等不受保护的渠道传输的数据量正以惊人的速度增长。此外，载有可能敏感记录的大型数据库的数目正在继续增加。不幸的是，这种发展伴随着未经授权访问（并可能修改）信息的威胁不断上升，这些信息被认为是安全的，不会受到此类入侵。可悲的是，公众已经习惯了关于电脑感染病毒、网络蠕虫、网站污损和身份盗窃案件的新闻。保护数据免受未经授权访问的最有效方法之一是在发送或存储数据之前对其进行加密，从而使其无法被窥探。只有授权方能够借助只有他们自己知道的密钥来解密信息。密码系统，或简称密码系统，是实现这一过程的一种方法。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代数函数场的数学结构。由于它们的快速运算和大量内置的数学难题，这些结构为高效和非常安全的密码系统提供了一个极好的框架。一些基于函数域的方案——即所谓的基于椭圆曲线的密码系统——已经在商业上用于各种电子设备，如计算机、个人数据管理器和手机。

The last few decades have seen an unprecedented increase in the use of computing and communication technology. Throughout the world, the amount of data that is being transmitted over unprotected channels, such as the internet and mobile telephones, is increasing at a staggering rate. In addition, the number of large scale databases containing possibly sensitive records is continuing to grow. Unfortunately, this development is coupled with an ever rising threat of unauthorized access to (and possibly modification of) information that was assumed to be safe from such intrusions. Sadly, the public has gotten quite accustomed to news flashes about virus-infected computers, internet worms, defaced web sites, and cases of identity theft. One of the most effective ways of protecting data against unauthorized access is to scrample (encrypt) it before sending or storing it, thereby rendering it unintelligible to prying eyes. Only authorized parties are able to unscramble (decrypt) the information with the aid of a secret key known only to them. A cryptographic system, or cryptosystem for short, is a method for effecting this process. 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 an algebraic function field. Due to their fast arithmetic and their wealth of built-in hard mathematical problems, these structures provide an excellent framework for efficient and very secure cryptosystems. Some function field based schemes -- namely the so-called elliptic curve based cryptosystems -- are already used commercially on a variety of electronic devices, such as computers, personal data organizers, and cell phones.