Algorithms in global fields, with applications

数论是数学的一个广泛分支，可以追溯到古希腊，它探索数字的性质和它们形成的结构。最初在纯抽象数学的保护伞下，由于计算技术的不断增强，近几十年来该学科的各个方面已经发生了变化。数论是加拿大最活跃和最具代表性的数学领域之一，加拿大在这一领域的研究和培训是国际公认的。数论已经被发现与数学内外的其他领域有许多重要的联系，包括计算机科学、物理学和工程学。它以量子现象为特征，在声学、现代卫星通信和密码学中都有应用，在网上银行和购物、电子商务、互联网通信和许多其他应用中，它是保护数字信息免受黑客和其他未经授权方攻击的基础。数据安全已成为我们社会最关注的问题之一，随着互联网上的信息越来越多，保护和验证数据以及保护个人身份的需求比以往任何时候都更加迫切。大多数现代信息保护方法的安全性依赖于某些计算数论问题的假定难度。具体来说，这些方案是以这样一种方式建立的，即攻击者破坏给定系统的唯一已知方法是解决潜在的困难数学问题的实例，这将需要远远超出他们能力的巨大计算能力。我的兴趣是在被称为全局场的重要数论结构中产生的计算难题。在黑莓智能手机、蓝光播放器和其他现实世界的技术中提供数据保护的椭圆曲线的设置给出了一个特殊的情况。研究这些问题和它们的计算难度导致对潜在的数论结构和它们的行为有更深的理解。它还为它们在潜在的密码学应用中的适用性提供了新的见解，包括抵抗量子计算机攻击的新型未来方案。我的研究采用综合的方法，包括探索数学基础，算法设计和分析，高性能计算机编程，以及对计算机实现产生的大量数据集的科学解释。有形的实际成果包括一套有效的算法技术在全局领域和最先进的算法的集合数论和可能的密码学计算。在这项研究中训练的学生将获得解决问题、定量推理和计算机编程方面的宝贵技能，从而为他们未来有意义、有成效的职业生涯做好准备。

Number theory is a wide-ranging branch of mathematics dating back to the ancient Greeks that explores properties of numbers and the structures they form. Originally very much under the umbrella of pure abstract mathematics, aspects of the discipline have been transformed in recent decades due to the ever increasing power of computing technology. Number theory is one of the most active and well represented areas of mathematics in Canada, and the country is internationally recognized for its research and training excellence in this field. Number theory has been discovered to have many important connections to other areas both in and outside mathematics, including computer science, physics and engineering. It features in quantum phenomena and has uses in acoustics, modern satellite communication and cryptography, where it serves as the foundation for protecting digital information from hackers and other unauthorized parties in online banking and shopping, e-commerce, internet communication and many other applications. Data security has emerged as one of the foremost concerns in our society, and as information becomes increasingly accessible through the Internet, the need to secure and authenticate data and guard one's identity is more imperative than ever. The security of most modern methods for information protection relies on the supposed difficulty of some computational number theoretic problem. Specifically, these schemes are set up in such a manner that the only known way for an attacker to break the given system is to solve an instance of the underlying hard mathematical problem, which would require enormous computing power far beyond their means. My interest is in computationally difficult problems arising in important number theoretic structures referred to as global fields. A special case is given by the setting of elliptic curves which provide data protection in the Blackberry smartphone, Bluray players and other real world technologies. Investigating these problems and their computational hardness leads to a deeper understanding of the underlying number theoretic structures and their behaviours. It also offers new insights into their suitability for potential cryptographic applications, including novel future schemes that are resistant to attacks by quantum computers. My research takes a comprehensive approach comprised of exploration into mathematical foundations, algorithm design and analysis, high-performance computer programming, and scientific interpretation of the extensive data sets produced by our computer implementations. Tangible practical outcomes include a suite of efficient techniques for arithmetic in global fields and a collection of state-of-the-art algorithms for number theoretic and possibly cryptographic computations. Students trained under this research will gain valuable skills in problem solving, quantitative reasoning and computer programming, thus preparing them for meaningful productive future careers.