Algorithms in global fields, with applications

数论是数学的一个广泛分支，可以追溯到古希腊人，它研究数字的性质及其构成的结构。最初在纯粹的抽象数学的保护伞下，由于计算技术的不断增强的能力，该学科的各个方面在近几十年来已经发生了变化。数论是加拿大最活跃和最具代表性的数学领域之一，该国在这一领域的研究和培训成果享誉国际。数论已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，并在声学、现代卫星通信和密码学中使用，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的人攻击的基础。数据安全已成为我们社会最关心的问题之一，随着信息越来越多地通过互联网获得，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设难度。具体地说，这些方案的建立方式是，攻击者破坏给定系统的唯一已知方法是解决基础数学问题的一个实例，这将需要远远超出他们能力范围的巨大计算能力。我感兴趣的是在被称为全球域的重要数论结构中出现的计算困难问题。通过椭圆曲线的设置给出了一个特例，它在黑莓智能手机、Bluray播放器和其他现实世界的技术中提供数据保护。研究这些问题及其计算难度有助于更深入地理解数论的基本结构及其行为。它还为它们是否适合潜在的密码应用提供了新的见解，包括未来可以抵抗量子计算机攻击的新方案。我的研究采取了一种全面的方法，包括探索数学基础、算法设计和分析、高性能计算机编程，以及对我们的计算机实现产生的大量数据集的科学解释。切实的实际成果包括一套用于全球领域的高效算术技术，以及一系列用于数论和可能的密码计算的最先进的算法。根据这项研究培训的学生将在解决问题、定量推理和计算机编程方面获得宝贵的技能，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.