Algorithms in global fields, with applications

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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.