Algorithms in number theory and cryptography

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In the past few decades, cryptography, the study of data security, has emerged as an important concern in our society. As our personal information becomes increasingly accessible through the Internet, the need to protect and authenticate data is more imperative than ever. The security of many methods in modern public-key cryptography relies on the supposed difficulty of certain computational problems in mathematics and, in particular, number theory. These schemes are set up in such a way that presumably the only way for an attacker to break the given system is to solve an instance of one such problem. Currently, very few number-theoretic problems are widely accepted as foundations for secure public-key protocols. Furthermore, most of these problems have strong theoretical relationships. In many cases, being able to solve one efficiently implies the ability to solve another efficiently as well. In other words, one dramatic mathematical discovery could render many current public-key cryptographic techniques insecure. I intend to investigate the suitability of certain problems in algebraic number theory and algebraic geometry as alternatives for public-key cryptography. This will involve improving the efficiency of algorithms for computing with the basic objects of interest, studying theoretically and algorithmically the difficulty of the supposed hard problems, and conducting general investigations into the associated algebraic number theoretic objects. For example, there are a number of extremely difficult computational problems in this area, of interest in computational mathematics in their own right, that have also been proposed for cryptographic applications, but much more investigation is needed in terms of their security and efficiency. By attempting to devise new, more efficient methods to solve these problems, I will provide other researchers with a much firmer foundation on which to base their assessments of these cryptographic protocols. The end result will be a suite of secure and efficient cryptographic protocols, whose performance characteristics and trade-offs are well-quantified and understood, that can be used to protect Internet communications, even if currently-used protocols are found to be insecure. Canadian security agencies and companies will be able to use these results to better inform their decisions on security policy, and the trainees working on these projects will develop valuable skills for future employment in both the private and public sectors.