Algorithms in number theory and cryptography

在过去的几十年里，密码学，数据安全的研究，已经成为我们社会的一个重要问题。随着我们的个人信息越来越多地通过互联网访问，保护和验证数据的需求比以往任何时候都更加迫切。现代公钥密码学中许多方法的安全性依赖于数学中某些计算问题的假定难度，特别是数论。这些方案的设置方式假定攻击者破坏给定系统的唯一方法是解决此类问题的一个实例。目前，很少有数论问题被广泛接受作为安全公钥协议的基础。此外，这些问题大多具有很强的理论关系。在许多情况下，能够有效地解决一个问题意味着能够有效地解决另一个问题。换句话说，一个引人注目的数学发现可能会使许多当前的公钥加密技术变得不安全。***我打算研究代数数论和代数几何中某些问题作为公钥密码学替代方案的适用性。这将涉及提高算法的效率，用于计算感兴趣的基本对象，从理论上和算法上研究假定困难问题的难度，并对相关的代数数论对象进行一般调查。例如，在这个领域中有许多极其困难的计算问题，它们本身对计算数学很感兴趣，也被提出用于密码学应用程序，但就其安全性和效率而言，需要进行更多的研究。通过尝试设计新的、更有效的方法来解决这些问题，我将为其他研究人员提供一个更坚实的基础，以此为基础来评估这些加密协议。最终的结果将是一套安全有效的加密协议，其性能特征和权衡是很好的量化和理解，可以用来保护互联网通信，即使目前使用的协议被发现是不安全的。加拿大安全机构和公司将能够利用这些结果更好地为他们的安全政策决策提供信息，而参与这些项目的学员将为未来在私营和公共部门的就业发展宝贵的技能

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.**