Bruhat-Tits theory in the representations of p-adic groups, and post-quantum cryptography

p-adic 群表示中的 Bruhat-Tits 理论和后量子密码学

• 批准号: RGPIN-2020-05020
• 负责人: Nevins, Monica
• 金额: $ 1.75万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759210
• 关键词: Bruhat; Tits; theory; representations; adic

My research program centers on an overarching vision on how to apply new ideas in algebra to solve problems in both pure and applied mathematics. Primary theme: representation theory of p-adic groups. This area of mathematics has become a focal point of research largely due to the celebrated Langlands program (1970s-). This is a vast series of conjectures that bring the power of representation theory to bear on some of the most fundamental questions about the nature of numbers. Here, the p-adic numbers, for any prime p, enter the picture as the myriad of different lenses that bring this theory into focus. My work centers on branching rules: breaking up complex representations into an infinite collection of smaller pieces, and then identifying those that retain answers to the original number-theoretic questions. What makes my research exciting is my discovery and development of new tools, using a combinatorial object called the Bruhat-Tits building of G, to give a geometric structure to this infinite collection and thus potentially, to the number-theoretic questions themselves. This is a field rich in fascinating open questions that can be made accessible at multiple levels, and is ideal for training my students in abstraction, critical thinking, and conducting research. Secondary theme: post-quantum mathematical cryptography. Our internet economy relies utterly on public-key infrastructure (PKI), which is the mathematical cryptography that allows us to initiate secure communications over an insecure channel. As quantum computers become a reality, the vast majority of existing PKI will be rendered obsolete because they are based on problems vulnerable to attack by quantum algorithms. Post-quantum cryptography (PQC) refers to a class of cryptosystems intended to be secure against all known attacks, using either classical and quantum computers. The development and thorough scrutiny of these mathematically sophisticated algorithms is arguably the most urgent and compelling mathematical challenge of the coming decade. My work is focussed on meeting this challenge. In the past, I have trained students in the use of algebraic methods to analyse, evaluate or attack new PQC. Going forward, I will additionally embark of the development of quantum algorithms, accelerated by an interdisciplinary project through New Frontiers in quantum information and quantum computing. One particular new direction of my research relates to the so-called Hidden Subgroup Problem, which is strongly related to my first theme (representation theory) and is, excitingly, a source for both quantum attacks and new PQC. Although high-risk to its core, research in PQC serves students extremely well, both as an entry to more theoretical mathematical research, and as an excellent opportunity to develop highly marketable skills in coding and cybersecurity.

我的研究计划集中于关于如何在代数中应用新想法来解决纯数学和应用数学问题的总体愿景。主要主题：P-ADIC群体的表示理论。这一数学领域已成为著名的Langlands计划（1970年代）的研究重点。这是一系列庞大的猜想，它使代表理论的力量对数字性质的一些最根本的问题承担。在这里，对于任何主要P，P-ADIC数字作为将该理论焦点的无数不同镜片输入。我的工作集中在分支规则上：将复杂的表示形式分解为无限的较小零件集合，然后确定那些保留原始数字理论问题的答案。使我的研究令人兴奋的是，我使用称为bruhat-tits构建G的组合对象的新工具的发现和开发，以给这个无限收藏，从而为数字理论问题本身提供几何结构。这是一个充满着令人着迷的开放问题的领域，可以在多个层面上访问，非常适合培训我的学生抽象，批判性思维和进行研究。次要主题：量词后数学加密。我们的互联网经济完全依赖于公钥基础设施（PKI），这是数学加密，使我们能够通过不安全的渠道启动安全的通信。随着量子计算机成为现实，绝大多数现有的PKI将被淘汰，因为它们是基于容易受到量子算法攻击的问题。量词后加密（PQC）是指使用经典计算机和量子计算机来保护所有已知攻击的一类密码系统。 这些数学上复杂的算法的发展和彻底审查可以说是未来十年中最紧迫，最引人注目的数学挑战。我的工作专注于应对这一挑战。过去，我曾培训过使用代数方法来分析，评估或攻击新PQC的学生。展望未来，我还将踏上量子算法的开发，并通过跨学科项目通过量子信息和量子计算的新边界加速。我的研究的一个新方向与所谓的隐藏子组问题有关，该问题与我的第一个主题（表示理论）密切相关，并且令人兴奋地是量子攻击和新PQC的来源。 尽管高风险是其核心，但PQC的研究既可以很好地为学生提供了很好的服务，又是对更多理论数学研究的参与，也是发展高度可销售的编码和网络安全技能的绝佳机会。