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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717501
• 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.

我的研究计划围绕着如何应用代数中的新思想来解决纯数学和应用数学中的问题的总体愿景展开。 主要主题：p-进群的表示理论。 这一数学领域已经成为研究的焦点，这在很大程度上要归功于著名的朗兰兹计划(1970年代-)。这是一系列巨大的猜想，将表象理论的力量运用到有关数字本质的一些最基本的问题上。在这里，对于任何素数p，p进位数以无数不同的透镜进入画面，这些不同的透镜使这一理论成为焦点。 我的工作集中在分支规则上：将复杂的表示法分解成无限的小块集合，然后确定那些保留了原始数论问题的答案的表示法。让我的研究令人兴奋的是我发现和开发了新的工具，使用一个称为G的Bruhat-Tits建筑的组合对象，给这个无限集合一个几何结构，从而潜在地给数论问题本身。 这是一个充满引人入胜的开放式问题的领域，可以在多个层次上访问这些问题，并且是训练我的学生抽象、批判性思维和进行研究的理想选择。 次要主题：后量子数学密码学。 我们的互联网经济完全依赖于公钥基础设施(PKI)，这是一种数学密码学，允许我们在不安全的渠道上启动安全通信。随着量子计算机成为现实，现有的绝大多数PKI将被淘汰，因为它们基于的问题容易受到量子算法的攻击。后量子密码学(PQC)指的是一类密码系统，旨在使用经典计算机和量子计算机来抵御所有已知的攻击。 对这些数学上复杂的算法的开发和彻底审查可以说是未来十年最紧迫和最引人注目的数学挑战。我的工作就是集中精力迎接这一挑战。过去，我曾培训学生使用代数方法来分析、评估或攻击新的PQC。展望未来，我将另外开始量子算法的开发，通过量子信息和量子计算的新前沿的一个跨学科项目加速。我研究的一个特别的新方向与所谓的隐子群问题有关，它与我的第一个主题(表象理论)密切相关，令人兴奋的是，它是量子攻击和新PQC的来源。 尽管PQC的研究对其核心具有很高的风险，但它为学生提供了非常好的服务，既是进入更多理论数学研究的入门，也是发展编码和网络安全方面极具市场潜力的技能的绝佳机会。