Automorphic Distributions, L-functions, and Cryptography

该奖项涉及两个主要研究项目：与威尔弗里德·施密德(哈佛大学)关于自同构分布的长期联合项目，以及与Ramarathnam Venkatesan(微软研究院)在数论和离散群在密码学中的应用方面的合作。第一个项目涉及研究自同构分布及其分析性质。PI和Schmids发展了另一种方法来证明朗兰兹L-函数的全纯，从而在新的情形下建立了它们的全纯。它还被用来发展Voronoi型求和公式，这些公式后来被用于得到GL(3)L函数的第一个次凸性结果。该项目涉及将他们的配对自同构分布的核心分析技术扩展到更一般的情况。第二个项目寻求利用现代数论的信息开发新的算法和密码原语，范围从离散群到L函数。L函数是现代数论的一个中心主题，从用整数解多项式方程的经典问题到研究曲面上波的性质，都是由问题引起的。所提出的研究主要涉及建立L函数的进一步的解析性质，主要是应用于朗兰兹的全纯猜想。就像在第二个项目中一样，这些信息可以用来给出密码系统的显式界限和参数估计。PI和Venkatesan在这个主题上的工作已经在许多Microsoft产品中找到了加密应用程序。事实上，现代密码系统通常基于复杂的数学问题，如大整数的因式分解。该提案寻求从SAT解算器和编码理论的角度来研究这些问题，并为其他应用创建密码算法。

This award involves two main research projects: a long-term joint project with Wilfried Schmid (Harvard University) on automorphic distributions, and collaborations with Ramarathnam Venkatesan (Microsoft Research) on applications of number theory and discrete groups to cryptography. The first project involves the study of automorphic distributions, and their analytic properties. The PI and Schmid have developed an alternative method for proving the holomorphy of Langlands L-functions, which establishes their holomorphy in new cases. It also has been used to develop Voronoi-style summation formulas, which were later applied to obtain the first subconvexity results for GL(3) L-functions. The project involves extending their core analytic technique of pairing automorphic distributions to more general situations. The second project seeks to develop new algorithms and cryptographic primitives using information from modern number theory, ranging from discrete groups to L-functions.L-functions are a central topic in modern number theory, arising from problems as diverse as classical questions about solving polynomial equations with integer solutions, to studying the properties of waves on curved surfaces. The proposed research mainly involves establishing further analytic properties of L-functions, primarily with applications to Langlands' holomorphy conjectures in mind. Such information can be used, as in the second project, to give explicit bounds and parameter estimates for cryptosystems. Work of the PI and Venkatesan on this topic has already found cryptographic applications in numerous Microsoft products. Indeed, modern cryptosystems are often based on hard mathematical problems such as factoring large integers. The proposal seeks to study these problems from the point of view of SAT solvers and coding theory, and to create cryptographic algorithms for other applications.