Automorphic Distributions, L-functions, and Cryptography

该奖项涉及两个主要研究项目：与Wilfried Schmid（哈佛大学）关于自守分布的长期联合项目，以及与Ramarathnam Venkatesan（微软研究院）关于数论和离散群在密码学中的应用的合作。 第一个项目涉及研究自守分布及其分析性质。 PI和Schmid发展了另一种方法来证明朗兰兹L-函数的全纯性，在新的情况下建立了它们的全纯性。 它也被用来发展Voronoi式求和公式，后来被应用于GL（3）L-函数的第一个次凸性结果。 该项目涉及扩展他们的核心分析技术配对自守分布到更一般的情况。 第二个项目旨在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函数是现代数论的中心主题，从求解整数解的多项式方程的经典问题到研究曲面上的波的性质等各种问题。 拟议的研究主要涉及建立进一步的分析性质的L-函数，主要是应用到朗兰兹的全纯理论。 这样的信息可以用来，在第二个项目中，给出明确的界限和参数估计的密码系统。 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。 事实上，现代密码系统通常基于困难的数学问题，例如大整数的因式分解。 该提案旨在从SAT求解器和编码理论的角度研究这些问题，并为其他应用程序创建加密算法。

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.