Automorphic Distributions, L-functions, and Cryptography

该奖项包括两个主要研究项目：与Wilfried Schmid（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）关于数论和离散群在密码学中的应用的合作。第一个项目涉及自同态分布及其解析性质的研究。PI和Schmid开发了一种证明朗兰兹l函数全纯性的替代方法，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，这些公式后来被用来获得GL(3) l -函数的第一个次凸性结果。该项目涉及到将他们的核心分析技术配对自同构分布扩展到更一般的情况。第二个项目旨在利用现代数论的信息，从离散群到l函数，开发新的算法和密码原语。l -函数是现代数论中的一个中心话题，从用整数解求解多项式方程的经典问题到研究曲面上的波的性质，都引起了l -函数。提出的研究主要涉及建立l -函数的进一步解析性质，主要是考虑到朗兰兹全纯猜想的应用。如在第二个项目中，可以使用这些信息来给出密码系统的显式边界和参数估计。PI和Venkatesan在这个主题上的工作已经在许多微软产品中找到了加密应用。事实上，现代密码系统通常是基于复杂的数学问题，比如分解大整数。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.