Automorphic L-functions and Cryptography

该资助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）关于解析数论的密码学应用的合作。 第一个项目试图使用自守形式的边界值分布来建立朗兰兹L-函数的解析性质。 PI和Schmid开发了另一种方法来证明Langlands L-函数的全纯性，绕过了现有方法面临的一些障碍，并产生了整个Langlands L-函数的新例子（例如GL（n，Z）\GL（n，R）上的外平方L-函数）。 该项目涉及扩展这些结果更广泛的家庭，并推广到一些领域和nonarchimedean的地方。 第二个项目使用解析数论和表示论的结果来分析和创建加密方案，特别是与椭圆曲线，isogenies，扩展图和模块形式相关的方案。 L-函数是现代数论中的一个中心话题，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，L-函数都有其广泛的应用。 朗兰兹的深度函性定理断言这些L-函数都与自守形式，矩阵群上的高度对称函数相连接。特别是，他们预测L-函数应该在复平面上解析。 拟议的研究涉及建立更多的情况下，这样的分析继续。 分析性一旦被证明，就意味着关于原始对象的强有力的结果。 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。 事实上，PI和Venkatesan在第二个项目上的初步工作最近已经被微软公司发布到安全产品中。 现代密码系统通常基于困难的数学主题，例如有限域上的整数和椭圆曲线的因子分解。 解析数论的使用，特别是L-函数的使用，对于获得对它们的安全性的具体和理论上的理解通常是必不可少的。

This grant 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 cryptographic applications of analytic number theory. The first project seeks to use the boundary value distributions of automorphic forms to establish analytic properties of Langlands L-functions. The PI and Schmid have developed an alternative method for proving the holomorphy of Langlands L-functions which bypasses some obstacles existing methods face, and which has resulted in new examples of entire Langlands L-functions (such as the exterior square L-functions on GL(n,Z)\GL(n,R)). The project involves extending these results to wider families, and generalizing to number fields and nonarchimedean places. The second project uses results from analytic number theory and representation theory to analyze and create cryptographic schemes, in particular ones related to elliptic curves, isogenies, expander graphs, and modular forms. 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. Langlands' deep functoriality conjectures assert that these L-functions are all connected to automorphic forms, highly symmetric functions on matrix groups. In particular, they predict the L-functions should be analytic on the complex plane. The proposed research involves establishing more cases of such analytic continuations. The analyticity, once proven, is then known to imply strong results about the original object. Such information can be used, as in the second project, to give explicit bounds and parameter estimates for cryptosystems. In fact, preliminary work of the PI and Venkatesan on the second project has recently been released into security products by the Microsoft Corporation. Modern cryptosystems are very often based on difficult mathematical topics, such as factoring integers and elliptic curves over finite fields. The use of analytic number theory and in particular L-functions is typically essential in gaining both concrete and theoretical understanding of their security.