Automorphic L-functions and Cryptography

这项资助包括两个主要的研究项目：与Wilfried Schmid（哈佛大学）关于自同态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。第一个项目寻求使用自同构形式的边值分布来建立朗兰兹l函数的解析性质。PI和Schmid开发了一种替代方法来证明朗兰兹l函数的全纯性，该方法绕过了现有方法面临的一些障碍，并产生了完整朗兰兹l函数的新例子(例如GL（n，Z)\GL（n，R）上的外部平方l函数）。该项目包括将这些结果扩展到更广泛的家庭，并推广到数字领域和非阿基米德地区。第二个项目使用解析数论和表示理论的结果来分析和创建密码方案，特别是与椭圆曲线、等同构、展开图和模形式相关的方案。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。Langlands的深泛函猜想断言这些l -函数都与自同构形式，即矩阵群上的高度对称函数相连接。特别是，他们预测l函数在复平面上应该是解析的。建议的研究包括建立更多的这种分析延拓的案例。分析性一旦被证明，就意味着对原始对象的强有力的结论。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。事实上，PI和Venkatesan在第二个项目上的初步工作最近已经由微软公司发布为安全产品。现代密码系统通常基于困难的数学主题，例如在有限域上分解整数和椭圆曲线。解析数论，特别是l函数的使用对于获得对其安全性的具体和理论上的理解是必不可少的。

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.