CAREER: Research and Education: Number Theory, Geometry and Cryptography

这个项目促进了对数论、几何学和密码学的理解。数论和几何是数学中最古老和最核心的主题之一，而它们在密码学中的应用是现代网络安全的基础。该项目专注于数论信息与几何结构之间的关系，如椭圆曲线、圆填充和格子。该项目的一个主要密码学重点是研究后量子密码学提案的安全性，后量子密码学是指可以利用量子计算机保护信息不受对手攻击的密码协议。此外，该项目将在科罗拉多大学博尔德分校建立一个数学实验室，其目的是让不同的本科生群体参与创造性、实验性、计算性和可视化的数学研究体验，并通过外联和社区参与传播这项研究的成果。参与数学实验室的学生将继续在教学、工业、医学等各个领域担任数学大使。该项目还支持女性在数学方面的指导，并促进数学家和计算机科学家之间的跨学科合作。该项目涉及三个分支机构。首先，PI将在描述虚二次场的某些几何方面的比安奇群的轨道和具有复乘法的椭圆曲线、薄群、类群和阿贝尔沙堆之间建立新的和已知的联系。在第二章中，PI将椭圆网络的理论推广到交换变种，并研究其应用(包括在基于配对的密码学中)。第三个主题是作为后量子密码学基础的环学习带错误问题的安全性研究。这个问题是数域中出现的格的几何结构的一个应用。PI将根据这个数字理论结构调查已知和潜在的新攻击的程度，并将这些与已知的安全参数和建议的实施进行比较。数学实验室将由本科生团队组成，由教职员工和研究生领导，专注于探索数学中的一个开放问题和推广应用。外展将包括数学可视化、艺术、软件和面向所有年龄段的互动研讨会。

This project advances the understanding of number theory, geometry, and cryptography. Number theory and geometry are among the oldest and most central topics in mathematics, while their application to cryptography underlies modern cybersecurity. The project focuses on the relationships between number-theoretic information and geometric structures such as elliptic curves, circle packings, and lattices. A primary cryptographic focus of the project is to study the security of proposals for post-quantum cryptography, that is, cryptographic protocols that can secure information against an adversary with a quantum computer. In addition, the project will create a Mathematics Lab at the University of Colorado, Boulder, whose aim is to involve a diverse undergraduate population in creative, experimental, computation- and visualization-based mathematics research experiences, and to disseminate the products of this research through outreach and community involvement. Students involved in the Mathematics Lab will go on to act as ambassadors of mathematics in all fields: teaching, industry, medicine, etc. The project also supports the mentoring of women in mathematics and the furtherance of cross-disciplinary collaboration between mathematicians and computer scientists.The project involves three branches. In the first, the PI will develop new and known connections between the orbits of Bianchi groups, which describe certain geometric aspects of imaginary quadratic fields, and elliptic curves with complex multiplication, thin groups, class groups, and abelian sandpiles. In the second, the PI will extend the theory of elliptic nets to abelian varieties and investigate applications (including to pairing-based cryptography). The third topic is the study of the security of the Ring-Learning-with-Errors problem as a basis for post-quantum cryptography. This problem is an application of the geometric structure of lattices appearing in number fields. The PI will investigate the extent of known and potential new attacks based on this number theoretical structure and compare these with known security parameters and suggested implementations. The Mathematics Lab will consist of teams of undergraduates led by faculty and graduate students, focusing on the exploration of an open problem in mathematics and an outreach application. Outreach will include mathematical visualizations, art, software, and interactive workshops for all ages.