EAGER: Number Theory and Cryptograpghy

该奖项支持首席研究员在数论及其密码学应用方面的研究。 数论是现代密码学和互联网安全的基础。 椭圆曲线和整数分解的基本数学理论已经研究了几个世纪。 该项目包括基础研究的组成部分，涉及几何和数论之间的关系，以及加密应用的基础研究，着眼于量子计算机的出现。 特别是，加密社区目前正在寻找对量子计算安全的密码系统（目前使用的系统不共享的属性），PI将从数论的角度研究其中一个主要候选者。 该项目更广泛的影响包括通过早期研究合作，对学生和妇女进行数学辅导。 这将通过本科生和早期研究生的夏季研究经验，以及妇女的指导研究合作会议进行。 PI还将通过艺术和软件接触公众。 该项目的科学部分有三个部分。 首先，PI将通过开发和解决有关数域及其子域和理想的格属性的问题来评估基于格的加密环学习错误问题的安全性。 她将确定已知数论攻击对问题的程度，因为它涉及可证明安全性的已知参数和建议的实施参数。 第二个关注比安奇群，作为提供一个几何视图的虚二次领域。 PI将开发从比安奇组的轨道到类组，连续分数，阿波罗圆填充，二次型，薄组，阿贝尔沙堆和椭圆曲线的连接。 最后，PI还将建立在她之前定义椭圆网的工作基础上。 椭圆网为椭圆曲线提供了一个不同的计算框架，并为基于配对的密码学带来了新的算法。 PI将扩展这项工作的上下文中的阿贝尔品种，并调查应用。

This award supports the principal investigator's research in number theory and its cryptographic applications. Number theory serves as the basis for modern cryptography and internet security. The underlying mathematical theories, of elliptic curves and integer factorization, have been studied for centuries. This project includes components of basic research, concerning the relationship between geometry and number theory, as well as foundational research in cryptographic applications, with an eye toward the advent of quantum computers. In particular, the cryptographic community is currently searching for cryptosystems secure against quantum computation (a property not shared by the systems currently in use), and the PI will investigate one of the prime candidates from a number-theoretical perspective. The broader impacts of the project involve the mentoring of students and women in mathematics through early research collaboration. This will take place through summer research experiences for undergraduate and early graduate students, and a mentoring research collaboration conference for women. The PI will also reach out the general public through art and software. The scientific component of the project has three parts. In the first, the PI will evaluate the security of the lattice-based cryptographic Ring Learning with Errors problem by developing and addressing questions concerning the lattice properties of number fields and their subfields and ideals. She will determine the extent of known number-theoretical attacks on the problem, as it relates to known parameters of provable security and suggested parameters for implementation. The second concerns Bianchi groups, as providing a geometric view of imaginary quadratic fields. The PI will develop connections from orbits of the Bianchi group to class groups, continued fractions, Apollonian circle packings, quadratic forms, thin groups, abelian sandpiles and elliptic curves. Finally, the PI will also build on her prior work defining elliptic nets. Elliptic nets provide a different computational framework for elliptic curves and have given rise to new algorithms for pairing-based cryptography. The PI will extend this work to the context of abelian varieties, and investigate applications.