EAGER: Number Theory and Cryptograpghy

EAGER：数论和密码学

基本信息

批准号：
1643552
负责人：
Katherine Stange
金额：
$ 20万
依托单位：
University of Colorado at Boulder
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1643552&HistoricalAwards=false
关键词：
EAGER Number Theory Cryptograpghy

项目摘要

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.

该奖项支持主要研究者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被研究了几个世纪。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，以及密码应用的基础研究，着眼于量子计算机的出现。特别是，密码学界目前正在寻找针对量子计算（当前使用的系统不具备的属性）安全的密码系统，而 PI 将从数论角度研究主要候选者之一。该项目更广泛的影响包括通过早期研究合作对学生和女性进行数学方面的指导。这将通过本科生和早期研究生的夏季研究经历以及针对女性的指导研究合作会议来实现。 PI 还将通过艺术和软件接触公众。该项目的科学部分分为三个部分。首先，PI 将通过开发和解决有关数域及其子域和理想的格子属性的问题来评估基于格的密码环学习问题的安全性。她将确定对该问题的已知数论攻击的程度，因为它与可证明安全性的已知参数和建议的实施参数相关。第二个涉及比安奇群，它提供了虚二次场的几何视图。 PI 将开发从比安奇群的轨道到类群、连分式、阿波罗圆堆积、二次型、薄群、阿贝尔沙堆和椭圆曲线的联系。最后，PI 还将以她之前定义椭圆网的工作为基础。椭圆网络为椭圆曲线提供了不同的计算框架，并催生了基于配对的密码学的新算法。 PI 将把这项工作扩展到阿贝尔簇的背景下，并研究应用。