Arithmetic of Thin Groups and Isogeny-Based Cryptography

稀疏群算法和基于同源的密码学

• 批准号: 2401580
• 负责人: Katherine Stange
• 金额: $ 35万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401580&HistoricalAwards=false
• 关键词: Arithmetic; Thin; Groups; Isogeny; Based

In this project, the PI studies a class of questions relating number theory and geometry which have certain mathematical underpinnings in common. These questions concern basic research in the arithmetic of group orbits (which are collections of integers arising from the recursive application of certain symmetries) and the underlying mathematics of certain new cryptographic schemes. In particular, the latter aspect of the project is directly in service of the development of post-quantum cryptography, namely, cryptography which will be secure against the eventual development of quantum computers to scale. The project will support the training of graduate students, as well as the Experimental Mathematics Lab at the University of Colorado Boulder, which aims to broaden undergraduate participation in mathematical research, including students who will go on to many roles in society. It will also support the Numberscope project, which is an outreach project aimed at scientists, artists and the general public.In the first branch of research, the PI studies certain families of integers which arise in orbits of thin groups. Group orbits of various kinds have been studied throughout the history of number theory, including for example points on elliptic curves (upon which much of modern cryptography is based) and Pythagorean triples. The orbits studied in this project come from a class of groups (thin groups) for which effective tools are harder to create. These arise, for example, from the study of continued fractions. However, one expects certain high-level phenomena to occur in both the old and new settings. One such example is local-to-global phenomena, where the PI will study the extent to which knowledge of local information (with respect to individual primes) controls global information (the integers in the orbit). The second aspect of the project concerns cryptographic applications of number theory. One of the current candidates for post-quantum cryptography is isogeny-based cryptography, which is based on elliptic curves. The security of mathematical public-key cryptography is based on hard problems, and the hard problems of isogeny-based cryptography demand scrutiny as part of the development and eventual deployment (or breaking) of such schemes. This project studies the difficulty of these underlying hard problems, namely the path-finding and endomorphism ring problems for supersingular isogeny graphs, by studying the graphs themselves. As always, the scope of the project allows for further serendipitous discoveries.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在这个项目中，PI 研究一类与数论和几何相关的问题，这些问题具有某些共同的数学基础。 这些问题涉及群轨道（由某些对称性的递归应用产生的整数的集合）算术的基础研究以及某些新密码方案的基础数学。 特别是，该项目的后一个方面直接服务于后量子密码学的发展，即密码学将能够抵御量子计算机最终大规模发展的影响。 该项目将支持研究生以及科罗拉多大学博尔德分校实验数学实验室的培训，该实验室旨在扩大本科生对数学研究的参与，包括将在社会中担任多种角色的学生。 它还将支持 Numberscope 项目，该项目是一个针对科学家、艺术家和公众的外展项目。在第一个研究分支中，PI 研究在稀群轨道中出现的某些整数族。 在数论的整个历史中，人们对各种群轨道进行了研究，包括椭圆曲线上的点（现代密码学的大部分基础）和毕达哥拉斯三元组。 本项目中研究的轨道来自一类群（薄群），很难为其创建有效的工具。 例如，这些源于对连分数的研究。 然而，人们预计在新旧环境中都会发生某些高级现象。 其中一个例子是局部到全局现象，其中 PI 将研究局部信息知识（相对于单个素数）控制全局信息（轨道中的整数）的程度。 该项目的第二个方面涉及数论的密码学应用。 当前后量子密码学的候选之一是基于同源密码学，它基于椭圆曲线。 数学公钥密码学的安全性基于难题，而基于同源密码学的难题需要仔细审查，作为此类方案的开发和最终部署（或破解）的一部分。 该项目通过研究图本身来研究这些潜在难题的难度，即超奇异同​​源图的路径查找和自同态环问题。 与往常一样，该项目的范围允许进一步的偶然发现。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。