Arithmetic of Thin Groups and Isogeny-Based Cryptography

在这个项目中，PI研究了一类与数论和几何有关的问题，这些问题具有一定的数学基础。这些问题涉及群轨道算法的基础研究（群轨道是由某些对称性递归应用产生的整数集合）和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，即密码学将在量子计算机的最终发展中变得安全。该项目将支持研究生的培训，以及科罗拉多大学博尔德分校的实验数学实验室，该实验室旨在扩大本科生对数学研究的参与，包括那些将在社会上扮演多种角色的学生。它还将支持Numberscope项目，这是一个针对科学家、艺术家和普通公众的扩展项目。在第一个研究分支中，PI研究出现在薄群轨道上的某些整数族。在数论的整个历史中，各种各样的群轨道都被研究过，包括椭圆曲线上的点（现代密码学的许多基础）和毕达哥拉斯三元组。在这个项目中研究的轨道来自一类组（瘦组），它们很难创建有效的工具。例如，它们来自于连分式的研究。然而，人们期望某些高级现象在旧环境和新环境中都发生。一个这样的例子是局部到全局的现象，PI将研究局部信息（相对于单个素数）的知识在多大程度上控制全局信息（轨道上的整数）。该项目的第二个方面涉及数论的密码学应用。目前后量子密码术的候选之一是基于椭圆曲线的等基因密码术。数学公钥密码学的安全性基于难题，而基于同基因的密码学的难题需要作为此类方案的开发和最终部署（或破坏）的一部分进行审查。本课题通过研究超奇异同构图的寻径和自同态环问题，来研究这些潜在难题的难度。一如既往，该项目的范围允许进一步的偶然发现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.