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，这些问题具有某些共同的数学基础。 这些问题涉及群轨道算术的基础研究（群轨道是由某些对称的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，即密码学将安全地对抗量子计算机的最终发展。 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许多角色的学生。 它还将支持数字范围项目，这是一个面向科学家、艺术家和一般公众的外联项目。在第一个分支的研究，PI研究某些家庭的整数出现在轨道上的薄组。 在数论的历史上，各种群轨道都被研究过，包括椭圆曲线上的点（现代密码学的基础）和毕达哥拉斯三元组。 在这个项目中研究的轨道来自一类群体（瘦群体），有效的工具是很难创建。 例如，这些问题产生于对连分数的研究。 然而，人们期望某些高层次的现象在旧的和新的环境中都发生。 其中一个例子是局部到全局现象，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（轨道中的整数）。 该项目的第二个方面涉及数论的密码学应用。 后量子密码学的当前候选者之一是基于椭圆曲线的基于同源性的密码学。 数学公钥密码术的安全性是基于困难问题的，而基于同源性的密码术的困难问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。 本项目通过研究超奇异同构图本身来研究这些基本难题的难度，即超奇异同构图的寻路和自同态环问题。 该奖项反映了NSF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.