Arithmetic of Thin Groups and Isogeny-Based Cryptography

在这个项目中，PI研究一类与数论和几何有关的问题，这些问题有一些共同的数学基础。这些问题涉及群轨道(由某些对称的递归应用产生的整数集合)算术和某些新密码方案的基本数学方面的基础研究。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，即密码学将是安全的，不会最终发展成规模化的量子计算机。该项目将支持研究生的培训，以及科罗拉多博尔德大学的实验数学实验室，该实验室旨在扩大本科生对数学研究的参与，包括将在社会上担任许多角色的学生。它还将支持Numbercope项目，这是一个针对科学家、艺术家和普通公众的外展项目。在研究的第一个分支中，PI研究在细群轨道上出现的某些整数家族。在数论的整个历史上，人们一直在研究各种类型的群轨道，包括例如椭圆曲线上的点(现代密码学的许多基础)和毕达哥拉斯的三元组。这个项目中研究的轨道来自一类很难创建有效工具的群(瘦群)。例如，这些起源于对连分式的研究。然而，人们预计在新旧环境中都会出现某些高水平的现象。一个这样的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)的知识控制全局信息(轨道上的整数)的程度。该项目的第二个方面涉及数论的密码应用。基于同源的密码学是当前后量子密码学的候选方案之一，它基于椭圆曲线。数学公钥密码学的安全性是基于难题的，而基于同源的密码学的难题需要作为此类方案的开发和最终部署(或破解)的一部分来仔细检查。本课题通过对超奇异同源图本身的研究，研究了这些基本困难问题的难度，即超奇异同源图的寻路问题和自同态环问题。一如既往，该项目的范围允许进一步的偶然发现。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.