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研究了一类问题，这些问题与数字理论和几何形状有关，这些理论和几何形状具有某些数学基础。 这些问题涉及集体轨道算术的基础研究（这是由于某些对称性的递归应用而产生的整数集合）和某些新的加密方案的基本数学。 特别是，该项目的后一个方面是直接为量词后加密术的开发服务，即密码学将对最终开发量子计算机进行扩展。 该项目将支持研究生的培训，以及科罗拉多大学博尔德大学的实验数学实验室，该实验室旨在扩大本科生参与数学研究的参与，包括将继续在社会上扮演许多角色的学生。 它还将支持Numberscope项目，这是一个针对科学家，艺术家和公众的外展项目。 在整个数字理论的历史上，都研究了各种类型的轨道，例如椭圆曲线的点（许多现代密码学所基于的）和毕达哥拉斯的三元组。 该项目中研究的轨道来自一类小组（薄组），这些轨道很难创建有效工具。 例如，这些是由持续分数的研究而产生的。 但是，人们期望在旧环境和新环境中都会发生某些高级现象。 一个这样的例子是局部到全球现象，其中PI将研究局部信息的知识在多大程度上控制全球信息（轨道上的整数）。 该项目的第二个方面涉及数字理论的加密应用。 目前的量子后加密术之一是基于等速的密码学，它基于椭圆曲线。 数学公钥密码学的安全是基于严重问题，基于同性恋的密码学要求审查的严重问题是此类计划的开发和最终部署（或破坏）的一部分。 该项目研究了这些潜在的硬性问题的难度，即通过研究图形本身来解决超大等级图的路径调查和内态环问题。 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的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准通过评估来支持的。