Applications to Cryptography of the Construction of Curves from Modular Invariants

有两个主要原因，尽管我们目前拥有使用密码学保护通信的可靠方法，但科学家必须继续改进和发现数学密码学方法。首先，随着技术的进步，需要性能更快的算法，这些算法可以在内存和/或计算能力较小的芯片上运行，例如智能手机和智能手表。第二，新的技术发展和新发现的漏洞和攻击随时可能使某些加密方法变得不那么安全，这使得有必要准备好一系列可供部署的替代方法。这个项目在数学上进行基础研究，这将支持新的数学密码方法的发展。更准确地说，这个项目的主要目标是开发必要的理论框架，以给出给定的六次复乘法域，为其雅可比简单并且具有与该域的整数环(如果有的话)的复乘的亏格3的每条超椭圆曲线写一个精确的方程，采用在亏格2的情况下使用的技术。除了这项工作，这个项目的目的是刻画域K，使得存在一个简单的超椭圆雅可比矩阵，它与域K的整数环复乘。本项目还将研究一个复乘法域是否可以同时容纳一个超椭圆雅可比矩阵和一个平面四次雅可比矩阵，它与域K的整数环相乘。由于亏格3的超椭圆曲线的雅可比被认为是安全的--而平面四次曲线的雅可比则不是--并且对于使用离散对数问题的密码学来说可能是有效的，因此在亏格3的超椭圆雅可比可以应用于密码学应用之前，研究这些问题是必须采取的第一步。这个奖项反映了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。

There are two main reasons why, despite the fact that we currently have robust ways to secure communications with cryptography, scientists must continue to improve and discover mathematical cryptographic methods. The first is that as technology advances, there is need for ever faster-performing algorithms that can be run on chips with smaller memory and/or computing power, such as smart phones and smart watches. The second is that new technological developments and newly-discovered vulnerabilities and attacks can at any time make certain cryptographic methods less secure, making it necessary to have a wide suite of alternative methods ready to be deployed. This project conducts fundamental research in mathematics that will support the development of new mathematical cryptographic methods.More precisely, the main goal of this project is to develop the necessary theoretical framework to, given a sextic complex multiplication field, write an exact equation for every hyperelliptic curve of genus 3 whose Jacobian is simple and has complex multiplication by the ring of integers of that field (if any), adapting techniques used in the case of genus 2. In addition to this work, the project aims to characterize the fields K such that there exists a simple hyperelliptic Jacobian with complex multiplication by the ring of integers of this field K. The project will also investigate whether a complex multiplication field can admit both a hyperelliptic and a plane quartic Jacobian with complex multiplication by the ring of integers of the field. As Jacobians of hyperelliptic curves of genus 3 are considered to be safe -- whereas Jacobians of plane quartic curves are not -- and potentially efficient for cryptography using the discrete log problem, investigating these questions constitutes the very first step that must be taken before hyperelliptic Jacobians of genus 3 can be deployed in cryptographic applications.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.