Applications to Cryptography of the Construction of Curves from Modular Invariants

从模不变量构造曲线在密码学中的应用

• 批准号: 1802323
• 负责人: Christelle Vincent
• 金额: $ 19.43万
• 依托单位: University of Vermont & State Agricultural College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802323&HistoricalAwards=false
• 关键词: Applications; Cryptography; Construction; Curves; Modular

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.

有两个主要原因，尽管我们目前有强大的方法来保护加密通信，但科学家必须继续改进和发现数学加密方法。首先，随着技术的进步，需要更快的算法，这些算法可以在具有较小内存和/或计算能力的芯片上运行，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，因此有必要部署一套广泛的替代方法。该项目进行数学基础研究，以支持新的数学加密方法的发展。更确切地说，该项目的主要目标是开发必要的理论框架，以给定六次复数乘法域，为每个亏格3的超椭圆曲线写出精确方程，其Jacobian是简单的，并且与该域的整数环（如果有的话）相乘，适应在属2的情况下使用的技术。除了这项工作之外，该项目的目的是描述域K，使得存在一个简单的超椭圆雅可比矩阵，该矩阵与域K的整数环相乘。该项目还将研究一个复数乘法域是否可以同时容纳一个超椭圆和一个平面四次雅可比矩阵，并通过该域的整数环进行复数乘法。由于亏格3的超椭圆曲线的雅可比行列式被认为是安全的-而平面四次曲线的雅可比行列式不是-并且对于使用离散对数问题的密码学可能是有效的，研究这些问题是在将亏格3的超椭圆雅可比算子应用于密码学应用之前必须采取的第一步。该奖项反映了NSF的法定使命，并被认为是值得的。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来提供支持。

项目成果

An inverse Jacobian algorithm for Picard curves

皮卡德曲线的逆雅可比算法

• DOI: 10.1007/s40993-021-00253-1
• 发表时间: 2021
• 期刊: Research in Number Theory
• 影响因子: 0.8
• 作者: Lario, Joan-C.;Somoza, Anna;Vincent, Christelle
• 通讯作者: Vincent, Christelle

Isogeny classes of abelian varieties over finite fields in the LMFDB

LMFDB 中有限域上阿贝尔簇的同源类

• DOI: 10.1007/978-3-030-80914-0_13
• 发表时间: 2021
• 期刊: and Computation
• 作者: Dupuy, Taylor;Kedlaya, Kiran S.;Roe, David;Vincent, Christelle
• 通讯作者: Vincent, Christelle

Modular invariants for genus 3 hyperelliptic curves

属 3 超椭圆曲线的模不变量

• DOI: 10.1007/s40993-018-0146-6
• 发表时间: 2019
• 期刊: Research in Number Theory
• 影响因子: 0.8
• 作者: Ionica, Sorina;Kılıçer, Pınar;Lauter, Kristin;Lorenzo García, Elisa;Mânzăţeanu, Adelina;Massierer, Maike;Vincent, Christelle
• 通讯作者: Vincent, Christelle