CT-ISG: New Directions in Elliptic Curve, Pairing-Based, and Torus-Based Cryptography

本研究的目标是解决基于环面、基于配对、多重线性、椭圆曲线和阿贝尔簇密码学中的一些重要问题。 椭圆曲线加密有助于保护互联网安全，并被美国和其他政府和机构用来提供安全通信。 阿贝尔簇密码术包括椭圆和超椭圆曲线密码术，可以很好地扩展到高安全级别，并且在受限环境中特别有利。 基于配对的密码学是椭圆曲线的最新前沿应用，具有许多有用的安全应用。 基于环面的密码术和基于压缩的阿贝尔簇密码术提高了离散对数和基于配对的密码术的效率。 多线性密码学概括了基于配对的密码学，这是一个悬而未决的问题。高效且安全的解决方案将提供高效且安全的多方通信。这些系统的安全性以及它们所基于的数论问题的难度对于经济的健康以及国家和国际安全至关重要。该项目重点研究椭圆曲线离散对数问题和其他密码安全所基于的数论问题的安全性，以及构建高效、安全、可扩展的公钥密码系统。 这些技术将涉及一些深入而有趣的数学。该项目将通过支持研究生、促进代表性不足群体的参与、进一步开发尖端网络安全新课程以及促进公众教育的外展活动，教育和培训网络安全方面的多元化劳动力。

The goal of this research is to solve some important problems in nnnntorus-based, pairing-based, multilinear, and elliptic curve and abelian variety cryptography. Elliptic curve cryptography helps to secure the Internet and is used by the U.S. and other governments and institutions to provide secure communication. Abelian variety cryptography includes elliptic and hyperelliptic curve cryptography, scales well to high security levels, and is especially advantageous in constrained environments. Pairing-based cryptography is a recent cutting-edge use of elliptic curves, with numerous useful security applications. Torus-based cryptography and compression-based abelian variety cryptography improve the efficiency of discrete log and pairing-based cryptography. Multilinear cryptography, which generalizes pairing-based cryptography, is an open question; an efficient and secure solution would give efficient and secure multi- party communication. The security of these systems and the difficulty of the number theoretic problems on which they are based are crucial for the health of the economy and for national and international security.The project focuses on the security of the elliptic curve discrete log problem and other number theoretic problems on which cryptographic security is based, and on the construction of efficient, secure, and scalable public key cryptosystems. The techniques will involve some deep and interesting mathematics.This project will educate and train a diverse workforce in cybersecurity by supporting graduate students, promoting the participation of underrepresented groups, furthering the development of new courses in cutting-edge cybersecurity, and promoting outreach that educates the public.