CT-ISG: The Foundational Security of Elliptic Curve Cryptography

在不断扩大的网络社区中，公钥加密技术被部署在各种平台和网关上，以确保各种通信和交易的真实性、保密性和完整性。因此，公钥加密的安全性是网络空间安全的基础。椭圆曲线密码学近年来兴起，以应对无线和移动通信中带宽、功率和尺寸限制的挑战，并有可能成为无线领域公钥密码学的主流。本文主要研究椭圆曲线密码的基础安全问题。利用全局对偶理论，提出了一种统一的方法来研究椭圆曲线密码系统所依赖的离散对数问题。这项研究旨在识别和分类椭圆曲线群的弱情况，超出目前已知的范围。本文的研究结果将加深对椭圆曲线密码系统安全性的理解，特别是对特定类别椭圆曲线的相对密码强度的理解。所提出的研究中开发的技术和方法将为曲线密码学中类似问题的未来研究提供基础。本研究的长期目标是探索使用代数群构建未来几代公钥密码系统的前景。椭圆曲线密码学也是一门集数学和工程研究与教育于一体的优秀学科。该项目的一部分是通过精心设计的课程、研讨会和讲习班，向各个层次和跨学科的学生介绍这一领域的研究。

In the ever expanding cyber community public-key cryptography isdeployed at various platforms and gateways to ensure authenticity,confidentiality, and integrity in all kinds of communication andtransaction. Therefore the security of public-key cryptography isfoundational to the security of the cyberspace. Elliptic curvecryptography has risen in recent years to meet the challenge ofconstraints in bandwidth, power, and size in wireless and mobilecommunication, and is likely to become the mainstay for public-keycryptography in the wireless arena. This research focuses oncritical issues concerning the foundational security of ellipticcurve cryptography. A unified approach using the theory of globalduality is developed to study the discrete logarithm problem uponwhich elliptic curve cryptosystems are based. This research seeks toidentify and classify weak cases of elliptic curve groups beyondwhat is currently known. The findings of this research will deepenunderstanding of the security of the elliptic curve cryptosystems ingeneral, and relative cryptographic strength of specific classes ofelliptic curves in particular. The techniques and methodologydeveloped in the proposed research will provide the basis for futureresearch on similar issues in curve-based cryptography. Along-term goal of this research is to explore the prospect of usingalgebraic groups for building future generations of public-keycryptosystems. Elliptic Curve Cryptography is also an excellentsubject to integrate research and education in mathematics andengineering. Part of this project is to introduce students at alllevels and across disciplinary boundaries to research in this areathrough well-designed courses, seminars and workshops.