In the ever expanding cyber community public-key cryptography is deployed at various platforms and gateways to ensure authenticity, confidentiality, and integrity in all kinds of communication and transaction. Therefore the security of public-key cryptography is foundational to the security of the cyberspace. Elliptic curve cryptography has risen in recent years to meet the challenge of constraints in bandwidth, power, and size in wireless and mobile communication, and is likely to become the mainstay for public-key cryptography in the wireless arena. This research focuses on critical issues concerning the foundational security of elliptic curve cryptography. A unified approach using the theory of global duality is developed to study the discrete logarithm problem upon which elliptic curve cryptosystems are based. This research seeks to identify and classify weak cases of elliptic curve groups beyond what is currently known. The findings of this research will deepen understanding of the security of the elliptic curve cryptosystems in general, and relative cryptographic strength of specific classes of elliptic curves in particular. The techniques and methodology developed in the proposed research will provide the basis for future research on similar issues in curve-based cryptography. A long-term goal of this research is to explore the prospect of using algebraic groups for building future generations of public-key cryptosystems. Elliptic Curve Cryptography is also an excellent subject to integrate research and education in mathematics and engineering. Part of this project is to introduce students at all levels and across disciplinary boundaries to research in this area through well-designed courses, seminars and workshops.
