Research And Application Of Elliptic Curve Cryptography

Security issues are playing an important role in computer networks. As the Internet becomes more and more accessible to the public, many kinds of commercial and social transactions can be performed electronically through the network; cryptographic algorithms are needed in order to insure the security of these activities. In public-key algorithms, Elliptic Curves Cryptography (ECC) algorithm has virtues of higher security and higher speed, allowing shorter key lengths, and requiring fewer computational resources. The theory of ECC has being well-studied and it is the hot spot in the field of cryptography study, especially the techniques of the implementation of ECC. A Zero-Knowledge proof (ZKP) is a powerful tool which can be used and already be used for many cryptographic applications. While because of the completeness property and the soundness property, the existing Zero-Knowledge proofs are iterative in nature. The multiple communication rounds make ZKPs unsuitable for practice.In this dissertation we studied the basic theory of Elliptic Curve Cryptography, including Galois field, finite group theory, the reduced form of elliptic curve equation, and the representation of the points that are on the elliptic curves. We proposed two new point multiplication algorithms which improved on Sliding Windows technology after studying and analyzing some classic point multiplication algorithms. The new algorithms, comparing with the old algorithm, not only improved the calculating speed but also reduced the memory consuming time. We applied the elliptic curves cryptographic to the ZKPs, and proposed a new ZKP protocol which runs in one-round while ensure the completeness property and the soundness property.