The Research Of Elliptic Curve Cryptosystem Over GF(2～n)

Being a milestone in the development of cryptography, public key crytosystem introduces a concept of double keys. In the Internet environment, this is of great importance to the management of key, the development of identity authentication and the applications of electronic commerce. Meanwhile, with the increasing applications of public key crytosystem and the strengthening of calculating capability, it is vital to provide public key crytosystem with high-speed encryption and decryption, with short key and adequate security. Elliptic curve cryptosystem answers for these requirements.This thesis analyzes in detail the security of the elliptic curve discrete logarithm problem, and proposes principles of establishing safe Elliptic curve cryptosystem against the present attacks.Elliptic curve cryptosystem is based on appropriate elliptic curves, this paper also explores the present approaches to finding appropriate elliptic curve in terms of algorithm, and finds that they are complex and unfit for choosing elliptic curves with higher security.Therefore, this thesis proposes a method of R&E (Random & Extension) with a relative less complex algorithm in time and space, and finds out a close elliptic curve accordingly. Then studies how to choose coefficients of elliptic curves.Based on the digital signature scheme of elliptic curve, this paper also designs a blind digital signature scheme and proves it. Then applies it to an electronic lottery scheme and studies the feasibility involved.