The Security Of Elliptic Curve Cryptosystem

With the rapid development of information technique, information security hasbecome an important subject. The propose of public key cryptosystem is to provide animportant encryption measure for computer network. Elliptic curve cryptosystem isbased on elliptic curve discrete log problem, and there is no any sub-exponential timealgorithm for solving general elliptic curve discrete log problem at present. As long asthe size of the finite field is large than 160 bits and the domain parameters are properlyselected, elliptic curve cryptosystem has the same security as the 1024-RSAcryptosystem. Hence, elliptic curve cryptosystem has obvious advantage over otherpublic key cryptosystem in security, implementation cost and usage effectiveness. It hasbeen accepted by several famous standard organizations, and become the public keycryptography standard of companies and organizations. A trend of replacing RSAcryptosystem with elliptic curve cryptosystem has been formed. Some countries(suchas the U.S.A, Japan, South Korea, and some Europian countries) have adopted ellipticcurve cryptosystem in their national cryptography standards.The known methods for solving elliptic curve discrete log problem can beclassified into two kinds: one is for general elliptic curve, e.g. Pollard's Rho method;another is for specific elliptic curve,e.g.MOV method.Only when we choose thedomain parameters properly, these attacks can be avoided. While we learn the attackresults,intensive study on the basic principle of all kinds of attack methods can help usavoid the existential and potential attacks. Comparably, all the public key cryptoschemealgorithms perform slowly, including the elliptic curve scheme. Hence, the research ofthe rapid implementation of elliptic curve scheme is of important significance. This thesis studies all the attack methods on elliptic curve discrete log problemdeeply,and proposes a method for choosing the domain parameters of elliptic curvecryptosystem in practice use.Meanwhile, we study and compare the main algorithmsfor implementing elliptic curve cryptosystem,present an optimized method for softwareimplementation of elliptic curve cryptosystem.