On Elliptic Curve Cryptosystem Algorithms And Their Applications

With the fast development of technology in computer network and wireless communications, the problem of information security has gone into each aspect of the daily life, thus becoming a hot point that people concerned. Cryptology is a core problem of information security, and modern cryptology researches two kinds of cryptosystems: the secret key cryptosystem and the public key cryptosystem. It is in addition to providing the function that the secret key cryptosystem keeps secret the correspondence, and the public key cryptosystem can provide some functions such as digital signature, identity authentication that the single key cryptosystem can't compare to.Elliptic curve cryptography is one of public key cryptosystems, which bases on the security of elliptic curve discrete logarithm. The author studied several kinds of attacks on elliptic curve discrete logarithms, and divided them into two types: attacks on general elliptic curve discrete logarithms and attacks on special elliptic curve discrete logarithms. For several kinds of special elliptic curves, their discrete logarithms are sub-exponential, and for general elliptic curves, their discrete logarithms are exponential. So the security of elliptic curve cryptography is the highest in the public key cryptosystems. It have already been got the extensive application, and a lot of nations and organizations adopt for the public key cryptosystem logarithm standard, becoming the public key cryptosystem of the next generation.In the implementation of the elliptic curve cryptosystem, we first have to select a secure elliptic curve. At the present time, selecting a random elliptic and computing its order is the most security way. In this paper, the efficient implementation of SEA algorithm is studied. For the methods of the SEA algorithm such as selecting modular polynomial, Elkies algorithm, Atkin algorithm, we made some researches and improvements and provide an efficient scheme of the combination of the methods in the SEA algorithm.