Research of the Elliptic Curves and Application in Cryptography

The security of the modern cryptography protocols is based on mathematical problems, For example: big integer factor decomposition, finite field of discrete logarithm problem. In the general case, there are no polynomial time algorithms for this problem, constant improvement of the attacking algorithm made these protocols require much larger key size, so that make sure the use of the security. But the larger key size increase the complexity of the algorithms, therefore, it is very important to find a resistant attack algorithm, small burden, the speed the discrete logarithm cryptographic algorithm, elliptic curve cryptography algorithms is just to meet this need. In this paper I reviewed the general protocols of the asymmetric cryptography system, and describe the elliptic curve cryptography and its application to may be vulnerable to attack. Finally, I mention other applications that elliptic curves have had upon the analysis of other cryptosystems not involving them.