With the development of network technology, computer cryptology become a important task. In the first paper, I discuss the development and basic concepts of cryptology. Emphases I introduce the theory and some representative arithmetic of public key cryptography. Then, first I introduce some basic concepts and properties of elliptic curve and elliptic curve discrete logarithm problem. Then I analyze the design idea of ECC and compare it's performance with other public key cryptography and discuss it's development. Then I analyze some arithmetic of ECC , for square overplus judging arithmetic, point adding arithmetic etc. , and design some optimized arithmetic, use some optimized measures.Then I discuss the whole implementation procedure at PC. At Last, I expect the development of application of cryptosystem.
