The Research Of ECC

Since Elliptic curve cryptography was proposed independently by Neal Koblitz and Victor Miller in 1985, the advantage of this Public key cryptosystem has been recognized. Compared to another famous public key algorithm based on RSA, ECC allows for shorter key length, has a higher safety property, faster speed, and requires fewer storage and bandwidth. Because of these advantages, many professionals think that elliptic curve crypto-system will be the most common standards of Public key cryptosystems. In recent years, ECC has received increased commercial acceptance as evidenced by its inclusion in standards by many accredited standards organizations such as IPsec and WAPI and so on.Scalar multiplication is the basic operation in Elliptic curve cryptography. In this paper, the general algorithms (such as binary method, NAF, sliding window method and so on) for scalar multiplication are analyzed and discussed, On the other hand, scalar multiplication consists of point addition and point double, on the basis of analyzing the speed of point addition and point double under different coordinate systems, we optimize operations in point addition and point double computation in mixed Jacobian coordinate. After optimization, in case of two multipliers, point double which costs 10 modular multiplication time could be finished in 5 modular multiplication time and point addition which cost 11 modular multiplication time could be finished in 6 modular multiplication time. On the other hand, point addition and point double can be implemented by modular multiplication in finite field. As early as 1985, P. L. Montgomery proposed Montgomery algorithm, this method replaced time-consuming division by addition and shift operations. In 1996, KOC discusses and analyzes several Montgomery multiplication algorithms. We find that a factor must be computed before computing another two large integers'multiplication, which hampers the parallelism of computation. A parallel algorithm is very important in design of cryptosystem chips. A high-speed chip could be designed using a good parallelism of the algorithm, by increasing certain of hardware resources and using parallelism and pipeline technique. In this paper we give improved algorithm. The analysis of application suggested that the efficiency of computation is obvious improved.