Study On Speeding Up Algorithm Of Elliptic Curve Cryptograhy Over GF(2～n)

Elliptic Curve Cryptography (ECC) has the highest safety strength of private key per bit in the Public-Key Cryptography recently. Under similar secure conditions, the ECC has the advantages such as: less computation amounts, shorter length of private key, smaller storing and bandwidth. Moreover, it has been declared as standard documents adopted by many international standard institutions and regarded as the most universally used public key system。This scalar multiplication is the most basic operation in elliptic curve cryptosystem. At recent, the researching hot point about elliptic curve cryptosystem is to find efficient algorithms to reduce the complexity of computing scalar multiplication on elliptic curve over GF ( 2n). The scalar multiplication is performed by iterative additions and doublings on the elliptic curve. Therefore, performing addition and doubling on an elliptic curve fast is crucial for efficient implementation of these cryptosystems.In this paper, firstly we presented some background about elliptic cure. Then, after we analyzed the researching work on the scalar multiplication algorithm over ( )GF 2 m,we did some work on the two aspects below:1.We brought forward a new mixed coordinates system to denote the point over EC, thereby reduce the number of field multiplication in scalar multiplication. In this paper, we introduced a new point addition equation basing on the new mixed coordinates. Then we describe how to combine the proposed mixed coordinates with all kinds of scalar multiplication. At last, through analyzing a special example, we drew a conclusion that the number of Field multiplication of the scalar multiplication algorithm will be reduced with the point of ECC being denoted by the advanced mixed coordinates.2.We brought forward an advanced scalar multiplication algorithm which was composed of a new sign-binary representation of integer k and a new algorithm to pre-computing the points. In this paper, we presented transforming algorithm to get a new sign-binary representation of integer k and a new algorithm to pre-computing the points. Then, We proofed its correctness. At last, through doing the 10 random...