Researches On Scalar Multiplication Fast Algorithm In Elliptic Curve

By the driving of the rapid development of network technology, computer network has gradually applied in various fields. With the rapid development of Internet, computer network security problem has become increasingly serious.Public key cryptography is still relatively expensive, which corresponds to relatively symmetric key, it is still an open challenge to further reduce the computational cost of the most time-consuming element of public key cryptography, to ensure its use for commercial communications and internet-based security applications. The elliptic curve cryptosystem is proposed makes algebraic geometry and number theory in the field of research for a long time time of elliptic curve on the stage of cryptography.Elliptic curve cryptosystem scalar multiplication and multi-scalar multiplication is to achieve elliptic curve cryptosystem major operation, the operation speed of the speed of the decision to implement efficient Elliptic curve cryptosystem speed as a whole.In this paper, scalar multiplication of elliptic curve cryptosystem is analyzed and studied, and then insufficient for scalar multiplication algorithm is related to an improved, so as to improve the operation efficiency of the algorithm. The main research contents and results are as follows:(1) A basic understanding of the weaknesses in the existing algorithms is achieved through researches and analyses of NAF scalar multiplication algorithm. Scalar k is expressed in the form of NAF in this method. Scalar k in that form may be longer than the binary representation form based on the nature of NAF. If zeroes can be made more concentrated in NAF form, then calculation efficiency can be improved by sliding technique. An improved NAF algorithm is proposed in this paper to solve these problems, the length of k can be reduced by using the improved algorithm. The data show that efficiency can be improved by about 9% using the improved algorithm, indicating that the improved algorithm is better than the original algorithm.(2) Through the analysis of Dimitrov’s scalar multiplication base on SMBR, because of need to compute P+Q,2P+Q many times in scalar multiplication based on MBNS. In the paper presents a new scalar multiplication base on SMBR. This algorithm only use one inversion based on trading in versions for multiplications, although the multiplication and square computing increase, but over all computation efficiency is improved. Bcause of analysis the optimal path, use the fast algorithm of direct computing 2k P,3k P make the efficiency of new algorithm is superior to algorithm given by Dimitrov scalar multiplication.