| Elliptic curve cryptosystems (ECC) has been the hot topic accounting for its many advantages such as high security, less storage, narrower bandwidth, especially suitable for smart cards and wireless applications etal. But when in the realization of the ECC, there are some key problems to solve, and one very important aspect is the fast realization of the ECC.The basic and most time-consuming comutation in elliptic curve cryptosystems is the scalar multiplication, so the computational performance of ECC schemes strongly depends on the efficiency of the scalar multiplication. Gerenrally speaking, there are two strategies to speed up the scalar multiplication:one is to seek fast algorithms over bottom filed, include multiplication, squares and inverse, the other is to investigate efficient the scalar K's representation, reduce point-add numbers. Based on the results of former researchers, we have done the research as follows:(1) Analysis multiplication, squares and inverse over bottom filed, and presents a new fast comb-multiplication without shift operation based on the existing algorithm. Compared with the original algorithm, the new algorithm in undertake no shift operation situation and need less pre storage.(2) Because of need to compute 5P many times in scalar multiplication based on MBNS, so this paper proposed an efficient algorithm to compute 5P directly over Fq in terms of affine coordinates. This algorithm only use one inversion based on trading inversions for multiplications, although the multiplication and square computing increase, but overall computation efficiency is improved.(3) Through the analysis of Dimitrov's scalar multiplication base on SMBR, combine with multi base number system and fast algorithms over bottom filed, presents a new scalar multiplication base on SMBR. Because of analysis the optimal path, and use the fast algorithm of direct computing 2kP,3kP and 5P, makes the efficiency of new algorithm is superior to algorithm given by Dimitrov scalar multiplication. |
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