一类j=0超奇异椭圆曲线的性质及其标量乘算法

Property and Scalar Multiplication Algorithm on Supersingular Elliptic Curves with j Invariant 0

针对非超奇异椭圆曲线上的标量乘算法已经有比较多的研究.与非超奇异曲线不同,超奇异椭圆曲线的自同态环是四元数代数的一个序模,为非交换环.本文主要针对特征大于3的有限域上一类j不变量为0的超奇异椭圆曲线,分析了曲线自同态环及其商环的结构.进而研究了此类曲线上整数表示的性质,并基于这种表示方法提出了一种针对此类曲线的标量乘算法.理论上证明了针对此类超奇异曲线,当选择合适系数集合时,此表示实质上为padic展开.实验结果表明:相较于4-NAF等方法,p-adic表示方法提高标量乘效率一倍以上.

The scalar multiplication algorithms for non-supersingular elliptic curves have been widely studied.In contrast,the endomorphism ring of supersingular elliptic curve is an order in a definite quaternion algebra,which is not commutative.This paper focuses on a class of supersingular elliptic curves of j-invariant zero in characteristic greater than 3.We make analysis of the structures of its endomorphism ring and quotient ring.Further we study the properties of integer expansion according to this class of curves.Based on this representation,a scalar multiplication algorithm is proposed.We demonstrate that the representation is essentially the p-adic expansion in theory when a suitable digit set is chosen.The experimental results show that compared with the method of 4-NAF,the p-adic method improves the efficiency of scalar multiplication of more than 100%.