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

作　　者：翁江 康晓春 豆允旗 马传贵[4] WENG Jiang;KANG Xiao-chun;DOU Yun-qi;MA Chuan-gui(Information and Navigation College,Air Force Engineering University,Xi’an,Shaanxi 710077,China;Information Engineering School,Communication University of China,Beijing 100024,China;State Key Laboratory of Mathematical Engineering and Advanced Computing,Zhengzhou,Henan 450001,China;Department of Basic,Army Aviation Institution,Beijing 101123 China)

机构地区：[1]空军工程大学信息与导航学院,陕西西安710077 [2]中国传媒大学信息工程学院,北京100024 [3]数学工程与先进计算国家重点实验室,河南郑州450001 [4]陆军航空兵学院基础部,北京101123

出　　处：《电子学报》2018年第9期2131-2138,共8页Acta Electronica Sinica

基　　金：国家自然科学基金项目(No.61379150);数学工程与先进计算国家重点实验室开放基金课题(No.2016A02);河南省重点科技攻关计划项目(No.122102210126;No.092101210502)

摘　　要：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%.

关 键 词：超奇异椭圆曲线 四元数代数 自同态环 Frobenius自同态 τ-adic展开 

分 类 号：TP309[自动化与计算机技术—计算机系统结构]