环Z_n上椭圆曲线的密钥交换协议  

The Elliptic Curves over Z_n and Key Exchange Protocol

作　　者：朱文余[1] 孙琦[1] 

机构地区：[1]四川大学数学学院,四川成都610064

出　　处：《电子学报》2005年第1期83-87,共5页Acta Electronica Sinica

基　　金：国家自然科学基金 (No .1 0 1 2 81 0 3)

摘　　要：设n =pq,p,q为奇素数 ,环Zn 上的椭圆曲线En(a ,b)的SOM密钥交换协议与QV密钥交换协议均选取En(a ,b)上的阶为Mn=lcm{ #Ep(a ,b) ,#Eq(a ,b) }的点G作为公钥 (称G为基点 ) ,并且限定其对应的Ep(a ,b)和Eq(a ,b)均为循环群 ,这就限制了这两个协议只能选择一类特殊的椭圆曲线En(a ,b)构作密钥交换协议 .本文指出 ,Ep(a ,b)和Eq(a ,b)均为循环群这一限定是不必要的 .本文给出了En(a,b)上存在阶为Mn 的点G的一个充分必要条件 ,并给出一个例子 ,其中Ep(a ,b)为循环群 ,Eq(a ,b)为非循环群 ,且对应的En(a ,b)上有阶为Mn 的点G .同时 ,本文选取En(a ,b)上阶为lcm{n1,m1}的点作为基点 ,这里n1,m1分别为Ep(a ,b)和Eq(a ,b)的最大循环子群的阶 .这样 ,就能够选择更多的椭圆曲线En(a ,b) ,用来构作密钥交换协议 (包括将两方之间的密钥交换协议扩展到三方 ) .SOM key exchange protocol and QV key exchange protocol were based on an elliptic curve En(a,b) over the ring Z n with a point G of order Mn=lcm{# Ep(a,b),# Eq(a,b)},where n=pq and p,q are odd primes.They pointed out that such a base point G exists if Ep(a,b) and Eq(a,b) are both cyclic groups.This restricts the choice of elliptic curves used to implement their protocols.In this paper we propose a necessary and sufficient condition under which En(a,b) has a point of order Mn=lcm{# Ep(a,b),# Eq(a,b)} and show by an example that En(a,b) may have a point G of order Mn even if Ep(a,b) is a cyclic group and Eq(a,b) is not.Our generalization makes it possible to choose more elliptic curves to establish key exchange protocol.And we give a new three or more users key exchange protocol with a point of order lcm{n 1,m 1} as base point,where n 1,m 1 are respectively the order of the maximal cyclic subgroups of Ep(a,b) and Eq(a,b).

关 键 词：密钥交换协议 环Zn 环Zn上的椭圆曲线 

分 类 号：TN918.1[电子电信—通信与信息系统]