The Elliptic Curves over Z_n and Key Exchange Protocol
Sun Qi · Dianzi xuebao · 2005
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).