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 restri

cts 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).