We first consider the case where [GAMMA] has no elliptic point nor irregular cusp.

We choose a normal subgroup [GAMMA]' [??] [GAMMA] of finite index that has no elliptic point nor irregular cusp.

Let p [member of] X be either an elliptic point or a cusp, and F = [pi]* p be the singular fibre over p.

Also Song [16] uses exponential operation and the time taken to perform an exponential operation is approximately 8 times than the time taken to perform one

elliptic point multiplication [40] i.e.

Note that the Levi form is 0 at any elliptic point p, since [V.sub.p] + [V.sub.p] spans all of [CT.sub.p.][R.sub.N].

In section 5 we discuss the existence of global first integrals at elliptic points in the situation where the elliptic region is assumed to be a Stein manifold.

Global first integrals at elliptic points. The conclusion of Theorem 3.5 guarantees for each point p with d(p) = 1 the existence of global solutions whose differentials are nonvanishing at p.

It might be mentioned that one elliptic point (7) exists surrounded by all the closed orbits in the cross-section of the channel.

It might be noted, however, that both [T.sub.cir] and [Delta][L.sub.axial] are infinity near the elliptic point, and [T.sub.cir] is infinity near the wall as shown in Fig.

18(a) and (b), [Mathematical Expression Omitted] and [Mathematical Expression Omitted] obtained from Eqs 17 and 18 have different results from those directly differentiated especially near the elliptic point and the wall (dimensionless [y.sup.*] = y/H).

The

elliptic points are used for generating the keys and whenever the new users are added or users leave, the keys are regenerated by considering the particular elliptic curve alone.

In other words, there are two

elliptic points and one hyperbolic point even in the B zone of the perturbed system.