On the elliptic curve analogue of the sum-product problem

Let E be an elliptic curve over a finite field of q elements and x(P) to denote the x-coordinate of a point P=(x(P),y(P)) E. Let denote the group operation in the Abelian group of -rational points on E. We show that for any sets at least one of the sets is large. This question is motivated by a series of recent results on the sum-product problem over .