On the elliptic curve endomorphism generator

For an elliptic curve $E$ over a finite field we define the point sequence $(P_n)$ recursively by $P_n=\vartheta (P_{n-1})={\vartheta }^n(P_0)$ with an endomorphism $\vartheta \in \mathrm{E}\mathrm{n}\mathrm{d}(E)$ and with some initial point $P_0$ on $E$. We study the distribution and the linear complexity of sequences obtained from $(P_n)$.