  For a given elliptic curve $\mathbf{E}$ over a finite field of odd
characteristic and a rational function $f$ on $\mathbf{E}$ we first study the
linear complexity profiles of the sequences $f(nG)$, $n=1,2,\dots$ which
complements earlier results of Hess and Shparlinski. We use Edwards coordinates
to be able to deal with many $f$ where Hess and Shparlinski's result does not
apply. Moreover, we study the linear complexities of the (generalized) elliptic
curve power generators $f(e^nG)$, $n=1,2,\dots$. We present large families of
functions $f$ such that the linear complexity profiles of these sequences are
large.
