  Let $\E$ be an elliptic curve over a finite field $\F_{q}$ of $q$ elements,
with $\gcd(q,6)=1$, given by an affine Weierstra\ss\ equation. We also use
$x(P)$ to denote the $x$-component of a point $P = (x(P),y(P))\in \E$. We
estimate character sums of the form $$ \sum_{n=1}^N \chi\(x(nP)x(nQ)\) \quad
\text{and}\quad \sum_{n_1, \ldots, n_k=1}^N \psi\(\sum_{j=1}^k c_j
x\(\(\prod_{i =1}^j n_i\) R\)\) $$ on average over all $\F_q$ rational points
$P$, $Q$ and $R$ on $\E$, where $\chi$ is a quadratic character, $\psi$ is a
nontrivial additive character in $\F_q$ and $(c_1, \ldots, c_k)\in \F_q^k$ is a
non-zero vector. These bounds confirm several recent conjectures of D. Jao, D.
Jetchev and R. Venkatesan, related to extracting random bits from various
sequences of points on elliptic curves.
