  Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in
$\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i
\in A \}$. Using certain constructions of expanders, we characterize all
polynomials $P$ for which the following holds
  \vskip2mm \centerline{\it If $|A+A|$ is small, then $|P(A)|$ is large.}
\vskip2mm
  The case $P=x_1x_2$ corresponds to the well-known sum-product problem.
