  We prove, using combinatorics and Kloosterman sum technology that if $A
\subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim
|A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim
\frac{{|A|}^{{3/2}}}{q^{{1/4}}$.
