Let $p$ be prime. Consider $\Bbb Z_{p}$, the cyclic multiplicative group.

Is it possible to choose a generator $c$ as small as $O(\log(p))$? (wiki shows $c$ as small as $O(\log^{6}(p))$ is possible under GRH http://en.wikipedia.org/wiki/Primitive_root_modulo_n#Upper_bounds)