KöMaL 2008 A462

Let $p$ be an odd prime and let 1<$a$<$p$ be an integer. Prove that $p$ divides $\sum_{k=0}^{p-2} {(-1)}^ka^{k^2}$ if and only if there exists an odd positive integer $k$ for which $a^k$ is congruent to 1 modulo $p$.