On a character sum problem of H. Cohn

Let $f$ be a complex valued function on a finite field $F$ such that $f(0) = 0$, $f(1) = 1$, and $|f(x)| = 1$ for $x \neq 0$. Cohn asked if it follows that $f$ is a nontrivial multiplicative character provided that $\sum_{x \in F} f(x) \bar{f(x+h)} = -1$ for $h \neq 0$. We prove that this is the case for finite fields of prime cardinality under the assumption that the nonzero values of $f$ are roots of unity.