Base in congruency

Let $p = 2017$ be a prime and $\mathbb F_p$ be the integers modulo $p$ . A function $f : \mathbb Z \to \mathbb F_p$ is called good if there is $\alpha \in \mathbb F_p$ with $p\nmid \alpha$ such that

$\large\ f(x)f(y) = f(x + y) + {\alpha}^{y}f(x - y)\pmod p$

for all $x, y \in \mathbb Z$ . How many good functions are there that are periodic with minimal period $2016$ ?