HMIC 2019 Problem 5

Let $p=2017$ be a prime and $\mathbb{F}_{p}$ be the integers modulo $p$. A function $f: \mathbb{Z} \rightarrow \mathbb{F}_{p}$ is called good if there is $\alpha \in \mathbb{F}_{p}$ with $\alpha \not \equiv 0(\bmod p)$ such that

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

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