Weird Functional Equation

Algebra Level 4

$f^{f(a)}(b) f^{f(b)}(a) = \big(f(a+b)\big)^2$

Let $f:\mathbb{N} \to \mathbb{N}$ be an injective function such that the above holds true for all $a,b \in \mathbb{N}$ . Let $S$ be the sum of all possible values of $f(2017)$ . Find $S \bmod{1000}$ .

Note: $f^{k} (n)$ means $\underbrace{f\big(f(f(\ldots f(n)\ldots))\big)}_{\text{number of }f \text{'s}\ = \ k}.$