Brilliant Number theoretical Functional Equations

Define a function $f : \mathbb{Z} \rightarrow \{1,2,...,2017 \}$ to be brilliant if for any integer $n$ where $1 \leq n \leq 2016$ there exists a certain integer $p(n)$ so that we have:

$f(m + p(n)) \equiv f(m +n) - f(m) \pmod{2018}$

Find the number of brilliant functions.

Details and assumptions - 2017 is prime. - This is not an original problem. This has been modified from a competition problem whose source I have forgotten.