NMC 2014 Problem 1

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ (where $\mathbb{N}$ is the set of the natural numbers and is assumed to contain 0 ), such that

$$f\left(x^{2}\right)-f\left(y^{2}\right)=f(x+y) f(x-y)$$

for all $x, y \in \mathbb{N}$ with $x \geq y$.