Dependence on Keyed Hash Function

Question

I'm reading the paper “Collision-Resistant Hashing? Towards Making UOWHFs Practical” which states:

With an ACR hash function $F$ the key $K$ is announced and the adversary wins if she manages to any collision $M,M'$ for $f = F_K$. The points $M$ and $M'$ may depend arbitrarily on $K$.

Could you give me an example of messages $M$ and $M'$, so that they depend arbitrarily on $K$?

_{My other questions related to the same paper:
Dependence on Keyed Hash Function • No Birthday Attack to TCR • Weaker Notion of Target Collision Resistance}

Answer 1

Say $F_K(x)$ = $M \bmod K$.

Then after $K$ is announced choose arbitrary $M$ and compute $M' = M + K$.