I'm reading the paper “Collision-Resistant Hashing? Towards Making UOWHFs Practical” , which compared TCR (Target Collision Resistant) and ACR (Any collision Resistant). It says
we wish to stress one important practical advantage of TCR over ACR: because $x$ must be specified before $K$ is known, birthday attacks to find collisions are not possible
I understood the definition of the notion of $TCR$, but my question is:
Why is a birthday attack is not possible?