No Birthday Attack to TCR 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?

My other questions related to the same paper:
Dependence on Keyed Hash Function • No Birthday Attack to TCR • Weaker Notion of Target Collision Resistance
I am literally quoting the paper here. You should really try to read the paper properly first before asking questions.


  In the notion of [22] the adversary does not get credit for finding any old collision. The adversary must still find a collision $M, M'$  but now $M$ is not allowed to depend on the key: the adversary must choose it before the key $K$ is known.


I can't really give an example of this, just as I can't give an example of why I can't produce light with a brick.