# 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 FunctionNo Birthday Attack to TCRWeaker Notion of Target Collision Resistance

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This question is not one that is answered with an example. –  K.G. Nov 10 '13 at 9:31
@K.G. Then Why birthday attack is not possible? –  juaninf Nov 10 '13 at 11:33

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.
@nightcracker jaja, thank by your reply, but Why the adversary can't make a birthday attack after the $K$ key es know? –  juaninf Nov 11 '13 at 16:24
@nightcracker thank by your reply, I'm reading birthday attack, wikipedia say: we expect to obtain a pair of different arguments $x_1$ and $x_2$ with $f(x_1)=f(x_2)$ after evaluating the function for about $1.25\sqrt{H}$ different arguments on average. I have a doubt in relation "different arguments", Can I evaluate $f(x_1)=f(x_2); f(x_1)=f(x_3),$.... or always all arguments have be differents, in other words do not repeat $f(x_1)$ in the example? –  juaninf Nov 11 '13 at 17:15