# How does the birthday attack work in AUTH and UF-CMA games?

In the AUTH and UF-CMA games, an adversary is required to forge a ciphertext or message/tag pair to win the game. Given an encryption scheme $E$ and a PRF $F$, let $\hat{E} = C || F_k(C)$ and $C = E(x)$ (encrypt-then-MAC with PRF as MAC), how is the birthday attack mounted? I understand it is sufficient to ask ~ $\sqrt{2^n}$ queries to find a collision in a MAC, but it's not clear how we can forge a new ciphertext/tag pair even if we have collision, because if we have found such a collision, we have already asked the oracle the two colliding messages. I also see how certain block ciphers can be exploited by finding internal collisions, but how is the birthday attack mounted in the general case?

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