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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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There is no generic attack that works against all PRFs (for exactly the reasons you mentioned -- if the PRF is secure beyond the birthday bound, there's no attack).

There is an attack on many of the specific PRF schemes that are used in practice, precisely because those specific ones are not secure beyond the birthday bound. These attacks involve taking advantage of specific details of the design of that particular PRF (e.g., finding internal collisions).

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