New answers tagged pseudo-random-function
Certainly yes. Proof A pseudorandom function must produce pseudorandom output for any given input. So, any seed value will provide a pseudorandom output. However, this might not work if the function has a limited input domain (list of possible inputs) causing incompatibility with the two functions. In your case, $F$ should be able to process the input ...
If you want to construct a PRF for arbitrarily-long inputs using AES, then just use CBC-MAC (while prepending the message length in the first block). I don't see any advantage in what you are proposing and therefore don't see any point in trying to analyze something non-standard.
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