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If a Pseudorandom Function (PRF) is supplied with a key with the same size of output block, can 2 or more keys generate the same output for a input?

pf flag

There a 2 examples:

A block cipher with 128 bits of block size taking a plaintext and a 128 bits key (AES-128).

A keyed hash function with 1024 bits of block size in its output, taking a message and a 1024 bits key (Skein-1024).

If I have a block or message M to process with a PRF and the key size is the same of output block, what are the chances of 2 or more keys among 2^N keys of keyspace generate the same output block for a message/block M?

I read this question and I know if the keysize is larger than the block size some keys will generate the same output for a fixed input: Block cipher with key longer than block size

fgrieu avatar
ng flag
Hint: under the model of a PRF, what' a model for it's output for constant input and two distinct keys? And then what's the probability of collision under that model? Note: the part of the question about key size being larger than some threshold needs not be taken into consideration for an ideal PRF. It may for a practical construction, but I don't know a precise reasoning where the output block size is a relevant threshold.
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