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Suppose we have a single, fixed plaintext block $p$ of 128 bits. Call the set of all possible 128 bit blocks $B$.

We could define fixed plaintext AES-128 as a function $f(k) \to B$ that takes a 128 bit key $k$ and outputs a 128 bit ciphertext block $c$.

As far as I know, AES-128 is a bijective function in this manner (is this assumption correct? edit: it is not). Additionally, the range spans $B$.

So how does AES with larger keys work? It must be a surjective function. What does this mapping look like? Are there, for every plaintext $p$, simply $2^{128}$ keys that map it to ciphertext $c$? Or is it unevenly distributed? Since the very same set must map $c$ back to $p$, it implies some sort of pattern in which bits are ignored/canceled out, dependent on the key.

Does any of this have any implications? Or are there no patterns here, and this is simply the way it is?

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