AES-128 keys unique for fixed plaintext/ciphertext pair?

[Matt Crawford]

> ... We can ask what is the 
> probability of a collision between f and g, i.e. that there exists 
> some value, x, in S such that f(x) = g(x)?

But then you didn't answer your own question.  You gave the expected
number of collisions, but not the probability that at least one
exists.

That probability the sum over k from 1 to 2^128 of (-1)^(k+1)/k!,
or about as close to 1-1/e as makes no difference.


But here's the more interesting question. If S = Z/2^128 and F is the
set of all bijections S->S, what is the probability that a set G of
2^128 randomly chosen members of F contains no two functions f1, f2
such that there exists x in S such that f1(x) = f2(x)?

G is a relatively miniscule subset of F but I'm thinking that the
fact that |G| = |S| makes the probability very, very small.

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