Is it theoretically possible for the output of SHAKE to be an infinite concatenation of the same 512-bit block?

Question

Let $X$ denote a particular 512-bit sequence.
Let $A,B,C$ denote sequences of bits such that: $$len(X || A) = len(X || B) = len(X || C) = 1600$$

$$\operatorname{Keccak-}f(X||A) = X||B$$ $$\operatorname{Keccak-}f(X||B) = X||C$$ $$\operatorname{Keccak-}f(X||C) = X||A$$

then we have a loop, so the output of SHAKE (if my understanding of how the sponge construction works is correct and if $X||A$ happens to be the state which was output after xoring the final block of input) is nothing but an infinite concatenation of the same 512-bit block. Is it possible?

Answer 1

Minor correction: SHAKE generates either 1088 or 1344 bits per squeeze; not 512. So, the example you give will give a repeating pattern, but it'd be either 3264 or 4032 bits long, not 512.

Other than that, yes, that is possible, as far as we know. In fact, a far more likely repeat is:

$$\operatorname{Keccak-}f(A||X) = A||X$$

This would give a repeating pattern of 1088 or 1344 bits long.

In addition, not only is this possible, there's a good likelihood that there actually is such a $A || X$ pattern. Such a $A || X$ is known as a fixed point of the permutation; a random permutation has at least one fixed point with probability approximately $1 - 1/e \approx 0.6321$; assuming that $\operatorname{Keccak-}f$ does, in fact, act like a random permutation, then it's more likely than not that there is such a fixed point.