The Index j in RC4 is not Pseudo-random due to Non-existence of Finney Cycle

In this very short note we prove that the pseudo-random index j of RC4 is indeed not pseudo-random. This is a simple result that missed our attention for quite a long time. We show that in long term Pr(j = i + 1) = 1 N − 1 N2 , instead of the random association 1 N and this happens for the non-existence of the condition S[i] = 1 and j = i + 1 that is mandatory for the non-existence of the Finney cycle.