Say I have a cyclic permutation $P$, a known input $x$, and a known output $y$ such that
$$y = P^a x$$
for some $a$. Is there a good way to search for $a$ (i.e. better than brute force)? Are some families of permutations easier to work this out for?
I'm ideally looking for a solution that would work when the elements of $x$ and $y$ are not distinct (e.g. there are duplicate entries).
My current thought is to represent the permutation as a matrix, diagonalise, and then try viewing the multiplications as additions to the phase. Not quite sure where that would lead me, though.