I'm looking into a secret sharing scheme that has a secret permutation $\theta$ which has the cycle structure (n/2)+(n/2) (i.e. two (n/2)-cycles).
The permutation $\theta$ is decomposed into two permutations $\alpha$ and $\beta$, where $\alpha$ is generated uniformly at random. So with knowledge of both $\alpha$ and $\beta$, we can find $\theta$, while with knowledge of $\alpha$ xor $\beta$, we cannot find $\theta$ (although, we could guess).
At this point, I want to make public $\beta\alpha(L)$ (L is actually a Latin square, but this is not too relevant for the question I want to ask). It is possible that an attacker could find $\beta\alpha$ from $\beta\alpha(L)$. However, I worry that knowledge of $\beta\alpha$ might give information about $\theta$.
If I know $\theta=\alpha\beta$, and I'm given the permutation $\beta\alpha$, what can I say about $\theta$? (without a priori knowledge of $\alpha$, $\beta$ or $\theta$)