If you iterate a cryptographic permutation long enough will you map the input to itself? Given a cryptographic permutation $\{0,1\}^n \rightarrow \{0,1\}^n$ does it follow that after some number of iterations you must eventually map the input to itself?
Any permutation $f$ (cryptographic or not) on a finite set $X$ satisfies $f^n=id$ for some $n&gt;0$. 

To see this, note that there's only finitely many functions from $X$ to itself. Hence $f,f^2,f^3,\dots$ aren't all distinct, so $f^k=f^l$ for some $k$ and $l$ with $l&lt;k$. Now $$id=f^{-l}f^l=f^{-l}f^k=f^{k-l}$$ so we can set $n=k-l$.