Past exam of the computer science course of the University of Oxford / 2015 / Computer security / 1 / d / ii

Take the positive integer $k$ whose existence is allowed by the question and writeModulo $p$, if $c=0$ then $c^{3^k}=c$ directly. Otherwise Fermat's little theorem applies, since $3^k-1$ is a multiple of $p-1$. Hence $c^{3^k}=c(\mathrm{mod}p)$. The same two-case argument works modulo $q$. By the Chinese remainder theorem,soThis also covers a ciphertext that is not coprime to $n$; applying Euler's theorem without checking that hypothesis would miss this detail. Equivalently, repeated application of a permutation on a finite set eventually returns to the starting point, although the arithmetic argument gives the requested sufficient exponent.