# Are there any encryption schemes that enable to permute homomorphically?

According to the Barrington's theorem, any circuit in NC1 can be converted to a branching program, whose main operation is the composition of permutations (along with the choosing of permutations according to inputs, which we omit here) . So I wonder whether there is a scheme that can directly encrypt the two-line notation of permutation, which also enables to do the composition of permutations on the ciphertext, except the general methods by using FHE/Multilinear mapping on permutation matrices?

Leveled FHE can evaluate arbitrary-length compositions of permutations, since iterated composition of permutations can be carried out in low depth. $\:$ (So, one does not need full FHE.)
• I don't have any hints about that, but their hint about that forms section 3.1, which starts on page 7. $\hspace{.4 in}$ – user991 Aug 24 '15 at 5:07