Can we design a block cipher based on SP-network without loss of E/D similarity theoretically?

Question

I understand what SP-network (Substitution-Permutation Network) and Feistel Network look like. And those ciphers based on Feistel (e.g. DES) often achieve so called E/D similarity, which indicates that the cipher itself follows the same pattern during either encryption or decryption.

SPN, comparing with Feistel Network, can bring more "confusion" into the cipher indeed. But the ciphers deriving from SPN (e.g. AES) may not be able to realize this E/D similarity. For instance, we have to put the inverse SBOX into the AES decryption process.

Thus, can we design a block cipher based on SP-network without loss of E/D similarity theoretically? Thanks!

Answer 1

Actually, to deal with the conflict between the inverse SBOX and E/D similarity, we can just expand the original SP network into something like "$S$->$P$->(entangled with key)->$P^{-1}$->$S^{-1}$". Then we can have a E/D similar block cipher. Some standard ciphers like PRINCE were designed this way.

However, it seems that we still can not make SPN itself to be of E/D similarity so we have to place extra components in it.