I don't have a deep mathematical background in cryptography. I am reading "The FFX Mode of Operation for Format-Preserving Encryption". Section D says the following.
Why feistel? The authors believe that, for FPE, there is, at present, no serious alternative to some form of Feistel. The approach benefits from being a classically known and extensively studied. For all their prescience in identifying FPE, the actual mechanistic ideas described by Brightwell and Smith do not possess any such history and are rather incoherent . Methods to achieve uniform shuffles on large domains, starting with a random function, remain completely impractical . Developing a new confusion/diffusion primitive seems out of that question in terms of delivering assurance and leveraging existing AES experience and implementations, while resurrecting an old confusion/diffusion primitive, say Hasty Pudding , is unrealistic for the same reasons. Only Feistel combines decades of history and a corpus of significant academic work.
Because a block cipher in the counter mode such as AES-CTR can encrypt a n bits plain text into a same sized n bits cipher text, it may be thought of a naive FPE cipher. But I guess it is not secure enough because they have invented many complex ciphers and finally standardized a few of them.
Can you explain the security aspects of this naive FPE cipher for n = 27 for example? I understand the concept of permutation, one-to-one mapping, key space, and message space.