Is a Feistel structure based cipher IND-CPA?

Using the format of the "game" to establish if a cipher is IND-CPA secure:

1. The adversary submits two distinct $M_0$, $M_1$ plain-texts to the challenger.
2. The challenger selects one of them at random and encrypts it with the symmetric key and gives cipher-text C.
3. The adversary has to guess which message was encrypted based on C with a probability greater than 1/2 for the cipher-text to be distinguishable.

If the challenger is using an encryption algorithm only based on Feistel structures, would it be possible to always correctly identify, which of the messages the challenger encrypted?

• As always with ciphers it depends on the details. A feistel-cipher can very well be IND-CCA2 (see Twofish for example), but it can also be horribly broken if the round function is crap. – SEJPM Sep 19 '16 at 13:45

And, given an attacker that is bounded to no more than $2^N-2$ queries, it is impossible to distinguish a random even permutation from a random permutation.
• @Kristian: I think the proof can be by examination for a $2\times2$-bit Feistel cipher (checking that the $16!/2$ even permutations are reached); then induction from a $2\times n$-bit to a $2\times(n+1)$-bit Feistel cipher. The details are slightly messy. – fgrieu Sep 21 '16 at 6:52