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Recent results by Alagic and Russell have given some evidence that theEven-Mansour cipher may be secure against quantum adversaries with quantumqueries, if considered over other groups than $(\mathbb{Z}/2)^n$. This promptsthe question as to whether or not other classical schemes may be generalized toarbitrary groups and whether classical results still apply to those generalizedschemes.  In this paper, we generalize the Even-Mansour cipher and the Feistel cipher.We show that Even and Mansour's original notions of secrecy are obtained on aone-key, group variant of the Even-Mansour cipher. We generalize the result byKilian and Rogaway, that the Even-Mansour cipher is pseudorandom, to superpseudorandomness, also in the one-key, group case. Using a Slide Attack wematch the bound found above. After generalizing the Feistel cipher to arbitrarygroups we resolve an open problem of Patel, Ramzan, and Sundaram by showingthat the $3$-round Feistel cipher over an arbitrary group is not superpseudorandom. Finally, we generalize a result by Gentry and Ramzan showing thatthe Even-Mansour cipher can be implemented using the Feistel cipher as thepublic permutation. In this last result, we also consider the one-key case overa group and generalize their bound.

How to Generate Pseudorandom Permutations Over Other Groups: Even-Mansour and Feistel Revisited