How to Generate Pseudorandom Permutations Over Other Groups: Even-Mansour and Feistel Revisited
This work addresses cryptographic security for quantum adversaries by extending classical schemes to arbitrary groups, though it appears incremental as it builds on prior results.
The paper generalizes the Even-Mansour and Feistel ciphers to arbitrary groups, showing that the one-key group variant of Even-Mansour achieves original secrecy notions and super pseudorandomness, and resolves an open problem by proving the 3-round Feistel cipher over arbitrary groups is not super pseudorandom.
Recent results by Alagic and Russell have given some evidence that the Even-Mansour cipher may be secure against quantum adversaries with quantum queries, if considered over other groups than $(\mathbb{Z}/2)^n$. This prompts the question as to whether or not other classical schemes may be generalized to arbitrary groups and whether classical results still apply to those generalized schemes. 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 a one-key, group variant of the Even-Mansour cipher. We generalize the result by Kilian and Rogaway, that the Even-Mansour cipher is pseudorandom, to super pseudorandomness, also in the one-key, group case. Using a Slide Attack we match the bound found above. After generalizing the Feistel cipher to arbitrary groups we resolve an open problem of Patel, Ramzan, and Sundaram by showing that the $3$-round Feistel cipher over an arbitrary group is not super pseudorandom. Finally, we generalize a result by Gentry and Ramzan showing that the Even-Mansour cipher can be implemented using the Feistel cipher as the public permutation. In this last result, we also consider the one-key case over a group and generalize their bound.