Luby-Rackoff Ciphers from Weak Round Functions?

The Feistel-network is a popular structure underlying many block-ciphers where the cipher is constructed from many simpler rounds, each defined by some function which is derived from the secret key. Luby and Rackoff showed that the three-round Feistel-network – each round instantiated with a pseudorandom function secure against adaptive chosen plaintext attacks (CPA) – is a CPA secure pseudorandom permutation, thus giving some confidence in the soundness of using a Feistel-network to design block-ciphers. But the round functions used in actual block-ciphers are – for efficiency reasons – far from being pseudorandom. We investigate the security of the Feistel-network against CPA distinguishers when the only security guarantee we have for the round functions is that they are secure against non-adaptive chosen plaintext attacks (nCPA). We show that in the information-theoretic setting, four rounds with nCPA secure round functions are sufficient (and necessary) to get a CPA secure permutation. Unfortunately, this result does not translate into the more interesting pseudorandom setting. In fact, under the so-called Inverse Decisional Diffie-Hellman assumption the Feistel-network with four rounds, each instantiated with a nCPA secure pseudorandom function, is in general not a CPA secure pseudorandom permutation. We also consider other relaxations of the Luby-Rackoff construction and prove their (in)security against different classes of attacks. ⋆ Most of this work was done while the author was a PhD student at ETH where he was supported by the Swiss National Science Foundation, project No. 200020103847/1. Currently the author is partially supported by the Commission of the European Communities through the IST program under contract IST-2002-507932 ECRYPT. ⋆⋆ This work was partially supported by the Zurich Information Security Center. It represents the views of the authors.