Balanced permutations Even–Mansour ciphers

The r-rounds Even–Mansour block cipher is a generalization of the well known Even–Mansour block cipher to r iterations. Attacks on this construction were described by Nikolić et al. and Dinur et al. for r = 2, 3. These attacks are only marginally better than brute force but are based on an interesting observation (due to Nikolić et al.): for a “typical” permutation P, the distribution of P(x) ⊕ x is not uniform. This naturally raises the following question. Let us call permutations for which the distribution of P(x) ⊕ x is uniformly “balanced” — is there a sufficiently large family of balanced permutations, and what is the security of the resulting Even–Mansour block cipher? We show how to generate families of balanced permutations from the Luby–Rackoff construction and use them to define a 2n-bit block cipher from the 2-round Even–Mansour scheme. We prove that this cipher is indistinguishable from a random permutation of {0, 1}²ⁿ, for any adversary who has oracle access to the public permutations and to an encryption/decryption oracle, as long as the number of queries is o(2^n/2). As a practical example, we discuss the properties and the performance of a 256-bit block cipher that is based on our construction, and uses the Advanced Encryption Standard (AES), with a fixed key, as the public permutation.