On the Indifferentiability of Key-Alternating Ciphers

The Advanced Encryption Standard (AES) is the most widely used block cipher. The high level structure of AES can be viewed as a (10-round) key-alternating cipher, where a t-round key-alternating cipher KAt consists of a small number t of fixed permutations Pi on n bits, separated by key addition: KAt(K,m) = kt ⊕ Pt(. . . k2 ⊕ P2(k1 ⊕ P1(k0 ⊕m)) . . . ), where (k0, . . . , kt) are obtained from the master key K using some key derivation function. For t = 1, KA1 collapses to the well-known Even-Mansour cipher, which is known to be indistinguishable from a (secret) random permutation, if P1 is modeled as a (public) random permutation. In this work we seek for stronger security of key-alternating ciphers — indifferentiability from an ideal cipher — and ask the question under which conditions on the key derivation function and for how many rounds t is the key-alternating cipher KAt indifferentiable from the ideal cipher, assuming P1, . . . , Pt are (public) random permutations? As our main result, we give an affirmative answer for t = 5, showing that the 5-round key-alternating cipher KA5 is indifferentiable from an ideal cipher, assuming P1, . . . , P5 are five independent random permutations, and the key derivation function sets all rounds keys ki = f(K), where 0 ≤ i ≤ 5 and f is modeled as a random oracle. Moreover, when |K| = |m|, we show we can set f(K) = P0(K) ⊕K, giving an n-bit block cipher with an n-bit key, making only six calls to n-bit permutations P0, P1, P2, P3, P4, P5.