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 KA_t consists of a small number t of fixed permutations P_i on n bits, separated by key addition: KA_t(K, m) = k_t ⊕ P _t(...k₂ ⊕ P₂(k₁ ⊕ P ₁(k₀ ⊕ m))...), where, (k₀..., k _t) 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 KA_t indifferentiable from the ideal cipher, assuming P₁,...,P_t 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 k_i = 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,P ₄,P₅.