Revisiting structure graphs: Applications to CBC-MAC and EMAC
Ashwin Jha, Mridul Nandi · Journal of Mathematical Cryptology · 2016
Abstract In [2], Bellare, Pietrzak and Rogaway proved an O ( ℓ q 2 / 2 n ) ${O(\ell q^{2}/2^{n})}$ bound for the PRF (pseudorandom function) security of the CBC-MAC based on an n -bit random permutation Π, provided ℓ < 2 n / 3 ${\ell<2^{n/3}}$ . Here an adversary can make at most q prefix-free queries each having at most ℓ ${\ell}$ many “blocks” (elements of { 0 , 1 } n ${\{0,1\}^{n}}$ ). In the same paper an O ( ℓ o ( 1 ) q 2 / 2 n ) ${O(\ell^{o(1)}q^{2}/2^{n})}$ bound for EMAC (or encrypted CBC-MAC) was proved, provided ℓ < 2 n / 4 ${\ell<2^{n/4}}$ . Both proofs are based on structure graphs representing all collisions among “intermediate inputs” to Π during the computation of CBC. The problem of bounding PRF-advantage is shown to be reduced to bounding the number of structure graphs satisfying certain collision patterns. In the present paper, we show that [2, Lemma 10], stating an important result on structure graphs, is incorrect. This is due to the fact that the authors overlooked certain structure graphs. This invalidates the proofs of the PRF bounds. In [31], Pietrzak improved the bound for EMAC by showing a tight bound O ( q 2 / 2 n ) ${O(q^{2}/2^{n})}$ under the restriction that ℓ < 2 n / 8 ${\ell<2^{n/8}}$