The Exact Security of PMAC

PMAC is a simple and parallel block-cipher mode of operation, which was introduced by Black and Rogaway at Eurocrypt 2002. If instantiated with a (pseudo)random permutation over n-bit strings, PMAC constitutes a provably secure variable input-length (pseudo)random function. For adversaries making q queries, each of length at most ` (in n-bit blocks), and of total length σ ≤ q`, the original paper proves an upper bound on the distinguishing advantage of O(σ2/2n), while the currently best bound is O(qσ/2). In this work we show that this bound is tight by giving an attack with advantage Ω(q2`/2n). In the PMAC construction one initially XORs a mask to every message block, where the mask for the ith block is computed as τi := γi · L, where L is a (secret) random value, and γi is the i-th codeword of the Gray code. Our attack applies more generally to any sequence of γi’s which contains a large coset of a subgroup of GF (2). We then investigate, if the security of PMAC can be further improved by using τi’s that are k-wise independent, for k > 1 (the original distribution is only 1-wise independent). We observe that the security of PMAC will not increase in general, even if the masks are chosen from a 2-wise independent distribution, and then prove that the security increases to O(q2/2n), if the τi are 4-wise independent. Due to simple extension attacks, this is the best bound one can hope for, using any distribution on the masks. Whether 3-wise independence is already sufficient to get this level of security is left as an open problem.