Improving the Security of MACs Via Randomized Message Preprocessing

"Hash then encrypt" is an approach to message authentication, where first the message is hashed down using an e-universal hash function, and then the resulting k-bit value is encrypted, say with a block-cipher. The security of this scheme is proportional to eq2, where q is the number of MACs the adversary can request. As e is at least 2-k, the best one can hope for is O(q2/2k) security. Unfortunately, such small e is not achieved by simple hash functions used in practice, such as the polynomial evaluation or the Merkle-Damgard construction, where e grows with the message length L. The main insight of this work comes from the fact that, by using randomized message preprocessing via a short random salt p (which must then be sent as part of the authentication tag), we can use the "hash then encrypt" paradigm with suboptimal "practical" e-universal hash functions, and still improve its exact security to optimal O(q2/2k). Specifically, by using at most an O(log L)-bit salt p, one can always regain the optimal exact security O(q2/2k), even in situations where e grows polynomially with L. We also give very simple preprocessing maps for popular "suboptimal" hash functions, namely polynomial evaluation and the Merkle-Damgard construction. Our results come from a general extension of the classical Carter-Wegman paradigm, which we believe is of independent interest. On a high level, it shows that public randomization allows one to use the potentially much smaller "average-case" collision probability in place of the "worst-case" collision probability e.