The Exact PRF-Security of NMAC and HMAC

NMAC is a mode of operation which turns a fixed input-length keyed hash function f into a variable input-length function. A practical single-key variant of NMAC called HMAC is a very popular and widely deployed message authentication code (MAC). Security proofs and attacks for NMAC can typically be lifted to HMAC. NMAC was introduced by Bellare, Canetti and Krawczyk [Crypto’96], who proved it to be a secure pseudorandom function (PRF), and thus also a MAC, assuming that (1) f is a PRF and (2) the function we get when cascading f is weakly collision-resistant. Unfortunately, HMAC is typically instantiated with cryptographic hash functions like MD5 or SHA-1 for which (2) has been found to be wrong. To restore the provable guarantees for NMAC, Bellare [Crypto’06] showed its security based solely on the assumption that f is a PRF, albeit via a non-uniform reduction. – Our first contribution is a simpler and uniform proof: If f is an ε-secure PRF (against q queries) and a δ-non-adaptively secure PRF (against q queries), then NMAC is an (ε + `qδ)-secure PRF against q queries of length at most ` blocks each. – We then show that this ε+`qδ bound is basically tight. For the most interesting case where `qδ ≥ ε we prove this by constructing an f for which an attack with advantage `qδ exists. This also violates the bound O(`ε) on the PRF-security of NMAC recently claimed by Koblitz and Menezes. – Finally, we analyze the PRF-security of a modification of NMAC called NI [An and Bellare, Crypto’99] that differs mainly by using a compression function with an additional keying input. This avoids the constant rekeying on multi-block messages in NMAC and allows for a security proof starting by the standard switch from a PRF to a random function, followed by an informationtheoretic analysis. We carry out such an analysis, obtaining a tight `q/2 bound for this step, improving over the trivial bound of `q/2. The proof borrows combinatorial techniques originally developed for proving the security of CBC-MAC [Bellare et al., Crypto’05]. We also analyze a variant of NI that does not include the message length in the last call to the compression function, proving a `q/2 bound in this case.