International Association
for Cryptologic Research

In this paper we study the security of hash-based MAC algorithms (such as HMAC and NMAC) above the birthday bound. Up to the birthday bound, HMAC and NMAC are proven to be secure under reasonable assumptions on the hash function. On the other hand, if an $n$-bit MAC is built from a hash function with a $l$-bit state ($l \\ge n$), there is a well-known existential forgery attack with complexity $2^{l/2}$. However, the remaining security after $2^{l/2}$ computations is not well understood. In particular it is widely assumed that if the underlying hash function is sound, then a generic universal forgery attack should still require $2^{n}$ computations and some distinguishing (e.g. distinguishing-H but not distinguishing-R) and state-recovery attacks should still require $2^{l}$ (or $2^k$ if $k < l$) computations.

In this work, we show that above the birthday bound, hash-based MACs offer significantly less security than previously believed. Our main result is a generic distinguishing-H and state-recovery attack against hash-based MACs with a complexity of only $\\tilde O(2^{l/2})$. In addition, we show a key-recovery attack with complexity $\\tilde O(2^{3l/4})$ against HMAC used with a hash functions with an internal checksum, such as GOST. This surprising result shows that the use of a checksum might actually weaken a hash function when used in a MAC. We stress that our attacks are generic, and they are in fact more efficient than some previous attacks proposed on MACs instantiated with concrete hash functions.

We use techniques similar to the cycle-detection technique proposed by Peyrin et al. at Asiacrypt 2012 to attack HMAC in the related-key model. However, our attacks works in the single-key model for both HMAC and NMAC, and without restriction on the key size.