Adventures in crypto dark matter: attacks, fixes and analysis for weak pseudorandom functions

A weak pseudorandom function (weak PRF) is one of the most important cryptographic primitives for its efficiency although it has lower security than a standard PRF. Recently, Boneh et al. (in: Theory of cryptography conference, Springer, pp 699–729, 2018) introduced two types of new weak PRF candidates, which are called a basic Mod-2/Mod-3 and alternative Mod-2/Mod-3 weak PRF. Both use the mixture of linear computations defined on different small moduli to satisfy conceptual simplicity, low complexity (depth-2 ACC) and MPC friendliness. In fact, the new candidates are conjectured to be exponentially secure against any adversary that allows exponentially many samples, and a basic Mod-2/Mod-3 weak PRF is the only candidate that satisfies all the features above. However, none of the direct attacks which focus on basic and alternative Mod-2/Mod-3 weak PRFs use their own structures. In this paper, we investigate weak PRFs from two perspectives; attacks, fixes. We first propose direct attacks for an alternative Mod-2/Mod-3 weak PRF and a basic Mod-2/Mod-3 weak PRF when a circulant matrix is used as a secret key. For an alternative Mod-2/Mod-3 weak PRF, we prove that the adversary’s advantage is at least 2 ^-0.105n, where n is the size of the input space of the weak PRF. Similarly, we show that the advantage of our heuristic attack on the weak PRF with a circulant matrix key is larger than 2 ^-0.21n, which is contrary to the previous expectation that ‘structured secret key’ does not affect the security of a weak PRF. Thus, for an optimistic parameter choice n= 2 λ for the security parameter λ, parameters should be increased to preserve λ-bit security when an adversary obtains exponentially many samples. Next, we suggest a simple method for repairing two weak PRFs affected by our attack. Moreover, we provide the first direct algorithm for a basic Mod-2/Mod-3 weak PRF with a random secret key even though it does not capture the current parameters.