Adventures in Crypto Dark Matter: Attacks, Fixes and Analysis for Weak Pseudorandom Function Candidates

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. (TCC’18) introduced two types of new weak PRF candidates, called a basic Mod-2/Mod-3 and alternative Mod-2/Mod-3 weak PRF. They 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 above features. However, none of direct attacks which focus on a basic and alternative Mod-2/Mod-3 weak PRFs uses their own structures. In this paper, we investigate weak PRFs in three perspectives; attacks, fixes, and a new analysis to support the hardness conjecture of weak PRFs. 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 input space of weak PRF. Similarly, we show that the advantage of our heuristic attack to the weak PRF with a circulant matrix key is larger than 2−0.21n, which is contrary to previous expectation that ‘a structured secret key’ does not affect the security of a weak PRF. Thus, for optimistic parameter choice n = 2λ for the security parameter λ, parameters should be increased to preserve λ-bit security when an adversary obtains exponentially many samples. Next, we provide a simple method for repairing two weak PRFs affected by our attack while preserving the depth-2 ACC circuit complexity and parameters. Moreover, we provide an observation and a new analysis to support the exponential hardness conjecture of a basic Mod-2/Mod-3 weak PRF when a secret key is uniformly sampled from {0, 1}m×n.