## Cryptology ePrint Archive: Report 2020/783

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

Jung Hee Cheon and Wonhee Cho and Jeong Han Kim and Jiseung Kim

Abstract: 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 ${\sf ACC^0}$) 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 $n = 2\lambda$ for the security parameter $\lambda$, parameters should be increased to preserve $\lambda$-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 ${\sf ACC^0}$ 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 $\{0,1\}^{m \times n}$.

Category / Keywords: secret-key cryptography / Cryptanalysis, weak PRF