Related-Key Differential Cryptanalysis of GMiMC Used in Post-Quantum Signatures

With the urgency of the threat imposed by quantum computers, there is a strong interest in making the signature schemes quantum resistant. As the promising candidates to ensure post-quantum security, symmetric-key primitives, in particular the recent MPC/FHE/ZK-friendly hash functions or block ciphers, are providing another choice to build efficient and secure signature schemes that do not rely on any assumed hard problems. However, considering the intended use cases, many of these novel ciphers for advanced cryptographic protocols do not claim the related-key security.

In this paper, we initiate the study of the ignored related-key security of GMiMC proposed by Albrecht et al. at ESORICS 2019, some versions of which are optimized and designed to be used in post-quantum secure signatures. By investigating the potential threats of related-key attacks for GMiMC intended to be deployed as the underlying building block in post-quantum signature schemes, we then construct two kinds of iterative related-key differentials, from which not only do we explore its security margin against related-key attacks, but also collision attacks on its key space can be performed. For example, for GMiMC instance that beats the smallest signature size obtainable using LowMC, we can find its key collision using only about $2^{10}$ key pairs. It worths noting that our current key collision attack is only applicable when the adversarial power is sufficiently strong (e.g., in the so-called multi-user setting), and it does not threaten the one-wayness of GMiMC. Furthermore, from the experiments of our related-key differentials, it can be observed that the differential clustering effect of GMiMC differs in both aspects: the choice of the finite field $\mathbb{F}$ being ${\mathbb{F}}_p$ or ${\mathbb{F}}_2^n$, and the size of the finite field $\mathbb{F}$.