International Association
for Cryptologic Research

Gröbner basis cryptanalysis of hash functions and ciphers, and their underlying permutations, has seen renewed interest recently. Anemoi (Crypto'23) is a permutation-based hash function that is arithmetization-friendly, i.e., efficient for a variety of arithmetizations used in zero-knowledge proofs. In this paper, exploring both theoretical bounds as well as experimental validation, we present new complexity estimates for Gröbner basis attacks on the Anemoi permutation over prime fields. We cast our findings in what we call the six worlds of Gröbner basis cryptanalysis. As an example, keeping the same security arguments of the design, we conclude that at least $23 /45$ instead of $17 / 33$ rounds would need to be used for $128 / 256$-bit security before adding a security margin.