Algebraic Cryptanalysis of the HADES Design Strategy: Application to Poseidon and Poseidon2

Arithmetization-Oriented primitives are the building block of advanced cryptographic protocols such as Zero-Knowledge proof systems. One approach to designing such primitives is the HADES design strategy which aims to provide an efficient way to instantiate generalizing substitution-permutation networks to include partial S-box rounds. A notable instance of HADES, introduced by Grassi et al. at USENIX Security ’21, is Poseidon. Because of its impressive efficiency and low arithmetic complexity, Poseidon is a popular choice among the designers of integrity-proof systems. An updated version of Poseidon, namely, Poseidon2 was published at AfricaCrypt ’23 aiming to improve the efficiency of Poseidon by optimizing its linear operations. In this work, we show some caveats in the security argument of HADES against algebraic attacks and quantify the complexity of Gröbner basis attacks. We show that the complexity of the attack is lower than claimed with the direct implication that there are cases where the recommended number of rounds is insufficient for meeting the claimed security. Concretely, the complexity of a Gröbner basis attack for an instance of Poseidon with 1024 bits of security is 731.77 bits and the original security argument starts failing already at the 384-bit security level. Since the security of Poseidon2 is derived from the security of Poseidon, the same analysis applies to the instances of Poseidon2. The results were shared with the designers and the security arguments were updated accordingly.