Paper 2024/310

A Zero-Dimensional Gröbner Basis for Poseidon

Abstract

In this paper we construct dedicated weight orders $>$ so that a $>$-Gröbner bases of Poseidon can be found via linear transformations for the preimage as well as the CICO problem. In particular, with our Gröbner bases we can exactly compute the ${\mathbb{F}}_q$-vector space dimension of the quotient space for all possible Poseidon configurations. This in turn resolves previous attempts to assess the security of Poseidon against Gröbner basis attacks, since the vector space dimension quantifies the complexity of computing the variety of a zero-dimensional polynomial system.