Paper 2025/1850

Almost Linear-Time Permutation Check

Abstract

Permutation and lookup arguments are at the core of most deployed SNARK protocols today. Most modern techniques for performing them require a grand product check. This requires either committing to large field elements (e.g. in Plonk) or using GKR (e.g. in Spartan) which has worse verifier cost and proof size. Moreover, both have a soundness error that grows linearly with the input size. We reduce permutation argument to sumcheck argument and present two permutation arguments that have $\mathsf{polylog}(n)/|\mathbb{F}|$ soundness error. BiPerm only requires the prover to perform $O(n)$ field operations on top of committing to the witness, but at the cost of limiting the choices of PCS. We show a general construction, MulPerm, which has no restriction on the PCS choice and its prover performs essentially linear field operations, $n\cdot \tilde O(\sqrt{\log(n)})$. Both permutation arguments generalize to lookups. We demonstrate how our arguments can be used to improve SNARK systems such as HyperPlonk (Eurocrypt 2023) and Spartan (Crypto 2020), and build a GKR-based protocol for proving non-uniform circuits. Further, MulPerm enables the fastest succinct argument with sublinear verification for arbitrary circuits over small fields. Plugging MulPerm into prior small field constructions overcomes the previous lack of an efficient permutation argument with sufficiently small soundness error.