Almost Linear-Time Permutation Check

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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 $\math