Sumcheck Arguments and Their Applications

We introduce a class of interactive protocols, which we call sumcheck arguments, that establishes a novel connection between the sumcheck protocol (Lund et al. JACM 1992) and folding techniques for Pedersen commitments (Bootle et al. EUROCRYPT 2016).

We define a class of sumcheck-friendly commitment schemes over modules that captures many examples of interest, and show that the sumcheck protocol applied to a polynomial associated with the commitment scheme yields a succinct argument of knowledge for openings of the commitment. Building on this, we additionally obtain succinct arguments for the NP-complete language R1CS over certain rings.

Sumcheck arguments enable us to recover as a special case numerous prior works in disparate cryptographic settings (discrete logarithms, pairings, groups of unknown order, lattices), providing one framework to understand them all. Further, we answer open questions raised in prior works, such as obtaining a lattice-based succinct argument from the SIS assumption for satisfiability problems over rings.