Improved Round-by-round Soundness IOPs via Reed-Muller Codes
Dor Minzer, Kai Zhe Zheng
TL;DR
This work develops new IOPPs for proximity to tri-variate Reed-Muller codes, achieving improved round-by-round soundness and query efficiency for security parameters in a practical regime, and then leverages these tests to construct IOPPs for the NP-complete language R1CS. The approach blends algebraization of R1CS into RM-encoded identities, proximity testing for both constant-rate and small-rate codes, and a Poly-IOP framework compiled via batched code membership tests with anchoring and quotienting to manage side conditions. Central technical innovations include proximity generators for Reed-Muller codes (with correlated agreement), a refined line-vs-point test analysis in higher dimensions, and quotienting techniques to remove side-conditions, enabling efficient composition across multiple code families. The resulting IOPP constructions offer improved round complexity and query complexity in certain parameter ranges and have potential practical impact for SNARK-like protocols and polynomial commitments, with avenues for concrete optimization discussed. Overall, the paper advances the state of IOPs by achieving better trade-offs between rounds, queries, and soundness through novel use of RM/RM-RS code proximity testing and poly-to-IOP compilations.
Abstract
We give an IOPP (interactive oracle proof of proximity) for trivariate Reed-Muller codes that achieves the best known query complexity in some range of security parameters. Specifically, for degree $d$ and security parameter $λ\leq \frac{\log^2 d}{\log\log d}$ , our IOPP has $2^{-λ}$ round-by-round soundness, $O(λ)$ queries, $O(\log\log d)$ rounds and $O(d)$ length. This improves upon the FRI [Ben-Sasson, Bentov, Horesh, Riabzev, ICALP 2018] and the STIR [Arnon, Chiesa, Fenzi, Yogev, Crypto 2024] IOPPs for Reed-Solomon codes, that have larger query and round complexity standing at $O(λ\log d)$ and $O(\log d+λ\log\log d)$ respectively. We use our IOPP to give an IOP for the NP-complete language Rank-1-Constraint-Satisfaction with the same parameters. Our construction is based on the line versus point test in the low-soundness regime. Compared to the axis parallel test (which is used in all prior works), the general affine lines test has improved soundness, which is the main source of our improved soundness. Using this test involves several complications, most significantly that projection to affine lines does not preserve individual degrees, and we show how to overcome these difficulties. En route, we extend some existing machinery to more general settings. Specifically, we give proximity generators for Reed-Muller codes, show a more systematic way of handling ``side conditions'' in IOP constructions, and generalize the compiling procedure of [Arnon, Chiesa, Fenzi, Yogev, Crypto 2024] to general codes.