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 $\lambda\leq \frac{\log^2 d}{\log\log d}$ , our IOPP has $2^{-\lambda}$ round-by-round soundness, $O(\lambda)$ 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 $O(\lambda \log d)$ and $O(\log d+\lambda\log\log d)$ 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.

翻译：我们提出了一种针对三变量Reed-Muller码的交互式邻近性预言证明（IOPP），该证明在特定安全参数范围内达到了已知最优的查询复杂度。具体而言，对于次数$d$和安全参数$\lambda\leq \frac{\log^2 d}{\log\log d}$，我们的IOPP具有$2^{-\lambda}$的逐轮可靠性、$O(\lambda)$次查询、$O(\log\log d)$轮交互以及$O(d)$的证明长度。这改进了针对Reed-Solomon码的FRI [Ben-Sasson, Bentov, Horesh, Riabzev, ICALP 2018]和STIR [Arnon, Chiesa, Fenzi, Yogev, Crypto 2024] IOPP方案——后两者的查询复杂度与轮数复杂度分别为$O(\lambda \log d)$和$O(\log d+\lambda\log\log d)$。我们利用该IOPP为NP完全语言Rank-1-Constraint-Satisfaction构建了具有相同参数的IOP。我们的构造基于低可靠性区域中的直线对点测试。与所有先前工作中使用的坐标轴平行测试相比，一般仿射直线测试具有更高的可靠性，这是我们可靠性提升的主要来源。使用该测试涉及若干技术难点，其中最显著的是向仿射直线的投影不保持单项次数，我们展示了如何克服这些困难。在此过程中，我们将现有的一些技术框架扩展到更一般的场景。具体而言，我们给出了Reed-Muller码的邻近性生成器，提出了一种更系统化的方法来处理IOP构造中的“边界条件”，并将[Arnon, Chiesa, Fenzi, Yogev, Crypto 2024]中的编译过程推广到一般编码。