Paper 2024/1512

Improved Soundness Analysis of the FRI Protocol

Abstract

We enhance the provable soundness of FRI, an interactive oracle proof of proximity (IOPP) for Reed-Solomon codes introduced by Ben-Sasson et al. in ICALP 2018. More precisely, we prove the soundness error of FRI is less than $\max\left\{O\left(\frac{1}{\eta}\cdot \frac{n}{|\mathbb{F}_q|}\right), (1-\delta)^{t}\right\}$, where $\delta\le 1-\sqrt{\rho}-\eta$ is within the Johnson bound and $\mathbb{F}_q$ is a finite field with characteristic greater than $2$. Previously, the best-known provable soundness error for FRI was $\max\left\{O\left(\frac{\rho^2}{\eta^7}\cdot \frac{n^2}{|\mathbb{F}_q|}\right), (1-\delta)^{t}\right\}$, as established by Ben-Sasson et al. in FOCS 2020. We prove the number of \emph{bad} folding points in FRI is linear in the length $n$ of codeword when it is $\delta$-far from the Reed-Solomon code. This implies the linear proximity gaps for Reed-Solomon codes and improves the provable soundness of batched FRI. Our results indicate that the FRI protocol can be implemented over a smaller field, thereby enhancing its efficiency. Furthermore, for a fixed finite field $\mathbb{F}_q$, we prove that FRI can achieve improved security.