Codes on any Cayley Graph have an Interactive Oracle Proof of Proximity

Abstract: Interactive Oracle Proofs of Proximity (IOPP) are at the heart of code-based SNARKs, a family of zeroknowledge protocols. The first and most famous one is the FRI protocol [BBHR18a], that efficiently tests proximity to Reed-Solomon codes. This paper generalizes the flowering IOPP introduced in [DMR25] for some specific (2, n)-regular Tanner codes to a much broader variety of codes: any code with symbols indexed on the edges of a Cayley graph. The flowering protocol of [DMR25] had a soundness parameter much lower than the FRI protocol [BCI + 23], and complexity parameters that could compete with the FRI [BBHR18a]. The lower soundness and the absence of restriction on the base field may lead to other practical speedups, however the codes considered in [DMR25] have an o(1) minimum distance. The generalization proposed in this paper preserves the soundness parameter with a slight decrease of the complexity parameters, while allowing being applied on codes with constant rate and constant minimum distance thanks to the good expansion properties of some families of Cayley graphs.