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Interactive Oracle Proofs of Proximity to Codes on Graphs

Hugo Delavenne, Tanguy Medevielle, Élina Roussel

TL;DR

This work introduces an Interactive Oracle Proof of Proximity (IOPP) tailored for codes on graphs, inspired by the FRI protocol, to test proximity to $\mathcal{C}[\Gamma,k]$ with oracle access and strong soundness. The core method, the Flowering protocol, folds the graph structure down to a single-vertex flower while preserving local views, enabling proximity testing against a Reed-Solomon base code $\mathsf{RS}[n,k]$ with improved soundness and relaxed field-size requirements $|\bbF|>\log N$. Key contributions include a formal dimension lower bound for graph-based codes, a commit-soundness analysis, and a constructive Cayley-multigraph instantiation that achieves positive rate with $o(1)$ minimum distance, along with concrete parameter trade-offs against the FRI protocol. The results offer a practical pathway to code-based SNARKs with flexible alphabets and scalable graphs, potentially enabling efficient arithmetization in frameworks like PlonK or R1CS. Future work aims to broaden the flowering construction beyond two-way cuts and explore richer equivalence classes of cut-graphs to further enhance applicability.

Abstract

We design an Interactive Oracle Proof of Proximity (IOPP) for codes on graphs inspired by the FRI protocol. The soundness is significantly improved compared to the FRI, the complexity parameters are comparable, and there are no restrictions on the field used, enabling to consider new codes to design code-based SNARKs.

Interactive Oracle Proofs of Proximity to Codes on Graphs

TL;DR

This work introduces an Interactive Oracle Proof of Proximity (IOPP) tailored for codes on graphs, inspired by the FRI protocol, to test proximity to with oracle access and strong soundness. The core method, the Flowering protocol, folds the graph structure down to a single-vertex flower while preserving local views, enabling proximity testing against a Reed-Solomon base code with improved soundness and relaxed field-size requirements . Key contributions include a formal dimension lower bound for graph-based codes, a commit-soundness analysis, and a constructive Cayley-multigraph instantiation that achieves positive rate with minimum distance, along with concrete parameter trade-offs against the FRI protocol. The results offer a practical pathway to code-based SNARKs with flexible alphabets and scalable graphs, potentially enabling efficient arithmetization in frameworks like PlonK or R1CS. Future work aims to broaden the flowering construction beyond two-way cuts and explore richer equivalence classes of cut-graphs to further enhance applicability.

Abstract

We design an Interactive Oracle Proof of Proximity (IOPP) for codes on graphs inspired by the FRI protocol. The soundness is significantly improved compared to the FRI, the complexity parameters are comparable, and there are no restrictions on the field used, enabling to consider new codes to design code-based SNARKs.
Paper Structure (10 sections, 10 theorems, 28 equations, 1 table)

This paper contains 10 sections, 10 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Scientific context
  3. Techniques and results
  4. Definitions
  5. General properties
  6. Flowering protocol
  7. Protocol and properties
  8. Proof of soundness
  9. Cayley multigraph over $(\bbF_2^r,+)$
  10. Conclusion

Key Result

Proposition 1

Let $\Gamma=(V,E)$ be a $n$-RIM. Let $f,f'\in W(\Gamma,\bbF)$. For $v\in V$ and $\ell\in[n]$, let $|{\overline{(v,\ell)}}|$ be the cardinal of the equivalence class of $(v,\ell)$ by $\sim_E$, let $m:=\max_{v\in V}\sum_{\ell\in[n]}\frac{1}{|{\overline{(v,\ell)}}|}$. Then $\Delta_V(f,f')\geq\frac{|{\t

Theorems & Definitions (27)

  • Definition 1: Regular indexed multigraph (RIM)
  • Definition 2: Word on graph, code on graph
  • Definition 3: Graph isomorphism
  • Definition 4: Cut-graph, cut-word
  • Definition 5: Flowering cut
  • Definition 6: Folding
  • Definition 7: Blossoming graph sequence
  • Definition 8: Vertex distance, Hamming distance
  • Proposition 1
  • proof
  • ...and 17 more