Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code $C = C(\mathcal{C}, \mathcal{P}, D)$ is a vector space associated to evaluations on $\mathcal{P}$ of functions in the Riemann-Roch space $L_{\mathcal{C}}(D)$. The problem of testing proximity to an error-correcting code $C$ consists in distinguishing between the case where an input word, given as an oracle, belongs to $C$ and the one where it is far from every codeword of $C$. AG codes are good candidates to construct short proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap. We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes introduced by Ben-Sasson, Bentov, Horesh and Riabzev, known as the FRI protocol. We identify suitable requirements for designing efficient IOPP systems for AG codes. In addition to proposing the first proximity test targeting AG codes, our IOPP admits quasilinear prover arithmetic complexity and sublinear verifier arithmetic complexity with constant soundness for meaningful classes of AG codes. We take advantage of the algebraic geometry framework that makes any group action on the curve that fixes the divisor $D$ translate into a decomposition of the code $C$. Concretely, our approach relies on Kani's result that splits the Riemann-Roch space of any invariant divisor under this action into several explicit Riemann-Roch spaces on the quotient curve. Under some hypotheses, these spaces behave well enough to define an AG code $C'$ on the quotient curve so that a proximity test to $C$ can be reduced to one to $C'$. Iterating this process thoroughly, we end up with a membership test to a code with significantly smaller length.