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.

中文翻译：

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代数几何代码接近度的交互式 Oracle 证明

在这项工作中，我们开始研究代数几何 (AG) 代码的邻近测试。AG 代码 $C = C(\mathcal{C}, \mathcal{P}, D)$ 是与 Riemann-Roch 空间 $L_{\ 中函数的 $\mathcal{P}$ 求值相关联的向量空间数学{C}}(D)$。测试与纠错码 $C$ 的接近度的问题在于区分输入字（作为 oracle 给出）属于 $C$ 的情况和远离 $C$ 的每个码字的情况。AG 代码是构建短证明系统的良好候选者，但没有针对它们的有效邻近测试。我们的目标是填补这一空白。我们通过推广由 Ben-Sasson、Bentov、Horesh 和 Riabzev 引入的 Reed-Solomon 代码的 IOPP（称为 FRI 协议），为一些 AG 代码系列构建了一个交互式 Oracle 邻近证明（IOPP）。我们确定了为 AG 代码设计高效 IOPP 系统的合适要求。除了提出第一个针对 AG 代码的邻近测试之外，我们的 IOPP 还承认准线性证明算术复杂度和次线性验证算术复杂度，并且对于有意义的 AG 代码类别具有恒定的稳健性。我们利用代数几何框架，该框架使固定除数 $D$ 的曲线上的任何群动作转化为代码 $C$ 的分解。具体而言，我们的方法依赖于 Kani 的结果，该结果将在此操作下的任何不变因数的 Riemann-Roch 空间拆分为商曲线上的几个显式 Riemann-Roch 空间。在某些假设下，这些空间的表现足以在商曲线上定义一个 AG 代码 $C'$，这样对 $C$ 的邻近测试就可以减少到 $C'$ 的一个。