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Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Authors Sarah Bordage, Mathieu Lhotel, Jade Nardi , Hugues Randriam

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Sarah Bordage
  • LIX, CNRS UMR 7161, Ecole Polytechnique, Institut Polytechnique de Paris, France
  • Inria, Palaiseau, France
Mathieu Lhotel
  • Laboratoire de Mathématiques de Besançon, UMR 6623 CNRS, Université de Bourgogne Franche-Comté, France
Jade Nardi
  • Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
Hugues Randriam
  • ANSSI, Paris, France
  • Institut Polytechnique de Paris, Télécom Paris, Palaiseau, France


The authors are very appreciative to the anonymous reviewers whose comments and suggestions helped to improve and clarify this manuscript. The third author thanks Marc Perret for his precious advice in the early days of this project. The authors are grateful to Daniel Augot for suggesting to work on this project and for many valuable discussions. They also thank Eli Ben-Sasson and Alessandro Chiesa for their helpful insights.

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Sarah Bordage, Mathieu Lhotel, Jade Nardi, and Hugues Randriam. Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 30:1-30:45, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code C = C(𝒳, 𝒫, D) over an algebraic curve 𝒳 is a vector space associated to evaluations on 𝒫 ⊆ 𝒳 of functions in the Riemann-Roch space L_𝒳(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 probabilistic 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, known as the FRI protocol [Eli Ben-Sasson et al., 2018]. We identify suitable requirements for designing efficient IOPP systems for AG codes. Our approach relies on a neat decomposition of the Riemann-Roch space of any invariant divisor under a group action on a curve into several explicit Riemann-Roch spaces on the quotient curve. We provide sufficient conditions on an AG code C that allow to reduce a proximity testing problem for C to a membership problem for a significantly smaller code C'. As concrete instantiations, we study AG codes on Kummer curves and curves in the Hermitian tower. The latter can be defined over polylogarithmic-size alphabet. We specialize the generic AG-IOPP construction to reach linear prover running time and logarithmic verification on Kummer curves, and quasilinear prover time with polylogarithmic verification on the Hermitian tower.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Interactive proof systems
  • Algebraic geometry codes
  • Interactive oracle proofs of proximity
  • Proximity testing


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