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Conference Papers Year : 2022

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Abstract

In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code C = C(X , P, D) is a vector space associated to evaluations on P 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 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, known as the FRI protocol [BBHR18a]. 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 thus provide a framework in which a proximity test to C can be reduced to one to a simpler code C. Iterating this process thoroughly, we end up with a membership test to a code with significantly smaller length. 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.
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Dates and versions

hal-03832439 , version 1 (27-10-2022)

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Sarah Bordage, Mathieu Lhotel, Jade Nardi, Hugues Randriam. Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes. CCC 2022 - 37th Computational Complexity Conference, Jul 2022, Philadelphie, United States. pp.30:1--30:45, ⟨10.4230/LIPIcs.CCC.2022.30⟩. ⟨hal-03832439⟩
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