International Association for Cryptologic Research

International Association
for Cryptologic Research

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Paper: On Codes and Learning with Errors over Function Fields

Authors:
Maxime Bombar , LIX, École Polytechnique and INRIA
Alain Couvreur , INRIA and LIX, École Polytechnique
Thomas Debris-Alazard , INRIA and LIX, École Polytechnique
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Conference: CRYPTO 2022
Abstract: It is a long standing open problem to find search to decision reductions for structured versions of the decoding problem of linear codes. Such results in the lattice-based setting have been carried out using number fields: Polynomial–LWE, Ring–LWE, Module–LWE and so on. We propose a function field version of the LWE problem. This new framework leads to another point of view on structured codes, e.g. quasi-cyclic codes, strengthening the connection between lattice-based and code-based cryptography. In particular, we obtain the first search to decision reduction for structured codes. Following the historical constructions in lattice–based cryptography, we instantiate our construction with function fields analogues of cyclotomic fields, namely Carlitz ex- tensions, leading to search to decision reductions on various versions of Ring-LPN, which have applications to secure multi party computation and to an authentication protocol.
BibTeX
@inproceedings{crypto-2022-32207,
  title={On Codes and Learning with Errors over Function Fields},
  publisher={Springer-Verlag},
  author={Maxime Bombar and Alain Couvreur and Thomas Debris-Alazard},
  year=2022
}