Paper 2025/531

Understanding the new distinguisher of alternant codes at degree 2

Axel Lemoine, French Institute for Research in Computer Science and Automation, Direction Générale de l'Armement
Rocco Mora, Helmholtz Center for Information Security
Jean-Pierre Tillich, French Institute for Research in Computer Science and Automation
Abstract

Distinguishing Goppa codes or alternant codes from generic linear codes [FGO+11] has been shown to be a first step before being able to attack McEliece cryptosystem based on those codes [BMT24]. Whereas the distinguisher of [FGO+11] is only able to distinguish Goppa codes or alternant codes of rate very close to 1, in [CMT23a] a much more powerful (and more general) distinguisher was proposed. It is based on computing the Hilbert series of a Pfaffian modeling. The distinguisher of [FGO+11] can be interpreted as computing . Computing still gives a polynomial time distinguisher for alternant or Goppa codes and is apparently able to distinguish Goppa or alternant codes in a much broader regime of rates as the one of [FGO+11]. However, the scope of this distinguisher was unclear. We give here a formula for corresponding to generic alternant codes when the field size satisfies , where r is the degree of the alternant code. We also show that this expression for provides a lower bound in general. The value of corresponding to random linear codes is known and this yields a precise description of the new regime of rates that can be distinguished by this new method. This shows that the new distinguisher improves significantly upon the one given in [FGO+11].

Metadata
Available format(s)
PDF
Category
Attacks and cryptanalysis
Publication info
Preprint.
Keywords
McEliece cryptosystemalgebraic attacksdistinguishers
Contact author(s)
axel lemoine @ inria fr
rocco mora @ cispa de
jean-pierre tillich @ inria fr
History
2025-03-23: approved
2025-03-21: received
See all versions
Short URL
https://ia.cr/2025/531
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2025/531,
      author = {Axel Lemoine and Rocco Mora and Jean-Pierre Tillich},
      title = {Understanding the new distinguisher of alternant codes at degree 2},
      howpublished = {Cryptology {ePrint} Archive, Paper 2025/531},
      year = {2025},
      url = {https://eprint.iacr.org/2025/531}
}
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