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On the dimension and structure of the square of the dual of a Goppa code

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Abstract

The Goppa code distinguishing (GD) problem asks to distinguish efficiently a generator matrix of a Goppa code from a randomly drawn one. We revisit a distinguisher for alternant and Goppa codes through a new approach, namely by studying the dimension of square codes. We provide here a rigorous upper bound for the dimension of the square of the dual of an alternant or Goppa code, while the previous approach only provided algebraic explanations based on heuristics. Moreover, for Goppa codes, our proof extends to the non-binary case as well, thus providing an algebraic explanation for the distinguisher which was missing up to now. All the upper bounds are tight and match experimental evidence. Our work also introduces new algebraic results about products of trace codes in general and of dual of alternant and Goppa codes in particular, clarifying their square code structure. This might be of interest for cryptanalysis purposes.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and their valuable comments or suggestions for simplifying or reorganizing proofs which helped to improve significantly the quality of the manuscript.

Funding

This work was supported in part by the ANR CBCRYPT project, Grant ANR-17-CE39-0007 and by the ANR BARRACUDA project, Grant ANR-21-CE39-0009, both of the French Agence Nationale de la Recherche.

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Authors and Affiliations

  1. Team COSMIQ, Inria, 2 rue Simone Iff, Paris, 75012, France

    Rocco Mora & Jean-Pierre Tillich

  2. Sorbonne Universités, UPMC Univ Paris 06, 4 place Jussieu, Paris, 75005, France

    Rocco Mora

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  1. Rocco Mora
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  2. Jean-Pierre Tillich
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Correspondence to Rocco Mora.

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Communicated by O. Ahmadi.

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Mora, R., Tillich, JP. On the dimension and structure of the square of the dual of a Goppa code. Des. Codes Cryptogr. 91, 1351–1372 (2023). https://doi.org/10.1007/s10623-022-01153-w

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