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

TL;DR

This paper provides a rigorous algebraic analysis of the dimension of square codes of the duals of Goppa and alternant codes, offering tight bounds and extending understanding relevant for cryptanalysis.

Contribution

It introduces a rigorous upper bound for the dimension of the square of the dual of Goppa and alternant codes, extending to non-binary cases and clarifying their algebraic structure.

Findings

Upper bounds match experimental data

Extension to non-binary Goppa codes

New algebraic results on trace code products

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.