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The Goppa Code Distinguishing (GD) problem asks to distinguish efficiently agenerator matrix of a Goppa code from a randomly drawn one. We revisit adistinguisher for alternant and Goppa codes through a new approach, namely bystudying the dimension of square codes. We provide here a rigorous upper boundfor the dimension of the square of the dual of an alternant or Goppa code,while the previous approach only provided algebraic explanations based onheuristics. Moreover, for Goppa codes, our proof extends to the non-binary caseas well, thus providing an algebraic explanation for the distinguisher whichwas missing up to now. All the upper bounds are tight and match experimentalevidence. Our work also introduces new algebraic results about products oftrace codes in general and of dual of alternant and Goppa codes in particular,clarifying their square code structure. This might be of interest forcryptanalysis purposes.

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