In this article, we continue the analysis started in \cite{CMT23} for the matrix code of quadratic relationships associated with a Goppa code. We provide new sparse and low-rank elements in the matrix code and categorize them according to their shape. Thanks to this description, we prove that the set of rank 2 matrices in the matrix codes associated with square-free binary Goppa codes, i.e. those used in Classic McEiece, is much larger than what is expected, at least in the case where the Goppa polynomial degree is 2. We build upon the algebraic determinantal modeling introduced in \cite{CMT23} to derive a structural attack on these instances. Our method can break in just a few seconds some recent challenges about key-recovery attacks on the McEliece cryptosystem, consistently reducing their estimated security level. We also provide a general method, valid for any Goppa polynomial degree, to transform a generic pair of support and multiplier into a pair of support and Goppa polynomial.

翻译：摘要：本文延续了文献\cite{CMT23}中对与Goppa码关联的二次关系矩阵码的分析。我们揭示了矩阵码中新的稀疏低秩元素，并根据其形态进行分类。基于该描述，我们证明与无平方二元Goppa码（即Classic McEliece所使用的码型）关联的矩阵码中，秩为2的矩阵集合远大于预期规模，至少在Goppa多项式次数为2的情况下如此。我们依托文献\cite{CMT23}中提出的代数行列式建模方法，推导出针对这类实例的结构性攻击。该方法能在数秒内破解近期关于McEliece密码系统密钥恢复攻击的若干挑战，持续降低其预估安全等级。此外，我们还提出一种适用于任意Goppa多项式次数的通用方法，可将通用的支撑与乘子对转化为支撑与Goppa多项式对。