A long standing open question is whether the distinguisher of high rate alternant codes or Goppa codes \cite{FGOPT11} can be turned into an algorithm recovering the algebraic structure of such codes from the mere knowledge of an arbitrary generator matrix of it. This would allow to break the McEliece scheme as soon as the code rate is large enough and would break all instances of the CFS signature scheme. We give for the first time a positive answer for this problem when the code is {\em a generic alternant code} and when the code field size $q$ is small : $q \in \{2,3\}$ and for {\em all} regime of other parameters for which the aforementioned distinguisher works. This breakthrough has been obtained by two different ingredients : (i) a way of using code shortening and the component-wise product of codes to derive from the original alternant code a sequence of alternant codes of decreasing degree up to getting an alternant code of degree $3$ (with a multiplier and support related to those of the original alternant code); (ii) an original Gr\"obner basis approach which takes into account the non standard constraints on the multiplier and support of an alternant code which recovers in polynomial time the relevant algebraic structure of an alternant code of degree $3$ from the mere knowledge of a basis for it.

翻译：长期存在的公开问题是：高码率交替码或Goppa码的区分器\cite{FGOPT11}能否转化为一种算法，仅通过任意生成矩阵的知识恢复此类码的代数结构。这将允许在码率足够高时破解McEliece方案，并破坏CFS签名方案的所有实例。我们首次为以下情况给出肯定答案：当码为{\em 一般交替码}且码域大小$q$较小时（$q \in \{2,3\}$），以及上述区分器适用的{\em 所有}其他参数范围内。这一突破通过两种不同方法实现：（i）利用码缩短和码的分量积，从原始交替码导出一系列次数递减的交替码，直至得到次数为$3$的交替码（其乘子和支撑与原始交替码相关）；（ii）一种原创的Gröbner基方法，考虑交替码乘子和支撑的非标准约束，仅通过次数为$3$的交替码的基即可在多项式时间内恢复其相关代数结构。