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A long standing open question is whether the distinguisher of high ratealternant codes or Goppa codes \cite{FGOPT11} can be turned into an algorithmrecovering the algebraic structure of such codes from the mere knowledge of anarbitrary generator matrix of it. This would allow to break the McEliece schemeas soon as the code rate is large enough and would break all instances of theCFS signature scheme. We give for the first time a positive answer for thisproblem when the code is {\em a generic alternant code} and when the code fieldsize $q$ is small : $q \in \{2,3\}$ and for {\em all} regime of otherparameters for which the aforementioned distinguisher works. This breakthroughhas been obtained by two different ingredients : (i) a way of using codeshortening and the component-wise product of codes to derive from the originalalternant code a sequence of alternant codes of decreasing degree up to gettingan alternant code of degree $3$ (with a multiplier and support related to thoseof the original alternant code);  (ii) an original Gr\"obner basis approach which takes into account the nonstandard constraints on the multiplier and support of an alternant code whichrecovers in polynomial time the relevant algebraic structure of an alternantcode of degree $3$ from the mere knowledge of a basis for it.

Polynomial time key-recovery attack on high rate random alternant codes