  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.
