The main practical limitation of the McEliece cryptosystem is probably the size of its public-key. To overcome this issue, a famous trend is to decrease the public-key size by focusing on subclasses of alternant/Goppa codes which admit a compact parity-check or generator matrix. For instance, a key-size reduction is obtained by taking alternant/Goppa codes which have quasi-cyclic (QC) or quasi-dyadic (QD) generator matrices. We show that the use of such compact alternant/Goppa codes introduced a fundamental weakness. It is possible to reduce the key-recovery on the original public-code C to the key-recovery on a (much) smaller code C'. To this end, we use a new operation on codes which exploits the automorphism group.