The main practical limitation of the McEliece public-key encryption scheme is probably the size of its key. A famous trend to overcome this issue is to focus on subclasses of alternant/Goppa codes with a non-trivial automorphism group. Such codes display then symmetries allowing compact parity-check or generator matrices. For instance, a key-reduction is obtained by taking quasi-cyclic (QC) or quasi-dyadic (QD) alternant/Goppa codes. We show that the use of such symmetric alternant/Goppa codes in cryptography introduces a fundamental weakness. It is indeed possible to reduce the key-recovery on the original symmetric public-code to the key-recovery on a (much) smaller code that has no symmetry anymore. This result is obtained thanks to an operation on codes called folding that exploits the knowledge of the automorphism group. This operation consists in adding the coordinates of codewords which belong to the same orbit under the action of the automorphism group. The advantage is twofold. The reduction factor can be as large as the size of the orbits, and it preserves a fundamental property: folding the dual of an alternant (respectively, Goppa) code provides the dual of an alternant (respectively, Goppa) code. A key point is to show that all the existing constructions of alternant/Goppa codes with symmetries follow a common principal of taking codes whose support is globally invariant under the action of affine transformations (by building upon prior works of Berger and Dür). This enables not only to present a unified view but also to generalize the construction of QC, QD, and even quasi-monoidic Goppa codes. Finally, our results can be harnessed to boost up any key-recovery attack on McEliece systems based on symmetric alternant or Goppa codes, and in particular algebraic attacks.

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.