## Section: New Results

### Compact McEliece Keys from Algebraic-geometry codes

In 1978, McEliece [31], introduced a public key cryptosystem based on linear codes and suggested to use classical Goppa codes which belong to the family of alternant codes. This proposition remains secure but leads to very large public keys compared to other public-key cryptosystems. Many proposals have been made in order to reduce the key size, in particular quasi-cyclic alternant codes. Quasi-cyclic alternant codes refer to alternant codes admitting a generator matrix made of severals cyclic bloks. These alternant codes contains weakness because they have a non-trivial automorphism group. Thanks to this property we can build, from a quasi-cyclic alternant code, an alternant code with smaller parameters which has almost same private elements than the original code. Faugère, Otmani, Tillich, Perret and Portzamparc [29] showed this fact for alternant codes obtained by using supports $x\in {\mathbb{F}}_{{q}^{m}}^{n}$ globally stable by an affine map $\phi :z\mapsto az+b$, with $a,b\in {\mathbb{F}}_{{q}^{m}}^{n}$. E. Barelli has extended this proof to the non-affine case: for all codes obtained by using supports $x\in {\mathbb{F}}_{{q}^{m}}^{n}$ globally stable by a map $\phi :z\mapsto \frac{az+b}{cz+d}$, with $a,b,c,d\in {\mathbb{F}}_{{q}^{m}}^{n}$.

In order to suggest compact keys for the McEliece cryptosystem E. Barelli and A. Couvreur studied quasi-cyclic alternant gemeotric codes. Alternant geometric codes means a subfield subcode of an algebraic-geometry codes. To build these codes, we need curves with automorphisms. In particular, we studied Kummer cover of plane curves.