Resumen de Skew polynomials, error correcting codes and mceliece cryptosystem
.In the last years, skew polynomials have been used in the design of error correcting codes with good distances and their corresponding fast algebraic decoding algorithms. Concretely, the σ-codes use similarity of the arithmetic of polynomials and skew polynomials to study σ-cyclic structures on block and concolutional codes. Non commutative key equations have also been derived, which opens the door to design Sugiyama’s like decoding algorithms. The Peterson-Gorenstein-Zierler approach to decode cyclic codes can be adapted to the skew cyclic framework too. The correspondence of skew and linearized polynomials allows to connect σ-cyclic codes with Gabidulin codes and the rank metric. Since these families of codes can be efficiently decoded, they are candidates to replace Goppa codes in the McEliene cryptosystem. We will explore current succesful attacks to some of them.
