We generalize the Gabidulin codes based encryption scheme presented by Loidreau in 2017, by combining the original idea with an idea proposed by Gabidulin, Rashwan and Honary in 2009. We then adapt the combinatorial attack proposed by Briaud and Loidreau in 2023 to evaluate the state of the art complexity of an algorithm recovering a decoder from the public-key. This enables to design, for a same security, schemes with smaller parameters than for the original scheme and to analyse the security of another modification of Loidreau's encryption scheme already published.

We investigate the decoding of the sum of Gabidulin codes. We show that there exists a probabilistic polynomial-time decoder up to some bound. We then give some potential applications of constructing and decoding a sum of Gabidulin codes. This approach can lead to a new insight in designing rank-metric based cryptographic schemes.

We show that the algebraic structure of codes can exploited to improve significantly the efficiency of code reconstructions techniques especially in the case of quasi-cyclic LDPC codes with large and smooth block sizes. Such codes are widely implemented in various standards. We investigate the case where the transmission is eavesdropped. The eaves dropper has access to noisy codewords which transited through a binary symmetric channel.

We transpose the theory of rank metric and Gabidulin codes to the case of fields of characteristic zero. The Frobenius automorphism is then replaced by any element of the Galois group. We derive some conditions on the automorphism to be able to easily transpose the results obtained by Gabidulin as well and a classical polynomial-time decoding algorithm. We also provide various definitions for the rank-metric.

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

Maximum rank distance codes are the equivalent in rank-metric of Reed-Solomon codes whose subcodes have been widely studied. In this paper we characterize subspace subcodes of MRD codes and we show that it is possible to construct efficient polynomial-time encoding-decoding procedures for these subcodes. In a second part we show that subfield subcodes of maximum rank distance codes can be represented in some sense by the direct sum of maximum rank distance codes of smaller length and same minimum distance. We then derive an algorithm correcting some error-patterns beyond the error-correcting capability of the codes

The Groebner basis calculation algorithms were successfully applied to construct new sequences analytically. New unimodular perfect sequences with 6 phases were proposed for various sequence lengths. For perfect root-of-unity sequences and for binary sequences with ideal autocorrelation this new approach was used to find sequences analytically. Although this approach was not able to find previously unknown sequences in both cases, it is still better than any other analytical method and almost on par with exhaustive search

We show that it is possible to know whether the secret Goppa code of an instance of the McEliece public-key cryptosystem was chosen with a binary generator polynomial. Furthermore, whenever such a weak key is used, we present an attack which can be completed, for codes of length 1024 and dimension 524, with a large, but feasible amount of computation.