First, we work on a construction of MDS Fq-linear codes over Fqb based on the isomorphism between _fields Fq[C] and Fqb , where C is the companion matrix of a primitive polynomial of degree b in Fq[x]. If the parameters of one of our codes are [n; k; d], we can recover up to n-k erasures. We propose an algorithm to recover the lost information symbols just solving a linear system with tb unknowns, where t in the number of erased information symbols. We would like to deeply study this algorithm to make it e_cient and compare these codes with other MDS codes. We would also like to find good cryptographic applications of these codes, such as the construction of optimal linear di_usion layers in block ciphers.At the same time, we study the performance of the SPC (single parity-check) simple product codes. These codes have a small minimum distance and, thus, their error correction capability is very limited. However, they are able to recover a higher number of erasures in special cases. We would like to count and analyse these cases in order to study the performance of these codes. Furthermore, SPC product codes obtained with more than two SPC codes have never been studied. Solving this problem can help us to solve graph theory problems, since an erasure pattern representing a codeword of an SPC product code can be also seen as a bipartite graph, where the erasures are the edges.Regarding cryptography, we model some cryptographic non-linear sequence generators, called shrinking generators, using linear cellular automata (CA). The sequences produced by these generators can be obtained as one of the output sequences generated by a family of regular CA. We can take advantage of this linearity and propose an e_cient cryptanalysis of these generators. We would also like study the cryptographic properties of the other sequences generated by the CA and try to model model other generators using CA. Besides, we want to connect CA with Neural Networks (NN) and then study our cryptographic problem from a new perspective never considered before.
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