We prove that any variant of the GPT cryptosystem which uses a right column scrambler over the extension field as advocated by the works of Gabidulin et al. with the goal to resist to Overbeck's structural attack are actually still vulnerable to that attack. We show that by applying the Frobenius operator appropriately on the public key, it is possible to build a Gabidulin code having the same dimension as the original secret Gabidulin code but with a lower length. In particular, the code obtained by this way correct less errors than the secret one but its error correction capabilities are beyond the number of errors added by a sender, and consequently an attacker is able to decrypt any ciphertext with this degraded Gabidulin code. We also considered the case where an isometric transformation is applied in conjunction with a right column scrambler which has its entries in the extension field. We proved that this protection is useless both in terms of performance and security. Consequently, our results show that all the existing techniques aiming to hide the inherent algebraic structure of Gabidulin codes have failed.

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Abstract. We define the notion of a residuated lattice valued function on a set as Jun and Song have done in BCK-algebras. We also investigate related properties of residuated lattice valued function. We establish the codes generated by residuated lattice valued function and conversely we give residuated lattice valued function and residuated lattice obtained by the giving binary block-code.

We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with identity is an MTL-ring if and only if it is an arithmetical ring. It is shown that a noetherian commutative ring R with an identity is an MTL-ring if and only if ideals of the localization R M at a maximal ideal M are totally ordered by the set inclusion. Remarking that noetherian MTL-rings are again BL-rings, we work outside of the noetherian case by considering non-noetherian valuation domains and non-noetherian Prüfer domains. We established that non-noetherian valuation rings are the main examples of MTL-rings which are not BL-rings.This leads us to some constructions of MTL-rings from Prüfer domains: the case of holomorphic functions ring through their algebraic properties and the case of semilocal Prüfer domains through the theorem of independency of valuations. We end up giving a representation of MTL-rings in terms of subdirectly irreducible product.

The purpose of this work is to extend the study of the commutative rings whose lattice of ideals can be a structure of BL-algebra as carry out by Heubo et al in 2018, to non commutative rings appointed in the work as pseudo BL-rings. We study and characterize rings whose ideals form a pseudo BL-algebra, we describe them in terms of their subdirectly irreductible factors. We obtain that these are (up to isomorphism) to a subring of a direct sums of unitary special primary rings and discrete valuation ring.

This article handles a particular type of trace σ-basis of order m, which has been studied in [3]. These basis were introduced for the first time by Wolfmann in