A MODAL Θ-VALENT APPROACH OF CONVOLUTIONAL CODES AND AN APPLICATION TO THE MC-ELIECE CRYPTOSYSTEM

Résumé

Error-correcting codes are used to reconstitute digital data, which is subject to alterations during storage and transmission. They are also used in cryptography to encrypt and authenti- cate data. Among the classes of codes proposed in the literature to efficiently deal with these aspects mentioned above, we have the modal Θ-valent codes which are a class of codes defined from the alphabets based on chrysippian modal Θ-valent logic. A modal Θ-valent code is a chrysippian modal Θ-valent extension of a classical code: therefore an enrichement on the set and logical aspect of classical codes. In this thesis, we study algebraically modal Θ-valent convolutional codes and we apply them to algebraic cryptography. To do this, we define the modal Θ-valent field of the Lau- rent series ((F_2Z((D)), F_α^'), +, ×), with cœfficients in the modal Θ-valent field (F_2Z, F_α), the modal Θ-valent chrysippian extension of F_2. We define an injective modal Θ-valent linear map f Θ : F_2Z((D))^n→ F_2Z((D))^n, for k, n ∈ N with k ≤ n. From this map, we have the modal Θ-valent F2Z-vector subspace (Im(f Θ); F ′nα|Im(f Θ)) of (Fn2Z((D)), F ′nα ) whose subset of invariants is a F_2-vector subspace of F_2 ((D))^n. Note that (Im(f Θ); F ′nα|Im(f Θ)) is called modal Θ-valent convolutional code on (F_2Z; F_α) whose subset of invariants is a binary convolutional code. We propose, via α-modalities, a process for decoding the new modal Θ-valent convolutional code (Im(f Θ); F_α^'n|Im(f Θ)) by a decoding algorithm of Viterbi. Using an approach similar to the one above, we define the generalized modal Θ-valent convolutional codes with rank metric on the modal Θ-valent extension (K_pZ; F_α^') of the finite field K = F_q with q elements, q = p^n, n ∈ N^∗ and p prime number. Finally, the modal Θ-valent convolutional codes defined above are used to set up a new cryptosystem seen as a modal Θ-valent extension of the MC-Eliece cryptosystem.

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