International audienc

International audienceWe generalize the notion of cyclic codes by using generator polynomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew-cyclic codes, this new class of codes allows to systematically search for codes with good properties. We give many examples of codes which improve the previously best known linear codes

International audienceIn this work the de nition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings whose multiplication is de ned using an automorphism and an inner derivation. This produces a more gen- eral class of codes which, in some cases, produce better distance bounds than skew module codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several ap- proaches to generalize Reed Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank matrix and derive families of MDS and MRD codes

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We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over GR(4, 2) are constructed. Euclidean self-dual codes give self-dual Z 4 −codes. Hermitian self-dual codes yield 3−modular lattices and quasi-cyclic self-dual Z 4 −codes.

In [4], starting from an automorphism θ of a finite field F q and a skew polynomial ring R = F q [X; θ], module θ-codes are defined as left R-submodules of R/Rf where f ∈ R. In [4] it is conjectured that an Euclidean self-dual module θ-code is a θ-constacyclic code and a proof is given in the special case when the order of θ divides the length of the code. In this paper we prove that this conjecture holds in general by showing that the dual of a module θ-code is a module θ-code if and only if it is a θ-constacyclic code. Furthermore, we establish that a module θ-code which is not θ-constacyclic is a shortened θ-constacyclic code and that its dual is a punctured θ-constacyclic code. This enables us to give the general form of a parity-check matrix for module θ-codes and for module (θ, δ)-codes over F q [X; θ, δ] where δ is a derivation over F q . We also prove the conjecture for module θ-codes who are defined over a ring A[X; θ] where A is a finite ring. Lastly we construct self-dual θ-cyclic codes of length 2 s over F 4 for s ≥ 3 which are asymptotically bad and conjecture that there exists no other self-dual module θ-code of this length over F 4 .

In this paper we introduce a new key exchange algorithm (Diffie-Hellman like) based on so called (non-commutative) skew polynomials. The algorithm performs only polynomial multiplications in a special small field and is very efficient. The security of the scheme can be interpretated in terms of solving binary quadratic equations or exhaustive search of a set obtained through linear equations. We give an evaluation of the security in terms of precise experimental heuristics and usual bounds based on Groebner basis solvers. We also derive an El Gamal like encryption protocol. We propose parameters which give 3600 bits exchanged for the key exchange protocol and a size of key of 3600 bits for the encryption protocol, with a complexity of roughly 223 binary operations for performing each protocol. Overall this new approach based on skew polynomials, seems very promising, as a good tradeoff between size of keys and efficiency

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