Codes in a Dihedral Group Algebra

Abstract

In 1978, Robert McEliece constructed the first asymmetric code-based cryptosystem using noise-immune Goppa codes; no effective key attacks has been described for it yet. By now, quite a lot of code-based cryptosystems are known; however, their cryptographic security is inferior to that of the classical McEliece cryptosystem. In connection with the development of quantum computing, code-based cryptosystems are considered as an alternative to number theoretical ones; therefore, the problem of seeking promising classes of codes to construct new secure code-based cryptosystems is relevant. For this purpose, noncommutative codes can be used, that is, ideals in group algebras \({{\mathbb{F}}_{q}}G\) over finite noncommutative groups \(G\). The security of cryptosystems based on codes induced by subgroup codes has been studied earlier. The Artin–Wedderburn theorem, which proves the existence of an isomorphism of a group algebra to the direct sum of matrix algebras, is important for studying noncommutative codes. However, the particular form of terms and the construction of the isomorphism are not specified by this theorem; thus, for each group, there remains the problem of constructing the Wedderburn representation. The complete Wedderburn decomposition for the group algebra \({{\mathbb{F}}_{q}}{{D}_{{2n}}}\) over the dihedral group \({{D}_{{2n}}}\) has been obtained by F.E. Brochero Martinez in the case when the cardinality of the field and the order of the group are relatively prime numbers. Using these results, we study codes in the group algebra \({{\mathbb{F}}_{q}}{{D}_{{2n}}}\) in this paper. The problem on the structure of all codes is solved, and the structure of codes induced by codes over cyclic subgroups of \({{D}_{{2n}}}\) is described, which is of interest for cryptographic applications.