The structure of finite group algebra of a semidirect product of abelian groups and its applications

In 1978 R. McEliece developed the first assymetric cryptosystem based on the use of Goppa's error–correctring codes and no effective key attacks has been described yet. Now there are many code–based cryptosystems known. One way to build them is to modify the McEliece cryptosystem by replacing Goppa's codes with other codes. But many variants of this modification were proven to be less secure.
In connection with the development of quantum computing code cryptosystems along with lattice-based cryptosystems are considered as an alternative to number-theoretical ones. Therefore, it is relevant to find promising classes of codes that are applicable in cryptography. It seems that for this non–commutative group codes, i.e. left ideals in finite non–commutative group algebras, could be used.
The Wedderburn theorem is useful to study non–commutative group codes. It implies the existence of an isomorphism of a semisimple group algebra onto a direct sum of matrix algebras. However, the specific form of the summands and the isomorphism construction are not explicitly defined by this theorem. Hence for each semisimple group algebra there is a task to explicitly construct its Wedderburn decomposition. This decomposition allows us to easily describe all left ideals of group algebra, i.e. group codes.
In this paper we consider one semidirect product $Q_{m,n} = (\mathbb{Z}_m \times \mathbb{Z}_n) \leftthreetimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$ of abelian groups and the group algebra $\mathbb{F}_q Q_{m,n}$. In the case when $n \mid q -1$ and $\gcd(2mn, q) = 1$, the Wedderburn decomposition of this algebra is constructed. In the case when field is of characteristic $2$, i.e. when this group algebra is not semisimple, a similar structure theorem is also obtained. Further in the paper, the primitive central idempotents of this group algebra are described. The obtained results are used to algebraically describe the group codes over $Q_{m,n}$.