Efficient Description of some Classes of Codes using Group Algebras
Abstract
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is fully specified by its first row. The ring of $n \times n$ circulant matrices can be identified with the quotient ring $\mathbb{F}[x]/(x^n1)$. In consequence, the strong algebraic structure of the ring $\mathbb{F}[x]/(x^n1)$ can be used to study properties of the collection of all $n\times n$ circulant matrices. The ring $\mathbb{F}[x]/(x^n1)$ is a special case of a group algebra and elements of any finite dimensional group algebra can be represented with square matrices which are specified by a single column. In this paper we study this representation and prove that it is an injective Hamming weight preserving homomorphism of $\mathbb{F}$algebras and classify it in the case where the underlying group is abelian. Our work is motivated by the desire to generalize the BIKE cryptosystem (a contender in the NIST competition to get a new postquantum standard for asymmetric cryptography). Group algebras can be used to design similar cryptosystems or, more generally, to construct low density or moderate density paritycheck matrices for linear codes.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.04061
 Bibcode:
 2022arXiv220804061C
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Cryptography and Security
 EPrint:
 A shortened version was submitted to MTNS 2022