Efficient Description of some Classes of Codes using Group Algebras

TL;DR

This paper explores the algebraic structure of group algebras related to circulant matrices, providing a new representation that preserves Hamming weight and classifies it for abelian groups, with applications in cryptography.

Contribution

It introduces an injective Hamming weight preserving homomorphism for group algebra representations and classifies this in the abelian case, extending algebraic tools for coding and cryptography.

Findings

Representation is an injective Hamming weight preserving homomorphism

Classification of the representation for abelian groups

Potential applications in cryptosystems and code design

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 $\mathbb{F}[x]/(x^n-1)$. In consequence, the strong algebraic structure of the ring $\mathbb{F}[x]/(x^n-1)$ can be used to study properties of the collection of all $n\times n$ circulant matrices. The ring $\mathbb{F}[x]/(x^n-1)$ 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 post-quantum standard for asymmetric cryptography). Group algebras can be used to design similar cryptosystems or, more generally, to construct low density or moderate density parity-check matrices for linear codes.