Reproducible families of codes and cryptographic applications

Abstract

Structured linear block codes such as cyclic, quasi-cyclic and quasi-dyadic codes have gained an increasing role in recent years both in the context of error control and in that of code-based cryptography. Some well known families of structured linear block codes have been separately and intensively studied, without searching for possible bridges between them. In this article, we start from well known examples of this type and generalize them into a wider class of codes that we call ℱ-reproducible codes. Some families of ℱ-reproducible codes have the property that they can be entirely generated from a small number of signature vectors, and consequently admit matrices that can be described in a very compact way. We denote these codes as compactly reproducible codes and show that they encompass known families of compactly describable codes such as quasi-cyclic and quasi-dyadic codes. We then consider some cryptographic applications of codes of this type and show that their use can be advantageous for hindering some current attacks against cryptosystems relying on structured codes. This suggests that the general framework we introduce may enable future developments of code-based cryptography.

Efficient Description of some Classes of Codes using Group Algebras

“…Note also that k[x]/(x n − 1) is the group algebra k [C n ] of the cyclic group C n with n elements. More generally, [Santini et al, 2021] study matrices whose rows are related through an arbitrary family of permutations rather than just cyclic shifts, and under which conditions these matrices form a ring. Group algebras themselves have a history in coding theory.…”