Abstract
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.
Highlights
Defining linear block codes that possess a certain inner structure and verify some regularity properties is a natural process in coding theory
BIKE and LEDAcrypt are two public-key encryption schemes based on, respectively, QC-Moderate-Density Parity-Check (MDPC) and QC-Low-Density Parity-Check (LDPC) codes, which naturally fit into the general framework we describe in this article
We have extended these notions to coding theory and have introduced the concept of reproducible and quasi-reproducible codes, which are codes described by a generator or a parity-check matrix yielding a compact representation
Summary
Defining linear block codes that possess a certain inner structure and verify some regularity properties is a natural process in coding theory. The work was motivated by its implications for the McEliece cryptosystem [2], and in particular by the necessity of having a family of codes whose generator and parity-check matrices can be represented in a compact way This is because, in code-based cryptography, the public key of an encryption (or signature) scheme. Previous efforts to reduce key size were centered on quasi-cyclic algebraic codes [3] and have been since extended to codes of a different nature, namely the Low-Density Parity-Check (LDPC) codes [4] and their recent generalization known as Moderate-Density Parity-Check (MDPC) codes [5] These codes are characterized by sparse parity-check matrices and admit matrices in quasi-cyclic form, formed by circulant square blocks.
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