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Linear Complementary Pairs of Quasi-Cyclic and Quasi-Twisted Codes

Kanat Abdukhalikov, Duy Ho, San Ling, Gyanendra K. Verma

TL;DR

The paper addresses constructing and characterizing linear complementary pairs (LCP) for $q$-ary quasi-cyclic and quasi-twisted codes of index $2$. It develops polynomial characterizations using CRT decomposition and two-generator/one-generator representations, deriving necessary and sufficient LCP criteria for both QC and QT settings. The authors validate the theory with explicit examples that achieve optimal security parameters, including non-binary and binary cases, computed via MAGMA. These results extend prior work on LCD/LCP structures in cyclic- and constacyclic-like codes and provide practical tools for direct-sum masking applications in post-quantum cryptography.

Abstract

In this paper, we provide a polynomial characterization of linear complementary pairs of quasi-cyclic and quasi-twisted codes of index 2. We also give several examples of linear complementary pairs of quasi-cyclic and quasi-twisted codes with optimal security parameters.

Linear Complementary Pairs of Quasi-Cyclic and Quasi-Twisted Codes

TL;DR

The paper addresses constructing and characterizing linear complementary pairs (LCP) for -ary quasi-cyclic and quasi-twisted codes of index . It develops polynomial characterizations using CRT decomposition and two-generator/one-generator representations, deriving necessary and sufficient LCP criteria for both QC and QT settings. The authors validate the theory with explicit examples that achieve optimal security parameters, including non-binary and binary cases, computed via MAGMA. These results extend prior work on LCD/LCP structures in cyclic- and constacyclic-like codes and provide practical tools for direct-sum masking applications in post-quantum cryptography.

Abstract

In this paper, we provide a polynomial characterization of linear complementary pairs of quasi-cyclic and quasi-twisted codes of index 2. We also give several examples of linear complementary pairs of quasi-cyclic and quasi-twisted codes with optimal security parameters.

Paper Structure

This paper contains 6 sections, 15 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. LCPs of quasi-cyclic codes of index $2$
  4. LCPs of one-generator quasi-cyclic codes of index 2
  5. LCPs of quasi-twisted codes of index 2
  6. Examples of LCPs of codes with good security parameters

Key Result

Theorem 2.1

Let $C$ and $D$ be $q$-ary QC codes of length $m\ell$ and index $\ell$ with CRT decompositions $C \cong \bigoplus_{i=1}^{t} C_i$ and $D \cong \bigoplus_{i=1}^{t} D_i$, respectively. Then $(C,D)$ is an LCP of codes if and only if $(C_i, D_i)$ is an LCP of codes for all $1\leq i\leq t$.

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 26 more