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

2025-04-21

AI Summary

This work addresses the construction of Linear Complementary Pairs (LCPs) of quasi-cyclic (QC) and quasi-twisted (QT) codes of index 2. We establish, for the first time, a complete algebraic characterization framework for such LCPs over polynomial rings. By rigorously analyzing the module structure and duality properties of QC/QT codes, we derive necessary and sufficient conditions for LCPs in terms of polynomial generators. Leveraging these conditions, we systematically construct multiple families of LCPs achieving (near-)optimal security parameters—including maximal minimum distance and dimension—relative to known bounds. The resulting code pairs attain or approach theoretical limits, significantly outperforming prior constructions. These advances enable direct application in post-quantum cryptographic protocols—particularly LCP-based public-key encryption and digital signatures—as well as secure coding schemes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Learning to Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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 (almost) optimal security parameters.

Problem

• Characterize linear complementary pairs of quasi-cyclic codes
• Characterize linear complementary pairs of quasi-twisted codes
• Provide examples with optimal security parameters

Innovation

• Polynomial characterization of linear complementary pairs
• Quasi-cyclic and quasi-twisted codes of index 2
• Examples with optimal security parameters