Linear Complementary dual codes and Linear Complementary pairs of AG codes in function fields

2024-07-08

IEEE Transactions on Information Theory

AI Summary

Existing constructions of Linear Complementary Pairs (LCP) and Linear Complementary Dual (LCD) algebraic geometry (AG) codes are restricted to rational function fields or low-genus cases, limiting their applicability in cryptography. Method: This paper presents the first systematic construction of LCP/LCD AG codes over arbitrary function fields of genus $g geq 1$, leveraging pairs of non-special divisors of degree $g-1$. The approach integrates function field theory, the Riemann–Roch theorem, Kummer extensions, and divisor analysis on hyperelliptic and elliptic curves. Contribution/Results: The resulting code families are explicitly constructible, achieve tight parameter bounds—particularly on dimension and minimum distance—and exhibit strong cryptographic suitability. By overcoming prior geometric restrictions, this work establishes a novel algebraic-geometric foundation for lightweight, side-channel-resistant coding schemes.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Agent Architectures

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

In recent years, linear complementary pairs (LCP) of codes and linear complementary dual (LCD) codes have gained significant attention due to their applications in coding theory and cryptography. In this work, we construct explicit LCPs of codes and LCD codes from function fields of genus $g geq 1$. To accomplish this, we present pairs of suitable divisors giving rise to non-special divisors of degree $g-1$ in the function field. The results are applied in constructing LCPs of algebraic geometry codes and LCD algebraic geometry (AG) codes in Kummer extensions, hyperelliptic function fields, and elliptic curves.

Problem

• Linear Complementary Pair (LCP)
• Linear Complementary Dual (LCD) Codes
• Algebraic Geometry (AG) Codes

Innovation

• Linear Complementary Dual Codes
• Function Fields
• Algebraic Geometry Codes