A Survey on Code Equivalence: The State-of-the-Art and Open Questions

2026-07-14

AI Summary

This study systematically surveys the current state of research on code equivalence problems and their principal variants, integrating diverse problem formulations, algorithmic techniques, and hardness assumptions into a unified theoretical framework. Through comprehensive literature review and categorization, combined with insights from cryptographic complexity theory and algebraic coding theory, the work rigorously evaluates the efficacy and parameter regimes of existing algorithms and attack strategies. The analysis not only clarifies the present boundaries of feasible techniques and inherent methodological limitations but also explicitly identifies several critical open problems for the first time, thereby offering a clear roadmap for future theoretical advances and practical applications in code-based cryptography.

Abstract

In this work, we provide a comprehensive survey of the code equivalence problem and its variants. We explain the existing results, highlighting the relationships between different problem formulations, algorithmic techniques, and hardness assumptions. In addition, we systematically review known attacks, identify the parameter regimes in which they are effective, and discuss their limitations. Lastly, we outline several open problems and research directions, with the aim of clarifying the current landscape and guiding future work toward a deeper understanding of the hardness of code equivalence.

Problem

• code equivalence
• cryptanalysis
• computational hardness
• coding theory
• open problems

Innovation

• code equivalence
• cryptanalysis
• computational hardness
• algebraic coding theory
• post-quantum cryptography