[Paper Review] NP-Complete Problems for Lee Metric Codes

Violetta Weger, Paolo Santini+2 more|arXiv (Cornell University)|Feb 27, 2020

Coding theory and cryptography29 references4 citations

TL;DR

This paper establishes the NP-completeness of the syndrome decoding problem (SDP) for codes over finite rings under the Lee metric, generalizes information set decoding (ISD) algorithms to this metric, and analyzes their complexity across a broad range of code parameters, providing foundational complexity results for Lee metric coding theory and code-based cryptography.

ABSTRACT

We consider codes over finite rings endowed with the Lee metric and prove the NP-completeness of the associated syndrome decoding problem (SDP). Then, we study the best known algorithms for solving the SDP, which are information set decoding (ISD) algorithms, and generalize them to the Lee metric case. Finally we assess their complexity for a wide range of parameters.

Motivation & Objective

• To prove that the syndrome decoding problem (SDP) is NP-complete for codes over finite rings under the Lee metric.
• To extend the framework of information set decoding (ISD) algorithms to the Lee metric setting.
• To analyze the computational complexity of the generalized ISD algorithms across diverse code parameters.
• To establish theoretical foundations for the security of code-based cryptosystems using Lee metric codes.

Proposed method

• Reduction from a known NP-complete problem to demonstrate the NP-completeness of the Lee metric SDP.
• Adaptation of classical ISD algorithms to operate under the Lee metric by redefining weight and error pattern selection criteria.
• Design of a generalized ISD algorithm that identifies and decodes information sets using Lee weight properties.
• Use of combinatorial analysis to estimate the expected number of trials needed to find a valid information set in the Lee metric.
• Complexity evaluation through asymptotic analysis of the algorithm's expected running time under varying code length, dimension, and minimum Lee distance.
• Parameterization of code families over finite rings to assess algorithmic performance across different metric regimes.

Experimental results

Research questions

• RQ1 Is the syndrome decoding problem NP-complete for codes over finite rings when the Lee metric is used?
• RQ2 How can information set decoding (ISD) algorithms be adapted to function efficiently under the Lee metric?
• RQ3 What is the computational complexity of the generalized ISD algorithm for Lee metric codes across different code parameters?
• RQ4 How does the Lee metric affect the efficiency and feasibility of decoding algorithms compared to the Hamming metric?
• RQ5 What are the implications of these findings for the security of code-based cryptosystems using Lee metric codes?

Key findings

• The syndrome decoding problem (SDP) for codes over finite rings under the Lee metric is proven to be NP-complete.
• The generalized ISD algorithm for the Lee metric achieves a non-trivial reduction in search space by leveraging Lee weight properties.
• The complexity of the generalized ISD algorithm is shown to scale with the Lee distance and code parameters, with exponential dependence on the error weight.
• The analysis reveals that Lee metric codes can offer comparable or improved decoding efficiency compared to Hamming metric codes under specific parameter regimes.
• The results suggest that Lee metric codes are viable candidates for post-quantum cryptography, with security rooted in the hardness of the SDP.
• The study provides a theoretical framework for evaluating the security of code-based cryptosystems using Lee metric codes.