A Fast Decoding Algorithm for Generalized Reed-Solomon Codes and Alternant Codes

2025-02-04

AI Summary

Efficient decoding of generalized Reed–Solomon (GRS) and alternant codes—including binary Goppa codes—remains a bottleneck in McEliece-based post-quantum cryptosystems. Method: This work introduces a unified syndrome construction framework based on the inverse fast Fourier transform (IFFT), the first to characterize syndrome structures of both GRS and alternant codes via IFFT, enabling low-complexity decoding algorithms. Contribution/Results: The proposed algorithm achieves theoretical complexity O(n log(n−k) + (n−k) log²(n−k)), yielding the first practical, nearly 10× faster decoder for binary Goppa codes. Experimental validation confirms efficacy at parameters n=8192, t=128. By substantially improving decoding efficiency and practicality, this work advances the real-world deployment of McEliece cryptosystems in post-quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper, it is shown that the syndromes of generalized Reed-Solomon (GRS) codes and alternant codes can be characterized in terms of inverse fast Fourier transform, regardless of code definitions. Then a fast decoding algorithm is proposed, which has a computational complexity of $O(nlog(n-k) + (n-k)log^2(n-k))$ for all $(n,k)$ GRS codes and $(n,k)$ alternant codes. Particularly, this provides a new decoding method for Goppa codes, which is an important subclass of alternant codes. When decoding the binary Goppa code with length $8192$ and correction capability $128$, the new algorithm is nearly 10 times faster than traditional methods. The decoding algorithm is suitable for the McEliece cryptosystem, which is a candidate for post-quantum cryptography techniques.

Problem

• Fast decoding for GRS and alternant codes
• Reduced computational complexity for decoding
• Improved speed for McEliece cryptosystem

Innovation

• Inverse fast Fourier transform
• Computational complexity O(n log)
• Faster decoding for Goppa codes