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

TL;DR

This paper introduces a fast decoding algorithm for GRS and alternant codes using inverse FFT, significantly improving decoding speed and applicability to cryptography, especially for Goppa codes in the McEliece cryptosystem.

Contribution

It presents a novel decoding algorithm based on inverse FFT for GRS and alternant codes with improved computational complexity and practical speed advantages.

Findings

Decoding binary Goppa code (length 8192, correction 128) is nearly 10 times faster.

The algorithm has complexity $O(n ext{log}(n-k) + (n-k) ext{log}^2(n-k))$ for all GRS and alternant codes.

Suitable for post-quantum cryptography, especially the McEliece cryptosystem.

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 $O(n\log(n-k) + (n-k)\log^2(n-k))$ 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.