A Fast Decoding Algorithm for Generalized Reed-Solomon Codes and Alternant Codes
Nianqi Tang, Yunghsiang S. Han, Danyang Pei, Chao Chen
TL;DR
Problem: efficiently decoding generalized Reed-Solomon and alternant codes across all parameters. Approach: define generalized syndromes as the high-degree part of the IFFT of the received vector, implement a four-step decoding pipeline using Lin-Chung-Han FFT transforms, solve a unified key equation, locate errors with Chien search, and recover values via Forney's formula. Contributions: a unified decoder with complexity $O(n\log(n-k) + (n-k)\log^2(n-k))$ applicable to all GRS and alternant codes, applicability to separable Goppa codes, and substantial practical speedups (e.g., near 10× for $n=8192$, $t=128$) in McEliece PQC contexts. Significance: provides a regular, hardware-friendly framework with broad applicability to post-quantum cryptography and potential for further FFT-based accelerations.
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.