New Constructions of Non-GRS MDS Codes, Recovery and Determination Algorithms for GRS Codes
Guodong Wang, Hongwei Liu, Jinquan Luo
TL;DR
Addresses the construction of non-GRS MDS codes with large lengths and provides efficient tools for recognizing and recovering GRS structure. It introduces modified and extended modified GRS codes, derives precise MDS/non-GRS conditions via Cauchy-matrix analysis, and maps relationships to Roth-Lempel, column-twisted, and twisted GRS codes. The work also presents practical recovery and determinant algorithms with favorable complexity, surpassing classical attacks like Sidelnikov–Shestakov. Together, these results expand the taxonomy of non-GRS MDS codes and enhance cryptanalytic and coding-theoretic capabilities for GRS-code frameworks.
Abstract
In this paper, we propose a new method for constructing a class of non-GRS MDS codes. The lengths of these codes can reach up to $\frac{q+3}{2}$ (for finite fields of odd characteristic) and $\frac{q+4}{2}$ (for even characteristic), respectively. Owing to their special structure, we can use the Cauchy matrix method to obtain the necessary and sufficient conditions for these codes to be MDS codes and non-GRS MDS codes. Additionally, the inequivalence between these codes and twisted GRS codes is analyzed. Furthermore, we analyze the relationships among several existing classes of codes used for constructing non-GRS MDS codes, propose explicit constructions, and discuss the lengths of non-GRS MDS codes based on these constructions. Finally, we design two efficient algorithms to address two main problems in GRS code research, i.e., determining whether an unknown code $C$ is a GRS code from its generator matrix $G$, and recovering the key vectors $\bmα$ and $\bm{v}$ such that $C = \GRS_{n,k}(\bmα, \bm{v})$ if $C$ is indeed a GRS code. A computational complexity comparison of the proposed algorithms ($O(nk+n)$) with that of the Sidelnikov-Shestakov attack (exceeding $O(qk^2n+qk^3)$) shows that our methods offer superior computational efficiency.