Roth-lempel and Twisted GRS Codes Enable Faster Decoding of Non-GRS MDS Codes

Maximising data reliability and security relies heavily on error-correcting codes, with generalized Reed-Solomon (GRS) codes currently dominating the field, yet their strong mathematical structure can present vulnerabilities in cryptographic applications. Runtian Zhu and Lingfei Jin, from Fudan University and the State Key Laboratory of Cryptology, alongside their colleagues, address this challenge by developing efficient decoding methods for two important alternatives to GRS codes, twisted generalized Reed-Solomon (TGRS) codes and Roth-Lempel codes. This research significantly advances the understanding of these non-GRS codes, offering algorithms that decode data much faster than previous approaches, reducing complexity from quadratic to near-linear time. Importantly, the team’s decoder for TGRS codes handles a far greater number of code variations than existing methods, and provides the first efficient decoding solution for Roth-Lempel codes, paving the way for more secure and robust data storage and transmission.

This research advances the understanding of these non-GRS codes, providing algorithms that decode data much faster than previous approaches, reducing computational complexity from quadratic to near-linear time.

Twisted and Generalized Reed-Solomon Code Constructions

This work investigates twisted Reed-Solomon (TRS) and generalized Reed-Solomon (GTRS) codes, exploring their construction and properties. A central theme is list decoding, where the algorithm identifies a list of possible codewords instead of a single correct one, crucial for noisy communication channels. The research connects these codes to algebraic geometry (AG) codes and Roth-Lempel codes, and focuses on achieving list decoding up to the Singleton bound, a theoretical performance limit.

Efficient Decoding of TGRS and Roth-Lempel Codes

Scientists have developed new decoding algorithms for TGRS codes and Roth-Lempel codes, significantly improving computational efficiency. The work refines the Guruswami-Sudan algorithm, a powerful technique for decoding these codes, to overcome limitations in existing methods. The team created a unique decoder for TGRS codes with near-linear time performance, supporting codes with a substantially larger number of twists than previous decoders.

Measurements confirm that this approach extends to Roth-Lempel codes, for which no efficient decoding algorithms were previously known. The resulting algorithms achieve a decoding time of O(n log²n log log n), representing a breakthrough in speed and efficiency, and the incorporation of algebraic manipulation detection (AMD) codes further improves reliability by enabling accurate recovery of the transmitted message.

Efficient Decoding of TGRS and Roth-Lempel Codes

This research presents advances in the decoding of TGRS codes and Roth-Lempel codes, both non-GRS maximum distance separable (MDS) codes. Scientists have developed list and unique decoding algorithms based on the Guruswami-Sudan algorithm, achieving near-linear time complexity, a substantial improvement over previous quadratic-time methods. The team’s decoder for TGRS codes supports codes with a significantly larger number of twists, and importantly, provides the first efficient decoder for Roth-Lempel codes.

These developments are noteworthy because they expand the practical applicability of non-GRS codes, increasingly relevant in cryptography where the strong algebraic structure of GRS codes can be a vulnerability. By surpassing the classical unique-decoding radius, the new list decoders offer enhanced performance, and the incorporation of algebraic manipulation detection codes further improves reliability by enabling accurate recovery of the transmitted message.