On the Hardness of the Lee Syndrome Decoding Problem
This addresses a foundational problem in coding theory and cryptography, providing incremental insights into metric-based decoding complexities.
The paper tackles the hardness of the Lee syndrome decoding problem over finite rings by proving its NP-completeness and analyzing solver complexities, showing asymptotic computational comparisons to Hamming metric algorithms.
In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric. We first prove that the decisional version of the problem is NP-complete, by a reduction from the $3$-dimensional matching problem. Then, we study the complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite fields to the Lee metric over finite rings, as well as proposing some novel solutions. For the analyzed algorithms, we assess the computational complexity in the asymptotic regime and compare it to the corresponding algorithms in the Hamming metric.