Low-Rank Parity-Check Codes over Galois Rings

TL;DR

This paper extends low-rank parity-check (LRPC) codes to Galois rings, proposing a decoding algorithm with a bounded failure probability, and demonstrates potential for faster decoding compared to Gabidulin codes over finite fields.

Contribution

It introduces LRPC codes over Galois rings, develops a decoding algorithm, and analyzes its failure probability and complexity, expanding the applicability of rank-metric codes.

Findings

Decoding over Galois rings is feasible with similar error correction capability.

The proposed decoder is faster than existing Gabidulin decoders.

Failure probability depends only on error rank, not free rank.

Abstract

Low-rank parity-check (LRPC) are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (2019), we define and study LRPC codes over Galois rings - a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.'s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.