Low-rank parity-check codes over Galois rings

Abstract

Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf Theory 65(12):7718-7735, 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.

Keywords: Algebraic coding theory; Galois rings; Low-rank parity-check codes; Rank-metric codes.

Fig. 1

Simulation results for $\lambda =2$, $k=8$ and $n=20$ over $S$ with $p=r=2$, $s=1$ and $m=21$. The markers indicate the estimated probabilities of not fulfilling the product condition (S: Prod), the syndrome condition (S: Synd), the intersection condition (S: Inter) and the decoding failure rate (S: Dec), where the black, blue and orange markers refer to errors of rank profile ${\varphi }_1(x)=t$, ${\varphi }_2(x)=tx$ and ${\varphi }_3(x)\in \{1,1+x,2+x,2+2x,3+2x,3+3x,4+3x\}$, respectively. The derived bounds on these probabilities are shown as lines