Bounded-Degree Low-Rank Parity-Check Codes

Abstract

Low-rank parity-check (LRPC) codes are the rank-metric analogue of low-density parity-check codes and they found important applications in code-based cryptography. In this paper we investigate a sub-family of LRPC codes, which have a parity-check matrix defined over a subspace ${\mathcal {V}}_{\alpha,d}=\langle 1,\alpha, \ldots, \alpha ^{d-1} \rangle _{\mathbb {F}_{q}}\subsetneq \mathbb {F}_{q^{m}} $ , where $\mathbb {F}_{q^{m}}$ is the finite field of $q^{m}$ elements, $\alpha \in \mathbb {F}_{q^{m}}$ is an element not in any proper subfield of $\mathbb {F}_{q^{m}}$ , and d is a positive integer significantly smaller than m. These codes are termed bounded-degree LRPC (BD-LRPC) codes. BD-LRPC codes are the same as the standard LRPC codes of density 2 when the degree $d=2$ , while for degree $d\gt 2$ they constitute a proper subset of LRPC codes of density d. Exploiting the structure of ${\mathcal {V}}_{\alpha,d}$ , the BD-LRPC codes of degree d can uniquely correct errors of rank weight r when $n-k \geq r + u$ for certain $u \geq 1$ , in contrast to the condition $n-k\geq dr$ required for the standard LRPC codes. This underscores the superior decoding capability of the BD-LRPC codes. Moreover, as the code length $n\rightarrow \infty $ , when $n/m\rightarrow 0$ , the BD-LRPC codes with a code rate of $R=k/n$ can be uniquely decodable with radius $\rho =r/n$ approaching the Singleton bound $1-R$ by letting $\epsilon =u/n\rightarrow 0$ ; and when $n/m$ is a constant, the BD-LRPC codes can have unique decoding radius $\rho = 1-R-\epsilon $ for a small $\epsilon $ , allowing for $\rho \gt (1-R)/2$ with properly chosen parameters. This superior decoding capability is theoretically proved for the case $d=2$ and confirmed by experimental results for $d\gt 2$ .