Generalized Low-Rank Parity-Check Codes

Let
$\mathbb {F}_{q}$
be the finite field with q elements and m be a positive integer. The
$\mathbb {F}_{q^{m}}$
-linear low-rank parity-check (LRPC) codes have been used in many cryptographic schemes. Motivated by recent attacks on those schemes, this paper generalizes LRPC codes based on 3-tensors in
$\mathbb {F}_{q}^{m \times m \times m}$
. The generalized LRPC codes are mostly
$\mathbb {F}_{q}$
-linear matrix codes, while a particular choice of the 3-tensor is isomorphic to the original
$\mathbb {F}_{q^{m}}$
-linear LRPC codes. We first introduce a bilinear T-product over
$\mathbb {F}_{q}^{m}$
associated with a 3-tensor
$T\in \mathbb {F}_{q}^{m \times m \times m} $
. Based on the T-product, we propose a generic method to expand
$\mathbb {F}_{q}$
-linear matrix code from dimension k to dimension
$km$
and then use the method to generalize LRPC codes. Finally, we propose two probabilistic polynomial-time decoding algorithms for the generalized LRPC codes under different circumstances. We provide estimates of their decoding failure rates, which, confirmed by experimental results, are almost the same as that of decoding
$\mathbb {F}_{q^{m}}$
-linear LRPC codes.