Low-Rank Parity-Check Codes Over Finite Commutative Rings and Application to Cryptography

Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder and derive an upper bound of the failure probability together with the complexity of the decoder. We then extend the definition to arbitrary finite commutative rings and also provide a decoder in this case. We end-up by introducing an application of the corresponding LRPC codes to cryptography, together with the new corresponding mathematical problems.