Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptography. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residual field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.

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.

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of publickey cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings. In this paper, we show that solving the rank decoding problem over finite principal ideal rings is at least as hard as the rank decoding problem over finite fields. We also show that computing the minimum rank distance for linear codes over finite principal ideal rings is equivalent to the same problem for linear codes over finite fields. Finally, we provide combinatorial type algorithms for solving the rank decoding problem over finite chain rings together with their average complexities.

The rank decoding problem has been the subject of much attention in this last decade. This problem, which is at the base of the security of public-key cryptosystems based on rank metric codes, is traditionally studied over finite fields. But the recent generalizations of certain classes of rank-metric codes from finite fields to finite rings have naturally created the interest to tackle the rank decoding problem in the case of finite rings. In this paper, we show that some combinatorial type algorithms for solving the rank decoding problem over finite fields can be generalized to solve the same problem over finite principal ideal rings. We study and provide the average complexity of these algorithms. We also observe that some recent algebraic attacks are not directly applicable when the finite ring is not a field due to zero divisors. These results could be used to justify the use of codes defined over finite rings in rank metric code-based cryptography.