Differentiation of polynomials in several variables over Galois fields of fuzzy cardinality and applications to Reed-Muller codes

Introduction. Polynomials in several variables over Galois fields provide the basis for the Reed-Muller coding theory, and are also used in a number of cryptographic problems. The properties of such polynomials specified over the derived Galois fields of fuzzy cardinality are studied. For the results obtained, two real-world applications are proposed: partitioning scheme and Reed-Muller code decoder.Materials and Methods. Using linear algebra, theory of Galois fields, and general theory of polynomials in several variables, we have obtained results related to the differentiation and integration of polynomials in several variables over Galois fields of fuzzy cardinality. An analog of the differentiation operator is constructed and studied for vectors.Research Results. On the basis of the obtained results on the differentiation and integration of polynomials, a new decoder for Reed-Muller codes of the second order is given, and a scheme for organizing the partitioned transfer of confidential data is proposed. This is a communication system in which the source data on the sender is divided into several parts and, independently of one another, transmitted through different communication channels, and then, on the receiver, the initial data is restored of the parts retrieved. The proposed scheme feature is that it enables to protect data, both from the nonlegitimate access, and from unintentional errors; herewith, one and the same mathematical apparatus is used in both cases. The developed decoder for the second-order Reed-Muller codes prescribed over the derived odd Galois field may have a constraint to the recoverable error level; however, its use is advisable for a number of the communication channels.Discussion and Conclusions. The proposed practical applications of the results obtained are useful for the organization of reliable communication systems. In future, it is planned to study the restoration process of the original polynomial by its derivatives, in case of their partial distortion, and the development of appropriate applications.