Information encryption by the GIL method using the symbols of multiple orthogonal quasigroups over a field of redundancy

Abstract

The paper explores the possibilities of improving the classical Gill cipher by using systems of multi-place orthogonal quasigroup operations built on finite residue fields of simple order. The proposed approach is based on establishing a connection between orthogonal n-ary quasigroups and invertible matrices, which allows the formation of new cryptographic transformations for information protection. The theoretical principles of constructing such structures, their properties and orthogonality conditions are considered. The article proves the criteria for the existence of invertible matrices whose elements belong to the residue field and are nonzero. Based on the obtained mathematical results, an algorithm for constructing systems of orthogonal linear n-ary quasigroups of arbitrary dimension is developed. The proposed algorithm ensures the uniqueness of the solution of the corresponding systems of equations, which is a necessary condition for the correct process of encrypting and decrypting messages. An example of constructing a system of five orthogonal quasigroups of arity five over a finite set of residues is given, which confirms the practical feasibility of the proposed approach. Special attention is paid to assessing the cryptographic stability of the developed method. It is shown that the number of possible key matrices increases rapidly with increasing their dimension, which significantly complicates the implementation of brute force attacks. The use of a set of matrices of different dimensions and a random order of their application additionally increases the level of system security. The results obtained can be used in the development of modern block cryptographic algorithms, data protection systems and information and communication networks.