Abstract
It is shown that it is reasonable to use Galois fields, including those obtained by algebraic extensions, to describe the position of a point in a discrete Cartesian coordinate system in many cases. This approach is applicable to any problem in which the number of elements (e.g., pixels) into which the considered fragment of the plane is dissected is finite. In particular, it is obviously applicable to the processing of the vast majority of digital images actually encountered in practice. The representation of coordinates using Galois fields of the form GF(p2) is a discrete analog of the representation of coordinates in the plane through a complex variable. It is shown that two different types of algebraic extensions can be used simultaneously to represent transformations of discrete Cartesian coordinates described through Galois fields. One corresponds to the classical scheme, which uses irreducible algebraic equations. The second type proposed in this report involves the use of a formal additional solution of some equation, which has a usual solution. The correctness of this approach is justified through the representation of the elements obtained by the algebraic expansion of the second type by matrices defined over the basic Galois field. It is shown that the proposed approach is the basis for the development of new methods of information protection, designed to control groups of UAVs in the zone of direct radio visibility. The algebraic basis of such methods is the solution of systems of equations written in terms of finite algebraic structures.