# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 5 Discussions  
Thus, it can be stated that when using the combined method of algebraic extensions, both the vectors corresponding to the discrete two-dimensional plane and the operators acting on them can be represented through one of the two possible extensions of the basic Galois field. It should be emphasized that in this paper, only linear transformations of discrete coordinates were considered, but this is already the first step toward solving more complex problems, for example, developing methods for solving quadratic equations written in terms of Galois fields. Any equation whose variables and coefficients are integers, including a square equation, can be considered as written in terms of some field \(GF(p)\), provided that the result of summing any set of its terms does not exceed p. The same is true for equations that involve two variables, provided that both these variables and coefficients take integer values. In this case, it is permissible to pass from a quadratic equation containing two variables to an equation in terms of an algebraic extension of the main field \(GF(p)\). The matrix of a quadratic equation corresponds to the matrix of a linear operator, so the proposed method for reducing operators to elements of algebraic systems obtained by algebraic extensions of the main field is, in the long term, the basis for solving quadratic equations with integer coefficients. Such a solution of quadratic equations is relevant, in particular, from the point of view of developing information security systems that make it possible to abandon the use of cryptographic methods that remain vulnerable to the human factor. In particular, it is possible to propose information security systems designed for use in the line-of-sight zone and based on identifying the location of the signal source, i.e., the operator sending the commands. In this case, there is no need to determine the operator's coordinates with high accuracy, which makes it possible to solve the corresponding equations in terms of discrete coordinates that can be put in correspondence with the elements of various algebraic structures, including those obtained by the method proposed in this paper. Such an approach is justified, in particular, from the point of view of increasing the efficiency of onboard computers. At present, electronic circuits are being actively developed that perform calculations in Galois fields of the simplest type (e.g., adders and multipliers modulo a prime number [87; 88; 89]), which also correspond to calculations in terms of multivalued logic [90], a similar conclusion can be made with respect to calculations in terms of algebraic rings [91]. The advantage here is a significant reduction in the number of operations, in particular, compared to the situation when fractional values are used in calculations. Consequently, the proposed approach can be used in the future as a basis for onboard computers built on simplified electronic circuits. Further, the presented results refer only to planar images arising in a wide variety of tasks, which relate not only to information security, but also, for example, to the processing of video information from agricultural land monitoring. There is, however, also a whole class of tasks involving three-dimensional images, related, for example, to the diagnostics of subsurface objects [92; 93; 94]. Accordingly, the next step is to generalize the proposed approach to the three-dimensional case. There is a basis for this: along with the representation (2), which arises due to the use of an irreducible cubic equation, one can presumably use a similar representation but obtained by the method proposed in this work. In the long run, it is really possible to raise the question of developing an analog of the theory of functions of a complex variable intended for solving problems in which digital (discrete) coordinates change in a finite range of values, but for the three-dimensional case.