Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods

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.

1. Introduction

Nowadays, objects corresponding to discrete Cartesian coordinates are used more and more frequently. A typical example in this respect is the tasks related to digital image processing [1,2,3]. In this case, the image is dissected into a set of pixels, each of which can be matched by two integers, which actually represent discrete Cartesian coordinates. For digital image processing, convolutional neural networks are now being increasingly used [4,5,6,7]. One of the advantages of using such networks is the possibility of staggered training. This approach, in particular, allows analyzing an image as consisting of individual fragments of simple geometric figures, i.e., straight-line segments, fragments of circles, ellipses, parabolas, etc. [8,9,10]. When Cartesian coordinates with continuously varying coordinate values are used (hereinafter referred to as “continuous coordinates”), different fragments of simple geometric figures can be reduced to each other using affine coordinate transformations. If discrete coordinates are used, especially in tasks involving the use of convolutional neural networks, the task becomes somewhat more complicated: ideally, a convolutional neural network oriented at dissecting an image into simple geometric elements should recognize, for example, any straight-line segments as the same geometric object, differing from each other only by coordinate transformations on the initial grid.

The above studies can be referred to the field of discrete geometry [11,12], various aspects of which have also been very intensively developed recently [13,14,15].

Another nontrivial area of application of discrete geometry becomes relevant due to the increasing use of unmanned aerial vehicles, including for military purposes [16,17,18]. As the current experience of using such vehicles shows, their vulnerability to electronic warfare methods becomes more and more significant. This fact forces us to pay attention to the use of UAV control channels protected physically. For this purpose, various approaches are currently proposed, including those using data series that are formed as a result of certain physical processes [19,20] (e.g., geophysical [21]). Another method of forming a secure data channel is the use of optical fiber. This approach has been previously discussed in [22,23,24], and according to the open press, FVP drones controlled via fiber optics are finding increasing use in the military conflict in Ukraine. Another of the alternative options for protecting information (at least the part of it related to executable commands) involves identifying the location of the radio signal source, which is interpreted as “one’s own” [25].

Let us take into account that in order to identify the location of a signal that is interpreted as “one’s own”, it is not necessary to determine its coordinates with great accuracy. It is enough to localize the area of its location. This allows one to move from the classical formulation of the problem to the formulation of the problem from the field of discrete geometry. Simplifying, a map on which the position of the control signal source is marked can be reduced to a discrete form with a sufficiently coarse partitioning into squares. These issues are discussed in more detail in Section 2; for now, let us return to the purely algebraic aspects of the problem at hand.

Description of coordinate transformations can be realized by various methods, especially when two-dimensional problems are considered. In this case, in particular, the coordinates of a point in the plane can be given through a complex variable. On this basis, in particular, the theory of comfort transformations, which are widely used in problems of hydrodynamics, electrostatics, etc., is constructed [26,27,28]. In this case, some operators describing the coordinate transformation can be reduced to the operation of multiplication by a complex number [29]; however, the set of such operators is finite.

As is known, the transition from real variables to complex variables can be considered as one of the examples of using the method of algebraic extensions [30], in which the root of an irreducible algebraic equation (in this case, the imaginary unit) is added to the initial mathematical object (in this case, the set of real numbers). It is essential that this method can be used also when the initial set is discrete. In particular, Galois fields $GF\left(p^n\right)$ can be constructed by this method. In this case, Galois fields $GF\left(p\right)$, which is a ring of subtraction classes of integers modulo prime number $p$, are considered as initial sets. It is appropriate to emphasize that the fields $GF\left(p^n\right)$ are currently used to solve a wide variety of applied problems [31,32,33].

In the case when the irreducible equation has degree two, the method of algebraic expansions allows one to pass from the field $GF\left(p\right)$ to the field $GF\left(p^2\right)$. An arbitrary element of such a field can be represented as

$$x=x_1+\theta x_2$$

(1)

where $\theta$ is the root of some equation, which is irreducible in the field $GF\left(p\right)$, and the variables $x_1$ and $x_2$ take values in the field $GF\left(p\right)$.

The analogy with the representation of a complex variable in the next conventional form is obvious.

$$z=x+iy$$

(2)

The difference is that in Formula (1), the variables $x_1$ and $x_2$ refer to a discrete (and, moreover, finite) set, while the variables $x$ and $y$ in Formula (2) change continuously.

This analogy allows one to assert that discrete Cartesian coordinates admit mapping through expressions of the form (1), and for clarity, the root of the irreducible equation $\theta$ can be treated as a logical imaginary unit of the first kind [34].

Note that the tasks in which it is acceptable to use a discrete coordinate grid are not only related to digital processing of such images as human faces, as well as with the information protection issues mentioned above. Another example is tasks related to remote sensing of the Earth (RSE) based on satellite images [35,36,37], monitoring of agricultural lands by means of overflight by unmanned aerial vehicles equipped with video recorders [38,39,40]. In the latter case, as well as in any mapping tasks, there arises a necessity to combine images acquired at different spatial locations of the recorder, which, from the mathematical point of view, is reduced to a coordinate transformation problem. Of course, such tasks can be solved by traditional methods [41,42]; however, the primary processing of video information using onboard computing systems, which are often based on convolutional neural networks, is of growing interest. As applied to remote sensing tasks, such primary processing is associated with the need to exclude images related to the case when the Earth’s surface is hidden by cloud cover [43,44], as applied to agricultural land monitoring, to exclude the same type of video information [45,46,47], etc. The solution to such tasks obviously requires use of onboard computing systems that perform primary data processing. There, reports [48,49], in which computing systems suitable for such purposes are proposed to be realized on the basis of chips with tunable logic structure (PLD), are known. However, in any case, the solution to this problem requires an appropriate algorithmization that allows to obtain a general picture while minimizing the amount of data coming from video recorders located on certain technical means to the data processing center, which makes it urgent to develop appropriate discrete geometry tools, which in the future will allow to simplify the logical structure of the used computational means. Tools allowing to simplify onboard computing systems are also relevant for realization of physical methods of information protection mentioned above.

The purpose of this paper is to develop a method for describing transformations of discrete Cartesian coordinates represented in the form (1) to a form complementary to such a representation.

Specifically, we were able to show that any linear operators acting on a “vector” of coordinates represented in the form (1) admit themselves to be represented through nonstandard algebraic extensions. As a result, the operators under consideration can also be reduced to elements corresponding to the set formed by extensions of the original field $GF\left(p\right)$.

2. Background: Technical Aspect

Let us demonstrate that the developed approach is indeed of interest for practical applications. For clarity, we will consider the simplest possible example, which, nevertheless, is of considerable interest from the point of view of developing methods of physical protection of information, allowing to refuse the use of cryptographic methods.

As noted above, the issue of information protection by physical methods is currently quite acute, which makes it possible to abandon the use of cryptographic methods, and also significantly complicates electronic warfare aimed at intercepting or suppressing UAVs. At the same time, there is a gradual transition to the use of UAVs as part of a group (swarm) [50,51,52]. One of the most important issues here is the development of control algorithms for the swarm that would allow it to act as a single unit [53,54,55]. It is appropriate to emphasize that such algorithms involve not only information exchange between the UAV and the operator, but also between the individual vehicles forming the group [56,57].

The approach to information protection proposed in [25] meets this trend. It is focused on identifying the meta-location of the coordinates of the point where the operator is located. The commands that are executed are those coming from the radio signal source located at such a point. This method is suitable for use only in the zone of direct radio visibility, but with the ever-wider use of UAVs, including for military purposes, this formulation of the issue is becoming increasingly important. This follows, among other things, from the increasing use of FPV drones controlled via ultra-thin optical fiber by both sides of the current conflict in Ukraine. It is obvious that the zone of direct radio visibility is comparable in scale to the zone for which information transmission via optical fiber can be used. It should also be noted that the identification of the source of the radio signal transmitting commands by its location is of interest, among other things, for the civil use of UAVs. As the airspace becomes saturated with such devices, the issue of the efficiency of using the corresponding radio frequency range will inevitably become relevant. The identification of the source of the radio signal by location is of interest from this point of view as well. Further, regardless of the technical implementation, the identification of the location of the signal source is reduced to solving a very specific geometric problem. This fact can be conveniently demonstrated using the “hyperbola method” used to establish the coordinates of an object exchanging signals with stationary radio stations [58,59]. Let us assume that there are two devices, separated in space, and equipped with radio signal receivers. Let us also assume that the communication channel between these devices operates in real time and allows, for example, to determine the phase difference between the harmonic signals coming from their source. This means that the two devices under consideration, acting together, allow us to determine the difference between the distances from each of these devices to the radio wave source. We emphasize that the registration of the phase difference is considered only as an example. We can also consider time delays between the arrival of pulses. This already depends on the hardware implementation of the computing systems used. But with any hardware implementation, a pair of devices by themselves will not be able to determine the coordinates of the radio signal source. This pair will allow us to record only the difference in distances, which was discussed above (Figure 1).

This figure emphasizes the fact used in the “hyperbola method.” If the above-mentioned distance difference is known, then it can be stated that the signal source lies on one of the hyperbola branches satisfying the equation. In Figure 1, the corresponding hyperbola branches are highlighted with red lines. To register the source coordinates, therefore, one more pair of devices must be used. (The hyperbola branches corresponding to the second pair are shown with blue lines.) The signal source coordinates correspond to the intersection of the corresponding hyperbola branches.

It is impossible not to see that the problem under consideration is reduced to a geometric one. The intersection point of the above hyperbolas corresponds to the solution of a system of two quadratic equations. Such systems of equations have been considered in many works; it is recognized that their solution is one of many nontrivial problems of classical algebra [60,61]. To solve applied problems of this kind, numerical methods are currently usually used [62,63]. There is, however, a significant nuance. The numerical solution of even a relatively simple system of quadratic equations requires fairly serious computing resources. At a minimum, onboard computing systems must provide for calculations using fractional values, which significantly complicates their algorithmic basis. It is much easier to implement calculations that correspond to integers, even in the case when a specific integer corresponds to a variable of multivalued logic. Moreover, in this case, there are quite a few opportunities to make the computational procedures more efficient (e.g., using methods using residue number systems (RNS) [64,65,66]).

In relation to the problem considered above, the transition-to-integer calculations correspond to the transition to a discrete coordinate grid. Indeed, to ensure the identification of a signal source as “one’s own”, there is no need to determine its coordinates with high accuracy. It is sufficient to determine the corresponding square on a discrete coordinate grid.

This, in turn, means it is permissible to move from the equations describing the intersection of hyperbola branches in terms of continuity to their discrete analog. In particular, such equations can be rewritten in terms of Galois fields. To do this, it is sufficient to move to the coordinate grid schematically shown in Figure 2.

This grid corresponds to the representation of the elements of the field $GF\left(p^2\right)$ in the form (1). One of the coordinates in this figure corresponds to the variable $x_1$ and the second to the variable $x_2$, and both variables belong to a discrete set, specifically the field $GF\left(p\right)$. (The corresponding formulas are considered in the next section.) For illustrative purposes, the case of three UAVs forming a group is considered. The figure emphasizes that in order to ensure information protection by the considered method, it is enough to define the square where the operator is located (highlighted in blue in Figure 2).

Classical methods of studying quadratic equations of two variables are associated with coordinate transformations. It is this approach that allows, among other things, to classify quadratic equations of the type under consideration (ellipse, parabola, and hyperbola). Consequently, the first task that arises when solving the problems under consideration in terms of finite algebraic structures is the task of optimizing the representations of coordinate transformations. One of the options for solving this problem is proposed in this paper.

Note also that the basis of the proposed approach is the fact that the “map” corresponding to real practical applications is always finite. This applies to the use of UAVs and many other applications. Thus, for digital image processing tasks generated by modern computers, their range of variation is also finite. This means that, in this case, it is acceptable to use Formula (1) to display the coordinates of each pixel. For example, if one of the existing Full HD (Full High Definition) standards is used, assuming a resolution of 1920 by 1080 pixels, then any Galois field can be used to enumerate pixels $GF\left(p^2\right),$ provided that $p>1920$.

It should also be noted that the problem of coordinate transformation also remains relevant from the point of view of those tasks that are solved, in particular, by convolutional neural networks. Many of these networks are focused on the “construction” of one or another image from elements of simple geometric figures [67,68]. With a standard approach oriented toward recognizing a specific image, elements corresponding, for example, to fragments of a parabola will be considered as different. If we use coordinate transformation, then, at a minimum, it becomes possible to use the corresponding classification features.

It should be emphasized that the use of nontrivial technical solutions in the field of UAV control (e.g., fiber-optic communication between UAVs that make up a group) is only a particular example demonstrating the relevance of developing nontrivial approaches to solving seemingly classical geometric problems, as well as to developing hardware that ensures their solution (as well as the corresponding algorithmic basis).

Currently, there is an active search for technical solutions aimed at increasing the efficiency of computing systems [69,70], as well as creating a fundamentally new element base for computing technology [71,72]. In addition to computers using optical signals [73,74], it is proposed to use systems built on organic transistors [75], optoelectronic polymers [76], nanomemristors [77], as well as on the basis of neuromorphic materials of various types [78,79,80], etc. As emphasized in [72], the development of an algorithmic basis for the functioning of computing technology built on the basis of innovative materials is relevant. There is every reason to believe that one of the promising areas of application of computing technology on a new element base may be onboard computing units of UAVs (e.g., kamikaze drones). Consequently, simplification of algorithms of special-purpose computing systems is relevant from this point of view as well.

3. Background: Theoretical Aspect

The main idea of the proposed method is based on the next fact. Some function-taking values in Galois fields [81] (or other finite algebraic structures, e.g., finite algebraic rings [82]) can serve as a model for any digital signal varying in a finite range of amplitudes. Indeed, functions of a real or complex variable, traditionally used in practice, are purely mathematical objects; hence, they represent nothing more than a model of a signal. The choice of such a mathematical object is nothing more than a matter of convention and convenience. The number of levels of a digital signal varying in a finite range of amplitudes is also known to be finite; hence, functions taking values in Galois fields (finite commutative algebraic bodies) can be used as a signal model.

Consequently, a function-taking value in Galois fields whose argument also takes values in the same field may be used for representation of any digital image. In particular, one can use the mapping

$$y=F\left(x\right); x,y\in GF\left(p^2\right)$$

(3)

With such mapping, the argument $x$ actually corresponds to a two-dimensional data array since the representation (1) has two components, i.e., one element of the Galois field uniquely characterizes, for example, a specific pixel.

Eligibility of such a mapping is determined by the fact that the set of elements of the field $GF\left(p\right)$ can be considered as a subset of the field elements $GF\left(p^2\right)$. This clarification is essential since the solutions of the algebraic problems discussed above assume that the values of the function have only one discrete dimension (the values of this function can be assigned a scale corresponding to a finite set of real numbers), while the coordinates of a pixel (if we consider a typical television screen) are two-dimensional. Accordingly, a separate element of the Galois field, represented in the form (1), defining, for example, the coordinates of a pixel, is also two-dimensional in the following sense. Each element of the Galois field $GF\left(p\right)$ can be assigned a natural number, but in order to uniquely characterize the element x, represented in the form (1), two such numbers are needed.

However, this does not cancel the possibility of using Formula (3), in which both the values of the function and its argument formally belong to the same set, specifically the field $GF\left(p^2\right)$. Indeed, all elements of the field $GF\left(p\right)$ are a subset of the field $GF\left(p^2\right)$, which justifies the use of Formula (3).

However, representation (1) can only be considered as a special case corresponding to the use of vectors with discrete coordinates. Along with Galois fields, algebraic rings can also be used for this purpose. As shown in this paper, this approach has quite definite advantages.

To demonstrate this, consider the matrix representation of the elements of Galois fields, taking into account that the most common way of representing coordinate transformations is based on the use of matrices. Let us consider the product of two elements, represented in the form (1), on each other.

$$C=AB=\left(A_1+\theta A_2\right)\left(B_1+\theta B_2\right)=A_1{B}_1-g{A}_2{B}_2+\theta \left(A_2{B}_1+A_1{B}_2\right)$$

(4)

where, for definiteness, it is assumed that $\theta$ is the root of the field irreducible $GF\left(p\right)$ equation

$$x^2+g=0$$

(5)

Finding elements $g$ that generate irreducible equations is, in general, an independent problem, the consideration of which is beyond the scope of this paper. From the point of view of practical applications, it is sufficient to point out that such elements can be found directly (by squaring the elements of the basic field).

We emphasize that there exists a number of Galois fields $GF\left(p\right)$ in which Equation (5) at $g=1$ has solutions. In particular, such a field is the field $GF\left(13\right)$ since there is

$$5^2\equiv -1\left(13\right)$$

(6)

Consequently, it is reasonable to consider the irreducible equation in a more general form (5). Formula (4) allows us to represent an element of the Galois field $A$ in the form of the following matrix:

$$A\leftrightarrow \left(\begin{array}{cc}{A}_1& -A_2\\ A_2& A_1\end{array}\right)$$

(7)

It is obvious, however, that the transition to the matrix representation of linear transformations (in particular, coordinate transformations) does not allow one to take full advantage of the features of the pixel coordinate representation in the form (1). Obviously, in Formula (4), the element $A$ can be considered as a representation of some linear operator acting on the element $B$. But directly from the form of Formula (7), it follows that multiplication by a certain element of the Galois field does not exhaust all possible linear operators. Indeed, the representation of an element of the field (1) contains two variables; a matrix composed of elements of the same Galois field contains four variables.

Consequently, the problem of representing any linear transformations (more precisely, corresponding linear operators) with the help of algebraic structures obtained by the method of algebraic extensions is of interest. In fact, the question arises whether it is possible to reduce linear operators to the operation of multiplying an element of some finite algebraic structure by an element belonging to the same structure. Formula (7) shows that Galois fields are not enough for this.

To solve this problem, this paper proposes a method of nonstandard algebraic extensions.

Namely, the traditional method of constructing algebraic extensions based on the use of irreducible equations, however, is not the only possible one. In particular, it was shown in [82] that it is possible to use another such method, starting from algebraic equations having solutions in the Galois field. However, in the cited work, a nonstandard method of forming algebraic extensions was considered in the particular example of a field $GF\left(3\right)$, which corresponds to ternary logic [83]. This logic is of interest for a number of applied problems [84,85,86], but the method proposed in [82] admits generalization.

In this paper, we consider its generalization and prove that this method allows us to reduce the representation of any linear operators, corresponding to matrices of dimension two, to the elements of the set obtained by algebraic extensions of the original field.

4. Results

4.1. Nonstandard Method of Algebraic Extensions

This method uses the following lemma, proved for the case when the field $GF\left(p\right)$ corresponds to an integer $p$, such that $p-1$ is divisible by 4. We emphasize that from the point of view of the practical applications indicated above, this limitation is not so significant since, when mapping the terrain, the discretization character can be chosen based on considerations of convenience.

Lemma 1. 

If the number$p-1$is divisible by 4, then the equation

$$x^2+1=0$$

(8)

has two solutions in the field$GF\left(p\right)$.

Proof. 

Let us represent the elements of the Galois field $GF\left(p\right)$ as the following set:

$$GF(p)\leftrightarrow \left\{-{\displaystyle \frac{p-1}{2}},-{\displaystyle \frac{p-1}{2}}+1,\dots ,-\mathrm{1,0},1,\dots ,{\displaystyle \frac{p-1}{2}}\right\}$$

(9)

This representation is adequate because, except for the case of $p=2$, which is not of interest, the number $p$ is odd. This representation, among other things, makes the use of the minus sign in all subsequent calculations completely correct.

In the most general case, all nonzero elements of the field $GF\left(p\right)$ satisfy the equation

$$x^{p-1}-1=0$$

(10)

By assumption, $p-1$ is divisible by 4. Consequently, Equation (10) can be rewritten in the form

$${\left(x^4\right)}^{{\displaystyle \frac{p-1}{4}}}-1=0$$

(11)

Or

$${\left(y\right)}^{{\displaystyle \frac{p-1}{4}}}-1=0$$

(12)

where $y=x^4$.

The number of roots of Equation (12) cannot exceed ${\displaystyle \frac{p-1}{4}}$, i.e., the elements satisfying Equation (12) cannot exhaust the considered Galois field. The remaining elements can therefore be considered as fourth roots of elements $y$ satisfying Equation (12). Element 1 is obviously a particular solution of Equation (12). The fourth roots of 1 must therefore satisfy Equation (10), i.e., correspond to some elements of the original Galois field. All fourth roots of 1 can be obtained by extracting the square root of 1 (this will give two elements of the field 1 and −1) and then extracting the square roots of both of these elements. Consequently, the field under consideration includes two square roots of −1. Consequently, Equation (8) has two solutions in the fields under consideration.

The lemma is proven. □

For example, for the field $GF\left(13\right)$, such solutions are $x=\pm 5$, Formula (7).

The basis for using the nonstandard method of algebraic extensions is the following theorem.

Theorem 1. 

A nonstandard algebraic extension of the field$GF\left(p\right)$that satisfies the conditions of the lemma proved above can be constructed by adjoining an algebraic element$i$, which is formally a solution of Equation (8) but does not coincide with the solutions$q_0$of this equation in the specified field.

Recall that the Galois fields $GF\left(p^n\right), n\ne 1$ are formed by the method of algebraic extensions. It consists in attaching one or more additional elements to the original field. An example of the use of this approach is the joining of an imaginary unit to a set of real numbers. Typically, in algebra, we use elements that are roots of equations that have no solutions in the original field. This theorem shows that this condition is not necessary.

Proof. 

The element $i$ under consideration satisfies the equation

$$i^2+1=0$$

(13)

but does not coincide with the solutions of Equation (8) in the main field, for which $q_0^2+1=0$ also takes place.

In the following, we will call the element $i$ a logical imaginary unit of the second kind.

Let us consider matrix representation for algebraic expansions of the proposed type. Using a formula similar to Formula (5), it can be seen that the components of the product of $C_1+i{C}_2$ of two elements $A_1+i{A}_2$ and $B_1+i{B}_2$ are represented in the form

$$C_1=A_1{B}_1-A_2{B}_2;C_2=A_1{B}_2+A_2{B}_1$$

(14)

or in matrix form

$$\left(\begin{array}{c}{C}_1\\ C_2\end{array}\right)=\left(\begin{array}{cc}{A}_1& -A_2\\ A_2& A_1\end{array}\right)\left(\begin{array}{c}{B}_1\\ B_2\end{array}\right)$$

(15)

In particular, for an element 1 and an element $i,$ the next matrix representations are valid.

$$1\leftrightarrow {\mathit{E}}_0=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right);i\leftrightarrow \mathit{I}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

(16)

The specificity of the Galois field used, in which Equation (8) has a solution $q_0$, consists of the following. Matrix representation of this solution has the form

$$q_0\leftrightarrow \left(\begin{array}{cc}{q}_0& 0\\ 0& q_0\end{array}\right)$$

(17)

By direct calculation, it is easy to show

$$\left(\begin{array}{cc}{q}_0& 0\\ 0& q_0\end{array}\right)\left(\begin{array}{cc}{q}_0& 0\\ 0& q_0\end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)\leftrightarrow -1$$

(18)

$$\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)\leftrightarrow -1$$

(19)

Comparison of expressions (18) and (19) shows that in fields of the type under consideration, there can indeed exist elements of different nature that satisfy the same Equation (8). This means the use of a logical imaginary unit of the second kind is legitimate.

The theorem is proven. □

Let us consider the nature of algebraic objects, generated by use of nonstandard algebraic extensions of the proposed type.

4.2. Construction of Algebraic Rings via Nonstandard Algebraic Extensions of Galois Fields

Let us make the following two linear combinations:

$$e_{10}=q_0+i,e_2=q_0-i$$

(20)

One can see that the elements $e_{\mathrm{1,2}}$ cancel each other out

$$e_{10}{e}_{20}=\left(q_0+i\right)\left(q_0-i\right)=-1+1=0,$$

(21)

since $q_0$ and $i$ are solutions of the same Equation (8).

Consequently, we can claim that the object obtained using algebraic extensions of the proposed type is not a field since it has divisors of zero. Let us define the elements

$$e_{\mathrm{1,2}}=\alpha \left(q_0\pm i\right)$$

(22)

where the multiplier $\alpha$ is determined on the basis of the condition

$$e_1+e_2=1$$

(23)

or

$$2\alpha q_0=1$$

(24)

With this choice of multiplier $\alpha ,$ elements $e_{\mathrm{1,2}}$ are idempotent. Indeed,

$$e_{\mathrm{1,2}}{e}_{\mathrm{1,2}}={\alpha }^2\left(q_0\pm i\right)\left(q_0\pm i\right)=2{\alpha }^2{q}_0\left(q_0\pm i\right)=\alpha \left(q_0\pm i\right)$$

(25)

This result corresponds to one of the general theorems of the theory of algebraic ideals [30]: there is a kind of ring $R$ that decomposes into a direct sum of ideals $r_k$.

$$R=r_1+r_2+\cdots +r_n$$

(26)

Each of these ideals is generated by idempotent elements $e_i$

$$r_k=R{e}_k$$

(27)

which cancel each other, and their sum is equal to the unit of the ring $R$.

$${\sum }_k{e}_k=1$$

(28)

For further calculations, it is convenient to represent idempotent elements (22) in the form

$$e_{\mathrm{1,2}}=\beta \pm \alpha i$$

(29)

where

$$\beta =\alpha q_0$$

(30)

$$\beta =2^{-1};\alpha =2^{-1}{q}_0^{-1}$$

(31)

The use of Formula (31) is legitimate since any nonzero element of a Galois field has an inverse. In particular, it can be shown by direct calculation that for Galois fields of the considered type, the element $\beta$ can be found in the general case as

$$\beta =2^{-1}=-{\displaystyle \frac{1}{2}}(p-1)$$

(32)

Starting from Formula (29), we can immediately specify the matrix representation for idempotent elements $e_{\mathrm{1,2}}$

$$e_1=\beta q_0+\alpha i \leftrightarrow {\mathit{E}}_1=\left(\begin{array}{cc}\beta & -\alpha \\ \alpha & \beta \end{array}\right)=\beta {\mathit{E}}_0+\alpha \mathit{I}$$

(33)

$$e_2=\beta q_0-\alpha i \leftrightarrow {\mathit{E}}_2=\left(\begin{array}{cc}\beta & \alpha \\ -\alpha & \beta \end{array}\right)=\beta {\mathit{E}}_0-\alpha \mathit{I}$$

(34)

where all matrix elements belong to $GF\left(p\right)$ and $p-1$ is divisible by 4.

As one would expect, the matrices (33) and (34) cancel each other out.

$$\left(\begin{array}{cc}\beta & -\alpha \\ \alpha & \beta \end{array}\right)\left(\begin{array}{cc}\beta & \alpha \\ -\alpha & \beta \end{array}\right)=\left(\begin{array}{cc}{\beta }^2+{\alpha }^2& 0\\ 0& {\beta }^2+{\alpha }^2\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$$

(35)

In deriving Formula (35), it is taken into account that $\beta =\alpha q_0$ and $q_0^2=-1$ and, hence, ${\alpha }^2=-{\beta }^2$. Similarly, it is proved that the matrices ${\mathit{E}}_i$ are idempotent since $2\beta =1$.

$${\mathit{E}}_{\mathit{i}}{\mathit{E}}_{\mathit{i}}=\left(\begin{array}{cc}{\beta }^2-{\alpha }^2& \mp 2\alpha \beta \\ \pm \alpha \beta & {\beta }^2-{\alpha }^2\end{array}\right)=\left(\begin{array}{cc}2{\beta }^2& \mp 2\alpha \beta \\ \pm 2\alpha \beta & 2{\beta }^2\end{array}\right)=\left(\begin{array}{cc}\beta & \mp \alpha \\ \pm \alpha & \beta \end{array}\right)$$

(36)

Thus, it can be seen that the classical method of algebraic expansions leads to the formation of fields, and the proposed method leads to the formation of algebraic rings.

Let us give for clarity concrete examples of the construction of idempotent elements in Galois fields $GF\left(17\right)$ and $GF\left(13\right),$ which satisfy the criterion formulated above (number $p-1$ is divisible by 4). In the field $GF\left(17\right)$, the element inverse to 2 is $\beta =-8$, and the root of Equation (8) is the element $q_0=4$. The element inverse to it is $q_0^{-1}=-4$.

Idempotent elements are represented in the form

$$e_{\mathrm{1,2}}=-8\pm 2i$$

(37)

The fact can be proven by direct calculation.

$$e_{\mathrm{1,2}}{e}_{\mathrm{1,2}}={\left(-8\pm 2i\right)}^2=64-4\mp 32i=-8\pm 2i\left(17\right)$$

(38)

These elements cancel each other out.

$$e_1{e}_2=64+4\equiv 0\left(17\right)$$

(39)

Similarly, in the field $GF\left(13\right)$, the element inverse to 2 is $\beta =-6$, and the root of Equation (8) is the element $q_0=5$. The element inverse to it is $q_0^{-1}=-5$, i.e., idempotent elements are

$$e_{\mathrm{1,2}}=-8\pm 4i$$

(40)

Direct calculation gives

$$e_1{e}_2=6^2+4^2=0\left(13\right)$$

(41)

$$e_{\mathrm{1,2}}{e}_{\mathrm{1,2}}={\left(-6\pm 4i\right)}^2=36-16\mp 48i\equiv -6\pm 4i\left(13\right)$$

(42)

The existence of idempotent elements, which can be constructed by means of the considered method of algebraic extensions, is proven.

Let us now show that the proposed method of algebraic extensions (in combination with the classical method based on solving the equations given) allows us to provide a representation of arbitrary linear operators (corresponding to 2 × 2 matrices) through multiplication by algebraic elements belonging to the resulting structures.

4.3. Representation of Operators Through Algebraic Extensions

In this paper, we consider Galois fields in which Equation (8) has a solution. This, however, does not exclude that the classical method of algebraic expansions cannot be applied to them. For this purpose, one can use, in particular, any equation of the same form but which has no solution.

$$x^2+g=0$$

(43)

The simultaneous use of the extensions generated by Equations (8) and (43) makes it possible to ensure that an arbitrary operator describing a coordinate transformation is represented in a form similar to (1) but containing two kinds of adjoint elements.

By attaching to the basic field the root $\theta$ of Equation (43), we can pass to the elements expressed by Formula (1). Passing again to the matrix representation in accordance with Formula (15), we can see that an arbitrary element $A$ obtained by using the element $\theta$ can be put in correspondence to the next matrix.

$$A\leftrightarrow \mathit{A}=\left(\begin{array}{cc}{A}_1& -g{A}_2\\ A_2& A_1\end{array}\right)=A_1\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+A_2\left(\begin{array}{cc}0& -g\\ 1& 0\end{array}\right)=A_1{\mathit{E}}_0+A_2\mathit{G}$$

(44)

Elements $\theta$ and $i$ are different, so it is acceptable to consider the combination

$$w=\left(A_1+A_2\theta \right)\left(\beta +\alpha i\right)+\left(B_1+B_2\theta \right)\left(\beta -\alpha i\right)$$

(45)

This combination satisfies Formulas (26) and (27) under the assumption that algebraic ideals $r_{\mathrm{1,2}}$ correspond to classical algebraic extensions of the basic Galois field.

Indeed, as follows from Formula (29), expression (45) can be rewritten in the form

$$w=\left(A_1+A_2\theta \right)e_1+\left(B_1+B_2\theta \right)e_2$$

(46)

The correctness of the representations (45) and (46) can be demonstrated by going to the matrix representation again. We have

$$\mathit{W}=\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right)=\left(A_1{\mathit{E}}_0+A_2\mathit{G}\right)\left(\beta {\mathit{E}}_0+\alpha I\right)+\left(B_1{\mathit{E}}_0+B_2\mathit{G}\right)\left(\beta {\mathit{E}}_0-\alpha \mathit{I}\right)$$

(47)

We will show that any matrix $\mathit{W}$ given over a Galois field of the considered type admits representation (47). This will prove that the ring of such matrices is isomorphic to an algebraic object, which is formed by simultaneous use of both classical and nonstandard methods of algebraic extensions.

Opening the brackets in Formula (47), we obtain

$$\mathit{W}=\beta \left(\left(A_1+B_1\right){\mathit{E}}_0+\left(A_2+B_2\right)\mathit{G}\right)+\alpha \left(\left(A_1-B_1\right)\mathit{I}+\left(A_2-B_2\right)\mathit{G}\mathit{I}\right)$$

(48)

We will use the following variables

$$x_1=\beta \left(A_1+B_1\right), x_2=\beta \left(A_2+B_2\right), x_3=\alpha \left(A_1-B_1\right), x_4=\alpha \left(A_2-B_2\right)$$

(49)

$$\mathrm{Then} \mathit{W}=\left(x_1{\mathit{E}}_0+x_2\mathit{G}\right)+\left(x_3\mathit{I}+x_4\mathit{G}\mathit{I}\right)$$

(50)

Formula (50) is equivalent to the following system of linear equations:

$$\left\{\begin{array}{c}{x}_1-g{x}_4=w_{11}\\ x_1-x_4=w_{22}\end{array}\right.$$

(51)

$$\left\{\begin{array}{c}-g{x}_2-x_3=w_{12}\\ x_2+x_3=w_{21}\end{array}\right.$$

(52)

It is not difficult to specify their explicit solutions.

$$x_4={\left(g-1\right)}^{-1}\left(w_{22}-w_{11}\right)$$

(53)

$$x_1={\left(g-1\right)}^{-1}\left({gw}_{22}-w_{11}\right)$$

(54)

$$x_2=-{\left(g-1\right)}^{-1}\left(w_{12}+w_{21}\right)$$

(55)

$$x_3={\left(g-1\right)}^{-1}\left(w_{12}+{gw}_{21}\right)$$

(56)

These solutions are elementary, but they clearly show that the simultaneous use of two different methods of algebraic extensions allows one to pass from the basic Galois field to an algebraic ring isomorphic to the set of linear operators acting on vectors, corresponding to discrete coordinates in the plane. In particular, instead of the matrix representation of the operator describing the coordinate transformation, we can use its representation in the form of

$$w=\left(x_1+x_2\theta \right)+\left(x_{3}i+x_4\theta i\right)$$

(57)

The product of two matrices corresponding to coordinate transformations can also be reduced to a representation of the form (57). For this purpose, it is enough to take into account the following commutation rule:

$$i\theta +\theta i=-\left(g+1\right)$$

(58)

which corresponds to the following equality:

$$\mathit{I}\mathit{G}+\mathit{G}\mathit{I}=\left(\begin{array}{cc}-g& 0\\ 0& -1\end{array}\right)+\left(\begin{array}{cc}-1& 0\\ 0& -g\end{array}\right)=-\left(\begin{array}{cc}g+1& 0\\ 0& g+1\end{array}\right)$$

(59)

It follows directly from (57) that an arbitrary operator describing the coordinate transformations can be represented also in the form containing idempotent elements.

$$w=\left(A_1+A_2\theta \right)e_1+\left(B_1+B_2\theta \right)e_2$$

(60)

Thus, the obtained results prove the constructiveness of using the method of nonstandard algebraic extensions, especially when it is used in parallel with the classical one. In particular, in this case, it becomes possible to represent an arbitrary operator describing coordinate transformations through an element of the ring corresponding to Formula (60).

4.4. Advantages of Representations of Linear Operators Using Idempotent Elements

Let us show that for each specific Galois field that meets the criterion formulated above, there exists a distinguished system of discrete coordinates on the plane in which the action of the operator on the coordinate vectors is factorized.

The element of the algebraic ring obtained by the combined method of algebraic expansions and represented in the form (60) corresponds to the representation of the matrix describing, in particular, the coordinate transformation in the following form:

$$\mathit{W}={\mathit{A}\mathit{E}}_1+{\mathit{B}\mathit{E}}_2$$

(61)

The detailed mapping between matrices and elements obtained by algebraic expansions is given by the formulas

$$\mathit{A}=A_1{\mathit{E}}_0+A_2\mathit{G}\leftrightarrow A_1+A_2\theta$$

(62)

$$\mathit{B}=B_1{\mathit{E}}_0+B_2\mathit{G}\leftrightarrow B_1+B_2\theta$$

(63)

We will show that such a representation does provide a simplification of the description of coordinate transformations for the case of the considered Galois fields. The use of Formula (1) based on the classical method of algebraic expansions is not the only possible one. Along with it is admissible to use the representation of two-dimensional coordinates using the element $i$, which corresponds to the nonstandard method of algebraic expansions.

In terms of ordinary vector expressions, this corresponds to the representation of a radius vector on a two-dimensional discrete plane as

$$\mathit{r}=r_1{\mathit{e}}_1+r_2{\mathit{e}}_2$$

(64)

where

$${\mathit{e}}_1=\left(\begin{array}{c}\beta \\ \alpha \end{array}\right); {\mathit{e}}_2=\left(\begin{array}{c}\beta \\ -\alpha \end{array}\right)$$

(65)

The following relations are valid:

$${\mathit{E}}_1\left(\begin{array}{c}\beta \\ \alpha \end{array}\right)=\left(\begin{array}{cc}\beta & -\alpha \\ \alpha & \beta \end{array}\right)\left(\begin{array}{c}\beta \\ \alpha \end{array}\right)=\left(\begin{array}{c}{\beta }^2-{\alpha }^2\\ 2\alpha \beta \end{array}\right)=\left(\begin{array}{c}\beta \\ \alpha \end{array}\right)$$

(66)

$${\mathit{E}}_1\left(\begin{array}{c}\beta \\ -\alpha \end{array}\right)=\left(\begin{array}{cc}\beta & -\alpha \\ \alpha & \beta \end{array}\right)\left(\begin{array}{c}\beta \\ -\alpha \end{array}\right)=\left(\begin{array}{c}{\beta }^2+{\alpha }^2\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$

(67)

$${\mathit{E}}_2\left(\begin{array}{c}\beta \\ \alpha \end{array}\right)=\left(\begin{array}{cc}\beta & \alpha \\ -\alpha & \beta \end{array}\right)\left(\begin{array}{c}\beta \\ \alpha \end{array}\right)=\left(\begin{array}{c}{\beta }^2+{\alpha }^2\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$

(68)

$${\mathit{E}}_2\left(\begin{array}{c}\beta \\ -\alpha \end{array}\right)=\left(\begin{array}{cc}\beta & \alpha \\ -\alpha & \beta \end{array}\right)\left(\begin{array}{c}\beta \\ -\alpha \end{array}\right)=\left(\begin{array}{c}{\beta }^2-{\alpha }^2\\ -2\alpha \beta \end{array}\right)=\left(\begin{array}{c}\beta \\ -\alpha \end{array}\right)$$

(69)

These relations mean that the action of an operator represented in the form (47) on a vector represented in the form (64) is factorized.

$$\mathit{W}\mathit{r}=\left({\mathit{A}\mathit{E}}_1+{\mathit{B}\mathit{E}}_2\right)\left(r_1{\mathit{e}}_1+r_2{\mathit{e}}_2\right)=r_1{\mathit{A}\mathit{e}}_1+r_2{\mathit{B}\mathit{e}}_2$$

(70)

Ratio (70), which can also be written in terms of elements obtained by the combined method of algebraic expansions, shows that each component of the vector describing the coordinates of a pixel in terms of the corresponding Galois field is subject to its «own» operator. Moreover, this operator is isomorphic to a separate element of the Galois field obtained by the method of classical algebraic expansions of the basic field.

The next task that arises when using the proposed approach is to bring the vectors ${\mathit{A}\mathit{e}}_1$ and ${\mathit{B}\mathit{e}}_2$ (or their corresponding algebraic elements) to the form of the form (64). This problem is obviously solvable since it reduces to the solution of linear equations. However, it is important to take into account that both the operator $\mathit{A}$ and the operator $\mathit{B}$ is also uniquely characterized by some vector on the considered plane with discrete coordinates, which corresponds to the representation (1).

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 $GF\left(p\right)$, 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 $GF\left(p\right)$. 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.

For this purpose, it is enough to use an irreducible equation of the third degree, which allows us to write down the analog of Formula (1) as

$$x=x_1+\theta x_2+{\theta }^2{x}_3$$

(71)

In this connection, it is appropriate to emphasize that complex numbers, corresponding to vectors in three-dimensional space, if we remain in terms of continuously varying values of current variables, cannot be constructed. Only Hamilton quaternions, which can be assigned to vectors in four-dimensional space, are known [95].

During processing of three-dimensional digital images, it is possible to use elements of discrete mathematical structures to display coordinate values. In particular, Formula (71) is applicable to display coordinates of elements of a digitized three-dimensional image.

It is essential that the functions that define, for example, a digital image in terms of functions whose arguments are elements of the Galois field (algebraic rings, etc.) can also be considered in terms of operations of multivalued logic [90], i.e., in the long run, this approach creates additional tools for solving the problems that are currently solved with the help of convolutional neural networks, the establishment of relationships between the elements of the image. Bringing transformations of discrete coordinates in a form using algebraic extensions is an element of such a theory.

6. Conclusions

Thus, for mapping digital image elements (pixels), as well as for mapping discrete Cartesian coordinates in all tasks where they are known to vary in a finite range, it is acceptable to use algebraic structures obtained by algebraic expansion of Galois fields $GF\left(p\right)$. Algebraic expansions, in turn, can be constructed in various ways. The classical method involves the use of irreducible equations. For this purpose, however, it is also possible to use a formal additional solution of the reduced equation. In the latter case, the algebraic expansion gives not a field but a ring containing divisors of zero. There is also a possibility of parallel use of extensions of both types mentioned above. The elements obtained by the simultaneous use of such extensions can be used to represent linear coordinate transformations described through the extension of the basic Galois field.

In the future, this approach may find application in digital image processing, but the simplest and most important area of its application is information protection systems intended for use in the direct radio visibility zone and based on determining the operator’s coordinates (signal source). In this case, the onboard computers with which the UAV group is equipped solve a system of equations that allow finding the coordinates of the signal source and comparing it with the specified ones. In case of a match, the incoming commands are accepted for execution. In this case, the considered system of equations can be solved in terms of the proposed algebraic structures since there is no need to calculate the operator’s coordinates with high accuracy. This approach, among other things, allows to significantly simplify the onboard electronic circuits that perform the calculations.