Algorithm for Obtaining Complete Irreducible Polynomials over Given Galois Field for New Method of Digital Monitoring of Information Space

Abstract

Irreducible polynomials are widely used in modern cryptography; however, algorithms for finding such polynomials remain quite complex and require significant computational resources. In this study, a new approach to finding irreducible equations over Galois fields $GF\left(p\right)$ is proposed. It is shown that such irreducible equations can be obtained by solving a system of linear equations over the base Galois field, generated by any element of the field $GF\left(p^K\right)$ that is distinct from the elements of the base field and from elements corresponding to lower-degree extensions. The connection of the proposed approach with algorithms based on the Frobenius automorphism is established. The case corresponding to the field $GF\left(3\right)$ and matrices over this field is examined in detail. It has been shown that the proposed method makes it possible to obtain complete sets of irreducible polynomials over a given Galois field. It has also been demonstrated that generating such sets is of particular interest for the development of new methods of digital monitoring of the information space, which are based on analogies with error-correcting coding techniques.

1. Introduction

The problem of constructing and classifying irreducible polynomials over Galois fields has consistently attracted the attention of researchers [1,2,3]. Such polynomials play a crucial role, in particular, in cryptography [4,5,6] and coding theory [7,8]. Specifically, this issue is of key importance in the development of algorithms for the advanced encryption standard (AES) [9,10] and elliptic curve cryptography (ECC) [11,12,13], as well as in the design of error-correcting codes such as Reed–Solomon codes [14,15]. Galois fields of comparatively small dimension are of interest for the development of lightweight cryptographic protocols used, for example, in devices that conform to the Internet of Things (IoT) concept [16,17]. Lightweight cryptographic protocols, such as SPONGENT and PRESENT, applied in such devices, indeed employ small-dimension finite fields in order to minimize computational costs on resource-constrained devices [18,19].

Classical methods for finding irreducible polynomials, such as the Berlekamp algorithm [20] (including its modifications [21,22]) or the Rabin algorithm [23,24], have proven highly effective in practice, particularly due to their reliability. However, their implementation requires substantial computational resources. This limits the applicability of such algorithms in systems such as RFID tags [25,26], SCADA system controllers [27,28], and other microsystems where stringent requirements are imposed on both processing speed and energy efficiency. To overcome these limitations, algebraic methods for generating irreducible polynomials are of interest, particularly those employing primitive elements of field extensions and Frobenius automorphisms [29,30]. It should also be noted that the above-mentioned algorithms do not guarantee obtaining a complete set of irreducible equations of a given degree over a specified Galois field.

Methods analogous to those used in the theory of error-correcting coding are also of considerable interest in the context of the current problem of improving psychological testing and the methodology of conducting sociological surveys [31,32]. Specifically, the task of psychological testing generally entails establishing a certain set of classification features that characterize a particular respondent [33,34]. For this purpose, tests are used that represent a set of questions. The answers to these questions can be mapped to a certain code sequence. In the simplest case, when the respondent is asked to choose between two options (“yes” or “no”), the set of answers corresponds to a sequence of binary symbols that can be interpreted as a code sequence.

Moreover, such a sequence can be regarded as a sequence containing errors. The “correction” of such errors in this case solves a classification problem, since the result of this procedure yields a sequence containing a smaller number of symbols. This approach was initially implemented in [35] using analogs of code sequences formed by permutations.

Such modernization of psychological testing gives possibility to overcome many difficulties connected to verification of psychological tests [36,37]. Classification features can be derived directly from large data arrays. A similar approach can be applied to sociological surveys. Moreover, in these cases, well-developed methods that ensure error correction can be used (Reed–Solomon codes [14,15], BCH codes [38,39], etc.).

However, this approach leads to the following problem. A set of responses to the test questions may be considered as a code containing errors, but the irreducible polynomial that provides the encoding generally remains unknown. Therefore, solving this problem requires, at a minimum, an algorithm for constructing a complete set of irreducible polynomials corresponding to a specific algebraic extension of a given Galois field. On this base the polynomial that allows solving the classification problem of the above type may be selected (including those based on experimental data).

In the long term, this approach also makes it possible to address problems related to the monitoring of the information space. It is unnecessary to emphasize that the modern information space contains enormous volumes of data accumulated, in particular, in social online networks, various Internet forums, and so forth. Adequate analysis of such information makes it possible, among other things, to analyze the processes taking place in society. Various methods have been proposed to solve this problem [40,41,42]. In Ref. [40], the information space was proposed to be viewed as a complex network in which messages, users, and thematic nodes are interconnected through graph-based and semantic relationships. The authors employ methods of graph mining, diffusion modeling, and content-based feature extraction to identify and classify false information, detect primary sources, etc. Such an approach makes it possible not only to record the facts of information propagation but also to reveal underlying patterns, predict topic evolution, and suppress anomalous flows.

The advantages of direct monitoring of the information space over traditional methods used in sociology are evident. First and foremost, sociological surveys require significant financial expenditures [43,44]. The main difficulty associated with the development of methods for direct monitoring of the information space lies in the fact that the nature of the information it contains is highly heterogeneous. For this reason, the use of methods analogous to those applied in the theory of error-correcting coding is of considerable interest from this perspective.

Specifically, in this case, it becomes possible to form code sequences (which are analogs of responses to test questions) of considerable length and then reduce them to a relatively small number of classification features. As analogs of responses to questions, one may consider, for example, users’ reactions to various resonant events (not necessarily of a political nature). In the simplest case, the reactions detected through monitoring—for instance, of social networks—correspond to ternary logic: positive reaction, negative reaction, or absence of reaction. Depending on the nature of the task being solved, it is also permissible to use five- and seven-valued logics.

The first step toward implementing such a methodology is the development of a simple and reliable algorithm for finding irreducible polynomials over various Galois fields.

The development of such an algorithm constitutes the aim of the present work.

The solution to this problem is based on a theorem proving that finding the complete set of irreducible polynomials of a given degree over a specified Galois field can be reduced to solving a system of linear equations over the base field.

Section 2 of this paper describes the main method used in this study—the method of representing algebraic complements in matrix form. Section 3.1 proves the main theorem for the case of fields $GF\left(p^3\right)$, which correspond to ternary logic. Section 3.2 presents a specific example of constructing the complete set of irreducible equations for the case $GF\left(3^3\right)$. Section 3.3 provides a generalization of this approach to the construction of irreducible cubic equations over an arbitrary Galois field. Section 3.4 analyzes the relationship between the developed approach and the Frobenius automorphism. Section 3.5 proves the main theorem for the general case.

Section 4 demonstrates that the developed method for obtaining irreducible polynomials can also be applied to the creation of new methods for monitoring the information space.

2. Methods

This study employs the well-known method [45] of matrix representation of algebraic extensions $GF\left(p^K\right)$ of Galois fields $GF\left(p\right)$. The approach is based on the fact that any element $x$ of the field $GF\left(p^K\right)$ can be regarded analogously to a vector of a linear space, which, in the case of the field $GF\left(p^3\right)$ is expressed as

$$x=a_2{\theta }^2+a_1\theta +a_0$$

(1)

where $\theta$ is the root of some irreducible cubic equation defining an algebraic extension, and $c_i$ are coefficients from the base field $GF\left(p\right)$. As pointed out in [46], for illustrative purposes the primitive element $\theta$ may be interpreted as a logical imaginary unit.

Expression (1), therefore, corresponds to a one-to-one correspondence of the form

$$x\leftrightarrow \left(\begin{array}{c}{a}_2\\ a_1\\ a_0\end{array}\right)$$

(2)

An irreducible cubic equation can, in the general case, be written as

$$x^3+A_2{x}^2+A_{1}x+A_0=0$$

(3)

where $A_0\ne 0$.

Relation (2) also makes it possible to express the third power of the element $\theta$ as

$${\theta }^3=-A_2{\theta }^2-A_1\theta -A_0$$

(4)

Consequently, the product of an arbitrary field element $x$ by the element $\theta$ is given by

$$\theta x=\left(a_1-a_2{A}_2\right){\theta }^2+\left(a_0-a_2{A}_1\right)\theta -a_2{A}_0$$

(5)

By considering ${\theta }^2,\theta ,1$ as analogs of basis vectors according to Expression (1) and the element $\theta$ as the analog of an operator acting on the element $x$, Expression (5) leads to the matrix representation of the element $\theta$ in the form

$$\theta \leftrightarrow \left(\begin{array}{ccc}-A_2& 1& 0\\ -A_1& 0& 1\\ -A_0& 0& 0\end{array}\right),$$

(6)

A similar $3\times 3$ matrix representation is valid for any other element of the field $GF\left(p^3\right)$. This work demonstrates that, using representations of the form (6), it is possible to construct the complete set of irreducible equations over the field of the considered type.

3. Results

3.1. The Basis of the Proposed Algorithm

In this section, the theorem forming the basis of the proposed algorithm is proved for the particular case of cubic equations over an arbitrary Galois field. The method of proof is chosen to ensure clarity while minimizing the use of abstract algebraic concepts. Its generalization is discussed in Section 3.5.

Theorem 1.

The coefficients $c_i$ of an irreducible equation over the field  $GF\left(p^3\right)$

$$x^3-c_2{x}^2-c_{1}x-c_0=0$$

(7)

can be obtained as the solution of a system of linear equations

$$\left(\begin{array}{ccc}{w}_{22}& w_{12}& 0\\ w_{21}& w_{11}& 0\\ w_{20}& w_{10}& 1\end{array}\right)\left(\begin{array}{c}{c}_2\\ c_1\\ c_0\end{array}\right)=\left(\begin{array}{c}{w}_{32}\\ w_{31}\\ w_{30}\end{array}\right),$$

(8)

where

$$\left(\begin{array}{c}{w}_{12}\\ w_{11}\\ w_{10}\end{array}\right)\leftrightarrow g;\left(\begin{array}{c}{w}_{22}\\ w_{21}\\ w_{20}\end{array}\right)\leftrightarrow g^2;\left(\begin{array}{c}{w}_{32}\\ w_{31}\\ w_{30}\end{array}\right)\leftrightarrow g^3,$$

(9)

$g$ is the predetermined element of the algebraic extension $GF\left(p^3\right)$ of the base field $GF\left(p\right)$, which does not coincide with any element of the base field.

Proof. 

Equation (8) can be written in the following two equivalent forms.

$$c_2\left(\begin{array}{c}{w}_{22}\\ w_{21}\\ w_{20}\end{array}\right)+c_1\left(\begin{array}{c}{w}_{12}\\ w_{11}\\ w_{10}\end{array}\right)+c_0\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}{w}_{32}\\ w_{31}\\ w_{30}\end{array}\right)$$

(10)

$$g^3-c_2{g}^2-c_{1}g-c_0=0$$

(11)

Relation (8) implies that the search for the coefficients of the irreducible equation can be carried out via a known solution $g$ that belongs to $GF\left(p^3\right)$ but does not coincide with any element of $GF\left(p\right)$. We will show that, under the above condition, the solutions of Equation (8) indeed yield the coefficients of the irreducible equation.

A cubic equation over an arbitrary Galois field $GF\left(p\right)$ is reducible only in two cases: either the corresponding polynomial factors into a product of a linear term and a quadratic polynomial, or into a product of three linear terms. In both cases, at least one root of the equation must belong to the base field.

Therefore, the question of whether Equation (11) constructed from a certain element $g$, which satisfies the conditions formulated above, is irreducible or not reduces to determining whether an element $g$ of this type can be a root of a quadratic equation.

This question can be answered in the general case. Consider the quadratic equation

$$g^2+ag+b=0,$$

(12)

For $p\ne 2$ it can be transformed into a perfect square in the usual manner:

$${\left(g+2^{-1}a\right)}^2=2^{-2}{a}^2-b$$

(13)

The element on the right-hand side of Equation (13) belongs to the base field. However, it can also be regarded as an element of its algebraic extension $GF\left(p^3\right)$. More precisely, elements of the base field $x$ can be expressed via the primitive element $\theta$ of the field $GF\left(p^3\right)$ as

$$x={\theta }^{m\frac{p^3-1}{p-1}}$$

(14)

where $m=\mathrm{0,1},2,\dots ,p-2$.

Expression (14) follows from the fact that all elements of the base field in the considered algebraic extension must be $p-1$ roots of unity, since they satisfy the equation

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

(15)

The following identity holds:

$$p^3-1=(p-1)(p^2+p+1)$$

(16)

The number $p^2$ is odd, since $p$ is odd (the case $p\ne 2$ is not considered here). Consequently, the sum $p^2+p+1$, being the sum of three odd numbers, is also odd. Therefore, the expression $\frac{p^3-1}{p-1}$ is odd. It follows that the possibility of extracting a square root from the right-hand side of Equation (14) is determined by the parity of the integer $m$. If $m$ is even, there exists a power of the element $\theta$ that enables the extraction of the square root; if $m$ is odd, such a power does not exist. Specifically, for even $m$, the following relation holds:

$$\sqrt{x}={\theta }^{\frac{m}{2} \frac{p^3-1}{p-1}}$$

(17)

Expression (17) shows that if the square root from the right-hand side of Equation (14) can be extracted, it can be extracted already within the base field. Indeed, for even $m$ Expression (17) has the same form as (14).

Therefore, if Equation (13) is reducible in the algebraic extension $GF\left(p^3\right)$, then the element $g$ must belong to the base field. Conversely, if $g$ does not belong to the base field, then the equation of the form (3) generated by $g$, is irreducible.

Consequently, an element $g$ that belongs to the algebraic extension $GF\left(p^3\right)$ but does not coincide with any element of $GF\left(p\right)$ indeed generates an irreducible equation. The theorem is proved.

On this basis, we can propose the following algorithm for generating irreducible equations in $GF\left(p^3\right)$ fields. One begins with any irreducible equation that defines an algebraic extension of $GF\left(p\right)$ to $GF\left(p^3\right)$. Then, to generate an irreducible equation, one can use any element of $GF\left(p^3\right)$ that does not coincide with any element of the base field. The coefficients of this equation are determined by solving the system of linear Equation (8) over the base field. □

3.2. Example of Constructing the Complete Set of Irreducible Equations

Let us consider a concrete example of applying the proposed method. As the base field, we take $GF\left(3\right)$. Its algebraic extension to $GF\left(3^3\right)$ can be constructed using the irreducible equation

$$x^3-x+1=0$$

(18)

This, in particular, means that the cube of a solution to (18) is expressed in terms of lower powers as

$${\theta }^3=\theta -1$$

(19)

This relation can be used to find the matrix representation of the element $\theta$. Specifically, according to the method described in Section 2, this element is represented as

$$\theta \leftrightarrow \left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ -1& 0& 0\end{array}\right),$$

(20)

Expression (20) follows from the relation, which is a special case of (5):

$$\theta \left(a_2{\theta }^2+a_1\theta +a_3\right)=a_1{\theta }^2+\left(a_3+a_2\right)\theta -a_2,$$

(21)

which, as noted above, can also be represented in matrix form:

$$\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ -1& 0& 0\end{array}\right)\left(\begin{array}{c}{a}_2\\ a_1\\ a_0\end{array}\right)=\left(\begin{array}{c}{a}_1\\ a_2+a_0\\ -a_2\end{array}\right)$$

(22)

Using Expression (22), one can obtain the representations for the powers of the element $\theta$. It is sufficient to start from the representation for $\theta$, given by the tuple $\left(\mathrm{0,1},0\right)$.

Table 1. Coefficients representing the powers of the element $\theta$.

$\mathit{n}$	${\mathit{a}}_2$	${\mathit{a}}_1$	${\mathit{a}}_0$
1	0	1	0
2	1	0	0
3	0	1	−1
4	1	−1	0
5	−1	1	−1
6	1	1	1
7	1	−1	−1
8	−1	0	−1
9	0	1	1
10	1	1	0
11	1	1	−1
12	1	0	−1
13	0	0	−1
14	0	−1	0
15	−1	0	0
16	0	−1	1
17	−1	1	0
18	1	−1	1
19	−1	−1	−1
20	−1	1	1
21	1	0	1
22	0	−1	−1
23	−1	−1	0
24	−1	−1	1
25	−1	0	1
26	0	0	1

presents the results of $n$-fold application of this matrix (values of $n$ are listed in the first column). The second, third, and fourth columns show the coefficients representing the powers of $\theta$ in the form (1). There are exactly three elements in the base field $GF\left(3\right)$ which can be listed as $-\mathrm{1,0},1$, as clearly demonstrated in [47]. It is evident that all 26 elements presented in Table 1 are distinct, confirming that $\theta$ is indeed a primitive element. It is also observed that the 13th power of this element equals −1, which is consistent with Expression (14).

Consequently, there exist 24 nonzero elements of the algebraic extension of the base field that can be used to construct irreducible equations in accordance with the proposed method.

The coefficients of these equations are straightforward to determine, since Equation (8) allows for a direct analytical solution, which is conveniently written in the form

$$c_2=D^{-1}\left({w_{11}w}_{32}-w_{12}{w}_{31}\right),$$

(23)

$$c_1=D^{-1}\left(w_{22}{w}_{31}-w_{21}{w}_{32}\right),$$

(24)

$$c_0=w_{32}-w_{20}{c}_2-w_{10}{c}_1,$$

(25)

where

$$D=w_{11}{w}_{22}-w_{12}{w}_{21},$$

(26)

The results of solving system (8) for each of the elements of the field $GF\left(3^3\right)$, which are not elements of the base field $GF\left(3\right)$, are presented in

Table 2. Coefficients of irreducible equations obtained by solving system (8).

n	c2	c1	c0	N
1	0	1	−1	1
2	−1	−1	1	2
3	0	1	−1	1
4	−1	0	1	3
5	1	−1	−1	4
6	−1	−1	1	2
7	−1	1	−1	5
8	1	1	1	6
9	0	1	−1	1
10	−1	0	1	3
11	−1	1	−1	5
12	−1	0	1	3
13	-	-	-	-
14	0	1	1	7
15	1	−1	−1	4
16	0	1	1	7
17	1	0	−1	8
18	−1	−1	1	2
19	1	−1	−1	4
20	1	1	1	6
21	−1	1	−1	5
22	0	1	1	7
23	1	0	−1	8
24	1	1	1	6
25	1	0	−1	8

.

The first column of this table contains the degree $n$ of the primitive element $\theta$ corresponding to each element of $GF\left(3^3\right)$, i.e., such that $g={\theta }^n$.

The second, third, and fourth columns contain the coefficients $c_i$ obtained using Formulas (23)–(25), i.e., the coefficients of an equation of the form (3) generated by a particular element $g$.

The fifth column lists the group numbers $N$ corresponding to the same equation.

These numbers have the following meaning: there exists a theorem [45] stating that all roots of an irreducible equation are distinct unless the corresponding polynomial can be reduced to a polynomial in $x^{p^k}$, where $k$ is an integer. For a cubic equation, such a situation can occur only in the degenerate case where the polynomial depends solely on $x^3$. Therefore, all three roots of any of the considered irreducible equations must be distinct.

Table 2 (column 5) shows that there are, in fact, not 24 different equations, but 24/3 = 8. Three of the obtained equations coincide—more precisely, the elements with indices $n$ split into eight groups of three such elements each. All elements in the same group generate the same irreducible equation. These three elements are the three distinct roots of that equation. This result, as further demonstrated in the following Section 3.4, also corresponds to the fact that the different roots of equations of the considered type are related to each other through a transformation corresponding to the Frobenius isomorphism [29,30] (the roots correspond to a cyclotomic class [48,49]), which leads to the aforementioned partition into subsets.

The validity of the results presented in Table 2 can be verified as follows.

The number of distinct cubic equations over the field $GF\left(3\right)$ whose root is not $x=0$ is relatively small. Specifically, there are exactly 18 such equations, since in the representation of the form (11), the coefficient $c_0$ can take only two values, whereas the coefficients $c_{\mathrm{2,1}}$ can each take three values.

Therefore, it is possible to enumerate all reducible equations and compare their coefficients with those given in Table 2.

Reducible cubic equations over the field under consideration have the form

$$(x\pm 1)(x^2+b_{1}x+b_2)=0$$

(27)

In this field, there exist only six equations of the form

$$x^2+b_{1}x+b_2=0,$$

(28)

whose root is not $x=0$. This follows from the fact that $b_2$ can take two possible values, while $b_1$ can take three. Among these equations, three are reducible:

$$\left(x+1\right)\left(x-1\right)=x^2-1=0$$

(29)

$${\left(x+1\right)}^2=x^2-x+1=0$$

(30)

$${\left(x-1\right)}^2=x^2+x+1=0$$

(31)

Therefore, the number of equations of the form (28) whose left-hand side is not factorable is three.

This corresponds to the existence of six equations of the form (27) in which the quadratic polynomial is irreducible over the base field.

Another four reducible equations correspond to the case where the polynomial is completely factorable into linear terms. These equations can be readily obtained and are provided explicitly in the Supplementary Materials.

Supplementary Materials also contains a classification of all cubic equations with coefficients in $GF\left(3\right)$. This classification shows that the entire set of equations of the type under consideration can be partitioned into the following three subsets:

−

Eight irreducible equations corresponding to Table 2;

−

Six reducible equations corresponding to Expression (27), in which the quadratic polynomial is irreducible over the base field;

−

Four reducible equations with nonzero roots belonging to $GF\left(3\right)$

This classification, in particular, confirms that the result presented in Table 2 is indeed valid.

3.3. Counting the Set of Irreducible Cubic Equations over an Arbitrary Galois Field

The obtained result can be generalized to arbitrary fields of the form $GF\left(p^3\right)$. To demonstrate this, we calculate the total number of irreducible equations that have no solution $x=0$ and whose coefficients belong to the base field $GF\left(p\right)$. As in the preceding case, this can be achieved by first determining the number of reducible equations of this type.

Both reducible and irreducible equations under consideration can be written in the form

$$x^3+A_2{x}^2+A_{1}x+A_0=0$$

(32)

where $A_0\ne 0$.

The number of cubic equations whose left-hand side in (32) factors completely into distinct linear terms is equal to the binomial coefficient $C_{p-1}^3$. The number of cubic equations whose left-hand side in (32) factors completely into terms, at least two of which coincide, is equal to ${\left(p-1\right)}^2$.

For convenience, we will hereafter refer to equations whose left-hand side in (32) factors completely into distinct terms as reducible equations of the first kind. Their number, $R_{13}\left(p\right)$, is given by

$$R_{13}\left(p\right)={\displaystyle \frac{1}{6}}\left(p-1\right)\left(p-2\right)\left(p-3\right)+{\left(p-1\right)}^2$$

(33)

There also exist reducible equations which, for convenience, we will refer to as reducible equations of the second kind, in which the left-hand side in (32) factors into two terms, one linear and the other quadratic. To determine the number of such equations, we use the same approach, namely, by first counting the number of equations of the form

$$x^2+B_{1}x+B_0=0$$

(34)

where $B_0\ne 0$ and whose left-hand side factors into linear terms.

The number of equations of the form (34) whose left-hand side factors into distinct linear terms is equal to the binomial coefficient $C_{p-1}^2$. The number of equations whose left-hand side factors into identical linear terms is $p-1$. The total number of equations of the form (34) is therefore $p\left(p-1\right)$. Consequently, the number of irreducible equations $Q_2\left(p\right)$ of the form (34) is

$$Q_2(p)=p\left(p-1\right)-{\displaystyle \frac{1}{2}}\left(p-1\right)\left(p-2\right)-\left(p-1\right)={\displaystyle \frac{1}{2}}p\left(p-1\right)$$

(35)

To obtain the number $R_{32}\left(p\right)$ of reducible cubic equations of the first kind from the number $Q_2$, this quantity must be multiplied by the number of possible linear factors, i.e., by $\left(p-1\right)$. We thus have

$$R_{23}\left(p\right)={\displaystyle \frac{1}{2}}p{\left(p-1\right)}^2$$

(36)

The total number of equations of the form (32) with $A_0\ne 0$ is

$$T_3\left(p\right)=\left(p-1\right)p^2$$

(37)

Therefore, the desired number of irreducible equations is

$$Q_3\left(p\right)=T_3\left(p\right)-R_{13}\left(p\right)-R_{23}\left(p\right)$$

(38)

The difference $T_3\left(p\right)-R_{23}\left(p\right)$ can be transformed into the form

$$T_3\left(p\right)-R_{23}\left(p\right)=\left(p-1\right)p\left[p-{\displaystyle \frac{1}{2}}\left(p-1\right)\right]={\displaystyle \frac{1}{2}}p(p^2-1)$$

(39)

The expression $R_{13}\left(p\right)$ can be rewritten as

$$R_{13}\left(p\right)=\left(p-1\right)\left[{\displaystyle \frac{1}{6}}\left(p-2\right)\left(p-3\right)+p-1\right]={\displaystyle \frac{1}{6}}p\left(p^2-1\right)$$

(40)

Consequently, the total number of irreducible equations is

$$Q_3\left(p\right)={\displaystyle \frac{1}{3}}p(p^2-1)$$

(41)

The number of elements of the field $GF\left(p^3\right)$ that are distinct from the elements of the base field is $p^3-p$. One third of this quantity coincides exactly with $Q_3\left(p\right)$. Thus, it has been proven that for arbitrary fields $GF\left(p^3\right)$, the same situation holds as was established above for the field $GF\left(p^3\right)$. Specifically, the number of irreducible equations that can be generated by solving equations of the form (8) equals $p^3-p$, whereas the total number of such equations is $Q\left(p\right)$, that is, three times smaller. These conditions can be satisfied simultaneously only in one case: when the elements of $GF\left(p^3\right)$ that are not contained in the base field are partitioned into subsets of three elements each, and the elements of each subset are the roots of the same equation. This conclusion also follows from the existence of cyclotomic classes [48,49], which is discussed in the following section.

It should be emphasized that $Q_3\left(p\right)$ is an integer for any $p$, since the numerator in fraction (40) is the product of three integers, $\left(p+1\right)p\left(p-1\right)$, one of which is necessarily divisible by three.

3.4. The Relationship Between the Set of Irreducible Equations over a Given Galois Field and the Frobenius Automorphism

The result shown in Table 2 can be expressed in terms of the Frobenius automorphism too. Recall that a Galois field automorphism is a mapping $\phi$ of the field onto itself such that

$$\phi \left(x+y\right)=\phi \left(x\right)+\phi \left(y\right); \phi \left(xy\right)=\phi \left(x\right)\phi \left(y\right)$$

(42)

The Frobenius automorphism is defined by the mapping

$$x\to x^p$$

(43)

This mapping is indeed an automorphism, since for fields of characteristic $p$ the following expression holds:

$${\left(x+y\right)}^p=x^p+y^p$$

(44)

Table 2 in fact shows that the set of roots of each irreducible equation over the field $GF\left(3^3\right)$ is connected precisely by such an automorphism, because if $q_1$ is an arbitrary root of an irreducible equation, then the other roots are given by $q_2=q_1^3$, $q_3=q_2^3$.

This conclusion is illustrated in more detail in

Table 3. Systematization of Galois field elements generating irreducible equations, from the perspective of the Frobenius automorphism.

n/N	1	2	3	4	5	6	7	8
1	1	2	4	5	7	8	14	17
3	3	6	12	15	21	24	16 (42)	25 (51)
9	9	18	10 (36)	19 (45)	11 (63)	20 (72)	22 (126)	23 (153)

. The first row of the table shows the numbers $N$, which appear in the last column of Table 2. The third and fourth rows show the third and ninth powers of the elements in the first row. The exponents are computed modulo 26, since all nonzero elements of $GF\left(3^3\right)$ satisfy

$$x^{26}=1$$

(45)

The “formal” exponents, i.e., the exponents before reduction modulo 26, are given in parentheses in Table 3.

Let us construct the following polynomial:

$$P\left(x\right)=(x-{\theta }^m)(x-{\theta }^{pm})(x-{\theta }^{p^{2}m})$$

(46)

It can be written in the form of (7):

$$P\left(x\right)=x^3-a_2{x}^2-a_{1}x-a_0=0$$

(47)

where

$$a_2={\theta }^m+{\theta }^{pm}+{\theta }^{p^{2}m}$$

(48)

$$a_1=-{\theta }^m{\theta }^{p^{2}m}-{\theta }^{pm}{\theta }^{p^{2}m}-{\theta }^m{\theta }^{pm}$$

(49)

$$a_0={\theta }^m{\theta }^{pm}{\theta }^{p^{2}m}$$

(50)

Using the Frobenius automorphism, we then have

$$P^p\left(x\right)=x^{3p}-a_2^p{x}^{2p}-a_1^{p}x-a_0^p=0$$

(51)

Under this automorphism, all coefficients of polynomial (47) remain unchanged; in particular,

$$a_2^p={\theta }^{mp}+{\theta }^{p^{2}m}+{\theta }^{p^{3}m}=a_2$$

(52)

Since ${\theta }^{p^3}=\theta$. Similar relations hold for the coefficients $a_0$ and $a_1$.

$$a_1^p=-{\theta }^{pm}{\theta }^{p^{3}m}-{\theta }^{p^{2}m}{\theta }^{p^{3}m}-{\theta }^{pm}{\theta }^{p^{2}m}=a_1$$

(53)

$$a_0^p={\theta }^{pm}{\theta }^{p^{2}m}{\theta }^{p^{3}m}=a_0$$

(54)

Therefore, the irreducible equation obtained starting from the element ${\theta }^m$ coincides with the equations obtained using the elements ${\theta }^{pm}$ and ${\theta }^{p^{2}m}$. This result is fully consistent with well-known theorems of abstract algebra [45]. Hence, Theorem 1 can also be proved by considering polynomials of the form (46), although the proof presented above is, in our view, more transparent.

3.5. Obtaining Complete Sets of Irreducible Equations of Arbitrary Degree over a Given Galois Field

The generalization of the proposed approach is given by a refined formulation of Theorem 1.

Theorem 2.

The coefficients of an irreducible equation of degree $K+1$ over a given Galois field are determined by solving the following system of linear equations. 

$$\left(\begin{array}{cc}\begin{array}{cc}{w}_{KK}& \dots \\ ⋮& \ddots \end{array}& \begin{array}{cc}{w}_{1K}& 0\\ \dots & ⋮\end{array}\\ \begin{array}{cc}{w}_{K1}& \dots \\ w_{K0}& \dots \end{array}& \begin{array}{cc}{w}_{11}& 0\\ w_{10}& 1\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{c}_K\\ ⋮\end{array}\\ \begin{array}{c}{c}_1\\ c_0\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{w}_{\left(K+1\right)K}\\ ⋮\end{array}\\ \begin{array}{c}{w}_{\left(K+1\right)1}\\ w_{\left(K+1\right)0}\end{array}\end{array}\right),$$

(55)

where the columns of the matrix correspond to the representations of the powers of the primitive element $g$ of the field $GF\left(p^{K+1}\right)$ in the form

$$x={\sum }_{n=0}^{n=K}{a}_n{\theta }^n$$

(56)

That is,

$$\left(\begin{array}{c}\begin{array}{c}{w}_{1K}\\ ⋮\end{array}\\ \begin{array}{c}{w}_{11}\\ w_{10}\end{array}\end{array}\right)\leftrightarrow g;\dots ;\left(\begin{array}{c}\begin{array}{c}{w}_{\left(K+1\right)K}\\ ⋮\end{array}\\ \begin{array}{c}{w}_{\left(K+1\right)1}\\ w_{\left(K+1\right)0}\end{array}\end{array}\right)\leftrightarrow g^{K+1},$$

(57)

This theorem makes it possible to obtain a complete set of irreducible equations of an arbitrary degree over a given Galois field, starting from any irreducible equation of the same degree over that field. If at least one such equation is known, it is possible to find the primitive element $\theta$ of the field $GF\left(p^{K+1}\right)$, whose powers allow determining all other primitive elements $g$. Recall that an element $g$ is primitive if there does not exist an integer $n_0<p^{K+1}-1$ such that

$$g^{n_0}=1$$

(58)

Proof. 

If the element $g$ is primitive, then the columns of matrix (55) are linearly independent, and therefore the system of Equation (55) has a solution. Expression (55) is equivalent to the equation

$$P\left(x\right)=x^{K+1}-a_K{x}^K-\cdots -a_{1}x-a_0=0$$

(59)

Therefore, it remains to be proven that if the element $g$ is primitive, then the corresponding equation is irreducible. It can be reducible only if the element $g$ is a root of an equation of degree $M<K+1$. For equations of degree $M$, a system of equations similar to (55) can be written. In this case, the vectors of the form (57) will naturally contain $M+1$ elements. Consequently, if a solution exists for such a case, the element $g$ must belong not only to the field $GF\left(p^{K+1}\right)$ but also to the field $GF\left(p^{M+1}\right)$. This is possible only if a condition of the form (58) is satisfied, which is excluded by the assumptions of the theorem.

The theorem is proved. □

As an example, consider the case of the field $GF\left(3^4\right)$. This field contains 80 nonzero elements, meaning their number is divisible by 8. Consequently, it contains 8 nonzero elements that can be represented in the form

$$x={\theta }^{10m}, m=\mathrm{0,1},\dots ,7$$

(60)

Among the four elements ${\theta }^{10m};{\theta }^{30m};{\theta }^{90m};{\theta }^{270m}$, only two are distinct, since $10\equiv 90\left(80\right)$ and $30\equiv 270\left(80\right)$. Therefore, such elements must be excluded from the set of nonzero elements of $GF\left(3^4\right)$ suitable for finding irreducible equations.

The elements (60) are also roots of the equation

$$x^8=1$$

(61)

That is, they correspond to an algebraic extension of lower degree, specifically to the field $GF\left(3^2\right)$, and therefore they allow forming irreducible equations of degree 2. An illustration of this conclusion, related to the algebraic extension $GF\left(3^4\right)$, is provided in Supplementary Materials.

It can also be shown that the irreducible polynomial (59) can be written in the form

$$P\left(x\right)={\prod }_{k=0}^{k=K}(x-{\theta }^{p^{k}m})$$

(62)

where ${\theta }^m$ is the primitive element of the algebraic extension $GF\left(p^{K+1}\right)$ of the base field $GF\left(p\right)$, and $p$ is the characteristic of the field.

Using (62), one can show that the coefficients of the irreducible equation satisfy an analog of Formulas (52)–(54) by virtue of the isomorphism (44)

$$a_i^p=a_i$$

(63)

Relation (63), in particular, indicates that the elements $a_i$ remain unchanged under all Frobenius isomorphisms. As follows from Galois theory, this means that they belong to the base field. Consequently, the proof of Theorem 2 can also be given directly on the basis of polynomial (62).

This also implies that, as in the specific case considered above, the solutions of a given irreducible equation, as expected, form a cyclotomic class. Therefore, if an irreducible equation is constructed on the basis of an element $g$, then the same equation will result from solving system (55) when the initial elements $g^{p^r}$ with $r=\mathrm{1,2},\dots ,K$ are used. This, in turn, means that the total number of irreducible equations of degree $K$ over the field $GF\left(p\right)$ is equal to the ratio of the number of primitive elements of the extension $GF\left(p^{K+1}\right)$ to $K$.

For the sake of reproducibility, the proposed method can be summarized by the following algorithmic outline (Algorithm 1). It shows the step-by-step procedure for generating all irreducible equations of degree n over GF(p) from a single known irreducible polynomial, highlighting the role of the matrix formulation (55) and the exclusion of Frobenius-related duplicates.

Algorithm 1. Construction of the complete set of irreducible equations over a given Galois field $GF\left(p\right)$

Input: p—Prime defining the base field $GF\left(p\right)$; n—Degree of the desired algebraic extension; f(x)—One known irreducible polynomial of degree n over $GF\left(p\right)$.    Output: Complete set of irreducible polynomials of degree n over $GF\left(p\right)$. Initialization. Construct the extension field GF(pn) using f(x) and determine its primitive element α. Element generation. Generate all nonzero elements of GF(pn) as powers αk, 1 ≤ k ≤ pn − 1. Equation formation. For each element β = αk; that does not belong to the base field GF(p): Represent the powers {β0, β1, …, βn−1} as column vectors according to (57). Form the matrix M whose columns correspond to these vectors. Solve the linear system M · c = −βn (over $GF\left(p\right)$) as given by (55) to obtain the coefficient vector c = [c0, c1, …, cn−1]. Construct the polynomial P(x) = xn + cn−1xn−1 + … + c0. Equivalence reduction. Exclude all polynomials related by the Frobenius automorphism β → βp, βp2, … since they yield identical irreducible equations. Output. Return the remaining unique polynomials as the complete set of irreducible equations of degree n over $GF\left(p\right)$.

This outline provides an explicit algorithmic description of the computational procedure, complementing the theoretical formulation of Equations (55) and (57).

4. Discussion

Thus, the algorithm for finding the complete set of irreducible equations of a given degree $K$ over a specified Galois field $GF\left(p\right)$ consists of the following steps. The starting point is any irreducible equation of degree $K$ over the given Galois field $GF\left(p\right)$. Using this equation, a primitive element of the extension $GF\left(p^{K+1}\right)$ is determined. Based on this element, all other primitive elements—that is, those for which condition (58) is not satisfied—are then identified. The obtained set of primitive elements is used to generate a set of irreducible equations as follows. The elements from this set and their powers are expressed in the form (57). On this basis, for each primitive element used, a system of equations of type (55) is constructed. This is a system of linear equations that can be solved using standard methods. Its solution yields the set of coefficients of the irreducible equation. Elements related to each other by the Frobenius automorphism are excluded from consideration, since they produce identical equations.

There is no doubt that further research in this direction is of significant interest for cryptography, as follows from the works cited in the Introduction. However, it should be emphasized once again that the development of an algorithm for obtaining complete sets of irreducible polynomials over a given Galois field is also of interest in terms of developing methods for digital monitoring of the information space.

The main idea of this approach is illustrated in Figure 1 of [50], which demonstrates the existence of analogies between classification problems currently solved using neural networks [51,52] and the error-correction methods employed in error-resistant coding.

This figure emphasizes that, in both of the aforementioned cases, there exists a surjection from set A of code sequences of a certain length (or images represented in digital form) onto set B, which corresponds to code sequences containing a smaller number of symbols. Figure 1 also highlights that any classification problem, whose initial data can be represented as sequences of symbols in a multivalued logic form, can be interpreted from the same perspective.

Consequently, the tools already developed in the theory of error-correcting coding (such as BCH codes and similar ones) can indeed be applied to solving classification problems. The solution of such problems is also of interest for improving methods of psychological testing.

Let us consider the following illustrative example using one of the simplest codes—the Hamming code [53]. When applying the Hamming (7,4) code, which allows the correction of a single error, the initial sequence of binary symbols $\left(a_1,a_2,a_3,a_4\right)$ is supplemented with three additional symbols according to a well-known rule.

$$\left(a_1,a_2,a_3,a_4\right) \stackrel{W}{\Rightarrow }\left(a_1,a_2,a_3,a_4,b_2,b_3,b_4\right)$$

(64)

Error correction in this case implies that the code sequence containing seven symbols is mapped onto a sequence containing four symbols. Such a mapping can be considered a particular case of the correspondence shown in Figure 1.

A sequence of seven binary symbols can, among other things, be formed on the basis of respondents’ answers to the questions of a particular psychological test. In this context, the operation that is treated as error correction in coding theory can be interpreted as the correlation between a set of answers to test questions and a set of classification features. There are solid grounds to assume that this approach may, in the future, eliminate many issues arising during the verification of new psychological tests. Traditionally, this procedure is built upon comparing new tests with the results of already established ones and/or relies on the expert assessment method, which does not guarantee the exclusion of subjective factors.

The operation inverse to operation (64) allows, at the very least, for the selection of the most relevant classification features based on the available body of experimental data (test results). Indeed, the classical approach used in error-correcting coding theory assumes that the division of symbols into source and check (redundant) symbols is known. In the case of psychological testing data, however, there is no such distinction between original and check symbols. Consequently, there arises a choice: which questions’ answers should be assigned to set $a_i$ and which to set $b_i$. By iterating through possible variants, one can determine the optimal classification features according to the following criterion: each of the above variants is characterized by a certain number of errors determined on the basis of experimental data. The optimal (or near-optimal) variant is the one corresponding to the minimal number of errors.

A similar approach can be applied to various psychological tests associated with different error-correction algorithms. The essence of the matter illustrated by the above example, however, remains unchanged. Specifically, it shows that we are essentially dealing with a problem that is inverse to that addressed by classical error-correcting coding theory.

This fact highlights the relevance of finding complete sets of irreducible equations of a given degree over specified Galois fields. Indeed, the foundation of many cryptographic methods lies in irreducible polynomials over particular Galois fields. These polynomials serve as the basis for algorithms ensuring error correction. However, traditional approaches developed in this field generally assume that a cryptographic key or its analog is either known or explicitly constructed.

When the aforementioned methods are applied to the processing of psychological test data, it is assumed that the “cryptographic key” (or, more precisely, its analog) must be derived from a large array of experimental data. Therefore, the first—and in many respects the principal—step in implementing the proposed approach is precisely the algorithm for finding the complete set of irreducible equations over a given Galois field. This is the exact problem solved in the present study.

A similar approach is also applicable to information-space monitoring—and in this case, it is even more suitable. Indeed, while in the verification of new psychological tests it is possible to test respondents using previously known methodologies, in the monitoring of social networks such a possibility is fundamentally absent. The classification features must be established directly from the available experimental data, since obtaining any additional information is difficult.

The simplest implementation of the proposed approach proceeds as follows. A dataset is formed reflecting the reactions of Internet users to various resonant events. As mentioned in the Introduction, there are reasons to assume that ternary logic is the most appropriate for tracking Internet users’ reactions to such events (positive reaction, no reaction, negative reaction). Sequences of this type of data can be processed using the methodology discussed above. In particular, in the simplest case, when ternary logic is employed, it is acceptable to use the Golay code [54].

In general, it can be stated that the use of Galois fields of the form $GF\left(3p_0\right)$ corresponds to a “batch” interpretation of the results of information-space monitoring. In this case, we are not dealing with a $3p_0$ valued logic but rather with segments of code sequences containing p₀ symbols, each corresponding to ternary logic. However, this does not exclude the use of other codes, including those conforming to existing standards based on binary encoding [55].

Moreover, this does not preclude the possibility of employing a more detailed classification of responses corresponding to five- or seven-valued logics, whose algebraic operations can be represented according to the methodology described in [56]. In the practical implementation of a more complex approach, tools of fuzzy logic may also be used; however, a detailed discussion of such aspects lies beyond the scope of the present study. Furthermore, it cannot be ruled out that the multivalued logics used for the algebraic representation of variable values may prove most promising for integrating the proposed information-space monitoring methodologies with artificial intelligence systems, since, as noted in [57], the scenarios of AI’s further development remain highly variable.

5. Conclusions

The improvement of algorithms for constructing irreducible polynomials over Galois fields remains a relevant task for cryptography, error-correcting code design, and other applications. This study demonstrates that if at least one irreducible equation of degree $K$ over a given Galois field $GF\left(p\right)$ is known, it becomes possible to obtain the complete set of irreducible equations of the same degree over that field. Moreover, the problem of determining the coefficients of such equations reduces to a linear one. This makes it possible to implement a simple algorithm for finding all equations from the given set by using matrices formed by the powers of primitive elements represented as column vectors.

The proposed approach is of interest not only for cryptography but also for the development of new methods for monitoring the information space. In particular, there is a prospect of developing methodologies for indirect testing of users of social online networks by tracking their reactions to various resonant events and representing these reactions as sequences of symbols in multivalued logic form. The systematization of the obtained results in this case can be performed using methods analogous to those developed in the theory of error-correcting coding (Reed–Solomon codes, etc.), which ensure the correction of errors in such sequences. The main analytical tool in this context is the complete set of irreducible equations over a particular Galois field. The next step in this direction involves the practical implementation of information-space monitoring methods based on analogies with error-correcting coding techniques, as well as the integration of the proposed methods with the capabilities provided by artificial intelligence.