Formation of Periodic Mosaic Structures Using Operations in Galois Fields

Abstract

Mosaic ornaments and periodic geometric patterns are deeply rooted in cultural heritage and contemporary design, where symmetry plays a fundamental role in both aesthetic and cognitive perception. This study develops an algebraic method for generating symmetrical and periodic mosaic structures using operations in Galois fields. The approach demonstrates that the intrinsic properties of finite fields naturally give rise to symmetry and periodicity, eliminating the need for specific initial patterns, even when applied to relatively simple algebraic expressions such as the Bernoulli lemniscate and the cissoid of Diocles. The proposed algorithm offers the advantages of simplicity and the ability to provide gradual transitions from one mosaic structure to another. Furthermore, it is demonstrated that standardization of algebraic expressions used for mosaic generation can be efficiently achieved through discrete logarithm operations. A novel method for computing discrete logarithms is introduced. The results confirm that symmetrical structures of high complexity can be obtained through simple expressions, and their periodicity becomes more pronounced with increasing field characteristics. This approach offers practical applications in textile and wallpaper design, smart materials, and psychological testing, while also suggesting new perspectives for the analysis of mosaic-like natural systems where symmetry is a defining property.

1. Introduction

Mosaic ornaments and geometric patterns, characterized by a high degree of symmetry, constitute an integral part of the cultural heritage of various nations across the world [1,2,3].

Such ornaments have been employed for centuries, including as elements of frescoes and in the architectural decoration of buildings, such as churches and cathedrals [4], mosques and mausoleums [5], as well as Hindu and Buddhist temples [6]. Geometric mosaic patterns remain widely used today, for instance in urban architectural design, as well as in the production of textiles, carpets, and related artifacts.

The advancement of technologies for the production of textiles, carpets, and other household items (including the increasing adoption of 3D printing [7,8,9]) requires an expansion of the spectrum of design methods available for the generation of patterns. An illustrative example is the development of smart textiles [10,11], which opens new opportunities for the creation of nontrivial and innovative designs.

These possibilities can be employed to enhance psychological comfort for consumers [11,12], including for the correction of psychological states [12,13]. Moreover, several hypotheses have been proposed suggesting that mosaics may be considered as a paradigm of visual construction that extends beyond the scope of decorative art [14]. This perspective is consistent with conclusions drawn in earlier studies [15,16].

At present, the costs of producing textile, wallpaper, and related patterns remain relatively high, as they require the involvement of highly paid professional designers [17]. In the literature, attempts have been reported to improve [18,19] and to algorithmize [20,21,22] the process of pattern generation. However, these approaches do not enable the efficient production of a wide variety of designs at high speed, nor do they facilitate the creation of sequences of patterns that gradually transform from one into another. These aspects are of particular importance for allowing consumers to select the most suitable designs, especially when patterns are employed for psychological correction.

Mosaic structures generated automatically (including in high-frame-rate modes) are of interest for the advancement of psychological testing methods, particularly in projective techniques [23,24]. One of the most widely used among these is the psychogeometric test developed by Susan Dellinger [25]. The application of such techniques is often met with criticism [26], which can be addressed by selecting appropriate geometric structures capable of evoking associations, including at the archetypal level [27]. The variability of patterns used in psychological testing also facilitates the refinement of association-based methodologies, the most renowned of which is the Rorschach test [28,29]. This is particularly relevant in the context of testing large population groups.

The objective of this study is to develop a relatively simple and efficient method for generating periodic mosaic structures, including those characterized by a high degree of symmetry, as well as the possibility of ensuring smooth transitions from one mosaic to another.

The algorithm employs Galois fields, which are increasingly applied in digital image processing [30,31] and in other digital technologies [32,33], including elliptic curve cryptography (ECC) [34,35]. For the purpose of mosaic generation, particularly in applications related to psychological testing, however, such algebraic structures have not previously been utilized. This problem is addressed for the first time in the present work. It should be emphasized that studies on mosaic structures, including two-dimensional ones, are well known (in addition to the works mentioned above, see also [36,37]). Nevertheless, these studies have not addressed the problem of high-speed generation of such structures, and especially not the creation of mosaics capable of gradually transforming from one into another.

The principal tool enabling sequential transitions between mosaic structures is the novel method for computing discrete logarithms in Galois fields, introduced here for the first time. This method is based on the application of an algebraic delta function [38]. In the cryptographic literature, it is widely acknowledged that until recently, the problem of computing discrete logarithms remained intractable [39]. In particular, it has been noted that the first practical public-key cryptosystem, the Diffie–Hellman key exchange algorithm, relies on the assumption that discrete logarithms are computationally hard. This assumption, characterized as a hypothesis, underpins the presumed security of various other public-key schemes and, as emphasized in [39], remains a formidable challenge. At the time of publication [39], this hypothesis was the subject of extensive debate, and the discussion has continued in subsequent research [40].

In the present study, a novel approach to discrete logarithm computation is proposed, complementing the algorithm for mosaic generation. This algorithm is applicable, at least, to relatively small Galois fields GF(p) (since the use of fields with very large values of p is not meaningful for mosaic construction) and is introduced here for the first time.

The study also discusses potential directions for further application of the proposed framework. One promising avenue arises from the observation that many natural objects, including those of critical importance for agricultural land monitoring, exhibit structures closely resembling mosaics [41,42,43].

2. Methods

Various functions $F\left(x,y\right)$ that take values in the field $GF\left(p\right)$ are employed. The arguments of these functions are also elements of the same Galois field. The elements of this field are represented in the following form (the validity of this representation is most clearly demonstrated in [44]):

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

(1)

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 [44].

A discrete two-dimensional Cartesian coordinate system was employed, in which each integer value of both coordinates is associated with an element of the field $GF\left(p\right)$. This approach corresponds to the partitioning of the image plane into individual pixels, each of which is assigned discrete values $n_x, n_y$, with 0 $\le n_{x,y}\le p-1$. It should be emphasized that pixel-based image partitioning is also employed in other studies on mosaic generation, in particular in [20]. For each pixel, a specific function $F\left(x,y\right)$, was computed, also taking values in this field. This operation is essentially equivalent to calculating the remainder of the division of $F\left(x,y\right)$ by the prime number $p$.

Subsequently, the values of the function were mapped onto the set $\left\{-1, 1\right\}$:

$$Q\left[F(x,y)\right]=\left\{\begin{array}{c}-1,F\left(x,y\right)<0\\ 1,F\left(x,y\right)\ge 0\end{array}\right.$$

(2)

It should be emphasized that Formula (2) provides a natural way to assign a specific discrete indicator to each pixel. In the case of two-color mosaics, this indicator assumes two values determined by the sign of the function defined over the Galois field. Furthermore, as will become evident in the subsequent discussion, it is precisely the properties of the Galois field that ensure the periodic nature of the generated mosaics, despite the fact that the employed functions, when considered over continuous variables, do not exhibit such periodicity.

The results of this mapping constitute mosaics (where the minus sign corresponds to white fields and the plus sign to colored ones), which are analyzed below. The principal distinction of the present method from approaches such as those described in [20,21,22] lies in the fact that, in this case, a single function $F\left(x,y\right)$ can be employed to describe the mosaic, and the computations are carried out using the simple expression (2), which significantly simplifies the algorithmic implementation of mosaic generation. The programming code is particularly straightforward when the function $F\left(x,y\right)$ is applied without additional correction. This code is provided in Supporting Information S1.

3. Results

Let us consider specific examples of mosaics that can be generated using mapping (2). For this purpose, the programming code provided in Supporting Information S1 was used. Figure 1, Figure 2, Figure 3 and Figure 4 correspond to the case where the field $GF\left(61\right)$ is used. In these figures, as well as in all other figures presented in this study, the size of the field along both the horizontal and vertical axes is equal to twice the value of $p$, corresponding to the utilized field $GF\left(p\right)$. Specifically, Figure 1 shows a mosaic formed by means of the following function:

$$f\left(x,y\right)=x+ay$$

(3)

for parameter values $a=5$ and $a=45$, respectively.

It can be observed that the simplest linear equation gives periodic structures that, up to discretization accuracy, resemble a system of equidistant parallel stripes. It should be noted that even this simplest example demonstrates the advantages of using Galois fields for constructing periodic mosaics. The original function, when considered as a function over the set of real numbers, does not inherently produce periodicity. However, when functions taking values in Galois fields are employed, such periodicity becomes apparent, even in the most elementary case.

Figure 2 presents a mosaic, characterized by a high degree of symmetry, generated using the following function:

$$f\left(x,y\right)={\left(x+ay\right)}^2$$

(4)

for $a=5$ and $a=45$, respectively.

It can be observed that in this case the primary structure corresponding to a system of equidistant stripes is preserved, although the internal structure of the stripes undergoes a transformation. It is worth emphasizing once again that the structure shown in Figure 2 reflects the distinctive properties of Galois fields. If the original function, including (4), is considered over the set of real variables, it is not periodic. Periodicity emerges when the function takes values in a Galois field, and the resulting mosaic structure becomes highly nontrivial, despite the fact that the underlying equation remains relatively simple.

Figure 3 presents a mosaic generated using the following function:

$$f\left(x,y\right)=x^2+ay$$

(5)

(the parameter values are indicated in the figure caption).

It can be observed that periodicity is again evident in this case, driven by the operation of taking the remainder modulo the prime number $p$, that defines the field $GF\left(p\right)$. The underlying structure of the mosaics shown in Figure 3 corresponds to a system of parabolas, shifted relative to one another along the vertical axis.

Figure 4 presents a mosaic generated using the Bernoulli lemniscate for parameter values $a=3$ $a=15$ and $a=30$ respectively.

$$f\left(x,y\right)=\left(x^2+y^2\right)-a^2(x^2-y^2)$$

(6)

It can be observed that a relatively minor complication of the employed function results in the emergence of a highly nontrivial mosaic structure, the characteristics of which strongly depend on the value of the control parameter $a$, while still retaining a high degree of symmetry.

This provides evidence that the proposed approach enables the generation of a wide variety of mosaic structures using comparatively simple means. Moreover, it also allows for a sequential transition from one mosaic to another by gradually varying the control parameter in small increments.

Figure 5 shows mosaics also generated using function (6), but for the field $GF\left(127\right)$. As in previous cases, the field size along each axis equals twice the value of $p$, corresponding to the employed Galois field.

It can be observed that when transitioning from the field $GF\left(61\right)$ to $GF\left(127\right)$, the characteristics of the mosaics generated using Equation (6) remain similar (for the same value of the single control parameter). The difference lies in the fact that, as the level of mosaic detail increases with the field characteristic, the periodic nature of the resulting structures becomes more pronounced. In particular, the structure shown in Figure 5c can be interpreted as the result of an “interference” between periodically repeating systems of concentric circles, whereas this feature is far less evident in Figure 4c. It can also be noted that for practical purposes such as generating patterns for textiles, floor tiles, wallpapers, and similar applications, it is sufficient to use Galois fields with relatively small characteristics, i.e., without excessively increasing the level of detail in the generated structures. The same applies to mosaics intended for psychological testing.

Thus, even the relatively simple algebraic expression, the Bernoulli lemniscate (6), allows for the generation of a wide variety of mosaics. Evidently, a considerable number of different algebraic expressions can be proposed, including those based on heuristic considerations. Therefore, it is reasonable to consider the possibility of unifying the generation of such mosaics. This task is feasible precisely due to the specific properties of Galois fields (finite commutative fields) and their algebraic extensions.

Indeed, if expression (6) is regarded as a polynomial dependent on $y$, then its factorization into linear factors can be considered by using Equation (6) in its transformed form:

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

(7)

It can be seen that this equation is a particular case of an equation of the following form:

$$y^2-f\left(x\right)=0$$

(8)

where the function $f\left(x\right)$, in general, depend on one or several additional parameters.

Consequently, the question of factorization of a polynomial of the form (8) reduces to the question of factorization of the following equation:

$$y^2-b=0, \forall b\in GF\left(p\right)$$

(9)

for arbitrary values of $b$.

Let us clarify why factorization is of interest in the context of mosaic generation using the proposed algorithm. When Equation (8) is expressed in terms of functions over real variables and $f\left(x\right)>0$, Equation (8) can be decomposed into two equations that may be considered independently. Specifically, in this case, Equation (8) can be written as

$$\left(y-\sqrt{f\left(x\right)}\right)\left(y+\sqrt{f\left(x\right)}\right)=0$$

(10)

Equation (10) is evidently satisfied if $y-\sqrt{f\left(x\right)}=0$ or $y+\sqrt{f\left(x\right)}=0$, i.e., one can reduce the problem to the consideration of equations linear in $y$. However, when the function $f\left(x\right)$ takes values in a Galois field, the analog of the square root operation exhibits well-defined specific characteristics. These characteristics are as follows.

All nonzero elements $b$ of the field $GF\left(p\right)$ satisfy the equation [45]

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

(11)

Each such element can be expressed as a power of a primitive element ${\theta }_0$, the number of which depends on the prime $p$.

$$b_n={\theta }_0^n;n=\mathrm{0,2},\dots ,p-2$$

(12)

Consequently, it is permissible to distinguish between elements that are even and odd powers of ${\theta }_0$

$$b_n={\theta }_0^k{\theta }_0^{2m}; k=\mathrm{0,1};m=\mathrm{0,1},\dots ,{\displaystyle \frac{p-1}{2}}$$

(13)

As a result, the set of equations of the form (9) splits into two subsets corresponding to the following formulas:

$$y^2-{\theta }_0^{2m}=0$$

(14)

$$y^2-{\theta }_0{\theta }_0^{2m}=0$$

(15)

Equation (14) is solved straightforwardly. Since division by nonzero elements is allowed in Galois fields, we achieve the following formula:

$${\left({\displaystyle \frac{y}{{\theta }_0^m}}\right)}^2=1; y=\pm {\theta }_0^m$$

(16)

Therefore, in this case, the following is true:

$$y^2-b=\left(y+{\theta }_0^m\right)\left(y-{\theta }_0^m\right)$$

(17)

Equation (15) has no solution within the base Galois field. Otherwise, the square root can be extracted only for half of the elements of the field $GF\left(p\right)$. However, the specific properties of Galois fields allow this problem to be resolved. Specifically, solutions of Equation (15) arise when passing to algebraic extensions, which are constructed precisely through irreducible algebraic equations [46,47]. This method is analogous to the construction of complex numbers, which are defined by the following equation:

$$y^2+1=0$$

(18)

whose solution is the imaginary unit $i$, such that $i^2=-1$.

An analog of the imaginary unit for Galois fields $GF\left(p\right)$ can be constructed using the following equation, which, as noted above, has no solution in the base field $GF\left(p\right).$

$$y^2-{\theta }_0=0$$

(19)

This analog, as discussed in [48], provides the possibility to interpret the solution of Equation (19) as a logical imaginary unit, since the set of elements of the Galois field $GF\left(p\right)$ can be put into correspondence with the set of values of a variable in $p$-valued logic. The same notation will be used for this analog, i.e.,

$$i^2={\theta }_0$$

(20)

It can be easily shown that any equation of the form (15) can be solved using this logical imaginary unit. Indeed, rewriting it as follows:

$${\left({\displaystyle \frac{y}{{\theta }_0^m}}\right)}^2={\theta }_0,$$

(21)

we obtain the following:

$$y=\pm i{\theta }_0^m$$

(22)

Thus, for the considered polynomial, factorization into linear factors can be indicated in this case as well:

$$y^2-b=\left(y+i{\theta }_0^m\right)\left(y-i{\theta }_0^m\right)$$

(23)

Combining expressions (17) and (23), it follows that any polynomial of the form (9) can be factorized as follows:

$$y^2-b\left(x\right)=\left(y+w\left(x\right)\right)\left(y-w\left(x\right)\right)$$

(24)

where the function $w\left(x\right)$ takes values in an algebraic extension of the base Galois field.

Although the reasoning underlying this result is elementary from the perspective of abstract algebra, it is significant for constructing mosaics of the considered type, since any algebraic expression involving $y^2$, can be brought to the form (24), which is an analog of the expression corresponding to two intersecting lines (see Figure 6):

$$f\left(x,y\right)=y^2-a^2{x}^2=\left(y+ax\right)\left(y-ax\right)$$

(25)

Furthermore, from Formulas (14) and (15), it follows that the key operation for finding the function $w\left(x\right)$ given a known function $b\left(x\right)$ is the operation of discrete logarithm calculation, which allows determining the exponent of the primitive element for any nonzero element of the field $GF\left(p\right)$. This operation, in particular, enables the extraction of the square root of any element of $GF\left(p\right)$ when passing to its algebraic extension corresponding to the introduction of a logical imaginary unit. This operation can be expressed by the following formula:

$$x={\theta }^n\to n$$

(26)

The discrete logarithm operation has been considered in numerous studies. Notably, Joux’s algorithm [49], the baby-step giant-step algorithm [50,51], and the Pohlig–Hellman algorithm [52,53], among others, are well known. In the realm of quantum computing, Shor’s algorithm is of particular interest [54,55]. As noted in the aforementioned studies [39,40], a significant limitation of existing methods for computing discrete logarithms is the substantial computational resources required. In the present work, the method previously proposed in [56] is employed, which is based on representing a primitive element as a product of factors, thereby substantially simplifying the computations. The method can be described as follows.

Let the number $p-1$ be factorized into the following product of powers of prime numbers $p_i$:

$$p-1=p_1^{q_1}{p}_2^{q_2}\dots p_k^{q_k}$$

(27)

Then, an arbitrary nonzero element of the field $GF\left(p\right)$ can be represented as the following product:

$$z=g_1^{s_1}{g}_2^{s_2}\dots g_k^{s_k}$$

(28)

where the exponents $s_i$ vary within the range from 0 to $s_{im}=p_1^{q_1}$, and $g_i$ are elements of the considered field satisfying the following condition:

$${g_i}^{p_1^{q_1}}=1$$

(29)

For illustration, let us consider the specific example of the field $GF\left(61\right)$. In this case, the following:

$$60=2^2·3·5$$

(30)

Based on Formula (30), it is straightforward to observe that any element of this field can be expressed in the form (28), specifically as follows:

$$z=g_1^{s_1}{g}_2^{s_2}{g}_3^{s_3}$$

(31)

where the elements $g_i$ can be chosen as follows $g_1=-11$; $g_2=13$; $g_3=-3$. Some arbitrariness in the choice of these elements exists because Equation (29) may be satisfied by various $g_i$. Furthermore, the following relations hold:

$$g_1^4=1; g_2^3=1; g_3^5=1$$

(32)

Consequently, the exponents $s_i$ vary within the ranges $s_1=0, 1, 2, 3$; $s_2=0, 1, 2$; $s_3=0, 1, 2, 3, 4$, since due to (32) the exponents in expression (31) are effectively computed modulo the powers of prime numbers appearing in the factorization (30), i.e., modulo 4, 3, and 5, respectively.

Let us consider the important particular case $s_{\mathrm{1,2},3}=1$. Then the product is computed modulo 61, as follows:

$$z_0=g_1{g}_2{g}_3=2$$

(33)

Let us consider the next expression in which the product is also computed modulo 61:

$${\left(z_0\right)}^r=g_1^r{g}_2^r{g}_3^r$$

(34)

By virtue of property (32), this expression can be rewritten as follows:

$${\left(z_0\right)}^r=g_1^{r\left(mod4\right)}{g}_2^{r\left(mod3\right)}{g}_3^{r\left(mod5\right)}$$

(35)

It can be observed that the exponents of the elements $g_i$ n the given expression vary within the same ranges as the exponents $s_i$ in Formula (31). Moreover, the set of powers of the element $z_0$ exhausts all possible combinations of the exponents $s_i$. Therefore, the element $z_0$, appearing in Formula (33) is primitive, i.e.,

$$z_0={\theta }_0$$

(36)

The obtained result is illustrated in

Table 1. The nature of the relationship between a field element and its discrete logarithm.

${\mathit{g}}_1^{\mathit{r}\left(\mathit{m}\mathit{o}\mathit{d}4\right)}$	${\mathit{g}}_2^{\mathit{r}\left(\mathit{m}\mathit{o}\mathit{d}3\right)}$	${\mathit{g}}_3^{\mathit{r}\left(\mathit{m}\mathit{o}\mathit{d}5\right)}$	${\left({\mathit{z}}_0\right)}^{\mathit{r}}$	$\mathit{r}$
−11	13	-3	2	1
−1	−14	9	4	2
11	1	−27	8	3
1	13	20	16	4
−11	−14	1	−29	5
−1	1	−3	3	6
11	13	9	6	7
1	−14	−27	12	8
−11	1	20	24	9
−1	13	1	−13	10
11	−14	−3	−26	11
1	1	9	9	12
−11	13	−27	18	13
−1	−14	20	−25	14
11	1	1	11	15
1	13	−3	22	16
−11	−14	9	−17	17
−1	1	−27	27	18
11	13	20	−7	19
1	−14	1	−14	20
−11	1	−3	−28	21
−1	13	9	5	22
11	−14	−27	10	23
1	1	20	20	24
−11	13	1	−21	25
−1	−14	−3	19	26
11	1	9	−23	27
1	13	−27	15	28
−11	−14	20	30	29
−1	1	1	−1	30
11	13	−3	−2	31
1	−14	9	−4	32
−11	1	−27	−8	33
−1	13	20	−16	34
11	−14	1	29	35
1	1	−3	−3	36
−11	13	9	−6	37
−1	−14	−27	−12	38
11	1	20	−24	39
1	13	1	13	40
−11	−14	−3	26	41
−1	1	9	−9	42
11	13	−27	−18	43
1	−14	20	25	44
−11	1	1	−11	45
−1	13	−3	−22	46
11	−14	9	17	47
1	1	−27	−27	48
−11	13	20	7	49
−1	−14	1	14	50
11	1	−3	28	51
1	13	9	−5	52
−11	−14	−27	−10	53
−1	1	20	−20	54
11	13	1	21	55
1	−14	−3	−19	56
−11	1	9	23	57
−1	13	−27	−15	58
11	−14	20	−30	59
1	1	1	1	60

. The first three columns of this table indicate the factors entering the product (35). The fourth column shows the element of the base Galois field formed according to Formula (35). The fifth column presents the corresponding power of the primitive element. It is evident that the values in the first three columns vary periodically with periods 4, 3, and 5, respectively, which fully corresponds to Formula (35). For clarity, the unit elements in the first three columns are highlighted in bold; these elements mark the periods of variation in the quantities $g_i^{r\left(mod4\right)}$. Table 1 also expresses the direct relationship between the considered field element and its discrete logarithm.

The method for calculating the discrete logarithm used in [56] is based on determining the exponents $s_i$. These exponents, corresponding to each element of the considered field, are presented in

Table 2. Relationship of partial discrete logarithms with imaginary and real values of the square root.

${\mathit{s}}_1$	${\mathit{s}}_2$	${\mathit{s}}_3$	$\mathit{u}$	$\mathit{D}\mathit{l}\mathit{u}$	$\mathit{R}\mathit{e}\left(\mathit{w}\right)$	$\mathit{I}\mathit{m}\left(\mathit{w}\right)$
1	1	1	2	1		0
2	2	2	4	2	1
3	0	3	8	3		1
0	1	4	16	4	2
1	2	0	−29	5		2
2	0	1	3	6	3
3	1	2	6	7		3
0	2	3	12	8	4
1	0	4	24	9		4
2	1	0	−13	10	5
3	2	1	−26	11		5
0	0	2	9	12	6
1	1	3	18	13		6
2	2	4	−25	14	7
3	0	0	11	15		7
0	1	1	22	16	8
1	2	2	−17	17		8
2	0	3	27	18	9
3	1	4	−7	19		9
0	2	0	−14	20	10
1	0	1	−28	21		10
2	1	2	5	22	11
3	2	3	10	23		11
0	0	4	20	24	12
1	1	0	−21	25		12
2	2	1	19	26	13
3	0	2	−23	27		13
0	1	3	15	28	14
1	2	4	30	29		14
2	0	0	−1	30	15
3	1	1	−2	31		15
0	2	2	−4	32	16
1	0	3	−8	33		16
2	1	4	−16	34	17
3	2	0	29	35		17
0	0	1	−3	36	18
1	1	2	−6	37		18
2	2	3	−12	38	19
3	0	4	−24	39		19
0	1	0	13	40	20
1	2	1	26	41		20
2	0	2	−9	42	21
3	1	3	−18	43		21
0	2	4	25	44	22
1	0	0	−11	45		22
2	1	1	−22	46	23
3	2	2	17	47		23
0	0	3	−27	48	24
1	1	4	7	49		24
2	2	0	14	50	25
3	0	1	28	51		25
0	1	2	−5	52	26
1	2	3	−10	53		26
2	0	4	−20	54	27
3	1	0	21	55		27
0	2	1	−19	56	28
1	0	2	23	57		28
2	1	3	−15	58	29
3	2	4	−30	59		29
0	0	0	1	60	30

. The first three columns of this table contain the three values of $s_i$, which can be referred to as partial discrete logarithms. For clarity, the unit values of $s_i$ are highlighted in bold in the table, marking the periods and emphasizing the periodic nature of their variation. The fourth column shows the element u of the considered Galois field corresponding to the given set of $s_i$ values. The fifth column presents the discrete logarithm of this element, denoted as $Dlu$. It can be seen that it is precisely the mismatch of the periods of $s_i$, that allows the computation of the discrete logarithms of all nonzero elements of the considered field. The fifth and sixth columns display the exponents corresponding to the real and imaginary parts of the value $w$, respectively. It can be observed that, in this case, the values in the first three columns also exhibit periodic variation. Additionally, this table enables the determination of the function $w\left(x\right)$ from the function $b\left(x\right)$.

The relationship between the exponents $s_i$ and the value of the discrete logarithm follows from the following considerations. The computation of the discrete logarithm represents a mapping from the Galois field to the ring of residue classes modulo $p-1$.

Elements of such a ring, in the most general case, admit the following representation through idempotent elements, as employed, among others, in [57]:

$$s\equiv e_1{s}_1+e_2{s}_2+\dots +e_N{s}_N, mod(p-1)$$

(37)

where $e_i$ are idempotent mutually annihilating elements, and $s_i=0,1,2,\dots ,p_i^{q_i}-1$.

Formula (37) is given for the set of exponents $s_i$, corresponding to the exponent $r$, i.e., the discrete logarithm of a certain element of the considered Galois field. The idempotent elements are formed according to the rule [57]:

$$e_i={\alpha }_i{\prod }_{i\ne j}^N{p}_j$$

(38)

where ${\alpha }_i$ is an integer. The choice of these integers is made under the following condition:

$$e_i{e}_i=1$$

(39)

The following is evident from the construction:

$$e_i{p}_i\equiv 0\left(P\right)$$

(40)

because any product of the form (40) contains the factorization $p-1=p_1{p}_2\dots p_N$.

For the particular case $GF\left(61\right)$ Formula (37) takes the following form:

$$r\equiv 45·s_1+40·s_2+36·u_3, mod\left(60\right)$$

(41)

whereas before, $s_1=\mathrm{0,1},\mathrm{2,3}$; $s_2=\mathrm{0,1},2$; $s_3=\mathrm{0,1},\mathrm{2,3},4$. The fact that the numbers 45, 40, and 36 are idempotent elements modulo 60 can be verified directly.

Therefore, as employed in [56], by determining the exponents $s_i$, one can compute the value of the discrete logarithm according to Formula (39). This methodology also enables deriving the explicit form of the function $w\left(x\right)$, which, according to Formula (24), defines a specific mosaic. To achieve this, it suffices to separate the even and odd values of $r$, then compute either ${\displaystyle \frac{r}{2}}$ (for even $r$) or ${\displaystyle \frac{r-1}{2}}$ (or odd $r$). The values of the function $w\left(x\right)$ are then calculated as either ${\theta }_0^{{\displaystyle \frac{r}{2}}}$, or $i{\theta }_0^{{\displaystyle \frac{r-1}{2}}}$.

The method to identify the values of $s_i$, is based on the following reasoning. In [38], the algebraic delta function was defined as follows:

$$\delta \left(x-x_i\right)=1-{\left(x-x_i\right)}^{p-1},$$

(42)

where $x_i$ is a fixed element of the field $GF\left(p\right)$.

It has the following property, which follows from relation (11):

$$\delta \left(x-x_i\right)=\left\{\begin{array}{c}1, x=x_i\\ 0, x\ne x_i\end{array}\right..$$

(43)

Next, let us consider the following polynomial:

$$F\left(x\right)={\sum }_{m=0}^{m=p-1}Dl\left({\theta }^m\right)\delta \left(x-{\theta }^m\right),$$

(44)

where $Dl\left({\theta }^m\right)=m$ corresponds to the discrete logarithm values (an example is contained in Table 1). When a particular element $x={\theta }^{m_0}$ of the corresponding Galois field is substituted into expression (44), all summands in the sum vanish except the one where $m=m_0$. Hence,

$$F\left(x\right)\to m.$$

(45)

It can be observed that the use of the algebraic delta function indeed allows the problem of computing discrete logarithms to be resolved in an exceptionally transparent manner. The elements $x_m={\theta }^m$ enter Formula (44) directly in the order corresponding to the computation of the discrete logarithm; therefore, when summing the polynomial in (44), its value is obtained automatically.

However, computing large powers according to Formula (42) is not an optimal procedure. It is much more convenient to consider the identifiers $s_i$, i.e., values, the number of which is significantly smaller than the total number of field elements. The corresponding function is given by the following:

$$S_i\left(x\right)={\sum }_{k=0}^{k=p-1}{s}_i\left({\theta }^m\right)\delta \left(x-{\theta }^m\right), i=\mathrm{1,2},3,$$

(46)

In this expression, $s_i\left({\theta }^m\right)$ represents the value of $s_i$ for the $k$-th nonzero field element, i.e., element $x_k={\theta }_0^k$ (the numbering of field elements can be chosen arbitrarily). As emphasized by Table 2, the functions $s_i\left(x_k\right)$, are periodic. For the specific field $GF\left(61\right)$ their periods are 4, 3, and 5, respectively. We will demonstrate that the periodic nature of these functions significantly simplifies the computation of expressions of the form (44) or (46).

In the theory of algebraic fields, the following is proved [45]:

$${\left(y\pm x\right)}^q=y^q\pm x^q$$

(47)

where $q$ is the characteristic of the field.

For fields $GF\left(p\right)$ the number $p$ coincides with the characteristic. Direct verification proves the validity of the following equality:

$$y^p-x^p=\left(y-x\right)\left(y^{p-1}+y^{p-2}x+\dots +y{x}^{p-2}+x^{p-1}\right)$$

(48)

Substituting the ratio (47) in the right part of Formula (48), we get the following:

$${\left(y-x\right)}^{p-1}=y^{p-1}+y^{p-2}x+\dots +y{x}^{p-2}+x^{p-1}$$

(49)

Consequently, the expression (44) can be represented as follows:

$$S_i\left(x\right)=\left(1-x^{p-1}\right){\sum }_{k=0}^{k=p-1}{s}_i\left(x_k\right)-{\sum }_{k=1}^{k=p-1}{x}^{p-1-k}{Q}_k$$

(50)

where

$$Q_k={\sum }_{l=1}^{l=p-1}{s}_i\left(x_l\right)x_l^k={\sum }_{l=1}^{l=p-1}{s}_i\left({\theta }^l\right){\theta }^{kl},$$

(51)

The first term in (50) vanishes by virtue of relation (11).

Sequences

$$w_m=\left(1,{\theta }^m,{\theta }^{2m},{\theta }^{3m},\dots ,{\theta }^{\left(p-2\right)m}\right)$$

(52)

form [57] a complete orthogonal basis on the interval containing p-1 cycles, i.e.,

$${\sum }_{j=0}^{j=p-2}{w}_{k_1}^{\left(j\right)}{w}_{k_2}^{\left(j\right)}=\left\{\begin{array}{c}1,k_1\equiv k_2\left(mod \left(p-1\right)\right) \\ 0,k_1\not\equiv k_2\left(mod\left(p-1\right)\right)\end{array}\right.$$

(53)

Therefore, the $Q_k$ values are non-binary Galois fields Fourier transform functions $s_i\left({\theta }^l\right)$:

$$Q_k=\widehat{F}\left[s_i\left(x\right)\right]={\sum }_{l=1}^{l=p-1}{s}_i\left({\theta }^l\right){\theta }^{k·l}$$

(54)

Importantly, the functions $s_i\left({\theta }^l\right)$ are periodic with a relatively small number of periods. For example, if a function has period 5, then the number of nonzero spectral components cannot exceed 5. Consequently, an expression of the form (51) contains no more than five terms in this case, which significantly simplifies computational procedures.

A block diagram illustrating the generalized procedure for computing the discrete logarithm using the proposed methodology is shown in Figure 7.

The initial step involves selecting the Galois field to be used for mosaic construction (“Initial field GF(p)”). Next, according to Formula (27), the number $p-1$ is factorized into prime factors (“Decomposition of number $p-1$”). Based on these values, the values $g_i$ are selected (“Numbers $g_i$”), which allow the primitive element to be represented as a product of roots of unity in accordance with Formula (31). Finding such values constitutes an independent task; however, as follows from the results presented below, for applied purposes related to mosaic generation, it is sufficient to use Galois fields with relatively small characteristics. Since the number of primes in the range from 1 to 300 is limited, pre-tabulated values can be used when implementing the algorithm.

Using the values of $g_i$, periodic functions (“Periodic functions”) entering Formula (46) are then constructed. Examples of such functions are presented in Table 2, columns 1–3. In parallel, an orthogonal basis (“Orthogonal basis”) is formed, determined by the employed Galois field according to Formula (53). Using this basis, the spectral components of the functions in Formula (46) are computed (“Spectral components”). On this basis, the quantities $s_i$ (“Partial values $s_i$”) are calculated, from which, using Formula (37), the discrete logarithm (“Discrete logarithm”) for a given number $x$ (“Initial $x$”) is determined. An example of programming code implementing the discrete logarithm operation is provided in Supporting Information S2.

Examples of the application of the discrete logarithm computation algorithm are illustrated in Figure 8 and Figure 9. Figure 8 shows examples of functions $b\left(x\right)$ corresponding to mosaics constructed using Formula (7), i.e.,

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

(55)

Figure 9 presents the functions corresponding to the solutions of Equations (15) for these particular cases. Figure 9a and Figure 9b correspond to the imaginary parts of the extracted square root for the cases of Figure 8a and Figure 8b, respectively.

Notably, in the calculation according to Formula (55), the inverse element of the sum $1+a^2$ within the sense of the employed Galois field is used (fractional values do not arise).

It can be observed that the considered algorithm indeed allows the description of rather complex mosaics to be reduced to relatively simple functions, albeit taking values in an algebraic extension of the base Galois field. On this basis, it is possible, in perspective, to develop a method for generating mosaics by directly specifying the functions $w\left(x\right)$. In particular, it is permissible to construct mosaics in which the functions $w\left(x\right)$ take only real values. The first step in this direction involves a method in which the functions $b\left(x\right)$, initially computed based on polynomial functions, are subsequently modified in various ways. At the next stage, it becomes possible to implement an algorithm that ensures a gradual (stepwise) transition from one type of mosaic to another. A preliminary experiment demonstrating the potential of this approach is presented in Supporting Information S3. In this experiment, the programming code provided in Supporting Information S3 was used. The code employs the same mosaic generation algorithm as that in Supporting Information S1, with the exception that every second computed value is artificially set to zero. The examples presented in Supporting Information S3 show that, in this case, the resulting mosaic structures approximate configurations that visually decompose into fragments corresponding to parallel ornamented bands. For generality, the code reads the functions $b\left(x\right)$ from a separate file.

Despite the simplicity of expression (25), this expression enables the generation of a wide range of nontrivial mosaics. In particular, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 present mosaics obtained using the following algebraic expression:

$$f\left(x,y\right)=y^2\left(a-x\right)-x^2$$

(56)

It can be seen that the polynomial on the right-hand side can also be brought to the form (25), since it can be transformed into the following equality:

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

(57)

In the construction of the mosaics presented in Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, smoothing of the resulting structures was also applied. An algorithm analogous to the moving average method was employed. For each pixel with coordinates $i,j$, a square centered at that pixel is considered. For Figure 11b, Figure 12b, Figure 13b, Figure 14b and Figure 15b, this square includes 9 pixels (including the central pixel), while for Figure 12c, Figure 13c, Figure 14c and Figure 15c, it includes 25 pixels. The resulting color is determined based on the majority of pixels (white or colored) within the selected square.

The resulting mosaics demonstrate that the smoothing procedure allows the periodic nature of mosaics generated using Galois fields with relatively large characteristics to be clearly revealed. Smoothing also provides an additional tool for mosaic generation, since increasing the field characteristic (even only up to 257) produces an excessively complex mosaic, which is unlikely to be of practical interest, particularly for applications such as textile or wallpaper pattern design. The use of excessively complex mosaics in psychological testing is also not justified.

For comparison, Figure 16 presents mosaics formed using the cissoid of Diocles.

$$f\left(x,y\right)=y^2\left(a-x\right)-x^3$$

(58)

The presented mosaics indicate that, in terms of mosaic generation, transitioning to more complex equations (even cubic equations) is not always justified. Visually, mosaics corresponding to an equation containing a variable to the third power are similar to those generated using simpler equations. This observation, of course, does not preclude potential applications of more complex equations in the future. However, for the purposes of psychological testing and for generating patterns used in the production of textiles, wallpapers, and similar materials, it is sufficient to employ relatively simple algebraic expressions.

4. Discussions

The use of Galois fields enables the development of a relatively simple algorithm for generating mosaics suitable for applied purposes, such as the production of textiles, including smart textiles [10], wallpapers, floor coverings, and similar applications. Modern manufacturing technologies allow for the production of wallpapers, carpets, and similar products tailored to individual preferences [12,17]. Many commonly used patterns are geometric in nature and closely resemble the mosaics examined in this study. Therefore, there is potential to select mosaic patterns not only for aesthetic appeal but also for enhancing psychological comfort in living environments.

Another promising area of application for such mosaics is the development of novel approaches to improving established psychological testing methods, in particular those such as the Rorschach test [28,29]. It should be noted that other commonly used tests also employ well-defined images (in particular, the Szondi Test [24] and Susan Dellinger’s psychogeometric test [25], both of which were mentioned above). These techniques have been subject to considerable criticism [26], which also applies to the Rorschach test [58]. The automatic generation of mosaics capable of evoking specific associations in test subjects offers a pathway to improving association-based methods, as suggested in [27]. Crucially, the link between mosaics and algebraic structures is essential for the statistical analysis and validation of the obtained results.

The main advantage of using Galois fields for these purposes lies in the possibility of employing extremely simple algebraic expressions to generate relatively complex mosaics. A comparison of the main characteristics of the proposed approach with previously known methods is presented in

Table 3. Comparison of the main characteristics of the proposed approach with previously known methods.

Criterion	Proposed Approach	Previously Known Approaches
Mosaic periodicity	Periodicity arises automatically due to the properties of Galois fields, even for simple algebraic expressions	Periodicity must be imposed artificially, typically requiring a special algorithm [15,18,20]
Need for an initial pattern	Not required	An initial pattern is necessary [15,18,20]
Mosaic modification and control	Stepwise transitions between different types of mosaics are possible	Mechanisms for transitioning between patterns of different types are either not provided [15,20] or require direct operator intervention [18,19,22]
Implementation complexity	The code is extremely simple (example—SI-1); computations use only integers	Algorithm implementation involves computational complexities [19,21,22] or requires searching for the initial pattern [15,18,20]
Scalability/detail level	Details can be enhanced straightforwardly by increasing the field characteristic	Detail enhancement is possible, but operations required for it are comparable in complexity to generating a new mosaic [15,18,19,20,21]
Post-processing/complexity management	Simple smoothing (analogous to 3 × 3 or 5 × 5 moving window) helps reveal periodicity and reduce excessive complexity	The use of smoothing filters is not provided [15,18,20,21]
Limitations	Requires a simple modulus pp; special cases for pp of the form 2 m −1 (quasi-Mersenne primes), and the possibility to relax the requirement via finite rings	Limitations depend on the method (tiling rules, geometric/numerical constraints, etc.)

. It should be emphasized that this table does not consider the metric of mosaic generation speed. This metric is not critical, as practical applications in the contexts under consideration only require reaching a rate of 24 frames per second, which corresponds to the existing standard. Achieving higher speeds is not meaningful.

The simplicity and convenience of mosaic generation using the proposed algorithm (Supporting Information S1) also enable the development of a comprehensive psychological testing methodology, simultaneously oriented toward analyzing the current psychological state of the subject and identifying factors corresponding to archetypal levels. As demonstrated in [1,2], cited in the Introduction, mosaic ornaments are an integral part of the cultural heritage of many nations worldwide. Accordingly, there is reason to believe that the corresponding patterns, in one way or another, correlate with archetypes. On this basis, a comprehensive psychological testing methodology could be developed in the future, in which both mosaics correlating with culturally traditional patterns and responses to specific questions are employed.

For the development of such methodologies (as well as for the refinement of psychological testing methods based on the findings of [27] and similar studies), the ability to generate mosaic structures that transition stepwise from one to another is of crucial importance. The feasibility of implementing such an approach is confirmed by the results presented in Supporting Information S3. It should also be noted that the transition from one mosaic structure to another is of interest for applications such as the advancement of smart textiles.

It is emphasized that this capability is enabled, in particular, by the properties of Galois fields, which allow mosaics to be realized using maximally simple algebraic expressions. The transition from one such expression to another can be defined algorithmically. Additional possibilities in this regard are provided by the transition to algebraic extensions of Galois fields, where the relevant polynomials decompose into linear factors. For equations of the form (8), all polynomial roots can be found using discrete logarithm computation. It should also be noted that equations of the form (8), corresponding to binomial polynomials, represent a specific case. However, the proposed approach can be readily generalized to equations of the following form:

$$y^N-b\left(x\right)=0$$

(59)

The use of relations of the type (59) for the aforementioned purposes generally requires the computation of discrete logarithms. This methodology is also presented in the current work. Regarding the operation of discrete logarithm computation, there is a specific nuance. For fields associated with primes of the form $p=2^k+1$, relation (27) becomes degenerate. These primes can be interpreted as quasi-Mersenne numbers [59]. However, in such cases, the discrete logarithm operation can be performed using the results from the cited work. Furthermore, the limitation on mosaic dimensions, determined by the requirement that p be prime, can be at least partially relaxed by employing the results of [60]. In this case, finite algebraic rings are used instead of Galois fields, and it can be anticipated that mosaic construction in the future may be implemented through logical operations, as described in the cited source. The solution to this problem, however, remains a matter for future research.

In a broader perspective, the proposed method for generating mosaic structures can be used to develop techniques for diagnosing structures similar to mosaics based on the use of radio waves. As shown in [41,42,43], cited in the Introduction, the analysis of mosaic-like structures is of interest for monitoring agricultural lands.

In [61], it was demonstrated that the description of any radiation transformer obeying the Helmholtz equation or its analog can be reduced to a discrete form. This conclusion is based on the fact that the classical Huygens–Fresnel principle can also be expressed in discrete form, which is particularly relevant for addressing problems in radio holography, an area currently undergoing active development [62,63,64].

For the development of such methods, especially those oriented toward practical applications (as dictated, for example, by mosaic-like structures occurring naturally [41,42,43]), it is expedient to employ certain test structures, which can be generated using the proposed method.

5. Conclusions

The generation of large arrays of diverse mosaics can be achieved through the use of an algorithm based on Galois fields and simple algebraic expressions (including classical expressions, such as the Bernoulli lemniscate or the Diocles’ cissoid). The advantage of this approach lies in the simplicity of implementing the computational algorithm, as well as the possibility of ensuring a gradual transition from one mosaic to another. The proposed method allows for generalization through the use of algebraic extensions of the base field, as well as the computation of discrete logarithms.

The proposed method for constructing complex mosaic structures holds promise for a variety of practical applications, including psychological testing. The most immediate application lies in expanding the design possibilities of wallpapers, fabrics, and related products—particularly with the aim of enhancing psychological well-being for consumers.