Property $\bar{A}$ of third-order noncanonical functional differential equations with positive and negative terms

1.   Introduction

In symmetric key cryptography, since 2000, the AES standard of 128 bits was adapted by the National Institute of Standard and Technology (NIST). For review, the original designers of AES, Joan Daemen, and Vincent Rijmen (known as Rijndael) submitted this designed approach along various other algorithms. There was a requirement to develop a strong enough algorithm that can be further used for encryption application. The reason that urges the cryptographers to build the AES is to retain the US Government document in private for a minimum of 20 years. Due to the advancement in computing power, the former methods of securing data i.e., Data Encryption Standard (DES) (developed by IBM in 1975 and break in 1998) and triple-DES has become weak for security purposes. Thus, the government of USA offered an open proposal to increase the standard of data encryption. Finally, the Rijndael cryptosystem (AES) was adapted for encryption scheme and it was circulated by FIPS in 26 NOV 2001 ^[1]. AES is largely being used for secure transactions through the internet and to transfer of money through banks ^[2].

S-box plays a seed role in AES. By understanding the concepts of its functionality, it comes to know that S-box is a non-linear factor in AES. It is a heart of any block cipher cryptosystem. The Symmetric block key algorithm uses substitution boxes that yield confusion and permutation boxes that yield diffusion in data when it is going to be encrypted. The significance of S-box is that it creates hurdles for cryptanalysis. Thus, no unauthorized person can get unlawful access to original messages ^[3]. Several attempts have been consequently followed to design valuable, highly secure, and robust S-boxes that many ciphers utilize to encrypt data. The usefulness of the original AES S-box is to provide substitute data, which is based on keys along with permutation to develop a substitution-permutation network. S-boxes have resulted in many interesting properties that are appropriate for different ciphers. Due to the shuffling of input bits, output bits also change. These changing of bits are analyzed to determine the confusion creating capability and the strength of S-boxes. Various S-boxes are presented in literature such as APA ^[4], Gray ^[5], S₈ and residue prime ^[6,7,8], and AES ^[3], which have good cryptographic properties and algebraic complexity ^[9].

In literature, there is an insight upon the building of S-boxes under three irreducible polynomials of $GF\left({2}^{4}\right).$ These three polynomials are used for the manufacture of small $4\times 4$ S-boxes ^[10]. Affine transformation is utilized that carries the best choice of 4×4 transformation matrices and $4\times 1$ constant matrices for all irreducible polynomials of $GF\left({2}^{4}\right)$ ^[11]. It was very challenging and hard for code breakers to break the code because numerous irreducible polynomials are practiced instead of single ones like the Rijndael algorithm, which worked only for single S-box. Ten Small S-boxes are structured and the permutation of symmetric group S₄ (consists of 24 permutations) ^[12] is operated on each of the small 10 S-boxes, and 240 new ones are obtained by applying permutation, one after the other. Formerly, 10 random S-boxes are selected. The number of choices for arbitrary selection is 240C10. Those innovative S-boxes play a very vital role for hiding data so that nobody can crack the code easily in a limited time.

In our projected algorithm, a new scheme is developed, which has very efficient algebraic complexity for safeguarding data. It safeguards the data in both text and image form. Our optimized research work will utilize the 10 S-boxes of $GF\left({2}^{4}\right)$ (Result of symmetric group permutation S₄) ^[10] and Cartesian permutations of Klein four-group V₄ ^[12] with itself. Our proposed algorithm seeks a new methodology that is sufficient for security purposes to develop 8×8 S-boxes using subfield $GF\left({2}^{4}\right)$ of $GF\left({2}^{8}\right)$. With the aid of this new stylist approach, 1600 independent S-boxes are obtained here and then random 10 S-boxes are picked. The key point is that we have billions of choices i.e., 1600C10≈(2.945764438E+37) to arbitrary pick any 10 S-boxes for utilization in AES. It is quite a large number that makes the process very impressive and safe, which nobody knows, instead of a receiver that is the choice of cryptographers to choose 10 S-boxes for encryption. For security purposes, the performance of our modified S-boxes is comparatively more accurate than other ones when we compare it with other S-boxes through different tests.

The structure of the paper layout is as follows. In Section 2, we briefly elaborate necessary algebraic expressions for AES S-boxes that use the Rijndael algorithm ^[1]. Section 3 displays the technique for already designed small substitution boxes ^[10]. In Section 4, we elaborate on the new scheme for the construction of modified S-boxes, permutation of Klein four group, and their use in proposed S-boxes. Section 5 depicts the random selection of S-boxes and pictorial representation of whole modified schemes. In Sections 6 and 7, message encryption and decryption by modified S-boxes, analysis of S-boxes, and image encryption by proposed schemes with their results is presented. In the last two sections, comparisons and conclusions are presented.

2.   Algebraic structure of the AES S-box

2.1.   Klein four group

The Klein four-group is the smallest noncyclic Abelian group in which every element has order 2. It is $\cong$ to the direct sum of two abelian groups ${\mathbb{Z}}_{2}\times {\mathbb{Z}}_{2}$ where Addition is defined component wise under mod (2) ^[12]. In group notation, the Klein four-group is defined by,

Permutation Representation: The permutation illustration of this group entails four points ^[12].

As ${V}_{4}$ is applicable only to four-bit data so for the application of this particular permutation on eight-bit data to increase the capability of diffusion of that cipher is to utilize the Cartesian structure of ${V}_{4}$. It comprises $16$ permutations that are signified by ${\mu }_{t}; 1\le t\le 16$ and the permutation chart according to their data type is given in Table 1.

2.2.   Galois field

Let $F = {F}_{q}$ be a field and $F\left[x\right]$ is the Euclidean domain. For the extension of field $F{\left[x\right]}^{m},$ the quotient rings

where the maximal ideal $< f\left(x\right) >$ is generated by $f\left(x\right)$ an irreducible polynomial of degree $m$ in $F\left[x\right]$. If we write $\alpha$ to denote the coset $x+\left(f\right(x\left)\right)$ , then $f\left(\alpha \right) = 0$ and

The field $F{\left[x\right]}^{m}$ is a Galois field ( $m$ -degree extension field of the field $F).$

2.3.   AES S-box functions

Two sub steps are discussed here for S-box function of input bytes that is utilized in AES for safeguarding data ^[3,13].

1) Multiplicative inversion: Let $a$ be a nonzero input byte. Taking its inverse in $GF\left({2}^{8}\right)$ and acquire output byte $f\left(a\right)$ .

2)Affine transformation: Next sub step is to use the affine transformation i.e.,

It is a required S-box function, where b is a stable byte and $M$ is constant bit matrix. Affine transformation is given below ^[14,15].

3.   Designing $4\times 4$ S-boxes

The small S-boxes comprises 16 elements that are defined over finite Galois field $GF\left({2}^{4}\right)$ ^[10] and they are designed under three distinct irreducible polynomials ^[13] through the best choice of 4×4 transformation matrices and $4\times 1$ constant matrices (Table 2). B represents each member of $GF\left({2}^{4}\right)$ in transformation T, which is written in the form of $4\times 1$ matrix.

The transformation matrices X and constant matrices C are fixed according to distinct irreducible polynomials P₁ (t), P₂ (t), and P₃ (t). The chart presented below represents the whole information about transformation and constant matrices according to respective polynomials.

After achieving 10 S-boxes, the symmetric group S₄ acts on them to permute the bytes S₄×S_i; 1≤i≤10 (Section 3). A total of 240 S-boxes are obtained under permutation of S₄ ^[12]. Thus, 10 S-boxes have been arbitrarily picked for utilization in a cryptographic area.

4.   Proposed S-boxes

For making practical and effective use, we demonstrate the configuration of proposed $8\times 8$ S-boxes in this section. This novel technique depends on four steps by utilizing $10$ small $4\times 4$ S-boxes of $GF\left({2}^{4}\right)$ (output of Section 3). Also, the Cartesian structure of Klein four-group is practiced here and the performance of proposed algorithm is much better for security purpose.

Step 1. The initial step for designing $8\times 8$ S-boxes is to follow the combination scheme of nibbles. As ${S}_{i}, {S}_{j}:GF\left({2}^{4}\right)\to GF\left({2}^{4}\right).$ So the mapping $\xi :{S}_{i}{S}_{j}\to {S}_{j}^{i}$ for combination is known as joint mapping that joins the nibbles of ${S}_{i}$ and ${S}_{j}$ for the formation of bytes of ${S}_{j}^{i}$. Where ${S}_{i}{S}_{j}:GF\left({2}^{4}\right)\to GF\left({2}^{8}\right)$ is defined as $\xi \left({S}_{i}\left(u\right){S}_{j}\left(v\right)\right) = {S}_{j}^{i}\left(uv\right) = {S}_{j}^{i}\left(t\right); 1\le i, j\le 10$. Here $u, v$ are nibbles and $t$ represents byte.

The process of combination is that first nibble of ${S}_{i}; 1\le i\le 10$ is joint one by one by all nibbles of ${S}_{j}; 1\le j\le 10$ and makes one row of ${S}_{j}^{i}$. By following this scheme $100$ new structured S-boxes ${S}_{j}^{i}; 1\le i, j\le 10$ of $GF\left({2}^{8}\right)$ are developed. The process of combination is presented in Table 3.

Step 2. In this step we utilize the change of $8\times 8$ basis matrix ^[14] for computing the S-box function of given byte. The algebraic expression for transformation $\beta :GF\left({2}^{8}\right)\to GF\left({2}^{8}\right)$ is defined by $\beta \left(t\right) = Xt.$ It is a multiplication of $8$ bit matrix of input block with this basis matrix $X$ for enhancing complexity. Where $t$ and $\beta \left(t\right)$ both are $8$. bit input matrix. The approach of using this transformation is to convert every input byte into output ones. The change of constant bit basis matrix $X$ ^[2] is represented as,

This transformation is applied to each byte of ${S}_{j}^{i}; 1\le i, j\le 10$ (Step 1). So as a consequence of this transformation $100$ modified S-boxes ${{S}_{j}^{i}}^{\text{'}}; 1\le i, j\le 10$ are achieved.

Step 3. Under this step, the affine linear transformation $eqn(1-2)$ is applied in quite consistent format to each of the distinct S-box ${S}_{j}^{i}; 1\le i, j\le 10$ (Step 1) and resulting S-boxes are ${{S}_{j}^{i}}^{\text{'}\text{'}}; 1\le i, j\le 10$. Then the SubBytes transformation is practiced and transform different 8 bit to another different 8 bit data ^[13]. The key point in this substitution process is that unique $100$ substitution boxes are utilized for subByte of modified $100$ S-boxes ${S}_{j}^{i}; 1\le i, j\le 10$. Thus ${{S}_{j}^{i}}^{\gamma }$ is obtained by subByte of ${{S}_{j}^{i}}^{\text{'}\text{'}}$ (substitution box) to ${{S}_{j}^{i}}^{\text{'}}$ (Step 2) for each $1\le i, j\le 10$. The subByte pattern that is depicted in Figure 1.

Step 4. Now for enhancing more the algebraic power of the outcome of Step 3, it is to break each and every byte of input blocks ${{S}_{j}^{i}}^{\gamma }; 1\le i, j\le 10$ into prefix and postfix nibbles for the evaluation of bijective transformation $f\left(z\right)$ defined on $GF\left({2}^{4}\right)$. Where $z$ represents an input byte. This specific bijective function is evaluated by the fixed values of ${a}^{\text{'}}, {b}^{\text{'}}, {c}^{\text{'}}\;and\;{d}^{\text{'}}$ chosen from $GF\left({2}^{8}\right)$ against the range of $z$ defined as $[0, 255]$. The respective transformation is given as,

where

In above equation $f\left(z\right)$ represents a byte and later on the group action of projective general linear group $\eta :PGL(2, GF\left({2}^{8}\right))\times GF\left({2}^{8}\right)\to GF\left({2}^{8}\right)$ ^[6] is defined on $GF\left({2}^{8}\right)$ referred as Mobius transformation. The expression for computational evaluation is $\eta \left(f\left(z\right)\right) = \frac{af\left(z\right)+b}{cf\left(z\right)+d}$ for fixed constants $a, b, candd$ chosen from $GF\left({2}^{8}\right)$. Large number of S-boxes have been synthesized by following this procedure but to make it easy for the reader the example is elaborated on this technique.

Example. This example will elaborate the whole technique of step 4 by fixed values ${a}^{\text{'}} = 1B,$ ${b}^{\text{'}} = 39,$ ${c}^{\text{'}} = 25$ and ${d}^{\text{'}} = 6B$ for evaluation of $f\left(z\right)$ and then particular Mobius transformation $\eta \left(f\left(z\right)\right) = \frac{35f\left(z\right)+15}{9f\left(z\right)+5}$ , where $1B, 39, 25, 6B, 35, 15, 9, 5 \in GF\left({2}^{8}\right)$ ^[17]. The resultant all entries after this transformation are from $GF\left({2}^{8}\right)$ and form $100$ S-boxes ${{S}_{j}^{i}}^{\neg }; 1\le i, j\le 10$. One of them is given in Table 4.

4.1.   Use of permutation in S-boxes

In this section, we will discuss how to get permuted S-boxes by diffusion process. The use of Cartesian permutation ${V}_{4}\times {V}_{4}$ in ${{S}_{j}^{i}}^{\neg }; 1\le i, j\le 10$ (Step 4) is quite important step in our research paper. The technique of permutation is

The whole description is tabulated as.

Here ${\mu }_{t}; 1\le t\le 16$ are the mixture of two permutations $\sigma$ and $\pi$. $\sigma$ is applicable to prefix ${(N}_{1})$ and $\pi$ is on postfix nibble ${(N}_{2})$ for each byte $x\in$ $GF\left({2}^{8}\right)$ of ${{S}_{\bf{1}}^{\bf{1}}}^{\neg }$. The application of ${8}^{th}$ permutation ${\mu }_{8}$ on $x$ is obtained as ${\mu }_{8}\left(x\right) = {\sigma }_{8}\left({N}_{1}\right(\mathrm{x}\left)\right){\pi }_{8}\left({N}_{2}\right(x\left)\right)$. As a consequence, 1600 S-boxes ${{S}_{j}^{i}}^{*}; 1\le \mathrm{i}\le 10, \;1\le \mathrm{j}\le 10$ are achieved by the action of $16$ permutations to each of ${{S}_{j}^{i}}^{\neg }; 1\le i, j\le 10$ (Step 4).

5.   Random selection of S-boxes

The main step that is the heart of our algorithm is to select arbitrarily $10$ S-boxes from $1600$. The total possible choice for this selection is ¹⁶⁰⁰C₁₀ $\approx \left(2.945764438E+37\right)$. The reason behind selection in this paper is to utilize in AES encryption that makes the system much more complicated for cryptanalysts. In our case selected $10$ S-boxes are ${{{{\boldsymbol{S}}_{\bf{2}}^{\bf{1}}}^{\mathrm{*}}, \boldsymbol{S}}_{\bf{5}}^{\bf{4}}}^{\mathrm{*}}{, {\boldsymbol{S}}_{\bf{7}}^{\bf{7}}}^{\mathrm{*}}{{, \boldsymbol{S}}_{\bf{10}}^{\bf{5}}}^{\mathrm{*}}, {{\boldsymbol{S}}_{\bf{1}}^{\bf{2}}}^{\mathrm{*}}{{, \boldsymbol{S}}_{\bf{10}}^{\bf{10}}}^{\mathrm{*}}{{, \boldsymbol{S}}_{\bf{10}}^{\bf{8}}}^{\mathrm{*}}{{, \boldsymbol{S}}_{\bf{7}}^{\bf{4}}}^{\mathrm{*}}{{, \boldsymbol{S}}_{\bf{9}}^{\bf{6}}}^{\mathrm{*}}{{, \boldsymbol{S}}_{\bf{9}}^{\bf{4}}}^{\mathrm{*}}$ , presented in Table 5.

6.   Redesigning the AES algorithm

The Modified AES algorithm is the most important section in our research paper for utilization of newly generated robust S-boxes. In our modified algorithm, all means of AES-128 ^[1] are changed to improve the complexity and make it unpredictable. Our case is key dependent, just like the original AES because if we build a key dependent framework ^[13], then cryptanalysis turns out to be more troublesome. Also, the Mixed column matrix is not fixed, so a better approach of shifting is adapted and S-box is not static. The elaboration is displayed below.

1) Add round key

The initial step add round key is the same as in the original AES, so that in each round, a new key is Xored with state. The State matrix and key ^[13] are both equal in order. These keys are originally built by AES key expansion.

2) Substitute bytes

A new strategy for byte substitution is accomplished here. As a consequence of (Step 5), ten S-boxes are utilized one by one in each round for substitution. The way of substitution is the same as in AES ^[1].

3) Shift bytes

In this step, bytes are moved by utilizing a new scheme known as shift box. Diverse shifts are attempted in various rounds. Shift boxes are not fixed for each round ^[15,18]. Their selection is based on a substitution box. If ${{\boldsymbol{S}}_{\boldsymbol{j}}^{\boldsymbol{i}}}^{\boldsymbol{*}}$ is used for substitution in Round 1 then ${S}_{i}$ will play the role of shift box for that specific one (presented in Table 6).

The way of shifting is that if the first component of shift box is 14, then the shifting is that the primary component of state moves towards the fourteenth position of state. Likewise, all elements are shifted. The technique is shown in the Figure 2.

4) Mix column

The mix column is an essential element of the encryption scheme. Like shift boxes ^[18], different mix columns are utilized in different rounds. Their selection is also based on the substitution box. If ${{\boldsymbol{S}}_{\boldsymbol{j}}^{\boldsymbol{i}}}^{\boldsymbol{*}}$ is used for substitution in Round 1, then ${S}_{j}$ will play the role of shift box for that specific one.

5) Round constant in key expansion

For each round, unique round constants (same as AES) ^[18], are used that are presented in Table 7.

The results of the message encryption and decryption by Modified AES structure are given in Table 8.

Moreover, the flow chart of modified encryption scheme are depicted in Figure 3.

7.   Analysis of S-boxes

Many standard analyses have been depicted in the literature to evaluate the strength of S-boxes with fitting confusion. For testing security of S-boxes, many analyses like Bijectivity ^[13], SAC ^[16], Non-linearity ^[19], and differential and linear probability ^[7] are used in the checking process ^[20].

7.1.   Bijectivity

S-boxes are bijective if all elements of $GF\left({2}^{8}\right)$ are given as an input to a Boolean function and corresponding to each value, and a unique output of $GF\left({2}^{8}\right)$ is obtained ^[21]. It is visible from the above 10 tables that those corresponding to a single input there is a single output. Thus, our all newly constructed $1600$ S-boxes that are designed by means of four steps are $100\%$ bijective.

7.2.   Nonlinearity

Checking for Nonlinearity is essential to identify how much our S-box creates hurdle for linear attack. A function $\varphi \left(x\right)$ is defined from $nth$ Boolean function $GF\left({2}^{n}\right)$ to $GF\left(2\right)$. So the nonlinearity for function $\varphi \left(x\right)$ is defined in the form of

Here, members of ${L}_{n}$ are all linear and affine functions as well, and ${d}_{\delta }(\varphi, t)$ represents the Hamming distance amongst $\varphi$ and $t$ .

The Nonlinearity through Walsh transform is predicted for function $\varphi$ is

For function $\varphi$ the notation ${S}_{ < \varphi > }\left(\sigma \right)$ is termed as cyclic spectrum and it is evaluated by ^[19],

where, $x.\sigma$ represents dot product for each $x, \sigma \in GF\left({2}^{n}\right)$. The radius bound of Nonlinearity is ${2}^{n-1}-{2}^{\frac{n}{2}-1}$. So, to counterattacks against linear cryptanalysis, the Nonlinearity of S-box should be close to its certain upper bound. It is represented in Table 9.

7.3.   Strict avalanche criterion (SAC)

SAC ^[16] interprets the evidence about altering bits in the output and that alteration is approximately half of the output bits that are altered. These bits are changed by changing one single bit of eight-bit input i.e., 0 to 1 or 1 to 0 ^[7]. This analysis is vital for examining the confusion aptitude of S-boxes.

7.4.   Bit independent criterion (BIC)

If Exclusive OR of a Boolean function ${\varphi }_{j}$ and two bits output of S-box ${\varphi }_{k}$ is extremely Non-linear and fulfills the properties of SAC, then for each and every pair of output bit, the correlation coefficient is near to 0 by inverting one input bit. Thus, for BIC, use ${\varphi }_{j}\;xor\;{\varphi }_{k}\;for\;(j\ne k)$ ^[19] to determine whether it meets the criteria of nonlinearity and SAC.

7.5.   Differential approximation probability

To conceptualize uniformity, apply the differential attack to input and notice the alteration in behavior and properties of output at the intermediate stage. Then, time linear and nonlinear responses due to differential attacks is observed. Here, corresponding input and output differentials are expressed by $\partial {x}_{i}$ and $\partial {y}_{i}$. Mathematical expression is ^[7]

The subsequent numerical values of $Dif.prob$ for the proposed S-box are inside a satisfactory range.

7.6.   Linear approximation probability

At the output bit, the estimation for unbalancing events is monitored through drawing changes at input bits. $\omega x$ and $\omega y$ are applied to each parity of individual bits and analyze each individual response and its effect at the output stage. Mathematically ^[19],

where ${2}^{n}$ represents cardinality and $x$ represents input. The results are tabulated in Table 9. As smaller the $Lin.prob$, the more grounded the capacity of S-box for fighting against direct cryptanalysis attack. Thus, the consequence of $Lin.prob$ is better. The analyses of the proposed S-boxes are displayed in Table 9.

8.   Comparison with prevailing S-boxes

The consequences of our proposed S-boxes and their average by means of comparison with others are depicted in Table 10. The proposed S-boxes' performance is concurrent with the aftereffects of the investigation for different S-boxes that are tried for comparison. The estimation of the proposed S-box is near to the ideal value that demonstrates its protection from attackers. The consequence of non-linearity ^[19] is near to Xyi and Skipjack S-box. An examination of SAC ^[16] for various S-boxes utilized as benchmarks is shown in this work. These outcomes run from a greatest value of 0.625 to at least 0.406, with an average estimation of 0.502. The average outcomes from SAC investigations yield esteems near the ideal estimation of 0.50. The subsequent numerical values of $Dif.prob$ ^[7] for the proposed S-box are inside a satisfactory range and bring it into comparison with various other S-boxes that are utilized in this paper for analysis. The smaller the $Lin.prob$ , the more grounded the capacity of S-box for fighting against direct cryptanalysis attack. Thus, the consequence of $Lin.prob$ ^[19] is better.

9.   Conclusions

It is concluded that this work is identified with the development of S-boxes. The whole process comprises a few stages i.e. Matrix multiplication, Affine Transformation, Substitution, Action of projective general linear group, and permutation with specific components of group ${V}_{4}\times {V}_{4}$. The proposed strategy has numerous advantages. The principal advantage is that $GF\left({2}^{4}\right)$ works behind the development of $8\times 8$ S-boxes, and when cryptanalysts apply different inverse methods for code break, they will use $GF\left({2}^{8}\right)$. Thus, they are not expected to break the code. Since the Galois field is cyclic, it implies one can develop every single other component of S-box with just a single generator. The second benefit is that it is known very well that diffusion is induced by permutation in a secure communication field. Thus, the shuffling by the permutations of cartesian product ${V}_{4}\times {V}_{4}$ enhances the diffusion capacity of the cipher. In Section 3, with the proposed technique, we developed $1600$ S-boxes subsequent to applying ${V}_{4}\times {V}_{4}$ permutation. At that point, we chose $10$ S-boxes arbitrarily, utilized it in AES calculations, and inspected their analyses in eminent cryptographic criteria. We inferred that our S-boxes have great cryptographic properties i.e., nonlinearity, differential probability, linear probability, SAC, and BIC. Since AES has used just a single S-box byte sub step and utilizes the same S-box in all, it relies on its key length. Hence, another advantage is that here we have 1600 boxes with great properties, chose 10 S-boxes, and use every one in distinctive rounds of AES. We have ¹⁶⁰⁰C₁₀ $\approx \left(2.945764438E+37\right)$ possible outcomes to choose $10$ S-boxes. Additionally, different steps of AES were changed. Therefore, we have new calculations and strong S-boxes that have crucial application in encryption algorithms of block cipher. We have given AES encryption and proposed encryption on which we have done work. In this way, it demonstrates that it is a profoundly secure framework, and in the event that somebody wants to break the code of this algorithm, they need to move through the opposite of all steps. That is the reason why it is difficult to break down the code of this proposed algorithm. Cryptanalysis requires numerous years to decode the message. Thus, they need to perform thousands or even millions of counts to register what $10$ S-boxes have been chosen to conceal the information? This strategy is most noteworthy for anchoring information when two gatherings build up a secret communication with each other.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their gratitude to the deanship of scientific research of King Khalid University, for funding this work through a research project under grant R.G.P.2/238/44.

Conflict of interest

The authors declare no conflicts of interest.