Optimal alternative for suitability of S-boxes to image encryption based on m-polar fuzzy soft set decision-making criterion

Our aim in this work is to improve the design and model of real-life applications. We put forward a standard based on m-polar fuzzy soft set decision-making criterion to examine the optimal alternative for the suitability of S-boxes to image encryption applications. The proposed standard studies the results of correlation analysis, entropy analysis, contrast analysis, homogeneity analysis, energy analysis, and mean of absolute deviation analysis. These analyses are applied to well-known substitution boxes. The algorithm of outcomes of these analyses is additional observed and a m-polar fuzzy soft set decision-making criterion is used to decide the optimal alternative for suitability of S-box to image encryption applications. All results taken by using the reality values for all S-boxes and experimental problems with reality values are discussed to show the validity of the optimal alternative for the suitability of S-box to image encryption.


Introduction
The block ciphers (symmetric key cryptosystem) present an essential job in the area of secure communications. The security of an encryption algorithm is related to the performance of the building block which is liable for producing uncertainty in the cipher. This functionality is attained by the use of an S-box, so this component is like a nucleus in an atom [1]. The perfection in the properties of an S-boxes has been a major problem of research in the area of cryptology. In this paper, we show the correlation analysis, entropy analysis, contrast analysis, homogeneity analysis, energy analysis, and mean of absolute deviation analysis for existing S-boxes. The correlation analysis is widely used to analyses the S-box's statistical properties [2]. The entropy analysis is a statistical method used to measure the uncertainty in image data. The amount of uncertainty in an encrypted image characterizes the texture of the image. In contrast analysis [3], the intensity difference between a pixel and its neighbor over the whole image is calculated. The elevated values of contrast analysis reflect the amount of randomness in encrypted images and results in enhanced security. The measure of closeness in the attribute single-valued neutrosophic decision-making problem. Sahin and Kucuk [35] defined a subsethood measure for SVNS and applied to MADM. Evaluation based on distance from average solution (EDAS), originally proposed by Ghorabaee et al. [36], is a new MADM method for inventory ABC classification. It is very useful when we have some conflicting parameters. In the compromise MADM methods such as TOPSIS and VIKOR [37], the best alternative is got by computing the distance from ideal and nadir solutions. The desirable alternative has lower distance from ideal solution and higher distance from nadir solution in these MADM methods. Ghorabaee et al. [38] extended the EDAS method to supplier selection. As far as we know, however, the study of the MADM problem based on EDAS method has not been reported in the existing academic literature. Hence, it is an interesting research topic to apply the EDAS in MADM to rank and determine the best alternative under the single-valued neutrosophic soft environment. Through a comparative analysis of the given methods, their objective valuation is carried out, and the method which maintains consistency of its results is chosen. For computing the similarity measure of two SVNSs, we propose a new axiomatic definition of the similarity measure, which takes in the form of SVNN. Comparing with the existing literature [31,32,39,40], our similarity measure can remain more original decision information. By means of level soft sets, Feng et al. [41] presented an adjustable approach to fuzzy soft sets based decision-making. By considering different Fig. 1 The image before and after encryption. This figure explained the difference between after and before encryption types of thresholds, it can derive different level soft sets from the original fuzzy soft set. In general, the final optimal decisions based on different level soft sets could be different. Thus, the newly proposed approach is, in fact, an adjustable method which captures an important feature for decision-making in an imprecise environment: some of these problems are essentially humanistic and thus subjective in nature. As far as we know, however, the study of the single-valued neutrosophic soft MADM problem based on level soft set has not been reported in the existing academic literature. Considering that different attribute weights will influence the ranking results of alternatives, we develop a new method to determine the attribute weights by combining the subjective elements with the objective ones. This model is different from the existing methods, which can be divided into two tactics: one is the subjective weighting evaluation methods and the other is the objective weighting determine methods, which can be computed by grey system theory [42]. Figures 1 and 2 explain the image before and after encryption. For more information about m-polar fuzzy sets and analyses of S-Box in image encryption applications based on fuzzy decision-making criterion. The remainder of this paper is organized as follows: firstly, Sections 1 and 2 introduced some background of image encryption, showed and analyzed the types of the Sboxes. AES, APA, Gray, Lui J, residue prime, S8 AES, SKIPJAC, and Xyi talked about the properties of these S-boxes (the correlation analysis, entropy analysis, contrast analysis, homogeneity analysis, energy analysis, and mean of absolute deviation). Also, these sections explained soft set, fuzzy soft set, and fuzzy polar soft set.
In Section 4, the analyses of S-Box in image encryption applications based on fuzzy sets and 2-polar fuzzy soft set decision-making criterion are studied, in problem statement chosen suitability of S-box to image encryption based on polar fuzzy soft set and construct an algorithm for a decision-making.
In Section 4.2, we developed a study to state decision-making based on 2-polar fuzzy soft set by using two measures. Also, in Section 4.2, the optimal alternative for the suitability of S-box to image encryption based on 2-polar fuzzy soft set and by using a join and meet for 2-polar fuzzy soft set are introduced. All the results are taken by using the reality values for all S-boxes, and experimental examples with reality values are discussed to show the validity of the proposed concept.
Section 5 is the conclusion and remarks.

Soft sets and m-polar fuzzy soft set
Let E be a non-empty finite set of attributes (parameters, characteristics, or properties) which the objects in U possess and let P(U) denote the family of all subsets of U. Then a soft set is defined with the help of a set-valued mapping as given below: Analyses of S-box in image encryption applications based on fuzzy sets and 2-polar fuzzy soft set decision-making criterion
Our aim is to examine the optimal alternative for suitability of S-boxes to image encryption. Correlation information plays the main role in stating the similarity of pixel patterns in the given image and its encrypted version by the use of techniques such as entropy analysis, contrast analysis, homogeneity analysis, energy analysis, and mean of absolute deviation analysis on the image. Now, we want to use the concept of soft set to choose the best S-box, so assume that we analyze S-boxes (AES, APA, Gray, Lui J, Residue, S8 AES, SKIPJACK, and Xyi ). There are form the set X = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 } of alternatives where x i {i = 1, 2, 3, 4, 5, 6,7,8,9) are S-boxes and the alternatives x i . To evaluate the S-boxes, we take the parameters I = {i 1 , i 2 , i 3 , i 4 , i 5 , i 6 } where i 1 stands for entropy, i 2 stands for contrast, i 3 stands for average correlation, i 4 stands for energy, i 5 stands for homogeneity, i 6 stands for MAD. These parameters are important with degree. (In Table 2 and Fig. 2 we explained the important values of Entropy, contrast, average eneragy, eneragy, homogeneity, mad of prevailing S-box).  where Aðx 1 Þði 1 Þ ¼ ð6:6733; 6:4325; 6:7325Þ means that the entropy of S-box of x 1 is given by group 1 (resp., by group 2, by group 3 ) is 6.6733 (resp., 6.4325,6.7325); meanings of Aðx s Þði t Þ can be explained similarly (s = 1, 2, 3, 4, 5, 6, 7, 8, 9; t = 1, 2, 3, 4, 5, 6).
Now, we compute m i ¼ P 4 k¼1 ðx k ÞðiÞ; x∈X; ði ¼ 1; 2; 3Þ and compute r i ¼ Table 5 Then, the maximum score is r 1 =1.62835354 and the optimal alternative for the suitability of S-box to image encryption based on 2-polar fuzzy soft set is to select x 1 (APA encryption ).

Suitability of S-boxes to image encryption based on two operations (∧ and∨) polar fuzzy soft sets
In this section, we study the problem by using two operations (∧ and ∨) polar fuzzy soft sets. So, we give S-boxes (Residue, Gray, AES, SKIPJACK, and Xyi). There are form the set X = {x 1 , x 2 , x 3 , x 4 , x 5 } of alternatives where x i {i = 1, 2, 3, 4, 5) are S-boxes and the alternatives x i . To evaluate the S-boxes, we take the important alternative parameters I = {i 1 , i 2 , i 3 } where i 1 stands for entropy, i 2 stands for homogeneity, i 3 stands for MAD. These parameters are important with degree (8.00, 1.0, and 60.0). Considering their own needs, the data for the optimal alternative for the suitability of S-box to image encryption based on 2-polar fuzzy soft set A∈ð½0; 60 2Â3 Þ IÂX defined by:  where Aðx 1 Þði 1 Þ ¼ fð6:6; 6:5Þ; ð7:6; 5:7Þ; ð7:6; 5:8Þg means that of the encryption x 1 (Residue ) of the parameter i 1 (entropy) increase and decrease of growth given by the first measure is the increase takes the value 6.6 and decrease takes the value 6.5 , by the second measure, the increase takes the value 7.6 and decrease takes the value 5.7, and by the third measure, the increase takes the value 7.6 and decrease takes the value 5.7 and decrease takes the value7.6 and decrease takes the value 5.8. Respectively; the meaning of Aðx s Þði t Þ can be explained similarly (s = 1, 2, 3.4,5; t = 1, 2, 3 ). Similarly, A subset B ¼ fB i g i : I→ð½0; 60 2Â3 Þ X is called also 3-polar fuzzy soft set on X, define  Similarly; (in Table 6, and Fig. 6 we define the values ofĈ ðxÞði; jÞ∈ð½0; 60 2Â3 Þ X ). Therefore, Now, we make a decision by two ways: (1) First way:  Table 7 and Fig. 7, we define the mapping ℂ M : X → R, by ℂ M ðxÞ ¼ P ði; jÞ∈I 2 βðxÞði; jÞ) Since ℂ M ((x 1 ) = 120.5688 = max ℂ M ,then the optimal alternative for the suitability of S-box to image encryption X based on 3-polar fuzzy soft set is x 1 (Residue encryption).
Motivated from the above problem, we give the following algorithm for decisionmaking problem: (2) Second way: Compute m i ¼ P 5 k¼1 ðx k Þði; jÞ; x∈X; ði; jÞ∈ðI Â IÞ and compute r i ¼ P 9 j¼1 ðm i −m j Þ ði ¼ 1; 2; 3; 4; 5Þ, then, (in Table 8  Motivated from the above problem, we give the following algorithm for decisionmaking problem: Now, find the optimal alternative for the suitability of S-box to image encryption based on 3-polar fuzzy soft set. By using the operator∨, First, compute ℂ ¼ A∨B: So, compute ℂ.   Table 10 and Fig. 10, we compute ℂ M : X → R, by ℂ M ðxÞ ¼ P ði; jÞ∈I 2 βðxÞði; jÞ) Since ℂ M ((x 1 ) = 120.1776 = max ℂ M ,then the optimal alternative for the suitability of S-box to image encryption X based on 3-polar fuzzy soft set is x 1 (Residue encryption).