1. Introduction

The notion of fuzzy set was originally proposed by Zadeh (1965) [1] as a mathematical model to grip real world problems, uncertainty and vagueness in data analysis. This theory provided some new mathematical modeling to handle uncertainty and vagueness problems in the fields of computer science, management, economic, electronic, computer science and engineering. In 1986, Atanassov introduced the notion of an intuitionistic fuzzy set (IFS) [2] as a generalization of a fuzzy set. After the introduction of an IFS, researchers engaged in the developing of IFSs. An interval valued intuitionistic fuzzy set (IVIFS) was introduced by Atanassov and Gargov (1989) [3] as another tool to handle real life problems. It is natural to believe that IVIFSs show a momentous character in inspiring decision modelling. However, the generalization from particular numbers to interval values of the membership and non-membership functions of intuitionistic fuzzy sets [4,5,6] poses considerable tasks in working with IVIFSs. There is still at an inceptive stage to use IVIFSs to decision modelling and the limited literature exists in this specialized area. The most important model of interval valued intuitionistic fuzzy set is decision making problems. Decision making problems have been studied by many researchers in different papers [7–16]. There are several methods for decision making, like multicriteria decision making [7,8], group decision making [9,10], TOPSIS method [11–16]. The most fruitful method in these models is TOPSIS method. There is a lot of qualitative information in complex decision making, where decision maker easily evaluated the results by using TOPSIS method. The decision maker’s limited attention and information processing capabilities. There are several extensions of the TOPSIS for interval-valued intuitionistic fuzzy environment. Here is being used the approach of [33]. But I suggest that we propose a variant of this proposal. In this paper, we use the interval valued intuitionistic fuzzy TOPSIS method as criterion for the S-boxes in image encryption. We try to find suitable and best S-box for image encryption on base of interval valued intuitionistic fuzzy TOPSIS method as new criterion.

The block ciphers (symmetric key cryptosystem) play a vital character in the field of secure communications. The security of an encryption algorithm is connected with the depiction of the structure block which is responsible for generating fuzziness in the cipher. We use the S - box to achieve this functionality, so this element is identical to a nucleus in an atom [17]. The excellence in the assets of S-boxes has been a supreme imperative difficulty of research in the subject of cryptology. In this paper, we show the entropy analysis, contrast analysis, correlation analysis, homogeneity analysis, energy analysis, and mean of absolute deviation analysis for prevailing S-boxes. The correlation analysis is mostly applied to calculate the S-box’s statistical chattels [18]. The entropy analysis is a statistical technique, applied to evaluate the uncertainty in the image data. The amount of vagueness in an encrypted image portrays the consistency of an image. The intensity difference between a pixel and its neighbours over the whole image is evaluated by contrast analysis [19]. The imperative reading of contrast analysis imitates the quantity of randomness in encrypted images and possessions in enhanced security. In homogeneity [20], Jing et. al., calculated the closeness in the distribution of grey level co-occurrence matrix (GLCM) elements to the GLCM diagonal. The GLCM is the formulation of how frequently different amalgamations of pixel brightness values (grey levels) occur in an image [21]. In energy analysis, Gadelmawla measured the sum of squared elements in the GLCM.. This analysis is responsible for the merits and demerits of different S-boxes in terms of energy of the ensuing encrypted image. The final method that Avcibas et. al., appliance on the encrypted image is the mean of absolute deviation (MAD) analysis [22]. This analysis decides the alteration in the original and an encrypted image. Many methods are proposed for emerging encryption in literature. Although these algorithms perform to be favourable, their stoutness is not yet recognized and they are sprouting to become standards. Some of these algorithms, wealth, revealing, are the public key cryptosystems constructed on chaotic Chebyshev polynomials [23], advanced encryption standard (AES) cryptosystem are exhausting the features of montage image for immensely secure, high data rate [24] and image encryption via logistic map function and heap tree [25]. The most fruitful technique applied to analyse the statistical strength of S-boxes are the differential approximation, linear approximation, correlation analysis, probability, fuzzy set theory and strict avalanche criterion etc. We have incorporated correlation method as a benchmark for the remaining analysis used in this work. With the exception of correlation analysis, the solicitation and application of the results of statistical analysis, presented in this paper, have not been used to estimate the strength of S-boxes. The correlation analysis, entropy analysis, contrast analysis, homogeneity analysis, energy analysis, and mean of absolute deviation analysis are performed on AES [26], APA [27], gray [17], Lui J [28], residue prime [29], S8 [30], SKIPJACK [31], and Xyi [32] S-boxes. The results of these S-boxes based on these criteria are not clearly identified the most suitable and best S-box for image encryption. The results of these analyses are analysed by the proposed criterion and an interval valued intuitionistic fuzzy TOPSIS method. In this paper, we highlighted on “problem statement”, which properly describes the problems and compensations of the analyses given. Interval valued intuitionistic fuzzy TOPSIS criterion analyses the effectiveness of S-boxes of the anticipated criterion to identify the strength of an S-box. Statistical image analysis of S-boxes” describes the statistical analysis applied in this work. The details of experiments performed in order to verify the statistical analysis results are shown in simulation results and discussion.” Finally, the study presents “conclusions” and “future direction “related to this work.

2. Basic Concept

Learn more in this section we will start with the basic definitions of intuitionistic fuzzy set (IFS), interval valued intuitionistic fuzzy set (IVIFS) and score functions. Further study of this article we need the following terms

3. Problem Statement

An m × n S-box takes some number of input bits, m, and transforms them into some number of output bits, n, where n is not necessarily equal to m.[39] An m×n S-box can be implemented as a lookup table with 2^m words of n bits each, when n = m = 8, then m × n S-box is 8 × 8 S-box. In this manuscript, we analyze 8×8 S-boxes (AES, APA, Gray, Lui J, Residue Prime and S₈) used in popular block ciphers. Without loss of generality, the analysis can be extended to S-boxes of other sizes. The statistical analysis is used to determine the application and suitability of an S-box to image encryption application (Tran et al., 2008). The strength of an encryption based on the S-box can be calculated by examining various parameters generated by numerous statistical analyses. It is imperative to be familiar with the significance and relationship between the outcomes of different types of analyses. Therefore, we develop a criterion which carefully inspects and scrutinizes the available parameters and makes a decision based on interval valued intuitionistic fuzzy decision making assessment using score function.

We use the correlation information to examine the similarity of pixel patterns in the given image and its encrypted version. Although this scrutiny has been broadly used to estimate different image encryption algorithms, it is incorporated here with other methods due to its significance and acceptability in matching images and determining similarities. The correlation analysis under some circumstances does not provide sufficient information in determining the strength of encryption; therefore, in order to increase the reliability of the decision, we employ further techniques such as entropy analysis, contrast analysis, homogeneity analysis, energy analysis and mean of absolute deviation analysis on image. These analyses, when applied in combination, provide more brilliant results and accordingly assist in assessing the enactment of S-boxes. To the best of our knowledge, entropy analysis, contrast analysis, homogeneity analysis, energy analysis and mean of absolute deviation analysis, have not been extensively analyzed and studied for the evaluation of S-boxes to image encryption application. (See Table 1).

In Figure 1 clearly explains the results of entropy, contrast, average correlation, energy, homogeneity and mean of absolute deviation of plain image. Since it clears from Figure 1 that the date of the original image is quite utilized, our main objective is to depatternized the data of the plain image with the help of standard S-box transformations.

In Figure 2 clearly describes the reading of Entropy, contrast, average correlation, energy, homogeneity and mean of absolute deviation after the application of Affine-Power-Affine S-box transformation of the plain image of Lena Image. Now we check the effects of APA S-box transformation on plain image is as follows, the entropy is altered from 6.773 to 7.8183, the contrast of the plain image is altered from 0.2455 to 8.9114. This means that the APA S-box transformation create a huge disorder in the contrast of the plain image. The correlation of plain image is misleading from 0.8771 to 0.1258, the energy is transformed from 0.2917 to 0.0193, the homogeneity is altered from 0.9334 to 0.4665, at the end the difference between the original image and cipher image corresponding to APA S-box is dignified with the help of mean of absolute deviation analysis and we conclude that the cipher image is dissimilar from plain image with the reading of 43.544. It is pretty vibrant from Figure 1 and 2 that the APA S-box transformation create a huge disorder in plain image. Therefore, there exist some uncertainty and not clearly describes through these parameters.

Next we compare Figure 1 with Figure 3. We discuss the result of entropy in Figure 3, after applying the Advanced Encryption Standard S-box transformation on plane image, is altered from 6.733 to 7.9325 which is different from APA cipher image entropy. The effect of contrast of the AEA cipher image is transformed from 0.2455 to 7.2240, which is also changed from the contrast of the APA cipher image. From the above analysis, we conclude that in both transformation in Figure 2 and Figure 3 for entropy it is pretty clear that APA S-box transformation is better than AES S-box transformation. Next we discuss the evaluation of correlation of AEA cipher image is changing from 0.8771 to 0.0815. This concludes that the AES S-box transformation is better than APA S-box transformation in the sense to produce un-correlation in the pixels of the plain image. The energy of plain image is changed from 0.2917 to 0.0193. Also in this case, the energy of the APA cipher image is better than the energy of the AES cipher image. The homogeneity is transformed from 0.9334 to 0.4665. From this we come to know that the APA cipher image is better than the AES cipher image. By comparison of Figure 2 and Figure 3 for absolute deviation, we conclude that the APA S-box transformation brings a huge disorder in the pixels of the plain image with value 62.066 as matched to AES S-box transformation. Finally, we conclude with a comparison of Figure 2 and Figure 3 that there arises some uncertainty due to in some cases the AES S-box transformation is superior but in some situation APA S-box transformation shows good results.

Now we show the analysis among Figure 2, 3 and 4. In Figure 2, 3, 4, it’s clear that the entropy of AES S-box transformation and Liu J S-box transformation is identical and well than APA S-box transformation. The contrast analysis of the Liu J S-box is slighter than APA S-box transformation as show in Figure 2 and 4. The effect of the correlation analysis in Figure 4, the reading of Liu J S-box transformation is 0.1311. Its mean that the value of Liu J S-box transformation for correlation analysis is weak as we compare to the value of APA S-box transformation. From Figure 2 and Figure 4, if we compare the energy analysis of APA S-box transformation and Liu J S-box transformation, its clear depict that the energy analysis of APA S-box transformation is better than Liu J S-box transformation. Also, if we see the figure 2 and figure 4, the readings of the homogeneity and MAD analysis of APA S-box transformation are also reasonably better than Liu J S-box transformation. Finally we know that from the above discussion, the APA S-box transformation is moderately worthy as we compare with Liu J S-box and AES S-box transformation for image encryption applications. By comparison of Figure 5 with other S-box transformation, we conclude that APA S-box transformation is also enhanced than residue prime S-box transformation.

The values of the S₈ S-box transformation in figure 6 for these parameters is much better than every other S-box transformation that is discussed in this manuscript. But in some case it does not clear whether which one S-box is better. Finally, the construction of S₈ S-box depends on the symmetric group of permutation and Galois field. The basic condition for an S-box is to deliver confusion in the data, but this box delivers an added step of security which is diffusion. Diffusion complicates the relationship between the plaintext and cipher text. The diffusion in S₈ S-box transformation is due to the permutations of S₈ group.

If we compare Figure 7 with other Figures, we observe that the strength of Gray S-box transformation is better in some cases but in some analysis this transformation is below average.

The Xyi S-box transformation exhibits very good results for different analysis, but in case of correlation analysis this transformation is not as good, which is required for a good image encryption. The results of energy analysis of Figure 4 transformation are much better than many other transformations. Hence, overall this is a secure S-box transformation.

The Skipjack box is constructed for sequential circuits use, but its transformation shows some good results for image encryption applications. But in many analyses this transformation is not on the first position so we can make use of it for secure communication.

The encrypted image of the plain image through each S-box transformation shown in figure 8, 9, 10, 11, 12, 13, 14 respectively. Finally, we didn’t say for these S-box through these parameters to which one is better and suitable for image encryption everyone has own quality and dis-quality at different stages. Therefore, there arise some uncertainty and vagueness in these analyses. Now, we use the TOPSIS method through interval valued intuitionistic fuzzy set to determine the suitability and best S-box for image encryption applications.

4. TOPSIS Method with Interval Valued Intuitionistic Fuzzy Information

Decision Support Systems are a precise class of computer-based information systems that support your decision-making activities. A decision support system analyzes data and provide interactive information support to professionals during the decision-making process. Decision making implies selection of the best decision from a set of possible options. In some cases, this selection is based on past experience. Past experience is used to analyze the situations and the choice made in these situations. In 1981 the TOPSIS technique was established by Hwang and Yoon [11], is a humble classification technique in information and applications. The standard TOPSIS technique endeavours to select alternatives that instantaneously have the direct distance from the positive ideal solution and the farthest distance from the negative-ideal solution. The positive ideal solution minimizes the cost criteria and maximizes the benefit criteria, whereas the negative ideal solution minimizes the benefit criteria and maximizes the cost criteria. TOPSIS sort’s full procedure of attribute information, affords a prime ranking of alternatives, and does not require attribute preferences to be independent (Chen and Hwang, 1992; Yoon & Hwang, 1995). To apply this technique, attribute values must be numeric, monotonically increasing or decreasing, and have commensurable units. In this section we are using the concept of interval valued intuitionistic fuzzy TOPSIS Method to cryptography and through an interval valued intuitionistic fuzzy TOPSIS algorithm based on improved score function. We intend to choose only the very best possible S-box. The procedure of method description in flow chart: see in Fig

Let X = {A₁, A₂, A₃, A₄, A₅, A₆, A₇,} be a set of seven alternatives. The alternatives are: 1) A₁ is plain image; 2) A₂ is AES; A₃ is APA; A₄ is Lui; A₅ is S₈; A₆ is Prime; A₇ is Gray. The decision maker takes a decision according to the following five criteria: 1) C₁ is entropy; 2) C₂ is a contrast; 3) C₃ is an average correlation; 4) C₄ is energy; 5) C₅ is homogeneity. In this analysis, the decision maker carefully inspects and scrutinizes the available criteria and makes a decision based on interval valued intuitionistic fuzzy TOPSIS method. The procedure starts with correlation analysis and we use the correlation information to analysis the similarity of pixel patterns in the given image and encrypt version. The seven possible alternatives are to be evaluated using the interval valued intuitionistic fuzzy information from the decision maker under the above five criteria. The description of seven alternatives on basis of five criteria as give in Table 2:

Let suppose that the weights of alternatives; entropy, contrast, average correlation, energy, homogeneity are 0.1, 0.25, 0.2, 0.15 and 0.3, respectively. Then, decision maker utilizes the approach of developers to get most suitable and best S-box for image encryption by using an interval valued intuitionistic fuzzy TOPSIS method based on improved score function. We can first find a square matrix R_7×5 (I_ij(x_ij)), by converting interval valued intuitionistic fuzzy decision matrix D_7×5 (x_ij). Table 3 is the score matrix after applies the improved score function to decision matrix.

Now we can write equations for positive ideal solution for S-boxes (alternatives) are denoted by [33]

• A⁺ = {〈C_j, [1,1], [0,0]〉| C_j ∈ C}, j = 1,2,3,4,5,6,7 and negative ideal solution for S-boxes (alternatives) are denoted [33] by

• A⁻ = {〈C_j, [0,0], [1,1]〉| C_j ∈ C}, j = 1,2,3,4,5,6,7

Now we find the separation measure based on score function d⁺ (A⁺,A_i) and d⁻ (A⁻,A_i) of each S-box from positive ideal and negative solutions, respectively [33]. The separation measure Equations are given as follows [33]:

Compute the numerical values of separation measure of each S-box. In table 4, the separation measure of each S-box.

In Fig 1a, we describe the separation measure w.r.t positive ideal solution and w.r.t negative ideals solution of the each S-box. Now we compute the relative closeness of each S-box with respect to the positive ideal solution A⁺ are defined as the following equation.

where C_i(A_i) (i = 1,2, … , n) is the relative closeness coefficient of each S-box (Alternatives) w. r. t the positive ideal solution A⁺ and the value of C_i(A_i)∊[0,1] for each S-box. Therefore, we will choose the best S-box through the descending order of C_i(A_i). i.e., The best S-box will be those whose have high values of C_i(A_i). We write the relative closeness coefficient of each S-box in table 5. We describe the maximum value of the relative closeness coefficient of each S-box in Fig 2a. In table 5 the maximum value of each S-box is S₈ as shown in fig 2a the graph of S₈ is high among all other S-box.

5. New TOPSIS Method for IVIF Numbers

A set A = {a₁, a₂, a₃, … , a_m,} of alternatives, a set A = {c₁, c₂, c₃, … , c_m,} criteria, a weighted vector W = (w₁, w₂, w₃, … , w_m,) such that ∑j=1nwj , an parameter α ∈ [0,1] and a matrix M of dimension n × m such that M_ij (the content of the position (i, j) in the matrix M) is the IVIF number which reflect when the alternative a_i satisfy the criterion c_j.

• Step 1.

  Generate from M two score matrices as follows:

  1. a)

     The first matrix, M^l, is obtained by applying to each position of the decision matrix the following operator

     Kαl(([a,b],[c,d])=αa+(1−α)(1−d)

     that is Mijl=Kαl(Mij) ; and

  2. b)

     The second matrix M^r is obtained by applying to each position of the decision matrix the following operator

     Kαr(([a,b],[c,d])=αc+(1−α)(1−b)

     that is, Mijr=Kαr(Mij) ;.

• Step 2.

  Calculate the weighted normalised decision matrices MW^l and MW^r which is defined by the following equations; MWijl=wjMijl and MWijr=wjMijr.

• Step 4.

  Define the worst and the best alternatives based, respectively, on MW^l and MW^r as follows:

  Alw={max{MWijl:j∈J−}:i∈Nm}∪{min{MWijl:j∈J+}:i∈Nm}≡(Ajlw)j∈NnAlb={max{MWijl:j∈J+}:i∈Nm}∪{min{MWijl:j∈J−}:i∈Nm}≡(Ajlb)j∈Nn,Arw={max{MWijr:j∈J−}:i∈Nm}∪{min{MWijr:j∈J+}:i∈Nm}≡(Ajrw)j∈Nn, and Arb={max{MWijr:j∈J+}:i∈Nm}∪{min{MWijr:j∈J−}:i∈Nm}≡(Ajrw)j∈Nn

  where, J₋ = {j ∈ N_n : c_j}, and J₊ = {j ∈ N_n : c_j}

• Step 5.

  Calculate the distance between the alternative a_i to the worst alternatives A^lw, A^rw, A^lb and A^rb as follow:

  dilw=∑j=1n(MWijl−Ajlw)2dilb=∑j=1n(MWijl−Ajlb)2dirw=∑j=1n(MWijr−Ajrw)2dirb=∑j=1n(MWijr−Ajrb)2

  where, J₋ = {j ∈ N_n : c_j}, and J₊ = {j ∈ N_n : c_j}

• Step 6.

  Calculate

  siwl=dilbdilw+dilb and siwr=dirbdirw+dirbSiw=Siwl+Siwr2

• Step 7.

  Rank the alternatives according to S_iw with i ∈ N_m.

6. Implementation of New TOPSIS Method for IVIFS.

• Step 1. Find the two score matrices from table 2 of decision matrix by using the following two equations. The values of score matrixes are shown in Table 6a & 6b.

  M7×5l=[0.670.570.440.470.770.370.240.540.530.580.710.580.570.440.540.740.640.850.510.440.370.610.640.580.340.610.540.170.240.610.440.670.370.640.77]

  Table 6a:

  Score Matrix

  M7×5r=[0.420.60.470.50.80.40.270.60.80.670.770.70.70.50.60.80.70.50.60.50.40.670.670.70.40.640.60.20.70.70.50.70.40.670.8]

  Table 6b:

  Score Matrix

• Step 2. Find the weighted normalized decision matrices by using the following. The detail of weighted normalized decision matrixes are given in Table 7a & 7b.

  MW7×5l=[0.0670.14250.0880.07050.2310.0370.060.1080.7950.1740.710.1450.1140.0660.1620.0740.160.170.7650.1320.370.15250.1280.0870.1020.0610.1350.0340.0360.1830.0440.16750.0740.0940.231]

  Table 7a:

  Weighted Normalized Decision Matrix

  MW7×5r=[0.0420.150.0940.0750.240.040.06750.1200.1200.2010.0770.1750.1400.0750.1800.080.1750.1000.0900.1500.040.1680.0400.1050.1200.0640.1500.0400.1050.2100.050.1750.0800.1010.180]

  Table 7b:

  Weighted Normalized Decision Matrix

• Step 3. Define the worst and the best alternatives based, respectively, on MW^l and MW^r as follows. The detail of worst and the best alternatives ae given in Table 8.

  Alw	0.0740	0.1675	0.0740	0.0960	0.2310
  Alb	0.0370	0.0600	0.0340	0.0360	0.1020
  Arw	0.0800	0.1750	0.1400	0.1200	0.2400
  Arb	0.0400	0.0675	0.0400	0.0750	0.1200

  Table 8.

  Worst and the best alternatives

• Step 3. Calculate the distance between the alternative A_i to the worst alternatives A^lw and A^rw as follow. The detail of distance between the alternative A_i to the worst alternatives are given in Table 9.

  	dilw	dilw	dirw	dirw
  A1	0.6530	0.1687	0.0790	0.1553
  A2	0.6449	0.1120	0.1230	0.0918
  A3	0.6309	0.1389	0.0751	0.1629
  A4	0.5789	0.1114	0.1030	0.1542
  A5	0.6268	0.1414	0.1280	0.1049
  A6	0.7110	0.1130	0.1096	0.1403
  A7	0.0300	0.7553	0.0920	0.1131

  Table 9.

  Distance Measure

• Step 5. Calculate the relative closeness separation measure by using the equations in step 5. Detail values see in table 10.

  	Sil	Sir	Sil+Sir2
  A1	0.2053	0.6628	0.4341
  A2	0.1480	0.4274	0.2877
  A3	0.1804	0.6845	0.4325
  A4	0.1614	0.5995	0.3810
  A5	0.1841	0.4541	0.3191
  A6	0.1371	0.5614	0.3493
  A7	0.9618	0.5514	0.7566

  Table 10.

  Relative closeness

  The result of relative closeness worst and best alternative shown in fig 3a. The final relative closeness of each alternative shown in Fig 3b. The S8 S-box has the maximum value of averaging relative closeness. The final selection of S-box is S8 S-box.

Conclusion:

Application of intuitionistic fuzzy set has been studied in different applied branches of information science and computer science. Decision support system is a precise class of computer-based information systems that support your decision-making activities. A decision support system analyzes data and provide interactive information support to professionals during the decision-making processes. The result of Decision making systems based on intuitionistic fuzzy set and interval valued intuitionistic fuzzy systems is almost excellent with compared to other analysis. The aim of this manuscript to combine the decision support system based on interval valued intuitionistic fuzzy set and image encryption to analyze the different S-box transformation. In this work we have joined the concept of interval valued intuitionistic fuzzy sets (IVIFS) with secure communication. Basically an S-box is very important component in a block cipher cryptosystem which have the responsibility to induce confusion in the data. So there is a need to find the confusion capability of different 8 × 8 S-boxes for image encryption application and also which box is best among others. In this paper we have analyzed prevailing S-box and come to know that S₈ S-box have very good readings. Our study will start a new direction in the field of decision support systems and cryptosystems. In future we will focus some other types of S-box transformation based on other criterion.

Acknowledgement

The authors would like to express their gratitude to King Khalid University Saudi Arabia for providing administrative and technical support.