New combination of simple additive and entropy weighting criteria for the selection of best substitution box

Abstract

Substitution boxes (S-boxes) are among the most widely recognized and fundamental component of most modern block ciphers. This is on the grounds that they can give a cipher fortifying properties to oppose known and possible cryptanalytic assaults. We have suggested a novel tool to select nonlinear confusion component. This nonlinear confusion component added confusion capability which describes to make the connection among the key and the cipher as complex and engaging as possible. The confusion can be obtained by using substitution box (S-box) and complex scrambling algorithm that relies on key and the input (plaintext). Various statistical and cryptographic characteristics were introduced to measure the strength of substitution boxes (S-boxes). With the help of the present objective weight methods and ranking technique, we can select an ideal S-box among other constructed confusion component to make our encryption algorithm secure and robust against various cryptographic attacks.

Keywords

Simple additive weighting entropy weighting S-boxes

1 Introduction

Sending enormous measure of secret data over the correspondence media has raised the security challenges. The interest for a more secure and more dependable cryptosystem has made new examination issues in the field of cryptography. For the safe transmission of information to web, images have an incredible significance. Image encryption gives secure transmission of digital images by changing over recognizable type of image into an unrecognizable structure.

The block ciphers play a pivotal role in sharing secret information. The capability of encryption algorithm aimed at creating confusion during enciphering process is responsible for the performance of a block cipher. The confusion is attained by using nonlinear confusion component involved in many block ciphers. The substitution box (S-box) is generally utilized in symmetric encryption algorithms to add confusion to scrambled digital information. A wise choice of S-box decides necessary protection from cryptanalysis and therefore delivers the first information immense in the cipher text. The investigation and examination of the characteristics of S-box identified with its encryption strength is advantageous in enciphering applications. Several construction techniques were developed in literature so far to add confusion capability based on different selection criteria.

In the field of encryption, the development of algebraic structures and S-box statistics has become a focus of attraction. This paper has its basis on image encryption using S-boxes. Substitution box is a unique and nonlinear part of block ciphers. The best way to design an input S-box can increase the quality of cipher documents is a unique and nonlinear core component of block ciphers. S-box is a significant unit to give higher security properties. A variety of S-boxes including advanced encryption standard (AES), affine power affine (APA), Prime, S₈, Gray, Xyi and SKIPJACK were used for substitution in modern block ciphers. These nonlinear confusion components are based on algebraic techniques which mainly includes Galois fields and its different modifications. Several new algebraic structures such as Galois ring and chain ring were proposed in order to construct this nonlinear confusion component. In 2013, Husain et. al., projected a new apparatus which utilized projective general linear groups and symmetry group S₈ for the construction of S-boxes. This new construction is further improved by adding various other techniques with linear fractional transformation (LFT) to increase cryptographic characteristics of S-boxes [1 –13]. Due to similar characteristics of chaos and cryptography, researchers utilized chaos theory to develop nonlinear confusion component of block ciphers [14 –27]. The chaos theory is further combined with optimization in order to construct nonlinear component. Now various other mathematical structures likewise Latin square, LA-semigroup and LA- associative were utilized to design new mechanism which uses minimum number of properties to construct nonlinear confusion component [28 –31]. In 2012, Iqtadar et. al., proposed a new technique for S-box construction using nonlinear chaotic algorithm (NCA) such S-box has the optimal values of cryptographic attributes [58].

Multi-criteria decision making methodology assists the conclusion maker in the identification of most suitable alternative from a collection of probable criteria. With the procedure with increase of decision techniques along with their variations, one must have a cognizance of their relative worth. Each strategy utilizes numeric procedures and helps the decision maker pick midst a discrete arrangement of elective choices. It gets accomplished on the premise of the effect of the options on specific models and subsequently on the in general utility of the decision maker(s). The trouble that consistently happens while attempting to look at choice techniques and pick the best one is that a mystery is reached, i.e., What dynamic technique ought to be utilized to pick the best decision making method.

Multi-criteria decision making (MCDM) is widely used to explore many conflicting approaches to decision-making. The contraindication criteria is the same in evaluating options: cost or price is usually one of the main criteria and a certain level of quality is generally another determinant that conflicts with costs [57]. Multi-criteria decision making involves three basic steps:

Determination of appropriate procedure and alternatives.

Connect mathematical steps with the general significance of the measures and the results of alternatives in these measures.

Analyze mathematical attributes to determine the positioning of every other option.

Some Multi-criteria decision making methods have been utilized by many researchers. The Weighted Sum Model (WSM) is considered the very first and perhaps the most extensively utilized technique. The Weighted Product Model (WPM) is considered as a WSM alteration, and is suggested for the weakness of the prior one. Another method known as Analytical Hierarchy Process (AHP) has gained popularity recently. Professor Belton and Gear suggested a change in AHP in 1981 that seems much more powerful than the original method. Something more methods used by ELECTRE and TOPSIS [59].

To find out the best S-box among all the nine S-boxes, we need to use any method of MCDM. In each S-box there are some beneficial criteria and non-beneficial criteria as well. Here energy, homogeneity and average correlation are non-beneficial criteria whereas contrast, mean of absolute deviation analysis (MAD) and entropy are beneficial criteria. Each MCDM method assigns weights to both advantageous and non-advantageous criteria [32 –53]. The Fig. 1 depicts some well known MCDM which are utilized in various systems to classify the best and worst component in any give data set.

Fig. 1

Some well-known multi-criteria decision making methods.

The principal purpose of the article is to suggest a new mechanism for the selection of best nonlinear confusion component of block ciphers by using MCDM. We have used entropy method for the selection of weight objectively. These weights were further used in SAW method for the selection of best cryptographic nonlinear component. Since the encryption scheme is highly depends on its nonlinear confusion component to add confusion characteristic in encryption scheme. According to Claude Shannon, one of the most important constituent of modern block ciphers design is confusion.

The rest article is structured as follow. In section 2, we have added various cryptographic characteristics of authenticating standard S-boxes. In section 3 is devoted to objective weight calculations namely entropy method. In simple additive weighting is used to select best S-box among other in section 4. Finally, conclusion is added in section 5.

2 Cryptographic analysis for standard S-boxes

To substantiate the cryptographic robustness of non-linear confusion component few techniques have been used which have been presented in the following section [55]. We have added some fundamental statistical and algebraic definitions related to nonlinear confusion component of block cipher. The standard cryptographic characteristics of S-boxes are nonlinearity, strict avalanche criterion, bit independent criteria, linear approximation probability and differential approximation probability. We have encrypted a standard Lena image with all available set of standard S-boxes and then analyzed its statistical characteristic which includes entropy, correlation, energy, contrast, mean absolute error and homogeneity.

2.1 Energy

It measures the energy of ciphered images administered by different S-boxes. Energy gives the total number of squares in the gray level co-event matrix gray. The mathematical expression for energy is given by the following relation: $\sum_{i, j} p {(i, j)}^{2},$ where p (i, j) represents the number of grey level co-occurrence matrices.

2.2 Homogeneity

It is a measure of nearness among the dispersal of elements in the grey level co-occurrence matrix (GLCM), also known as grey tone spatial dependency matrix (GTSDM) to GLCM diagonal. It represents the statistics of a combination of values of pixel vividness or gray levels in tabular form. The homogeneity is calculated as: $H = \sum_{i, j} \frac{p (i, j)}{1 + | i - j |},$ where p (i, j) is representing the gray-level co-occurrence matrices in GLCM.

2.3 Contrast

It is used to identify objects in image formation. Encrypted image has high levels of contrast due to the high random rate brought by using the S-box in the encryption process. Often, a contrast analysis enables the viewer to know clearly point to items which are present in the composition of the image. A ciphered image has high levels of contrast which is because of randomness presented with the use of the S-box in enciphering. The mathematical formula for contrast is given as follow: $C = \sum_{i, j} | i - {j |}^{2} p (i, j),$ where the total amount of gray-level co-occurrence matrices is represented by p (i, j).

2.4 Mean of absolute deviation

It is a measure of the alteration among an actual image and an enciphered image. It calculates the change between the actual image and the enciphered image. The definition of absolute deviation is limited to examining the differences among the two images. Mathematically it is given as: $MAD = \frac{1}{L \times L} \sum_{j = 1}^{L} \sum_{i = 1}^{L} | a_{ij} - b_{ij} |,$ where a_i,j represents the pixels of original image, b_i,j denotes the pixels of the enciphered image, and L the dimensions of the image.

2.5 Entropy

Entropy is a mathematical method that calculates the uncertainty in an image. Entropy is a measure of random statistics which is used to display the texture of an image. The calculation method of information entropy is given as: $K (s) = - \sum_{i = 1}^{n} p (i) {log}_{2} p (i) .$

The degree of randomness for each standard S-boxes is presented in Table 1.

Table 1
Entropy value of Lena image with different S-boxes

Images Entropy

Plain Image 7.4767

AES 7.4235

APA 7.4235

Prime 7.4235

S ₈ 7.4211

Gray 7.4235

Xyi 7.4235

SKIPJACK 7.4235

Images	Entropy
Plain Image	7.4767
AES	7.4235
APA	7.4235
Prime	7.4235
S ₈	7.4211
Gray	7.4235
Xyi	7.4235
SKIPJACK	7.4235

2.6 Correlation

It is a value that shows the relationship between the nearest pixel values in an encrypted image. The small amount of merging indicates that the encryption process achieved the highest randomness between adjacent pixels in the encrypted image. This analysis consists of three parts. First, the interaction between local and vertical pixels is calculated. To test the common similarity between the explicit image and the encrypted image, the complete image fusion is analyzed in one step. This helps us to find similarities or differences from a global perspective (See Table 2, Figs. 3 –10).

Table 2
Correlation coefficient of two adjacent pixels of plain image and ciphered image for different S-boxes

Images Vertical correlation Horizontal correlation Diagonal correlation

Plain Image 0.9782 0.9749 0.9593

AES 0.0972 0.0702 0.0627

APA 0.0935 0.0843 0.0717

PRIME 0.0674 0.0501 0.0368

S₈ 0.0908 0.0756 0.0630

Gray 0.0911 0.0725 0.0609

Xyi 0.1198 0.1039 0.0992

SKIPJACK 0.0612 0.0553 0.0498

Images	Vertical correlation	Horizontal correlation	Diagonal correlation
Plain Image	0.9782	0.9749	0.9593
AES	0.0972	0.0702	0.0627
APA	0.0935	0.0843	0.0717
PRIME	0.0674	0.0501	0.0368
S₈	0.0908	0.0756	0.0630
Gray	0.0911	0.0725	0.0609
Xyi	0.1198	0.1039	0.0992
SKIPJACK	0.0612	0.0553	0.0498

Fig. 2

Encryption of Lena image with standard S-boxes with one round.

Fig. 3

Correlation diagram of plain Lena image in different directions.

Fig. 4

Correlation diagram of encrypted images with AES S-box.

Fig. 5

Correlation diagram of encrypted images with APA S-box.

Fig. 6

Correlation diagram of encrypted images with Prime S-box.

Fig. 7

Correlation diagram of encrypted images with S₈ S-box.

Fig. 8

Correlation diagram of encrypted images with Gray S-box.

Fig. 9

Correlation diagram of encrypted images with Skipjack S-box.

Fig. 10

Correlation diagram of encrypted images with Xyi S-box.

2.7 Nonlinearity

The nonlinearity N_h of a Boolean function h is defined as the lowest value of distance from any affine function. It is given by: $N_{h} = \frac{1}{2} (2^{n} - {WHT}_{max} (h)),$ where WHT (h) the Walsh-Hadamard transformation of a Boolean function is h defined as: $\hat{F} h (α) = \sum_{x \in B^{n}} \hat{h} (x) {\hat{L}}_{α} (x),$ where $\hat{h} (x) = {(- 1)}^{h (x)}$ .

2.8 Strict avalanche criterion

Strict avalanche criterion is regarded as complete, on the off chance that the likelihood of change in output bits is 0.5 for single information bit change.

2.9 Bit independence criterion

Bit independence criteria of strict avalanche criteria means that the avalanche variables are pairwise autonomous for a given arrangement of avalanche vectors produced by supplementing of only the slightest bit.

2.10 Linear approximation probability

Linear approximation probability is a degree which determines the degree of unevenness of an occurrence. Mathematically written as: $LP = max_{Γ x, Γ y \neq 0} {\frac{# x | x . Γ_{x} = p (x) . Γ_{y}}{2^{n}} - \frac{1}{2}}$

2.11 Differential approximation probability

The formula of differential approximation probability is given by: $DP = {\frac{# x \in X | p (x) \oplus p (x \oplus Δ x) = Δ y}{2^{n}}},$ where Δx and Δy denote input and output differential. A minimal value of differential approximation probability indicates the power of S-box.

We can see in Figs. 11 and 12, the hierarchy for the weighting and ranking of our nonlinear confusion component. Initially have alternatives and criteria are decided which form the decision matrix. Next step is to identify which multi-criteria decision making technique should be adopted. Some methods require weight allocation which can be allocated using other weight assignment method. We have used entropy method for weight allocation. Next the beneficial and non-beneficial criteria are classified by the decision maker. In the end alternatives are ranked using some multi-criteria decision making method.

Fig. 11

Selection mechanism of proposed MCDM scheme.

Fig. 12

Flowchart for entropy method.

3 Entropy method for assigning weights

Entropy weight method (EWM) is a usually utilized estimation strategy that determines the distribution of value in decision-making. As the level of dispersion increases, the level of fragmentation increases, and more details can be obtained. In the meantime, high weight ought to be given a sign, and the other way around. Compared to the various subjective weight measuring techniques, the main benefit of EWM is to avoid distortion of the human material in the load of the displays, thus improving the observation of the full test results. Subsequently, EWM has been extensively utilized used in decision-making in recent years [56]. In the case of MCDM problems one of the most difficult tasks is to allocate weights accurately to conditions with respect to alternatives that can be ranked. Weight allocation to the beneficial criteria such as contrast, mean of absolute deviation and entropy, and non-beneficial criteria such as energy, homogeneity, and average correlation by utilizing entropy method. Table 3 shows the decision matrix of S-boxes AES, APA, Lui, Prime, S₈, Gray, Xyi and SKIPJACK which are the possible alternatives of the decision maker. Let Y₁ : Energy, Y₂ : Homogeneity, Y₃ : Contrast, Y₄ : Mean absolute error, Y₅ : Averagecorrelation, and Y₆ : Entropy be the criteria.

Table 3
Decision Matrix

Images Y₁ Y₂ Y₃ Y₄ Y₅ Y₆

AES 0.0211 0.4701 7.224 43.455 0.0815 7.9325

APA 0.0193 0.4665 8.9114 62.066 0.1258 7.8183

Lui 0.0211 0.4701 7.224 43.456 0.1311 7.9325

Prime 0.0198 0.4728 6.9646 53.089 0.2769 7.8811

S₈ 0.019 0.4552 8.1274 58.389 0.0734 7.9447

Gray 0.0198 0.4567 7.7961 49.723 0.1014 7.9299

Xyi 0.0188 0.4605 7.8942 57.238 0.1413 7.9127

SKIPJACK 0.0232 0.5004 5.4255 52.733 0.3123 7.8939

SUM 0.1621 3.7523 59.5672 420.149 1.2437 63.2456

Images	Y₁	Y₂	Y₃	Y₄	Y₅	Y₆
AES	0.0211	0.4701	7.224	43.455	0.0815	7.9325
APA	0.0193	0.4665	8.9114	62.066	0.1258	7.8183
Lui	0.0211	0.4701	7.224	43.456	0.1311	7.9325
Prime	0.0198	0.4728	6.9646	53.089	0.2769	7.8811
S₈	0.019	0.4552	8.1274	58.389	0.0734	7.9447
Gray	0.0198	0.4567	7.7961	49.723	0.1014	7.9299
Xyi	0.0188	0.4605	7.8942	57.238	0.1413	7.9127
SKIPJACK	0.0232	0.5004	5.4255	52.733	0.3123	7.8939
SUM	0.1621	3.7523	59.5672	420.149	1.2437	63.2456

3.1 Mathematical formulation of entropy method

Step 1: This step involves normalization of the decision matrix. The mathematical relation is given as: $t_{ij} = \frac{x_{ij}}{\sum_{i = 1}^{m} x_{ij}} .$

Step 2: Entropy is obtained by using following mathematical expression: $e_{j} = - k \sum_{i = 1}^{m} t_{ij} ln t_{ij}, j = 1, 2, \dots . ., n$

Step 3: The weight vector w_j is calculated which assigns weight to each beneficial and non-beneficial criterion. It uses the sum $\sum_{j = 1}^{n} (1 - e_{j})$ and is divided by 1 - e_j. The weight vector w_j is given by: $w_{j} = \frac{1 - e_{j}}{\sum_{j = 1}^{n} (1 - e_{j})}, j = 1, 2, 3, \dots ., n$

On the basis of these notions entropy method consists of following steps:

In this method decision matrix is normalized, then entropy is calculated which is being utilized for the calculation of weights of beneficial and non-beneficial criteria. The steps involved in this method are as under:

Step 1: Normalize the decision matrix

In this step each criteria value x_ij is divided by the columnar sum $\sum_{i = 1}^{m} x_{ij}$ . The normalized value of each criteria is represented by t_ij and is given below: $t_{ij} = \frac{x_{ij}}{\sum_{i = 1}^{m} x_{ij}} .$

Table 3 represents the normalized values of alternatives i.e., S-boxes of AES, APA, Lui, Prime, S₈, Gray, Xyi and SKIPJACK against each criterion i.e., Y₁: Energy, Y₂: Homogeneity, Y₃: Contrast, Y₄: Mean of absolute deviation, Y₅: Average correlation, Y₆: Entropy. The normalized values of these alternatives against each criterion are represented by {t₁, t₂, t₃, t₄, t₅, t₆}.

Step 2: Compute Entropy

Table 5 shows the entropy values of S-boxes. The value of entropy is obtained with the help of mathematical expression given as:

$e_{j} = - k \sum_{i = 1}^{m} t_{ij} ln t_{ij}, j = 1, 2, \dots . ., n$ where $k = \frac{1}{ln (p)}$ and p is number of alternatives. In our case p = 8, therefore $k = \frac{1}{ln (8)} = 0.480898$ . In Table 4 each entropy value e_j is calculated by finding the product of t_ij with ln t_ij and are represented by s₁, s_2, s_3, s_4, s_5, s₆. The following table shows the calculation of entropy for the given standard statistical characteristics of each S-box.

Table 4
Normalized decision matrix

t ₁ t ₂ t ₃ t ₄ t ₅ t ₆

AES 0.130167 0.125283 0.121275 0.103428 0.06553 0.125424

APA 0.119062 0.124324 0.149602 0.147724 0.10115 0.123618

Lui 0.130167 0.125283 0.121275 0.10343 0.105411 0.125424

Prime 0.122147 0.126003 0.11692 0.126358 0.222642 0.124611

S ₈ 0.117212 0.121312 0.136441 0.138972 0.059017 0.125617

Gray 0.122147 0.121712 0.130879 0.118346 0.081531 0.125383

Xyi 0.115978 0.122725 0.132526 0.136233 0.113613 0.125111

SKIPJACK 0.143122 0.133358 0.091082 0.12551 0.251106 0.124813

	t ₁	t ₂	t ₃	t ₄	t ₅	t ₆
AES	0.130167	0.125283	0.121275	0.103428	0.06553	0.125424
APA	0.119062	0.124324	0.149602	0.147724	0.10115	0.123618
Lui	0.130167	0.125283	0.121275	0.10343	0.105411	0.125424
Prime	0.122147	0.126003	0.11692	0.126358	0.222642	0.124611
S ₈	0.117212	0.121312	0.136441	0.138972	0.059017	0.125617
Gray	0.122147	0.121712	0.130879	0.118346	0.081531	0.125383
Xyi	0.115978	0.122725	0.132526	0.136233	0.113613	0.125111
SKIPJACK	0.143122	0.133358	0.091082	0.12551	0.251106	0.124813

Table 4

Entropy value calculated

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆
AES	–0.0626	–0.06025	–0.05832	–0.04974	–0.03151	–0.06032
APA	–0.05726	–0.05979	–0.07194	–0.07104	–0.04864	–0.05945
Lui	–0.0626	–0.06025	–0.05832	–0.04974	–0.05069	–0.06032
Prime	–0.05874	–0.06059	–0.05623	–0.06077	–0.10707	–0.05993
S ₈	–0.05637	–0.05834	–0.06561	–0.06683	–0.02838	–0.06041
Gray	–0.05874	–0.05853	–0.06294	–0.05691	–0.03921	–0.0603
Xyi	–0.05577	–0.05902	–0.06373	–0.06551	–0.05464	–0.06017
SKIPJACK	–0.06883	–0.06413	–0.0438	–0.06036	–0.12076	–0.06002
t_ij ln t_ij	–0.48089	–0.4809	–0.4809	–0.4809	–0.4809	–0.4809
e _j	0.231255	0.231259	0.231259	0.231259	0.231259	0.231259

Table 5

Weight vectors assigned to each criterion

	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆
AES	–0.0626	–0.06025	–0.05832	–0.04974	–0.03151	–0.06032
APA	–0.05726	–0.05979	–0.07194	–0.07104	–0.04864	–0.05945
Lui	–0.0626	–0.06025	–0.05832	–0.04974	–0.05069	–0.06032
Prime	–0.05874	–0.06059	–0.05623	–0.06077	–0.10707	–0.05993
S₈	–0.05637	–0.05834	–0.06561	–0.06683	–0.02838	–0.06041
Gray	–0.05874	–0.05853	–0.06294	–0.05691	–0.03921	–0.0603
Xyi	–0.05577	–0.05902	–0.06373	–0.06551	–0.05464	–0.06017
SKIPJACK	–0.06883	–0.06413	–0.0438	–0.06036	–0.12076	–0.06002
t_ij ln t_ij	–0.48089	–0.4809	–0.4809	–0.4809	–0.4809	–0.4809
e_j	0.231255	0.231259	0.231259	0.231259	0.231259	0.231259
1-e_j	0.768745	0.768741	0.768741	0.768741	0.768741	0.768741
w _j	0.166667	0.166667	0.166667	0.166667	0.166667	0.166667

Step 3: Compute the Weight Vector

Based on previous Table 4, we calculated the weights corresponding to each statistical properties of encrypted image after passing it through standard S-boxes. The tabulated results of step 3 is given in Table 5.

The weight vector w_j assigns weight to each beneficial and non beneficial criterion. It uses the sum $\sum_{j = 1}^{n} (1 - e_{j})$ and is divided by 1 - e_j. The weight vector w_j is given by: $w_{j} = \frac{1 - e_{j}}{\sum_{j = 1}^{n} (1 - e_{j})}, j = 1, 2, 3, \dots ., n .$

Table 5 represents the weight vector w_j which comprises the value of e_j’s calculated in step 2. We can see this weight vector has given equal importance to all the criteria. This means Energy, Homogeneity, Contrast, Mean of Absolute deviation, Average correlation and Contrast have been assigned an equal weightage which is 0.166667 in our case.

After assigning weight next step is to figure out that which alternative is best among all the alternatives i.e. the best S-box in this case. Here simple additive weighting (SAW) technique is acquired for finding the best S-box among all the eight S-boxes namely AES, APA, Lui, Prime, S₈, Gray, Xyi and SKIPJACK.

4 Simple additive weighting (SAW)

Simple Additive Weighting (SAW) which is otherwise called weighted direct mix or scoring strategies is a basic and frequently utilized multi attribute decision technique. The strategy depends on the weighted average. An assessment score is determined for every option by duplicating the scaled worth assigned to the option of that characteristic with the loads of comparative significance straightforwardly relegated by chief followed by adding of the items for all standards. The upside of this strategy is that it is a analogous straight change of the basic information which implies that the general significant degree of the normalized scores stays equivalent.

The test rating is calculated separately by multiplying the estimated value given some of that value and the estimated value given directly by the decision maker followed by summarizing the products of all the processes. The advantage of this method is that it is a uniform conversion of raw data which means that the corresponding order of the size of the guaranteed schools remains the same. The procedure of SAW consists of these steps:

4.1 Mathematical formulation of simple additive weighting

Step 1: In the first step b_ij and a_ij is calculated using the following formulas: $b_{ij} = {\begin{matrix} \frac{x_{ij}}{({max}_{i} x_{ij})}, for benefical criteria \\ \frac{1 / x_{ij}}{{max}_{i} (\frac{1}{x_{ij}})}, for non - beneficial criteria \end{matrix}$ where each a_ij is evaluated by: $a_{ij} = (1 / x_{ij}) .$

Step 2: This step involves ranking of criteria based upon S_i value. $S_{i} = \sum_{j = 1}^{M} w_{j} b_{ij i = 1, 2, 3, \dots ., N}$

Step 1: Calculation of b_ij and a_ij:

The value of b_ij for beneficial and non-beneficial criteria is calculated using the following mathematical relation of SAW technique are listed in Table 6.

Table 6
Calculation of w_j for beneficial criteria

s ₁ s ₂ b _ij b _ij s ₅ b _ij

w _j 0.166 0.166 0.166 0.166 0.166 0.1667

AES 0.021 0.470 7.224 43.45 0.081 7.9325

APA 0.019 0.466 8.911 62.06 0.125 7.8183

Lui 0.021 0.470 7.224 43.45 0.131 7.9325

Prime 0.019 0.472 6.964 53.08 0.276 7.8811

S₈ 0.019 0.455 8.127 58.38 0.073 7.9447

Gray 0.019 0.456 7.796 49.72 0.101 7.9299

Xyi 0.018 0.460 7.894 57.23 0.141 7.9127

SKIPJ 0.023 0.500 5.425 52.73 0.312 7.8939

MAX 0.023 0.500 8.911 62.06 0.312 7.9447

	s ₁	s ₂	b _ij	b _ij	s ₅	b _ij
w _j	0.166	0.166	0.166	0.166	0.166	0.1667
AES	0.021	0.470	7.224	43.45	0.081	7.9325
APA	0.019	0.466	8.911	62.06	0.125	7.8183
Lui	0.021	0.470	7.224	43.45	0.131	7.9325
Prime	0.019	0.472	6.964	53.08	0.276	7.8811
S₈	0.019	0.455	8.127	58.38	0.073	7.9447
Gray	0.019	0.456	7.796	49.72	0.101	7.9299
Xyi	0.018	0.460	7.894	57.23	0.141	7.9127
SKIPJ	0.023	0.500	5.425	52.73	0.312	7.8939
MAX	0.023	0.500	8.911	62.06	0.312	7.9447

Table 6 contains b_ij values for both beneficial and non-beneficial criteria such as contrast, mean of absolute deviation and entropy, and non-beneficial criteria such as energy, homogeneity, and average correlation. Table 7 contains a_ij values for beneficial and non-beneficial criteria.

Step 2: Ranking of Alternatives:

Table 7

Calculation of b_ij for non-beneficial criteria and a_ij

a _ij	a _ij	b_ij	b_ij
47.39	2.1272	0.0230	12.269
51.81	2.1436	0.0161	7.9491
47.39	2.1272	0.0230	7.6277
50.50	2.1150	0.0188	3.6114
52.63	2.1968	0.0171	13.624
50.50	2.1896	0.0201	9.8619
53.19	2.1715	0.0174	7.0771
43.10	1.9984	0.0189	3.2020
53.19	2.1968	0.0230	13.624

In this step the alternatives are ranked based upon the value of S_i. A minimum value of S_i is considered optimum. S_i value is calculated using the following mathematical form: $S_{i} = \sum_{j = 1}^{M} w_{j} b_{ij i = 1, 2, 3, \dots ., N}$

In Table 8, the weight vector w_j is product with each b_ij of beneficial and non-beneficial criteria and the row sum of the result is calculated. It can be clearly seen that S-box of S₈ is the best S-box among all the other S-boxes with the S_i value 0.973711.

Table 8

Ranking of the S-boxes

w _j	0.166667	0.166667	0.166667	0.166667	0.166667	0.166667
	Y₁	Y₂	Y₃	Y₄	Y₅	Y₆	S_i	RANK
AES	0.148500	0.161384	0.000395	0.116691	0.150103	0.166411	0.733483	7^th
APA	0.162349	0.162630	0.166667	0. 67000	0.097244	0.164015	0.919573	2^nd
Lui	0.148500	0.161384	0.135108	0.116693	0.093313	0.166411	0.821409	5^th
Prime	0.158249	0.160463	0.130257	0.142561	0.044180	0.165333	0.801042	6^th
S ₈	0.164913	0.166667	0.152004	0.156793	0.166667	0.166667	0.973711	1^st
Gray	0.158249	0.166120	0.145808	0.133522	0.120645	0.166357	0.890700	3^rd
Xyi	0.166667	0.164749	0.147643	0.153702	0.086577	0.165996	0.885334	4^th
SKIPJACK	0.135058	0.151612	0.101471	0.141605	0.039172	0.165601	0.744519	8^th

5 Conclusion

In this paper, an image encryption technique using some standard S-boxes has been studied. Preliminary a standard Lena image is encrypted using standard S-boxes such as AES, APA, Prime, S₈, Prime, Gray, Xyi and SKIPJACK. The confusion is obtained by using substitution box (S-box) and complex scrambling algorithm. Furthermore, statistical analyses are performed such as correlation, entropy, energy, contrast, homogeneity and mean of absolute deviation of the original Lena image and encrypted image are calculated. Secondly, MCDM is applied for finding the best S-box among all, which ranked the S-box of AES as optimum. We can conclude that S-box of S₈ (S₈-AES) is the best choice for encrypting the standard Lena image. To achieve better encryption effects, this work can be extended to a combination of two or more MCDM methods. The strength of this technique can be analyzed through different cryptographic analyses. MCDM methods can be used to select the best algorithm for lightweight cryptographic security and in other domains like transportation, business, engineering, and banking.

6 Future recommendation

To achieve better encryption effects this work can be extended to a combination of two or more multi-criteria decision making methods. The robustness of these methods can be analyzed through different cryptographic analyses. Multi-criteria decision making methods can be opted for the selection of best algorithm for lightweight cryptographic security and in other domains like transportation, business, banking, and engineering.

7 Conflict of interest

It is declared that authors have no conflict regarding the publication of this article.

Footnotes

Acknowledgments

The second author acknowledges Ajman University for supporting the research, Internal Research Grant No: [DGSR Ref. 2020-IRG-HBS-01].

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New combination of simple additive and entropy weighting criteria for the selection of best substitution box

Abstract

Keywords

1 Introduction

2.1 Energy

2.2 Homogeneity

2.3 Contrast

2.4 Mean of absolute deviation

2.5 Entropy

Table 1 Entropy value of Lena image with different S-boxes Images Entropy Plain Image 7.4767 AES 7.4235 APA 7.4235 Prime 7.4235 S 8 7.4211 Gray 7.4235 Xyi 7.4235 SKIPJACK 7.4235

2.8 Strict avalanche criterion

2.9 Bit independence criterion

2.10 Linear approximation probability

2.11 Differential approximation probability

4.1 Mathematical formulation of simple additive weighting

6 Future recommendation

7 Conflict of interest

Footnotes

Acknowledgments

References

Table 1
Entropy value of Lena image with different S-boxes

Images Entropy

Plain Image 7.4767

AES 7.4235

APA 7.4235

Prime 7.4235

S ₈ 7.4211

Gray 7.4235

Xyi 7.4235

SKIPJACK 7.4235