# A Novel Technique for the Construction of Safe Substitution Boxes Based on Cyclic and Symmetric Groups

Authors: Abdul Razaq; Hanan A. Al-Olayan; Atta Ullah; Arshad Riaz; Adil Waheed
Journal: Security and Communication Networks
 (2018)

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## Abstract

In the literature, different algebraic techniques have been applied on Galois fieldGF(28) to construct substitution boxes. In this paper, instead of Galois field GF(28), we use a cyclic group C255 in the formation of proposed substitution box. The construction proposed S-box involves three simple steps. In the first step, we introduce a special type of transformation T of order 255 to generate C255. Next, we adjoin 0 to C255 and write the elements of C255∪0 in 16×16 matrix to destroy the initial sequence 0,1,2,…,255. In the 2nd step, the randomness in the data is increased by applying certain permutations of the symmetric group S16 on rows and columns of the matrix. In the last step we consider the symmetric group S256, and positions of the elements of the matrix obtained in step 2 are changed by its certain permutations to construct the suggested S-box. The strength of our S-box to work against cryptanalysis is checked through various tests. The results are then compared with the famous S-boxes. The comparison shows that the ability of our S-box to create confusion is better than most of the famous S-boxes.


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## Body

## 1. Introduction

The foundation of modern cryptography was laid by Shannon [1]. Cryptography is the science of converting the secret information into dummy data so that it could reach the destination safely without leakage of the information. The modern cryptography is divided into several branches. However, symmetric key cryptography and public key cryptography are the two main areas of study. In symmetric key cryptography, the same key is used at both ends to encrypt and decrypt data/information, but in public key cryptography two different keys, public and private keys, are used. It is well-known that, in symmetric key cryptography the substitution box is a standout and basic ingredient, which performs substitution. In block ciphers, it is widely used to make the relationship between the ciphertext and the key unclear and vague. Due to these important applications of substitution box many algorithms have been developed to construct safer and more reliable S-boxes. Substitution boxes are used for the strong design of block encryption algorithms. S-box is the only nonlinear component for most of the block encryption algorithms such as international data encryption algorithm (IDEA), advanced encryption standard (AES), and data encryption standard (DES) [2]. Substitution boxes yield a DES-like cryptosystem with the perplexity property depicted by Shannon. In [3], it is shown that for weaker S-boxes, DES can be easily broken. It means that the security of DES-like cryptosystems is merely determined by the quality of the S-boxes used. Thus, in order to develop secure cryptosystems, the formation of safe S-boxes is a main focus of the researcher. To examine the strength of S-boxes, nonlinearity test, bit independent criterion, strict avalanche criterion, linear approximation probability analysis, differential uniformity test, and majority logic criterion are used. In the literature, there are many S-box construction methods such as inversion mapping, power polynomial, heuristic methods, and pseudorandom methods [4]. Incursions on the S-box component of data encryption standard (DES) damage the design process of advanced encryption standard (AES) [3, 5]. Therefore, the substitution box component of AES is designed to ensure the security of the data/information in the presence of differential and linear cryptanalysis attacks [6].Recently, since proposed algebraic attacks have been succeeded in some loops of AES, researchers have focused on alternative construction methods for substitution box [21]. Therefore, substitution box construction techniques based on group theory have been applied for alternative substitution box designs.


## 2. Algebraic Structure of Proposed Substitution Box

Let us denote a set of positive integers less than256 by Υ; that is, Υ=1,2,3,…,255. Consider a transformation T:Υ→Υ defined by(1)Tz=z16z+1mod257ifz∈Υ-16,136z32z+1mod257ifz=16,136.It can be easily verified that T has order 255; that is, for any z∈Υ, T255z=z. Thus for all z∈Υ, Tz generates a cyclic group C255=Tz,T2z,T3z,…,T254z,z. In this paper, we have taken z=1.Step I.First we simply present the elements of(2)C255∪0=T1,T21,T31,…,T2541,1,0in 16×16 matrix (see Table 2). Cayley graph of C255∪0 is shown in Figure 1. In this way, the initial sequence 0,1,2,…,255 is destroyed. If this matrix is conceded as S-box, its nonlinearity is 103.75, which is acceptable. Now we move to step II to create more randomness.Figure 1
Cayley graph ofC255∪0.Step II.Since we have presented our data in 16×16 matrix, that is, a matrix with 16 rows and 16 columns, the randomness can be increased by interchanging the positions of the rows and columns. Algebraically, it is achieved by applying permutations of the symmetric group S16 on the matrix. Since order of S16 is 16!, therefore corresponding to one matrix (S-box) formed after applying one permutation on rows, 16! number of new S-boxes can be created by applying all the permutations on columns. Thus by this technique, we can construct 16!2 different S-boxes. We choose two particular types of permutations of the symmetric group S16 such that one of them is applied on the rows and the other on columns. This action increases the diffusion capability of the cipher. The permutations are as follows.(3)12,7,113,6,10,13,5,84,9,12141516: applied on rows1,5,10,13,15,3,7,122,6,9,144,8,11,16: applied on columnsThe resulting S-box (see Table 3) has nonlinearity of 106.25. In step III, we further enhance its working capability.Step III.Recently, we have noticed that certain permutations of the symmetric group S256 are amazingly constructive. In this step, we apply a permutations of S256 (see Table 1) on the data/matrix obtained after step II to construct a very strong S-box (see Table 4).Table 1
The permutation ofS256 used in step 3.
(1 225 221 169 78 255 136 173 62 146 56 119 229 114 117 174 143 247 105 16 197 139 201 205 124 15 103 80 133 228 74 13 166 127 226 53 219 181 209 45 251 60 43 232 160 239 71 9 64 231 208 18 98 115 254 213 150 75 82 27 111 230 11 227 184 30 212 241 248 170 17 235 32 249 207 77 69 95 252 81 222 149 92 73 57 23 162 61 89 220 211 175 91 33 157 223 68 159 14 54 83 191 193 0 102 24 90 183 126 116 134 37 144 244 192 35 253 233 216 187 196 198 104 84 47 155 178 106 34 128 101 206 50 148 94 245 19 93 40 97 171 165 125 189 195 63 121 3 164 4 29 137 129 52 203 79 123 177 182 176 39 96 215 238 67 107 210 25 179 141 242 31 243 41 8 200 186 110 199 152 108 65 12 237 59 85 118 113 46 120 142 185 20 147 190 28 36 153 140 5 135 99 21 49 7 38 188 55 42 240 109 167 145 2 236 151 122 224 218 132 163 86 180 48 131 194 88 10 26 156 246 168 214 100 58 6 66 204 22 130 51 202 158 172 234 161) (44 72 154 76 138 217 112) (70 250 87)Table 2
16 × 16 matrix evolved after 1st step.
121 148 21 87 165 53 116 2 39 174 106 4 91 8 16 241 249 166 253 151 83 218 255 141 204 92 170 236 109 136 120 199  175 102 244 46 44 85 133 118 47 183 221 68 157 128 202 60  95 228 201 216 250 51 219 122 161 23 131 22 27 171 178 195  40 59 247 152 224 220 144 239 52 34 84 207 167 64 150 101  111 30 186 176 188 114 123 229 130 108 160 125 208 154 187 238  6 61 24 209 245 140 98 194 254 11 138 142 35 214 180 89  153 226 66 20 177 158 203 252 192 76 14 112 67 110 200 72  163 248 169 26 164 17 242 42 73 232 78 212 182 32 225 75  45 179 25 184 215 15 240 93 231 88 9 94 185 57 147 190  145 243 181 65 5 54 99 80 237 191 31 104 168 77 43 222  115 119 246 3 63 159 117 12 48 233 196 251 19 70 103 49  132 97 149 127 28 134 143 69 81 71 227 146 156 107 193 90  50 173 223 205 18 113 37 33 105 10 198 217 62 79 86 230  235 126 234 96 135 38 206 7 41 56 29 162 197 55 129 100  189 36 74 210 139 124 172 213 211 13 155 82 58 137 1 0Table 3
16 × 16 matrix obtained after 2nd step.
4 8 16 241 121 148 21 87 53 165 2 116 174 39 91 106 104 77 43 222 145 243 181 65 54 5 80 99 191 237 168 31  112 110 200 72 153 226 66 20 158 177 252 203 76 192 67 14  251 70 103 49 115 119 246 3 159 63 12 117 233 48 19 196  146 107 193 90 132 97 149 127 134 28 69 143 71 81 156 227  68 128 202 60 175 102 244 46 85 44 118 133 183 47 157 221  236 136 120 199 249 166 253 151 218 83 141 255 92 204 109 170  207 64 150 101 40 59 247 152 220 224 239 144 34 52 167 84  22 171 178 195 95 228 201 216 51 250 122 219 23 161 27 131  125 154 187 238 111 30 186 176 114 188 229 123 108 130 208 160  142 214 180 89 6 61 24 209 140 245 194 98 11 254 35 138  212 32 225 75 163 248 169 26 17 164 42 242 232 73 182 78  94 57 147 190 45 179 25 184 15 215 93 240 88 231 185 9  217 79 86 230 50 173 223 205 113 18 33 37 10 105 62 198  162 55 129 100 235 126 234 96 38 135 7 206 56 41 197 29  82 137 1 0 189 36 74 210 124 139 213 172 13 211 58 155Table 4
Proposed S-box evolved after3rd step.
142 125 220 89 219 63 251 158 149 46 126 146 28 208 144 218 245 9 189 17 120 240 159 166 79 165 128 73 241 26 137 7 118 83 78 99 228 21 138 183 1 246 117 170 217 207 60 75 145 231 171 22 55 39 242 154 134 199 56 213 214 11 147 53 255 148 41 62 71 244 197 203 133 100 30 188 185 140 93 253 172 69 119 151 12 180 139 57 233 65 35 111 43 238 132 66 20 77 201 173 84 155 91 179 74 32 193 176 29 164 80 113 59 235 136 52 64 175 3 192 19 186 156 88 6 169 61 110 51 243 14 18 227 101 121 58 191 143 45 114 225 152 254 153 24 48 222 70 105 50 206 25 72 127 67 5 112 215 90 96 135 181 195 16 194 174 92 36 10 210 236 130 216 40 86 248 239 229 54 102 33 212 44 129 161 184 205 226 34 187 202 0 182 178 232 42 106 190 204 87 122 103 49 107 15 249 124 234 163 141 37 237 211 209 221 38 250 198 115 85 162 68 108 224 4 167 2 95 247 109 196 252 13 98 104 8 116 223 160 177 230 23 168 131 47 123 27 82 31 97 76 157 200 150 81 94


## 3. Security Analysis

In this section, a point by point exploration of the suggested S-box is presented. Furthermore, we have made a comparison with the famous S-boxes, such as AES S-box, Xyi S-box, Skipjack S-box, S8 AES S-box, Residue Prime S-box, APA S-box, and Gray S-box. The illustration of various analysis applied on these substitution boxes is given. It is seen that our S-box meets all the standards near the ideal status.


### 3.1. Nonlinearity

The key objective of the substitution box is to provide assistance in giving nonlinear change from unique data to the encoded information. The measure of nonlinearity presented by the cipher considered as the most important part in the entire process of encryption. It is defined as(4)Nf=2r-11-2-rmax⁡Wfz.Here (5)Wfz=∑zϵF2r-1fx⊗⍭.zis the Walsh Spectrum. The average values of the nonlinearity of newly constructed S-box is 112. A comparison between the nonlinearity of the suggested S-box and multiple renowned substitution boxes is given in Table 5.Table 5
The nonlinearity test outcomes of different substitution boxes.
S boxes 0 1 2 3 4 5 6 7 Ave Suggested S-box 112 112 112 112 112 112 112 112 112 Coset Diagram S-box [7] 108 106 108 108 108 104 106 106 106.75 Gray [8] 112 112 112 112 112 112 112 112 112 Arun [9] 108 106 104 98 102 102 98 74 99 Prime [10] 94 100 104 104 102 100 98 94 99.5 S8 AES [11] 112 112 112 112 112 112 112 112 112 Xyi [12] 106 104 106 106 104 106 104 106 105 AES [6] 112 112 112 112 112 112 112 112 112 Skipjack [13] 104 108 108 108 108 104 104 106 106.75 Alkhaldi [14] 108 104 106 106 102 98 104 108 104 Chen [15] 100 102 103 104 106 106 106 108 104.3 Tang [16] 100 103 104 104 105 105 106 109 104.5 Khan [17] 102 108 106 102 106 106 106 98 104.25 Belazi [18] 106 106 106 104 108 102 106 104 105.25


### 3.2. Bit Independence Criterion

Webster and Tavares firstly demonstrated bit independence criterion [22]. A function h:0,1n→0,1n fulfils the BIC requirements if ∀i,j,k∈1,2,3,…,n, the output bitsj andk, where j≠k, change independently by inverting the input bit i. In cryptographic systems, the BIC is a very important characteristic because by increasing independence between bits, it is very hard to decipher and predict the scheme of the system. The outcomes of nonlinearity of BIC are presented in Table 6. In order to find the independence properties a comparison of the bits, created by the eight basic functions, with each other is established. The relationship between the outcomes of change in ith input bit and the change in jth and kth output bits is identified. In the first phase the ith bit is varied from 1 ton by keeping jth and kth bits fixed. Next, the values ofj andk are altered from 1 ton. Furthermore, the minimum and average values of BIC along with square deviation of the proposed S-boxes are presented in Table 7. The average and minimum values of BIC of the proposed S-box are 112. The square deviation of the newly created substitution box is 0. All these results are better than most of the well-known S-boxes and similar to AES, S8 AES, and Gray S-boxes.Table 6
BIC nonlinearity for the suggested S-box.
Rows/Columns 0 1 2 3 4 5 6 7 0 - 112 112 112 112 112 112 112 1 112 - 112 112 112 112 112 112 2 112 112 - 112 112 112 112 112 3 112 112 112 - 112 112 112 112 4 112 112 112 112 - 112 112 112 5 112 112 112 112 112 - 112 112 6 112 112 112 112 112 112 - 112 7 112 112 112 112 112 112 112 -Table 7
BIC results of different S-boxes.
S-boxes Minimum value Average Square deviation Suggested S-box 112 112 0 Gray 112 112 0 Arun 92 103 3.5225 Prime 94 101.71 3.53 S8 AES 112 112 0 Xyi 98 103.78 2.743 AES 112 112 0 Skipjack 102 104.14 1.767


### 3.3. Strict Avalanche Criterion Analytically

Tavares and Webster introduced strict avalanche criterion [22]. In this criterion, the output bits are examined after changing a single input bit. In ideal condition, by changing a single input bit, half of the output bits change their shape. In [23] an effective technique is presented to check whether a complete substitution box satisfies the SAC or not. The results of SAC of the suggested S-box (see Table 8) are nearly equal to 1/2, which shows its strength.Table 8
Values of SAC for the suggested S-box.
Rows/Columns 0 1 2 3 4 5 6 7 0 0.4844 0.4688 0.4844 0.5469 0.4688 0.5625 0.5469 0.5313 1 0.4531 0.5313 0.5156 0.5469 0.4688 0.4688 0.4844 0.5469 2 0.5000 0.4688 0.4609 0.4688 0.5156 0.4531 0.5469 0.5625 3 0.5313 0.5234 0.5313 0.5156 0.4531 0.4688 0.5234 0.4375 4 0.5625 0.4844 0.4688 0.5156 0.5469 0.5469 0.5625 0.4844 5 0.5000 0.4844 0.5156 0.5625 0.4844 0.5469 0.4844 0.5156 6 0.4844 0.5156 0.5000 0.4844 0.4844 0.4844 0.4688 0.5625 7 0.5469 0.5625 0.4531 0.4688 0.5156 0.4844 0.5313 0.4844


### 3.4. Linear Approximation Probability

In this analysis, the imbalance of an event is examined. It is useful in finding the maximum value of an imbalance of the output in an event. Let us denote the input and output masks byTx and Ty, respectively. Then mathematically, linear approximation probability is defined as follows.(6)LP=maxTx,Ty≠0⁡#x∈I/x•Tx=Sx•Ty2n-12In above expression I denotes the set of all possible values in domain and 2n is the number of elements of the S-box.The maximum LP value is 0.0625, which is matching with the best known S-boxes such as Gray, APA, and AES. In Table9, a comparison of the results of this analysis, between our S-box and some famous S-boxes, is given.Table 9
Linear approximation probability analyses of different S-boxes.
S-boxes Suggested S-box AES Skipjack Prime Gray Arun S8 AES Xyi Max value 144 144 156 162 144 164 144 168 Max LP 0.062 0.062 0.109 0.132 0.062 0.2109 0.062 0.156


### 3.5. Differential Uniformity

Differential uniformity is another important method of block cipher cryptanalysis. It was introduced by Biham and Shamir to break block ciphers [3]. It exploits certain events of I/O differences and represents the maximum likelihood of generating an output differential Δk = Ki⊕Kj when the input differential is Δh = Hi⊕Hj. In this analysis, the XOR distribution between the inputs and outputs of substitution box is computed. Mathematically, it is defined as(7)DU=#hεHSh⨁Sh⨁Δh=Δkwhere # denotes cardinality and H is set of all inputs h [3, 24, 25]. By using the approach introduced in [3], an input/output XOR distribution matrix of size 16×16 is calculated for suggested S-box and is provided in Table 10. As a general S-box design guideline, the maximum differential uniformity has to be kept as low as possible to withstand differential attacks. The highest value of differential uniformity for suggested S-box is 4, which is compared with some well-known S-boxes in Table 11 to show the strength of suggested S-box.Table 10
Differential uniformity of proposed S-box.
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 - - - - -Table 11
Maximum differential uniformity of various S-boxes.
S-boxes Suggested S-box AES Gray Skipjack Chen Khan S8 AES Tang Xyi Max DU 4 4 4 12 12 16 4 10 12


## 3.1. Nonlinearity

The key objective of the substitution box is to provide assistance in giving nonlinear change from unique data to the encoded information. The measure of nonlinearity presented by the cipher considered as the most important part in the entire process of encryption. It is defined as(4)Nf=2r-11-2-rmax⁡Wfz.Here (5)Wfz=∑zϵF2r-1fx⊗⍭.zis the Walsh Spectrum. The average values of the nonlinearity of newly constructed S-box is 112. A comparison between the nonlinearity of the suggested S-box and multiple renowned substitution boxes is given in Table 5.Table 5
The nonlinearity test outcomes of different substitution boxes.
S boxes 0 1 2 3 4 5 6 7 Ave Suggested S-box 112 112 112 112 112 112 112 112 112 Coset Diagram S-box [7] 108 106 108 108 108 104 106 106 106.75 Gray [8] 112 112 112 112 112 112 112 112 112 Arun [9] 108 106 104 98 102 102 98 74 99 Prime [10] 94 100 104 104 102 100 98 94 99.5 S8 AES [11] 112 112 112 112 112 112 112 112 112 Xyi [12] 106 104 106 106 104 106 104 106 105 AES [6] 112 112 112 112 112 112 112 112 112 Skipjack [13] 104 108 108 108 108 104 104 106 106.75 Alkhaldi [14] 108 104 106 106 102 98 104 108 104 Chen [15] 100 102 103 104 106 106 106 108 104.3 Tang [16] 100 103 104 104 105 105 106 109 104.5 Khan [17] 102 108 106 102 106 106 106 98 104.25 Belazi [18] 106 106 106 104 108 102 106 104 105.25


## 3.2. Bit Independence Criterion

Webster and Tavares firstly demonstrated bit independence criterion [22]. A function h:0,1n→0,1n fulfils the BIC requirements if ∀i,j,k∈1,2,3,…,n, the output bitsj andk, where j≠k, change independently by inverting the input bit i. In cryptographic systems, the BIC is a very important characteristic because by increasing independence between bits, it is very hard to decipher and predict the scheme of the system. The outcomes of nonlinearity of BIC are presented in Table 6. In order to find the independence properties a comparison of the bits, created by the eight basic functions, with each other is established. The relationship between the outcomes of change in ith input bit and the change in jth and kth output bits is identified. In the first phase the ith bit is varied from 1 ton by keeping jth and kth bits fixed. Next, the values ofj andk are altered from 1 ton. Furthermore, the minimum and average values of BIC along with square deviation of the proposed S-boxes are presented in Table 7. The average and minimum values of BIC of the proposed S-box are 112. The square deviation of the newly created substitution box is 0. All these results are better than most of the well-known S-boxes and similar to AES, S8 AES, and Gray S-boxes.Table 6
BIC nonlinearity for the suggested S-box.
Rows/Columns 0 1 2 3 4 5 6 7 0 - 112 112 112 112 112 112 112 1 112 - 112 112 112 112 112 112 2 112 112 - 112 112 112 112 112 3 112 112 112 - 112 112 112 112 4 112 112 112 112 - 112 112 112 5 112 112 112 112 112 - 112 112 6 112 112 112 112 112 112 - 112 7 112 112 112 112 112 112 112 -Table 7
BIC results of different S-boxes.
S-boxes Minimum value Average Square deviation Suggested S-box 112 112 0 Gray 112 112 0 Arun 92 103 3.5225 Prime 94 101.71 3.53 S8 AES 112 112 0 Xyi 98 103.78 2.743 AES 112 112 0 Skipjack 102 104.14 1.767


## 3.3. Strict Avalanche Criterion Analytically

Tavares and Webster introduced strict avalanche criterion [22]. In this criterion, the output bits are examined after changing a single input bit. In ideal condition, by changing a single input bit, half of the output bits change their shape. In [23] an effective technique is presented to check whether a complete substitution box satisfies the SAC or not. The results of SAC of the suggested S-box (see Table 8) are nearly equal to 1/2, which shows its strength.Table 8
Values of SAC for the suggested S-box.
Rows/Columns 0 1 2 3 4 5 6 7 0 0.4844 0.4688 0.4844 0.5469 0.4688 0.5625 0.5469 0.5313 1 0.4531 0.5313 0.5156 0.5469 0.4688 0.4688 0.4844 0.5469 2 0.5000 0.4688 0.4609 0.4688 0.5156 0.4531 0.5469 0.5625 3 0.5313 0.5234 0.5313 0.5156 0.4531 0.4688 0.5234 0.4375 4 0.5625 0.4844 0.4688 0.5156 0.5469 0.5469 0.5625 0.4844 5 0.5000 0.4844 0.5156 0.5625 0.4844 0.5469 0.4844 0.5156 6 0.4844 0.5156 0.5000 0.4844 0.4844 0.4844 0.4688 0.5625 7 0.5469 0.5625 0.4531 0.4688 0.5156 0.4844 0.5313 0.4844


## 3.4. Linear Approximation Probability

In this analysis, the imbalance of an event is examined. It is useful in finding the maximum value of an imbalance of the output in an event. Let us denote the input and output masks byTx and Ty, respectively. Then mathematically, linear approximation probability is defined as follows.(6)LP=maxTx,Ty≠0⁡#x∈I/x•Tx=Sx•Ty2n-12In above expression I denotes the set of all possible values in domain and 2n is the number of elements of the S-box.The maximum LP value is 0.0625, which is matching with the best known S-boxes such as Gray, APA, and AES. In Table9, a comparison of the results of this analysis, between our S-box and some famous S-boxes, is given.Table 9
Linear approximation probability analyses of different S-boxes.
S-boxes Suggested S-box AES Skipjack Prime Gray Arun S8 AES Xyi Max value 144 144 156 162 144 164 144 168 Max LP 0.062 0.062 0.109 0.132 0.062 0.2109 0.062 0.156


## 3.5. Differential Uniformity

Differential uniformity is another important method of block cipher cryptanalysis. It was introduced by Biham and Shamir to break block ciphers [3]. It exploits certain events of I/O differences and represents the maximum likelihood of generating an output differential Δk = Ki⊕Kj when the input differential is Δh = Hi⊕Hj. In this analysis, the XOR distribution between the inputs and outputs of substitution box is computed. Mathematically, it is defined as(7)DU=#hεHSh⨁Sh⨁Δh=Δkwhere # denotes cardinality and H is set of all inputs h [3, 24, 25]. By using the approach introduced in [3], an input/output XOR distribution matrix of size 16×16 is calculated for suggested S-box and is provided in Table 10. As a general S-box design guideline, the maximum differential uniformity has to be kept as low as possible to withstand differential attacks. The highest value of differential uniformity for suggested S-box is 4, which is compared with some well-known S-boxes in Table 11 to show the strength of suggested S-box.Table 10
Differential uniformity of proposed S-box.
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 - - - - -Table 11
Maximum differential uniformity of various S-boxes.
S-boxes Suggested S-box AES Gray Skipjack Chen Khan S8 AES Tang Xyi Max DU 4 4 4 12 12 16 4 10 12


## 4. Majority Logic Criterion

In majority logic criterion, statistical analyses are performed to examine the statistical strength of the S-box in image encryption application [26]. The encryption process creates a distortion in the image, these kinds of distortions determine the strength of the algorithm. Therefore, it is necessary to investigate the statistical properties through various analyses. These analyses are correlation, entropy, contrast, homogeneity, and energy. The suggested S-boxes can further be used for encryption and multimedia security. We have used two JPEG images, Pepper and Baboon, for MLC analysis. The results of these analyses in comparison with the other well-known S-boxes are depicted in Table 12. Figure 2 shows the result of image encryption with proposed S-box. The histograms of the original image and the encrypted images of Baboon and Pepper are shown in Figure 3. These results indicate that the proposed S-box is suitable for encryption applications and is adequate enough to become part of the algorithms designed for the secure transmission of information/data.Table 12
Comparison of Majority logic criterion for S-box over various S-boxes.
S-boxes Correlation Entropy Contrast Homogeneity Energy Pepper Image  Plain Text 0.9383 7.5909 0.2760 0.9024 0.1288  Suggested S-box -0.0134 7.9842 8.6969 0.4045 0.0174  Atta [19] −0.0043 7.9823 8.6727 0.4076 0.0173  Skipjack 0.1205 7.7561 7.7058 0.4708 0.0239  Khan [20] 0.0103 7.9562 8.3129 0.4219 0.0180  Belazi −0.0112 7.9233 8.1423 0.4648 0.0286  Baboon Image  Plain Text 0.6782 7.1273 0.7179 0.7669 0.1025  Suggested S-box -0.0060 7.9820 8.6488 0.4062 0.0174  AES 0.0554 7.2531 7.5509 0.4662 0.0202  Prime 0.0855 6.9311 7.6236 0.4640 0.0202  Xyi 0.0417 7.2531 8.3108 0.4533 0.0196  Skipjack 0.1025 7.2531 7.7058 0.4689 0.0193  Khan [14] −0.0512 7.9612 8.1213 0.4011 0.0210  Belazi 0.0119 0 7.9252 8.0391 0.4428 .02219Figure 2
Original image and the encrypted images using two rounds of encryption: (a) Pepper and (b) Baboon.
(a) (b)Figure 3
Histogram of the original image and the encrypted images: (a) Pepper and (b) Baboon.
(a) (b)


## 5. Conclusion

In this study, we introduce a group theoretic technique to form strong S-boxes. The cyclic groupC255 instead of a Galois field is used to destroy the initial sequence 0,1,2,…,255. The construction of S-box involves three simple steps:(i)
First present the elements ofC255∪0=T1,T21,T31,…,T2541,1,0 in 16×16 matrix.(ii)
Next, apply two permutations ofS16 on rows and column of the matrix. It will significantly improve the performance of the S-box.(iii)
In the last step, a permutation ofS256 is applied on the matrix (obtained in step (ii)) to form proposed S-box.The results acquired from different analyses show that the performance of our S-box against various algebraic attacks is much better than most of well-known S-boxes and similar to AES, S8 AES, and Gray S-boxes. Therefore, our S-box meets all the requirements and is considered as a strong S-box for the secure communication.



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