Security and Communication Networks
Volume 2017 (2017), Article ID 5101934, 16 pages
https://doi.org/10.1155/2017/5101934

Research Article

A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map

Abdul Razaq,¹ Awais Yousaf,² Umer Shuaib,³ Nasir Siddiqui,⁴ Atta Ullah,⁵ and Adil Waheed⁶

¹Department of Mathematics, University of Education Lahore, Jauharabad Campus, Jauharabad, Pakistan
²Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan
³Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan
⁴Department of Basic Sciences, University of Engineering and Technology, Taxila, Punjab, Pakistan
⁵Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
⁶Department of Information Technology, University of Education Lahore, Jauharabad Campus, Jauharabad, Pakistan

Correspondence should be addressed to Abdul Razaq; makenqau@gmail.com

Received 15 August 2017; Accepted 10 October 2017; Published 20 November 2017

Academic Editor: Zheng Yan

Copyright © 2017 Abdul Razaq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The substitution box is a basic tool to convert the plaintext into an enciphered format. In this paper, we use coset diagram for the action of on projective line over the finite field to construct proposed S-box. The vertices of the cost diagram are elements of which can be represented by powers of , where is the root of irreducible polynomial over . Let denote the elements of which are of the form of even powers of . In the first step, we construct a matrix with the elements of in a specific order, determined by the coset diagram. Next, we consider defined by to destroy the structure of . In the last step, we apply a bijective map on each element of the matrix to evolve proposed S-box. The ability of the proposed S-box is examined by different available algebraic and statistical analyses. The results are then compared with the familiar S-boxes. We get encouraging statistics of the proposed box after comparison.

1. Introduction

In secure communication, the role of the nonlinear component for block ciphers (substitution box) is of significant importance. The concept of substitution box was given by Shannon in 1949 [1]. In order to create confusion during the process of enciphering the digital data, substitution box plays a central role [2]. If the S-box is not good, it means one has to compromise on the quality of encryption. The strength of the S-box affirms the capability of block ciphers. Several attempts have been made to increase the quality of the S-box. In order to assess the properties of well-known S-boxes, the cryptographers have drawn attention to the literature. Different techniques have been developed to inspect the statistical and algebraic structure of S-boxes. These analyses include linear approximation probability (LP) method, bit independence criterion (BIC), majority logic criterion (MLC), strict avalanche criterion (SAC), nonlinearity method, and differential approximation probability (DP) method.

In this paper, we establish a novel technique to construct substitution boxes by coset diagrams and bijective maps.

2. Coset Diagrams for Modular Group

The modular group, denoted by , has a finite presentation , where and are linear fractional transformations which map to and , respectively. Coset diagrams ([3–7]) are the graphical representation of the action of on , where is a prime. Since the order of is three, its three cycles are represented by triangles. The vertices of the triangles, which are elements of , are permuted anticlockwise by . Any two of vertices of the triangles are joined by an edge which represents . The heavy dots are used to denote fixed points of and , if they exist.

Consider the action of modular group on (Figure 1). The permutation representations of and can be calculated by and .

The action of is not possible on , because image of 0 under does not belong to . Therefore, we choose for the action of instead of .

Figure 1: The coset diagram for the action of modular group on .

3. Coset Diagram Used in the Construction of Proposed Substitution Box

Consider a primitive irreducible polynomial over ; then . The element of (see Table 1) can be represented by some power of , where is the root of . Consider the action of on . Then the permutation representations of and can be calculated by and , respectively. So

The coset diagram for this action has 86 orbits (fragments). It has one copy of (Figure 2) and 85 copies of (Figure 3). First, we find the orbit of coset diagram which contains (Figure 4).

Table 1: Representation of the elements of .

Binary values

Binary values

Binary values

Binary values

000000000

000000001

000000010

000000100

000001000

000010000

000100000

001000000

010000000

100000000

000010001

000100010

001000100

010001000

100010000

000110001

001100010

011000100

110001000

100000001

000010011

000100110

001001100

010011000

100110000

001110001

011100010

111000100

110011001

100100011

001010111

010101110

101011100

010101001

101010010

010110101

101101010

011000101

110001010

100000101

000011011

000110110

001101100

011011000

110110000

101110001

011110011

111100110

111011101

110101011

101000111

010011111

100111110

001101101

011011010

110110100

101111001

011100011

111000110

110011101

100101011

001000111

010001110

100011100

000101001

001010010

010100100

101001000

010000001

100000010

000010101

000101010

001010100

010101000

101010000

010110001

101100010

011010101

110101010

101000101

010011011

100110110

001111101

011111010

111110100

111111001

111100011

111010111

110111111

101101111

011001111

110011110

100101101

001001011

010010110

100101100

001001001

010010010

100100100

001011001

010110010

101100100

011011001

110110010

101110101

011111011

111110110

111111101

111101011

111000111

110011111

100101111

001001111

010011110

100111100

001101001

011010010

110100100

101011001

010100011

101000110

010011101

100111010

001100101

011001010

110010100

100111001

001100011

011000110

110001100

100001001

000000011

000000110

000001100

000011000

000110000

001100000

011000000

110000000

100010001

000110011

001100110

011001100

110011000

100100001

001010011

010100110

101001100

010001001

100010010

000110101

001101010

011010100

110101000

101000001

010010011

100100110

001011101

010111010

101110100

011111001

111110010

111110101

111111011

111100111

111011111

110101111

101001111

010001111

100011110

000101101

001011010

010110100

101101000

011000001

110000010

100010101

000111011

001110110

011101100

111011000

110100001

101010011

010110111

101101110

011001101

110011010

100100101

001011011

010110110

101101100

011001001

110010010

100110101

001111011

011110110

111101100

111001001

110000011

100010111

000111111

001111110

011111100

111111000

111100001

111010011

110110111

101111111

011101111

111011110

110101101

101001011

010000111

100001110

000001101

000011010

000110100

001101000

011010000

110100000

101010001

010110011

101100110

011011101

110111010

101100101

011011011

110110110

101111101

011101011

111010110

110111101

101101011

011000111

110001110

100001101

000001011

000010110

000101100

001011000

010110000

101100000

011010001

110100010

101010101

010111011

101110110

011111101

111111010

111100101

111011011

110100111

101011111

010101111

101011110

010101101

101011010

010100101

101001010

010000101

100001010

000000101

000001010

000010100

000101000

001010000

010100000

101000000

010010001

100100010

001010101

010101010

101010100

010111001

101110010

011110101

111101010

111000101

110011011

100100111

001011111

010111110

101111100

011101001

111010010

110110101

101111011

011100111

111001110

110001101

100001011

000000111

000001110

000011100

000111000

001110000

011100000

111000000

110010001

100110011

001110111

011101110

111011100

110101001

101000011

010010111

100101110

001001101

010011010

100110100

001111001

011110010

111100100

111011001

110100011

101010111

010111111

101111110

011101101

111011010

110100101

101011011

010100111

101001110

010001101

100011010

000100101

001001010

010010100

100101000

001000001

010000010

100000100

000011001

000110010

001100100

011001000

110010000

100110001

001110011

011100110

111001100

110001001

100000011

000010111

000101110

001011100

010111000

101110000

011110001

111100010

111010101

110111011

101100111

011011111

110111110

101101101

011001011

110010110

100111101

001101011

011010110

110101100

101001001

010000011

100000110

000011101

000111010

001110100

011101000

111010000

110110001

101110011

011110111

111101110

111001101

110001011

100000111

000011111

000111110

001111100

011111000

111110000

111110001

111110011

111110111

111111111

111101111

111001111

110001111

100001111

000001111

000011110

000111100

001111000

011110000

111100000

111010001

110110011

101110111

011111111

111111110

111101101

111001011

110000111

100011111

000101111

001011110

010111100

101111000

011100001

111000010

110010101

100111011

001100111

011001110

110011100

100101001

001000011

010000110

100001100

000001001

000010010

000100100

001001000

010010000

100100000

001010001

010100010

101000100

010011001

100110010

001110101

011101010

111010100

110111001

101100011

011010111

110101110

101001101

010001011

100010110

000111101

001111010

011110100

111101000

111000001

110010011

100110111

001111111

011111110

111111100

111101001

111000011

110010111

100111111

001101111

011011110

110111100

101101001

011000011

110000110

100011101

000101011

001010110

010101100

101011000

010100001

101000010

010010101

100101010

001000101

010001010

100010100

000111001

001110010

011100100

111001000

110000001

100010011

000110111

001101110

011011100

110111000

101100001

011010011

110100110

101011101

010101011

101010110

010111101

101111010

011100101

111001010

110000101

100011011

000100111

001001110

010011100

100111000

001100001

011000010

110000100

100011001

000100011

001000110

010001100

100011000

000100001

001000010

010000100

100001000

Figure 2: The orbit of the coset diagram for the action of modular group on .

Figure 3: The orbit of the coset diagram for the action of modular group on .

Figure 4: The copy of having the vertex .

In each , one can see that is a path from the vertex to the vertex , such that, during this journey, one has to pass each vertex of .

4. Algebraic Structure of Proposed Substitution Box

In the literature, algebraic techniques are being applied on Galois fields. In this novel technique, the initial sequence of Galois field is destroyed with the help of vertices of the coset diagram.

Let denote the elements of which are of the form of even powers of .

Our first aim is to write the vertices of coset diagram in a matrix. For this, we choose only those vertices which belong to .

Step 1. Our coset diagram has 86 orbits; we construct a matrix having entries from in the following way.
First, we find the orbit of coset diagram which contains Let us denote this orbit by and apply on , so that we reach . During this journey, we pass through , , , and and at the end reach . Write , and as first three elements of the first row of matrix.

After writing 3 vertices of any , in order to select the next copy from , we find a vertex , where , , , , , and are the vertices of , such that , where . If is already exhausted in previously chosen copies of , then we move to the copy of containing and so on. Apply on so that we pass through each vertex. Note that, in each , only three vertices belong to out of 6. Write these 3 vertices in matrix in order. The process continues until all the vertices of are exhausted. Next, pick from and write it as last element of the last row.

Step 2. Consider defined by ; the elements of can be represented by powers of (see Table 3), where is the root of irreducible polynomial over . In this step, we apply on each element of the matrix evolved in the first step. In this way, we obtain a matrix having entries from . Next, we convert each entry of the matrix into binary form and ultimately into decimal form (Table 4).

Before going to Step 3, let us define a map in the following way.

Let be the linear fractional transformation, obtained by the action of on ; that is, .

Clearly is of the form , where , and is a mapping from But, usually is not a bijective map.

(i) Let there be elements of missing in the range of (ii) Let be repeated in the range of .(iii) Let be the smallest elements in whose images are , respectively.

Suppose that is the set of all those elements, except , whose images are , and is the set of missing elements in the range of . Then

In this paper, we have taken .

Step 3. In this step, the bijective map defined by

is applied on the matrix obtained at the end of Step 2 to evolve proposed S-box (Figure 5 and Table 5). More details of this last step are given in Table 2.

Table 2: Construction of S-box using linear fractional transformation.

0

66

1

82

2

95

254

2

255

0

Table 3: Representation of the elements of .

Binary values

Binary values

Binary values

Binary values

00000000

00000001

00000010

00000100

00001000

00010000

00100000

01000000

10000000

00011101

00111010

01110100

11101000

11001101

10000111

00010011

00100110

01001100

10011000

00101101

01011010

10110100

01110101

11101010

11001001

10001111

00000011

00000110

00001100

00011000

00110000

01100000

11000000

10011101

00100111

01001110

10011100

00100101

01001010

10010100

00110101

01101010

11010100

10110101

01110111

11101110

11000001

10011111

00100011

01000110

10001100

00000101

00001010

00010100

00101000

01010000

10100000

01011101

10111010

01101001

11010010

10111001

01101111

11011110

10100001

01011111

10111110

01100001

11000010

10011001

00101111

01011110

10111100

01100101

11001010

10001001

00001111

00011110

00111100

01111000

11110000

11111101

11100111

11010011

10111011

01101011

11010110

10110001

01111111

11111110

11100001

11011111

10100011

01011011

10110110

01110001

11100010

11011001

10101111

01000011

10000110

00010001

00100010

01000100

10001000

00001101

00011010

00110100

01101000

11010000

10111101

01100111

11001110

10000001

00011111

00111110

01111100

11111000

11101101

11000111

10010011

00111011

01110110

11101100

11000101

10010111

00110011

01100110

11001100

10000101

00010111

00101110

01011100

10111000

01101101

11011010

10101001

01001111

10011110

00100001

01000010

10000100

00010101

00101010

01010100

10101000

01001101

10011010

00101001

01010010

10100100

01010101

10101010

01001001

10010010

00111001

01110010

11100100

11010101

10110111

01110011

11100110

11010001

10111111

01100011

11000110

10010001

00111111

01111110

11111100

11100101

11010111

10110011

01111011

11110110

11110001

11111111

11100011

11011011

10101011

01001011

10010110

00110001

01100010

11000100

10010101

00110111

01101110

11011100

10100101

01010111

10101110

01000001

10000010

00011001

00110010

01100100

11001000

10001101

00000111

00001110

00011100

00111000

01110000

11100000

11011101

10100111

01010011

10100110

01010001

10100010

01011001

10110010

01111001

11110010

11111001

11101111

11000011

10011011

00101011

01010110

10101100

01000101

10001010

00001001

00010010

00100100

01001000

10010000

00111101

01111010

11110100

11110101

11110111

11110011

11111011

11101011

11001011

10001011

00001011

00010110

00101100

01011000

10110000

01111101

11111010

11101001

11001111

10000011

00011011

00110110

01101100

11011000

10101101

01000111

10001110

Table 4: matrix evolved after 2nd step.

1
190
65
2
23
46
142
89
35
4
216
71
8
163
113
96

138
173
16
32
58
108
235
6
64
98
165
54
47
25
128
185

194
27
161
28
29
116
45
191
214
131
56
193
207
233
146
31

232
212
20
213
127
250
205
169
41
125
24
44
135
51
33
176

188
7
19
59
218
155
143
88
38
206
102
76
180
37
137
145

22
152
215
220
67
132
11
229
153
139
90
223
189
203
104
252

117
167
72
251
55
91
234
211
13
243
114
197
201
241
141
204

26
247
156
200
245
3
192
93
244
144
80
177
84
122
12
124

42
61
221
240
48
70
60
40
123
36
17
151
18
157
106
101

9
133
230
39
82
49
78
158
179
69
92
115
134
118
172
202

154
86
74
79
246
150
148
43
53
255
249
62
68
195
164
97

239
181
210
34
184
231
242
119
21
149
121
219
236
238
198
81

178
166
182
159
237
99
162
73
85
140
94
147
83
103
100
5

222
129
10
186
208
224
130
248
107
199
112
160
136
183
14
75

109
105
217
57
111
52
126
50
171
66
95
253
168
174
77
227

87
63
228
175
225
110
254
226
196
15
30
170
120
187
209
0

Table 5: Proposed S-box evolved after 3rd step.

82
114
50
95
120
28
196
91
44
242
70
153
205
147
73
40

204
136
250
48
247
166
29
183
157
237
145
243
53
122
225
121

109
65
149
253
221
144
203
112
76
219
93
110
88
33
181
63

34
79
171
78
182
7
92
140
213
227
132
160
210
52
174
133

116
226
47
234
67
162
191
151
155
90
42
233
127
246
207
185

71
175
72
61
13
217
39
37
169
202
80
57
115
97
9
5

55
142
26
6
30
74
32
81
4
17
86
105
99
19
197
94

75
12
161
100
15
46
111
235
16
188
208
130
69
163
255
229

244
164
60
21
230
49
154
223
54
212
195
176
165
159
184
102

64
214
36
252
59
117
194
158
128
240
11
45
211
198
137
98

167
168
224
239
14
177
179
1
77
0
8
87
89
108
146
248

22
126
83
193
123
35
18
186
96
178
199
62
25
23
104
206

129
143
125
152
24
20
148
27
218
200
238
180
190
232
216
228

58
222
189
119
85
56
220
10
103
101
173
150
209
124
254
107

231
31
68
170
236
251
113
245
138
215
192
3
141
135
172
41

187
201
38
134
51
131
2
43
106
241
249
139
156
118
84
66

Figure 5: Flowchart for the construction of the proposed S-box.

5. Statistical Analysis and Simulation Results

In this section, we implement various security performance tests on newly created S-box to examine its special properties. The assessment of the characteristics of proposed S-box determines its application in different encryption methods and for security purposes. We use five different security performance tests, namely, linear approximation probability (LP), differential approximation probability (DP), nonlinearity, bit independence criterion (BIC), and strict avalanche criterion, to assess the cryptographic competence of substitution box. The results obtained from proposed S-box are then compared with the well-known S-boxes. The description of different types of tests implemented on these S-boxes is given below.

5.1. Nonlinearity

The concept of nonlinearity was first introduced by Pieprzyk and Finkelstein in 1988 [8]. It is the basic tool to measure the strength of the S-box. An S-box with bigger nonlinearity is more secure than that with lesser nonlinearity. The nonlinearity is expressed as

where is Walsh spectrum and .

The average value of the nonlinearity of the proposed S-box is 106.75. In Table 6, nonlinearity of the proposed S-box is compared with multiple renowned substitution boxes. One can see that the nonlinearity of the proposed S-box is better than most of the familiar S-boxes.

Table 6: Nonlinearity of basic functions of various substitution boxes.

S-box

Ave

Gray [13]
112
112
112
112
112
112
112
112
112

Gautam et al. [14]
108
106
104
98
102
102
98
74
99

Prime [15]
94
100
104
104
102
100
98
94
99.5

S₈ AES [16]
112
112
112
112
112
112
112
112
112

Shi et al. [17]
106
104
106
106
104
106
104
106
105

AES [2]
112
112
112
112
112
112
112
112
112

Skipjac and Kea [18]
104
108
108
108
108
104
104
106
106.75

Alkhaldi et al. [19]
108
104
106
106
102
98
104
108
104

Chen et al. [20]
100
102
103
104
106
106
106
108
104.3

Tang et al. [21]
100
103
104
104
105
105
106
109
104.5

Khan et al. [22]
102
108
106
102
106
106
106
98
104.25

Belazi et al. [23]
106
106
106
104
108
102
106
104
105.25

Proposed
108
106
108
108
108
104
106
106
106.75

5.2. Bit Independence Criterion

According to bit independence criterion [9, 10], if any input bit is inverted, then the output bits and must change independently. In other words, the avalanche variables must be pairwise independent for a given set of avalanche vectors. We have tested the nonlinearity of bit independence criterion of S-box (Table 7). We also compared the minimum and average values of bit independence criterion along with square deviation of the proposed S-box with different renowned S-boxes (Table 8).

Table 7: Nonlinearity of bit independence criterion of the proposed S-box.

—
96
104
104
106
106
108
106

96
—
106
102
102
102
106
106

104
106
—
104
102
106
102
106

104
102
104
—
104
102
104
102

106
102
102
104
—
104
104
96

106
102
106
102
104
—
102
106

108
106
102
104
104
102
—
104

106
106
106
102
96
106
104
—

Table 8: Bit independence criterion of various substitution boxes.

S-boxes
Minimum value
Average
Square deviation

Proposed
96
103.643
2.7283

Gray [13]
112
112
0

Gautam et al. [14]
92
103
3.5225

Prime [15]
94
101.71
3.53

S₈ AES [16]
112
112
0

Hussain et al. [12]
98
103.78
2.743

AES [2]
112
112
0

Hussain et al. [15]
102
104.14
1.767

5.3. Strict Avalanche Criterion

Strict avalanche criterion (SAC) introduced by Tavares and Webster is founded on the ideas of the avalanche and completeness effect [9, 10]. It is a formalization of the avalanche effect. If by complementing a single input bit all the output bits are changed with a 0.5 probability, SAC is said to be satisfied. Table 9 displays the outcomes of the strict avalanche criterion.

Table 9: Strict avalanche criterion of the proposed S-box.

.4375
.5000
.5000
.4844
.4844
.4844
.5312
.4375

.5000
.4375
.5000
.5625
.5000
.5469
.4688
.5000

.5312
.5469
.5156
.4687
.4531
.5312
.5156
.4375

.4844
.4375
.5469
.5312
.4688
.5781
.5781
.5000

.4688
.5156
.4531
.5625
.5781
.5469
.4844
.5156

.5000
.5156
.5312
.5000
.4688
.5156
.5488
.5000

.5156
.5313
.4844
.4687
.4531
.5000
.4688
.4531

.5312
.4844
.5625
.5469
.4844
.4531
.5625
.5000

5.4. Linear Approximation Probability

In linear approximation probability method, we examine the imbalance of an event [11]. The analysis is used to calculate the highest value of imbalance of the outcome of the event. The uniformity of the input bits should be similar to that of the output bits. Each th input bit is analyzed individually and its outcomes are checked in the output bits. The masks which are applied on the parity of both input and output bits are denoted by and , respectively. Mathematically,

where represents the collection of all possible inputs and is the total number of elements. The results of this important analysis obtained from our S-box and different established S-boxes are given in Table 10. The comparison shows that our S-box is strong enough to deal with different linear attacks.

Table 10: Linear approximation probability analyses of different S-boxes.

S-boxes
AES
Gray
Skipjack
Prime
Proposed
Gautam et al. [14]
S₈ AES
Xyi

Max value
144
144
156
162
162
164
144
168

Max LP
0.062
0.062
0.109
0.132
0.1484
0.2109
0.062
0.156

5.5. Differential Approximation Probability

In this analysis, differential uniformity is determined by examining the mapping from the input bits to the output. The main focus of this test is to ensure differential uniformity; that is, the input differential must be associated with an output differential in a unique way.

It is represented by

where and are the input and output differentials, respectively. We have applied differential approximation probability test on our S-box. The results are presented in Table 11.

Table 11: Differential probability of the proposed S-box.

.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.01562
.02344
.03125

.02344
.02344
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.02344
.02344
.02344

.01563
.02344
.02344
.03125
.03125
.03125
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.01562
.02344

.03125
.03906
.02344
.03125
.02344
.02344
.03125
.02344
.03125
.03125
.03125
.02344
.02344
.02344
.02344
.02344

.03125
.04688
.03906
.02344
.02344
.02344
.03125
.02344
.01562
.02344
.03125
.02344
.02344
.02344
.02344
.03125

.03125
.03125
.02344
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.02344
.03125
.02344
.03125
.02344
.02344

.02344
.03125
.03125
.03906
.03125
.02344
.03125
.03906
.02344
.02344
.03125
.02344
.03125
.02344
.03125
.02344

.01563
.03125
.03125
.03125
.03125
.03125
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.03125
.02344
.02344

.03125
.03125
.03906
.02344
.03125
.02344
.02344
.03125
.03125
.03125
.02344
.03125
.02344
.03906
.03125
.03125

.03125
.02344
.03125
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.03125
.02344
.02344

.02344
.03125
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.02344
.03125
.02344
.02344
.02344
.03125

.02344
.02344
.03125
.02344
.02344
.02344
.02344
.02344
.02344
.02344
.03125
.03125
.02344
.02344
.02344
.02344

.02344
.02344
.02344
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.03125
.03125
.03125

.03125
.02344
.04688
.02344
.02344
.02344
.03906
.03125
.03125
.02344
.02344
.02344
.03125
.03125
.02344
.03125

.02344
.02344
.03906
.03125
.02344
.02344
.02344
.02344
.02344
.03125
.03125
.02344
.03125
.02344
.03906
.03125

.03125
.03125
.02344
.02344
.03125
.03125
.02344
.02344
.02344
.03125
.02344
.02344
.02344
.03125
.02344
—

6. Majority Logic Criterion

The majority logic criterion [12] is helpful in finding the premier candidate S-box fit for a certain kind of encryption application. In this criterion, image encryption strength of the S-box is investigated through statistical studies. A distortion in the image is created by encryption process; therefore, it is important to study the statistical characteristics. It is achieved with the help of various analyses such as entropy, contrast, correlation, energy, and homogeneity. The proposed S-box can further be used for encryption and multimedia security. In this paper, 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 6 shows the result of image encryption with proposed S-box. The histogram of the original image and the encrypted images of Baboon and Pepper are shown in Figure 7. These results show that our S-box fulfills all the requirements to be declared as a very suitable S-box for encryption applications. Thus, it is recommended to become a part of algorithms designed for the secure transmission of information/data.

Table 12: Comparison of MLC for proposed S-box over different S-boxes.

S-boxes
Entropy
Contrast
Correlation
Energy
Homogeneity

Pepper image

    Plaintext
7.5909
0.2760
0.9383
0.1288
0.9024

    Proposed
7.9532
8.5155
0.0079
0.0174
0.4100

    AES
7.9211
7.5509
0.0554
0.0202
0.4662

    Ullah et al. [24]
7.9823
8.6727
−0.0043
0.0173
0.4076

    Skipjack
7.7561
7.7058
0.1205
0.0239
0.4708

    Khan et al. [25]
7.9562
8.3129
0.0103
0.0180
0.4219

    Belazi
7.9233
8.1423
−0.0112
0.0286
0.4648

Baboon image

    Plaintext
7.1273
0.7179
0.6782
0.1025
0.7669

    Proposed
7.9551
8.5267

0.0174
.4088

    AES
7.2531
7.5509
0.0554
0.0202
0.4662

    Prime
6.9311
7.6236
0.0855
0.0202
0.4640

    Xyi
7.2531
8.3108
0.0417
0.0196
0.4533

    Skipjack
7.2531
7.7058
0.1025
0.0193
0.4689

    Khan et al. [25]
7.9612
8.1213
−0.0512
0.0210
0.4011

    Belazi
7.9252
8.0391
0.0119 0
.02219
0.4428

(a)

(b)

Figure 6: Original image and the encrypted images using two rounds of encryption: (a) Pepper and (b) Baboon.

(a)

(b)

Figure 7: Histogram of the original image and the encrypted images: (a) Pepper and (b) Baboon.

7. Conclusion

In the present study, a strong S-box is created with the help of coset graph for the action of modular group and a bijective map. According to our information, this is the first use of coset graphs in the construction of S-box. The proposed S-box is highly secure and the results obtained from different analyses are nearly equal to the ideal ones. Therefore, it is very useful for secure communication.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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