Security and Communication Networks
Volume 2018 (2018), Article ID 4987021, 9 pages
https://doi.org/10.1155/2018/4987021

Research Article

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

Abdul Razaq,¹ Hanan A. Al-Olayan,² Atta Ullah,³ Arshad Riaz,¹ and Adil Waheed⁴

¹Department of Mathematics, University of Education Lahore, Jauharabad Campus, Pakistan
²Department of Mathematics, King Saud University, Saudi Arabia
³Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan
⁴Department of Information Technology, University of Education Lahore, Jauharabad Campus, Pakistan

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

Received 4 June 2018; Accepted 19 September 2018; Published 4 October 2018

Academic Editor: Stelvio Cimato

Copyright © 2018 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.

In the literature, different algebraic techniques have been applied on Galois field to construct substitution boxes. In this paper, instead of Galois field , we use a cyclic group 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 of order 255 to generate . Next, we adjoin to and write the elements of in matrix to destroy the initial sequence . In the step, the randomness in the data is increased by applying certain permutations of the symmetric group on rows and columns of the matrix. In the last step we consider the symmetric group , 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.

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 than by ; that is, . Consider a transformation defined by

It can be easily verified that has order 255; that is, for any , Thus for all , generates a cyclic group =. In this paper, we have taken

Step I. First we simply present the elements of

in matrix (see Table 2). Cayley graph of is shown in Figure 1. In this way, the initial sequence 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 of .

Step II. Since we have presented our data in matrix, that is, a matrix with rows and 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 on the matrix. Since order of is , therefore corresponding to one matrix (S-box) formed after applying one permutation on rows, number of new S-boxes can be created by applying all the permutations on columns. Thus by this technique, we can construct different S-boxes. We choose two particular types of permutations of the symmetric group 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.

The 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 are amazingly constructive. In this step, we apply a permutations of (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 of 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: matrix evolved after 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
0

Table 3: matrix obtained after 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
155

Table 4: Proposed S-box evolved after 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

Here

is 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

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 fulfils the BIC requirements if , the output bits j and k, where , change independently by inverting the input bit . 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 input bit and the change in jth and kth output bits is identified. In the first phase the ith bit is varied from to n by keeping and bits fixed. Next, the values of j and k are altered from to n. 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 . 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 , 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 by and , respectively. Then mathematically, linear approximation probability is defined as follows.

In above expression denotes the set of all possible values in domain and 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 Table 9, 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 = when the input differential is = . In this analysis, the XOR distribution between the inputs and outputs of substitution box is computed. Mathematically, it is defined as

where denotes cardinality and is set of all inputs [3, 24, 25]. By using the approach introduced in [3], an input/output XOR distribution matrix of size 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
.02219

(a)

(b)

Figure 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.

5. Conclusion

In this study, we introduce a group theoretic technique to form strong S-boxes. The cyclic group instead of a Galois field is used to destroy the initial sequence . The construction of S-box involves three simple steps:

(i)First present the elements of in matrix.(ii)Next, apply two permutations of on rows and column of the matrix. It will significantly improve the performance of the S-box.(iii)In the last step, a permutation of 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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

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

Acknowledgments

This research project was supported by a grant from the Research Center of the Center for Female Scientific and Medical Colleges, Deanship of Scientific Research, King Saud University.

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