Security and Communication Networks
Volume 2020 (2020), Article ID 8883884, 14 pages
https://doi.org/10.1155/2020/8883884

Research Article

The Effect of the Primitive Irreducible Polynomial on the Quality of Cryptographic Properties of Block Ciphers

Sajjad Shaukat Jamal,¹ Dawood Shah,² Abdulaziz Deajim,¹ and Tariq Shah²

¹Department of Mathematics, College of Science, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia
²Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Correspondence should be addressed to Sajjad Shaukat Jamal; shussain@kku.edu.sa

Received 23 May 2020; Revised 3 August 2020; Accepted 28 August 2020; Published 24 September 2020

Academic Editor: Tom Chen

Copyright © 2020 Sajjad Shaukat Jamal 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.

Substitution boxes are the only nonlinear component of the symmetric key cryptography and play a key role in the cryptosystem. In block ciphers, the S-boxes create confusion and add valuable strength. The majority of the substitution boxes algorithms focus on bijective Boolean functions and primitive irreducible polynomial that generates the Galois field. For binary field F2, there are exactly 16 primitive irreducible polynomials of degree 8 and it prompts us to construct 16 Galois field extensions of order 256. Conventionally, construction of affine power affine S-box is based on Galois field of order 256, depending on a single degree primitive irreducible polynomial over . In this manuscript, we study affine power affine S-boxes for all the distinct degree primitive irreducible polynomials over to propose 16 different substitution boxes. To perform this idea, we introduce 16 affine power affine transformations and, for fixed parameters, we obtained 16 distinct S-boxes. Here, we thoroughly study S-boxes with all possible primitive irreducible polynomials and their algebraic properties. All of these boxes are evaluated with the help of nonlinearity test, strict avalanche criterion, bit independent criterion, and linear and differential approximation probability analyses to measure the algebraic and statistical strength of the proposed substitution boxes. Majority logic criterion results indicate that the proposed substitution boxes are well suited for the techniques of secure communication.

1. Introduction

The exchange of digital data through the Internet has revolutionized the communication parameters over the years. But this rapid communication also provides opportunities to access this digital data illegally. For this reason, the security of this content on the Internet has become a serious challenge for the researchers of different fields. To counter the emerging challenges of security, cryptography and steganography are used to hide the secret information whereas watermarking is used for copyright protection. In this manuscript, we discuss cryptography and relevant aspects of this field. For convenience, cryptography is divided into two types named symmetric key cryptography and asymmetric key cryptography. In symmetric key cryptography, two parties share secret information and keys during encryption and decryption procedures. The private key is shared by both sender and receiver. In addition to this, block ciphers and stream ciphers are two main branches of symmetric key cryptography. In 1949, Shannon gave the idea of block cipher and some examples of block ciphers are Advanced Encryption Standard (AES) [1], Data Encryption Standard (DES), International Data Encryption Algorithm (IDEA), and many more [2, 3]. In AES, there is availability of three different key sizes such as 128, 192, and 256 bits, whereas in DES, the only available key size is 56 bits. The AES has 10, 12, and 14 rounds for key sizes of 128, 192, and 256 bits, respectively. All these rounds have four basic steps, that is, subbyte, shift row, mix column, and add round key. Subbyte is the step which substitutes the plaintext data with substitution box (S-box). This S-box is the only nonlinear part of block cipher used in different well-known cryptosystems. It is used to create confusion to make plaintext data obscure for any attacker and hence S-box is an integral part of any cryptosystem. S-box is a function which has input and output from the Galois field. The Galois field is a finite field having order 256 and denoted by .

1.1. Related Work

S-box is used to create confusion as observed in AES, International Data Encryption Algorithm (IDEA), DES, and many more cryptosystems [4]. It is an established fact that the strength of block cipher depends on the standard and quality of S-box. Due to the necessary immersion of S-box to generate nonlinearity, intricacy persuades different researchers to design strong S-boxes to enhance the security level of cryptosystems. Among different available methods, the algebraic structure-based construction of S-boxes has much more attention. These S-boxes have strong cryptographic features and are robust against linear and differential cryptanalysis.

In the literature, different structural advancements are viewed to improve the quality of S-boxes. The algebraic complexity of AES S-box has been improved with the extension of this S-box, that is, affine power affine (APA) [5]. Furthermore, the symmetric group S₈ has also been applied to AES S-box to improve the quality and numbers of S-boxes [6]. Similarly, the application of transformation using binary gray codes on AES S-box gives Gray S-box [7]. In [8], S-boxes are constructed by using the projective general linear group (PGL). Moreover, the construction scheme of chaotic S-boxes using DNA sequence and chaotic Chen system is given in [9, 10]. Different analytical, algebraic, and chaos-based techniques for the construction of S-boxes are given in [11–16]. Conventionally, AES uses a polynomial of 8 terms which have all the required properties and improves the security for AES. But the Gray S-box has a -term polynomial. Moreover, residue prime, Xyi, and Skipjack S-boxes are frequently used for the encryption and decryption schemes [17, 18].

It is assumed that the model of Boolean functions and primitive irreducible polynomial has an impact on the strength of S-box. In [19], different primitive irreducible polynomials have been used to identify the effect of primitive irreducible polynomial. To investigate this fact, we want to study all the primitive irreducible polynomials to understand whether there is an impact of irreducible polynomial or not. Archetypally in the synthesis of an S-box, the numbers and in affine transformation belong to Galois field . As the polynomial ring has 16 primitive irreducible polynomials of degree 8, it shows that only 16 opportunities are available for constructing Galois fields . In this paper, we have constructed 16 different robust S-boxes over the elements of these 16 irreducible polynomials. Firstly, we define 16 affine power affine transformations on these different Galois fields which can be given as ; here, for values, we would be able to get 16 distinct S-boxes.

1.2. Motivation

Due to the role of S-boxes in cryptosystems, it is essential to explore all of its aspects. The motivation behind this work is to study all primitive irreducible polynomials and their role in the construction of S-boxes.

(1)The Mobius transformation used in a different construction of S-boxes has certain limitations and restrictions in its structure [7]. For example, the condition on the parameters, i.e., squeezes the remaining cases. Hence, there is a need for any other transformation.(2)There are 16 primitive irreducible polynomials in the principal ideal domain whose impact was not studied yet regarding their impression on analyses of S-boxes.(3)By exploring all primitive irreducible polynomials, we have a better opportunity to obtain the cryptographically strong cryptosystems.

1.3. Our Contribution

In this manuscript, we studied all binary degree 8 primitive irreducible polynomials for the construction of S-boxes. The quality of the proposed work can be seen from the different security analyses and resistance against malicious attacks. This whole study can be summarized as follows:

(1)We constructed S-boxes associated with the 16 binary degree 8 primitive irreducible polynomials.(2)The APA transformation is used in this work, which is bijective and has no restrictions on the parameters.(3)To evaluate the strength of the proposed S-boxes, we have performed different analyses along with differential cryptanalysis. The outcomes of these analyses are compared with the well-known S-boxes.

The remaining part of the paper is planned as follows: Section 2 presents the preliminaries and construction scheme of the proposed S-boxes. In Section 3, algebraic and statistical analyses are calculated in detail. Section 4 presents definitions of the balanced Boolean function. Section 5 concludes the paper.

2. Primitive Irreducible Polynomials of Degree 8 and GF (2⁸)

2.1. The Galois Fields

We summarize here some well-known facts from the theory of rings and fields. Let be a commutative ring with identity. A nonempty subset of is called an ideal of if is an additive subgroup of and for every , where . If, furthermore, there does not exist a proper ideal of properly containing , then we say that is a maximal ideal of Besides; is said to be a field if each of its nonzero elements has a must inverse in . If is a field of prime characteristic , then is an extension of the prime field . A polynomial is said to be irreducible if it cannot be factored in into two polynomials of strictly smaller degrees. The principal ideal,

generated by a monic irreducible polynomial is a maximal ideal in . If is of degree , then the quotient ring,

is an extension field of of degree consisting of elements. This field is called a Galois field and is denoted by and is said to be the field extension of defined by the irreducible polynomial . A representative of each element of can be chosen to be of degree strictly less than . If is a root of in an algebraic closure of , then is isomorphic to the field:

and so we can identify the two fields. Furthermore, if is a generator of the cyclic finite multiplicative group of nonzero elements of , then we say that is primitive.

The Galois field is particularly of specific interest in cryptographic applications, especially in S-boxes constructions. For our cryptographic purposes, we are interested in such a field whose defining irreducible polynomial is “primitive” (of degree 8, of course). It is well known that there are such polynomials over , for example, , which we list in Table 1. In the following section, we construct 16 S-boxes out of the Galois fields corresponding to the aforementioned sixteen primitive irreducible polynomials.

Table 1: Primitive irreducible polynomials and their corresponding Galois fields.

Primitive polynomials roots
Galois field

2.2. The Proposed S-Box Construction Method

For each , consider the affine power affine map (APA):

where and are two affine maps with , and

Among other things, the map , which is obviously bijective, was introduced by [5] to produce confusion in the scheme. For our S-boxes, we choose , and and Figure 1 demonstrates the flow chart of the construction of the 16 different S-boxes. Moreover, the construction of S-boxes in correspondence to polynomial 1 (P₁) to polynomial 16 (P₁₆) is shown in Figure 1. All the S-boxes are given in Tables 2–17, corresponding to P₁ to P₁₆. These tables are before the conclusion section.

Figure 1: Flow chart for the construction of the proposed S-boxes and S-boxes corresponding to P₁ and P₁₆.

Table 2: S-box corresponding to P2.

Proposed S-box 1

176
141
139
249
75
179
4
69
48
62
243
197
49
105
167
250

240
60
189
182
188
230
178
101
236
21
153
110
38
155
127
207

91
213
29
55
88
78
244
215
81
221
158
219
74
201
210
255

119
90
37
42
43
147
67
83
96
22
99
253
144
11
56
9

63
223
205
80
140
71
121
125
120
54
168
187
82
202
10
70

26
19
161
186
183
65
232
64
172
41
93
44
40
137
34
128

164
241
163
25
87
214
76
196
1
31
45
8
198
97
102
246

2
77
235
233
51
57
118
100
117
208
143
30
138
217
66
157

211
245
15
134
180
126
114
142
89
148
254
218
123
85
154
169

103
165
6
195
84
224
184
107
203
200
145
229
0
7
231
226

58
3
53
212
191
159
35
242
5
220
216
247
192
204
136
228

79
115
92
227
248
73
50
112
132
108
225
16
252
23
130
185

33
133
237
61
173
171
27
86
113
181
175
116
251
177
170
234

12
150
18
14
68
174
149
39
17
124
199
32
131
28
162
20

160
111
135
13
104
129
239
98
222
109
146
72
152
206
190
238

52
95
166
24
209
46
47
59
122
156
106
151
194
94
36
193

Table 3: S-box corresponding to P2.

Proposed S-box 2

186
170
249
53
208
106
49
220
147
203
143
207
250
177
133
19

240
84
152
38
18
21
58
59
67
219
15
209
70
150
41
121

64
255
231
139
182
117
25
100
47
125
109
77
190
113
110
148

118
197
198
146
39
138
108
161
75
13
103
200
73
91
105
204

80
104
130
124
2
132
214
63
144
48
112
181
72
196
37
151

3
135
10
129
221
184
26
111
40
222
11
127
223
31
29
87

86
189
205
23
253
114
89
218
176
193
229
180
201
158
4
69

22
188
228
85
241
236
140
239
16
164
57
79
8
56
24
102

211
160
142
153
54
68
213
210
230
71
238
235
82
252
52
98

33
7
65
42
5
1
32
162
212
192
175
90
17
245
43
119

178
44
36
46
166
171
51
234
195
62
145
45
247
14
168
174

217
122
159
0
101
237
60
27
163
202
243
99
116
120
131
6

224
232
20
123
34
55
137
206
155
226
81
173
93
169
107
172

167
199
191
94
136
96
183
149
233
246
179
156
9
83
50
95

61
28
254
88
141
165
126
35
128
66
76
134
216
97
242
187

244
30
74
215
12
115
185
194
78
154
227
251
225
248
92
157

Table 4: S-box corresponding to P3.

Proposed S-box 3

126
250
162
102
32
143
129
192
28
200
47
42
155
131
177
221

240
142
29
49
138
120
94
46
20
64
5
99
116
27
50
167

67
165
117
133
108
175
96
0
174
54
58
251
223
164
181
44

154
88
238
69
89
194
201
193
22
75
61
137
123
72
86
169

121
71
144
18
159
90
16
189
105
31
51
26
8
149
176
38

65
110
48
183
210
196
217
59
163
237
118
93
13
241
140
208

173
247
119
33
233
228
66
170
80
222
166
161
160
146
236
224

179
63
115
81
98
231
150
97
243
229
178
215
17
6
14
245

211
199
253
197
85
41
43
76
3
62
128
55
91
40
124
60

73
134
77
21
1
107
218
180
204
112
15
230
246
104
151
130

168
220
227
145
103
185
191
2
182
82
52
202
34
190
25
152

139
114
187
186
141
30
122
113
23
92
111
101
12
239
127
213

87
156
188
216
184
153
37
249
70
10
226
255
95
4
84
206

148
214
172
225
254
45
135
24
35
11
53
39
195
9
252
248

203
36
205
100
242
244
235
209
125
74
232
207
212
158
147
19

109
56
219
157
171
7
106
57
234
83
136
79
132
68
198
78

Table 5: S-box corresponding to P4.

Proposed S-box 4

49
103
220
208
58
70
173
238
242
160
154
169
158
110
139
94

240
200
78
149
155
117
219
131
50
161
226
14
48
189
1
147

115
120
218
34
51
7
104
182
167
150
192
64
178
95
57
91

176
3
100
159
205
134
63
85
243
22
69
236
177
144
210
20

44
79
206
89
96
74
66
146
197
15
0
108
13
215
25
181

235
93
194
221
73
18
246
56
140
106
244
247
148
237
88
232

191
251
234
99
9
33
59
252
45
179
222
125
132
109
229
136

77
41
162
8
188
126
202
97
67
217
163
98
84
171
29
42

211
38
213
119
46
168
5
116
204
129
75
31
152
239
166
172

68
10
11
6
35
83
212
186
43
105
249
199
255
196
111
32

87
184
156
24
223
190
195
153
92
230
112
209
39
52
80
12

113
198
60
40
138
164
76
101
72
61
145
122
174
23
19
86

170
183
214
245
135
231
36
54
224
81
143
124
30
228
4
114

26
53
127
203
141
133
187
90
118
201
27
180
71
130
17
185

82
175
137
165
28
216
227
37
55
47
2
157
248
233
123
254

142
121
253
128
102
225
193
16
107
250
241
62
207
65
151
21

Table 6: S-box corresponding to P5.

Proposed S-box 5

61
163
105
177
30
219
248
58
41
7
9
127
151
118
169
196

240
34
202
51
77
191
126
233
215
39
254
197
20
48
93
192

122
128
97
95
181
217
65
64
173
23
91
0
198
43
90
16

175
87
70
162
168
92
2
49
100
245
193
249
205
107
55
139

108
33
11
114
74
81
53
15
6
110
206
137
200
104
247
116

19
182
14
46
60
115
158
167
130
113
17
204
4
218
141
176

147
1
138
216
213
179
226
150
253
31
85
66
231
187
243
57

71
236
244
225
45
38
165
40
172
201
188
119
224
242
29
232

211
214
54
185
42
227
190
82
159
44
157
250
235
221
171
28

155
35
52
124
120
84
78
136
59
123
36
88
251
83
134
21

3
143
153
223
102
129
209
111
131
140
144
184
98
80
178
220

12
148
186
99
149
210
239
94
234
37
152
230
63
241
67
189

22
194
238
203
56
207
117
26
86
112
18
237
68
47
212
229

166
146
199
228
121
101
180
174
76
72
32
160
69
208
10
125

246
73
252
103
164
8
79
109
25
27
24
106
142
154
183
62

96
255
132
156
75
145
50
135
13
133
222
161
170
5
195
89

Table 7: S-box corresponding to P6.

Proposed S-box 6

162
120
205
183
16
20
137
191
89
95
128
239
159
10
219
94

240
155
187
248
196
92
139
223
12
136
18
222
80
140
131
158

100
53
236
50
192
72
118
167
132
42
3
51
90
138
15
28

249
246
174
221
11
161
19
126
150
0
254
96
13
49
181
112

81
163
108
26
135
216
234
99
85
14
78
62
177
104
179
207

189
22
255
27
178
225
212
237
147
114
253
165
75
17
244
184

199
166
87
208
71
242
195
175
101
247
79
243
149
44
217
201

38
36
37
5
251
63
125
66
69
218
190
86
103
30
148
123

211
198
33
48
232
197
105
9
93
31
107
145
194
185
41
227

40
146
214
58
228
77
144
152
57
35
180
172
241
60
55
6

8
61
24
229
169
1
168
171
154
204
52
233
21
213
119
245

56
133
113
127
45
98
115
65
64
134
173
200
102
202
110
74

182
157
141
76
91
68
34
224
59
203
206
170
215
153
73
39

67
29
130
142
116
186
88
160
47
106
2
54
193
121
231
210

7
252
117
250
176
4
84
43
124
235
209
151
122
230
226
129

32
23
188
111
156
220
164
238
82
97
143
70
83
46
109
25

Table 8: S-box corresponding to P7.

Proposed S-box 7

86
101
177
89
199
188
236
165
198
145
112
232
92
164
76
137

240
244
5
87
151
98
181
129
117
94
230
108
39
29
184
206

103
123
56
180
35
142
246
168
48
36
64
222
187
111
196
15

54
190
120
104
173
1
208
105
162
224
251
12
253
172
57
170

114
3
153
159
185
204
19
245
128
50
97
140
227
127
214
44

176
218
118
209
4
53
247
68
186
249
6
93
25
150
28
88

0
254
115
147
85
154
221
99
69
45
70
136
130
32
8
134

171
139
250
243
248
174
191
20
31
2
43
33
255
22
152
27

211
71
61
109
95
47
9
226
135
67
143
124
160
125
228
18

7
49
26
79
220
179
231
223
148
210
51
241
141
113
200
16

192
197
239
146
24
183
252
82
167
126
225
242
58
219
52
107

233
55
149
235
77
178
157
201
65
40
216
74
207
133
131
78

83
122
73
102
66
213
138
80
11
182
34
46
195
10
193
17

237
75
119
110
238
156
21
72
203
175
234
41
189
158
116
215

205
13
91
202
106
38
229
30
81
23
37
63
144
96
59
42

62
100
84
166
14
60
217
161
169
132
163
90
212
194
155
121

Table 9: S-box corresponding to P8.

Proposed S-box 8

41
6
47
28
109
248
100
139
227
141
111
60
7
125
178
31

240
224
53
123
91
63
67
237
71
217
11
205
252
133
61
22

97
172
182
233
124
51
196
236
255
170
128
3
186
179
83
119

29
206
234
210
218
1
151
220
245
68
13
213
54
214
30
15

23
98
95
82
189
212
48
120
187
219
239
222
202
192
154
246

143
24
131
181
102
27
204
184
129
137
114
74
122
57
188
199

94
75
20
79
76
158
92
130
39
19
96
134
103
108
33
229

89
21
175
50
166
195
87
203
164
46
174
121
25
34
241
16

211
238
104
112
235
14
116
251
191
86
253
230
225
52
155
38

58
177
173
40
45
148
62
42
201
149
190
250
216
157
59
244

169
153
150
117
142
101
145
9
207
208
247
160
140
8
64
180

194
88
37
162
115
35
156
5
90
36
84
106
152
232
159
110

144
73
80
126
200
165
113
56
161
49
127
12
168
0
66
226

147
32
132
146
243
4
43
138
55
163
167
135
223
81
93
209

69
99
107
44
198
105
228
10
65
77
215
197
193
176
18
17

249
78
242
85
70
72
171
183
26
221
254
231
118
136
185
2

Table 10: S-box corresponding to P9.

Proposed S-box 9

164
17
78
216
114
139
98
51
31
89
58
243
221
249
159
18

240
64
151
225
120
210
180
135
57
115
204
72
62
155
227
195

109
95
127
141
76
229
178
246
12
147
47
235
166
217
122
212

205
26
27
138
148
70
171
130
140
215
203
156
146
11
106
233

242
162
6
206
142
16
119
218
253
169
104
213
102
41
245
231

197
250
65
86
90
152
237
241
167
254
4
14
208
182
71
101

232
68
111
10
134
55
209
103
189
61
74
116
20
40
118
186

42
192
131
226
43
113
63
157
38
69
8
60
149
44
255
53

211
82
224
110
32
94
49
236
80
160
196
222
185
108
188
154

176
123
238
117
30
247
88
150
181
144
132
35
37
198
96
97

136
92
23
75
121
81
112
83
125
239
87
2
143
19
105
201

137
214
15
56
172
179
133
252
36
220
85
184
50
234
39
194

73
99
173
79
124
46
25
100
168
3
33
228
191
145
165
161

199
77
126
28
183
59
170
22
54
202
223
66
163
219
187
174

177
244
175
5
128
67
230
7
93
45
29
21
158
13
129
9

0
48
200
107
34
24
251
248
1
190
84
52
193
153
91
207

Table 11: S-box corresponding to P10.

Proposed S-box 10

109
26
163
213
77
207
155
87
34
96
136
40
177
25
128
11

240
147
166
252
115
158
185
123
146
68
239
149
160
180
6
111

22
198
208
243
103
97
24
187
179
228
132
110
188
10
151
130

62
119
131
60
219
148
245
9
101
81
205
3
222
98
203
19

154
178
66
117
246
108
226
135
202
224
59
236
192
156
141
112

217
28
39
167
172
104
44
230
54
253
82
168
23
4
150
58

107
64
43
48
20
118
181
102
173
254
200
76
237
209
143
204

63
193
221
214
14
88
16
126
223
90
157
85
46
169
196
191

211
42
1
199
176
100
216
152
183
89
37
99
75
233
72
122

184
52
79
206
235
194
225
71
15
162
50
210
242
127
153
234

57
70
165
248
164
5
67
55
2
129
124
189
21
171
241
86

251
159
125
134
32
93
116
12
255
106
175
73
170
35
139
84

41
120
18
27
78
17
182
174
53
215
83
247
94
227
31
137

29
61
113
145
212
238
138
105
30
47
121
92
74
7
80
142

232
95
197
220
195
231
49
161
36
8
69
91
65
13
244
218

51
0
250
144
229
114
190
45
249
133
140
38
33
186
201
56

Table 12: S-box corresponding to P11.

Proposed S-box 11

128
191
132
167
111
120
159
218
25
68
173
217
32
39
99
239

240
94
157
95
75
112
190
213
152
202
40
101
2
207
140
64

19
238
35
151
154
197
199
60
61
187
44
201
72
37
126
118

24
80
124
141
16
41
193
160
7
107
163
129
248
66
189
221

10
109
76
150
110
255
171
70
17
18
43
174
153
13
113
206

244
219
3
203
12
134
48
245
164
130
230
4
144
53
182
78

155
227
253
158
26
86
186
62
79
21
175
162
222
215
247
208

98
137
210
63
57
67
138
96
139
242
254
38
92
188
97
52

211
106
6
9
22
85
212
198
234
225
42
223
119
136
90
33

46
121
249
184
34
142
251
20
214
216
50
209
135
87
74
176

161
5
77
36
229
83
149
73
168
181
196
115
194
11
56
8

82
231
51
1
148
65
195
172
23
177
228
0
131
252
88
170

220
27
69
71
123
232
49
226
236
58
104
143
178
166
117
205

54
169
108
91
204
127
116
45
165
89
30
55
31
84
246
15

156
122
224
192
103
145
243
114
185
102
200
180
250
81
233
47

59
125
133
100
147
14
183
146
235
237
29
28
105
93
179
241

Table 13: S-box corresponding to P12.

Proposed S-box 12

140
249
110
3
253
6
19
160
102
247
235
219
17
166
184
34

240
198
221
159
212
199
188
148
27
0
52
73
176
15
4
93

1
28
32
214
195
238
216
24
229
92
241
36
209
207
12
9

25
245
139
97
109
41
225
37
105
106
71
100
206
165
129
59

171
70
208
126
111
242
181
119
215
22
183
233
14
222
204
150

226
144
82
121
40
87
67
228
116
234
81
13
180
5
96
237

42
51
95
16
99
79
21
69
31
30
125
55
89
143
123
178

122
158
80
66
10
133
46
252
33
112
210
157
131
203
127
43

211
54
132
217
58
155
152
192
53
128
84
44
88
236
194
168

170
239
231
218
232
205
103
151
2
173
26
145
20
251
136
117

146
49
83
35
202
90
243
164
78
108
201
167
175
191
68
163

65
85
185
255
104
18
227
177
75
39
161
182
187
57
98
186

64
246
179
196
174
38
118
135
101
193
77
223
94
248
72
250

107
62
189
153
8
48
7
56
169
60
23
11
149
141
29
156

147
120
154
254
134
130
244
200
172
213
197
162
190
113
45
86

114
230
47
138
61
115
220
50
137
224
76
63
142
124
91
74

Table 14: S-box corresponding to P13.

Proposed S-box 13

24
177
151
25
7
71
226
153
127
137
209
73
254
213
96
237

240
234
102
131
201
142
118
64
16
249
176
6
5
18
222
95

52
36
82
160
224
225
144
74
67
129
1
13
105
143
186
248

181
113
231
108
58
216
43
114
156
37
168
9
170
60
3
246

40
122
217
247
223
85
163
4
65
152
31
189
70
12
173
210

207
233
197
107
83
115
164
99
66
56
23
111
120
61
251
46

205
154
255
167
198
81
87
109
20
135
59
68
100
39
220
172

145
11
132
185
106
53
162
112
253
54
180
33
116
175
104
134

211
123
204
26
190
90
8
92
15
110
119
29
188
230
219
199

76
79
174
130
147
166
63
215
241
41
244
141
124
0
35
98

218
245
242
72
21
239
47
94
208
238
10
165
171
91
14
192

77
243
148
158
80
203
45
42
194
146
159
235
214
161
32
169

93
101
140
30
97
50
84
126
229
196
200
48
55
139
57
44

69
128
49
155
19
62
232
252
236
136
250
27
157
195
125
206

34
179
78
38
193
227
28
184
133
88
187
178
75
212
138
117

191
202
149
22
228
89
150
121
221
17
182
86
51
183
103
2

Table 15: S-box corresponding to P14.

Proposed S-box 14

93
13
118
253
48
210
140
163
100
34
18
128
11
90
53
114

240
227
242
106
112
138
147
92
250
206
141
56
152
57
94
212

50
149
235
21
233
254
102
131
156
142
173
194
31
45
244
195

237
43
183
168
121
95
9
71
37
120
204
62
119
174
122
186

123
29
105
136
191
196
91
40
229
86
4
41
125
22
224
162

155
30
223
161
39
169
14
63
52
133
199
36
241
124
59
65

190
127
219
88
214
115
126
96
38
239
55
7
197
73
2
42

23
144
47
46
68
72
6
185
109
176
1
98
201
234
222
203

211
175
157
113
110
217
85
3
249
255
116
104
25
82
179
97

158
135
44
78
117
164
160
221
66
236
145
103
60
230
232
134

182
67
177
188
15
8
107
79
216
12
246
51
213
180
58
17

189
252
215
245
69
218
192
10
225
74
54
83
84
61
166
193

16
139
81
132
19
101
76
208
231
151
181
49
187
99
202
228

205
220
154
198
111
167
20
26
200
64
165
247
248
209
207
148

226
0
28
184
33
159
27
153
24
172
171
75
238
243
77
87

35
170
32
137
143
251
129
178
146
150
80
89
108
130
5
70

Table 16: S-box corresponding to P15.

Proposed S-box 15

12
220
182
7
142
239
96
251
36
246
21
162
223
103
35
40

240
132
32
219
116
160
243
28
128
168
159
206
193
95
50
154

38
234
47
66
126
192
54
169
65
175
254
89
45
123
233
39

17
194
111
202
242
226
178
210
185
64
77
218
108
127
147
222

191
214
237
110
184
155
30
20
86
212
156
1
255
119
167
141

105
252
176
63
205
139
151
53
33
189
170
213
183
224
204
75

69
100
230
196
188
31
43
104
83
91
145
60
113
117
152
94

157
129
102
235
135
49
15
181
163
195
58
249
134
171
107
37

211
34
24
238
67
22
236
88
253
109
207
221
164
203
138
19

48
229
122
71
76
93
120
143
81
106
26
137
215
200
187
217

118
72
244
245
150
209
121
144
199
14
186
27
70
115
84
55

4
11
73
80
179
79
146
18
173
6
172
8
56
136
78
61

42
52
85
166
133
41
228
25
99
90
158
125
57
29
2
248

10
231
92
5
201
180
68
177
174
23
161
208
98
62
232
140

51
165
216
0
241
101
9
97
131
148
198
247
130
114
250
112

44
46
225
13
153
3
16
82
190
124
59
197
149
227
74
87

Table 17: S-box corresponding to P15.

Proposed S-box 16

6
136
68
218
166
118
48
114
2
173
30
39
204
81
144
134

240
46
115
83
91
202
1
203
10
63
145
4
89
112
213
17

47
8
176
149
117
181
38
133
56
87
162
20
199
78
60
157

254
14
192
29
200
178
140
69
24
121
160
148
158
252
67
228

111
103
102
194
33
122
44
205
61
100
229
215
80
187
217
189

28
124
147
92
31
76
45
171
137
239
95
154
42
64
51
55

119
236
77
242
0
12
75
216
84
151
220
49
101
150
129
53

155
198
174
146
90
233
88
225
54
179
5
74
246
138
110
230

211
153
208
214
40
184
13
255
71
163
62
130
175
232
231
128

156
141
34
152
15
212
165
41
106
97
167
72
177
169
234
107

170
120
247
104
245
195
7
172
19
65
85
36
70
108
209
222

197
132
244
57
201
182
96
52
125
210
183
127
241
126
235
26

58
227
143
11
238
86
135
191
66
168
237
3
59
123
32
27

105
207
221
22
248
251
243
50
23
16
186
98
82
250
180
164

193
219
93
79
35
196
249
188
139
253
161
99
37
223
224
226

131
21
109
25
73
116
43
206
94
113
190
142
185
159
9
18

In the proposed work, we present an APA S-box corresponding to each where the APA map gives the lookup tables. We, then, show that these S-boxes have strong cryptographic properties certified with the help of analyses such as nonlinearity, strict avalanche criterion (SAC), bit independent criterion (BIC), linear approximation probability (LP), and differential approximation probability (DP) [20].

3. Security Analysis

In this section, we present some algebraic and statistical analyses of S-box followed [21]. Such analyses indicate the strength of all the proposed S-boxes and give an idea for their application in image encryption and other modes of secure communication.

3.1. Nonlinearity

Nonlinearity analysis of a function is the minimum hamming distance between the Boolean function : and its all n-bit affine functions. In the truth table of Boolean function , the nonlinearity of represents the degree of dissimilarity between and all affine function. If the function has high minimum hamming distance, it indicates it has high nonlinearity. It is an established fact that high nonlinearity provides resistance to any kind of linear approximation attacks [22, 23]. The calculated upper bound of nonlinearity is so that, for, the optimal value of nonlinearity is. Table 18 shows the nonlinearity of 16 S-boxes corresponding to all primitive irreducible polynomials. From this table, it can be seen that the value of nonlinearity has not been affected due to background irreducible polynomial.

Table 18: Nonlinearity of the proposed S-box.

S-boxes
f₁
f₂
f₃
f₄
f₅
f₆
f₇
f₈
Average

S-box 1
112
112
112
112
112
112
112
112
112

S-box 2
112
112
112
112
112
112
112
112
112

S-box 3
112
112
112
112
112
112
112
112
112

S-box 4
112
112
112
112
112
112
112
112
112

S-box 5
112
112
112
112
112
112
112
112
112

S-box 6
112
112
112
112
112
112
112
112
112

S-box 7
112
112
112
112
112
112
112
112
112

S-box 8
112
112
112
112
112
112
112
112
112

S-box 9
112
112
112
112
112
112
112
112
112

S-box 10
112
112
112
112
112
112
112
112
112

S-box 11
112
112
112
112
112
112
112
112
112

S-box 12
112
112
112
112
112
112
112
112
112

S-box 13
112
112
112
112
112
112
112
112
112

S-box 14
112
112
112
112
112
112
112
112
112

S-box 15
112
112
112
112
112
112
112
112
112

S-box 16
112
112
112
112
112
112
112
112
112

3.2. Strict Avalanche Criteria

In [24], Webster and Tavares introduced the strict avalanche criteria (SAC) on the concepts of completeness and avalanche. If a single input bit changes, the output bits change with almost 0.5 probability. It helps to show that the resulting output vector is highly random, and no single pattern can be predictable by minor variation in the input vector [25]. By seeing the performance indexes of S-boxes, the proposed S-boxes successfully satisfy SAC. Table 19 depicts the value of SAC for all the proposed 16 S-boxes. It shows that the maximum value of SAC is 0.562500 for the first 9 S-boxes including 11th, 14th, and 16th S-boxes. Similarly, the minimum value of SAC is 0.453125 for the first 10 S-boxes including 12th and 14th S-boxes. The average value of SAC lies in the interval [0.4856, 0.509766].

Table 19: Strict avalanche criterion.

S-box
Max
Min
Average
Square deviation

S-box 1
0.562500
0.437500
0.496094
0.0172495

S-box 2
0.546875
0.453125
0.495117
0.0128725

S-box 3
0.562500
0.453125
0.494141
0.0152856

S-box 4
0.562500
0.453125
0.50708
0.0118748

S-box 5
0.546875
0.453125
0.503174
0.0153901

S-box 6
0.562500
0.453125
0.501709
0.016637

S-box 7
0.562500
0.453125
0.502441
0.0170951

S-box 8
0.562500
0.453125
0.503906
0.0165152

S-box 9
0.562500
0.453125
0.485596
0.0153978

S-box 10
0.546875
0.453125
0.509766
0.0123912

S-box 11
0.562500
0.4375
0.50415
0.0191487

S-box 12
0.5625
0.453125
0.501221
0.016475

S-box 13
0.546875
0.4375
0.500977
0.0127235

S-box 14
0.5625
0.453125
0.508301
0.0158654

S-box 15
0.546875
0.437500
0.498779
0.0143727

S-box 16
0.5625
0.437500
0.496582
0.0143171

3.3. Bit Independent Criterion

Another algebraic criterion (BIC) is used to evaluate the strength of S-box, which is presented by Detombe and Tavares in [26]. In Table 14, the outcomes of BIC to SAC and BIC for the proposed S-boxes are given. The minimum BIC to SAC value is 0.47070 for 12th S-box and the highest minimum value is 0.49219 for 2nd S-box. The average BIC to SAC lies between 0.49679 and 0.50739. Similarly, the square deviation values for all the proposed S-boxes are given in Table 20. The maximum and average value of BIC is 112 for all S-boxes. It is depicted that the proposed S-boxes give the nearest best value of BIC analyses.

Table 20: Bit independent criterion.

BIC-SAC
BIC

S-box
Min
Average
Square deviation
Max
Average
Square deviation

S-box 1
0.47461
0.50119
0.01132
112
112
0

S-box 2
0.49219
0.50600
0.00845
112
112
0

S-box 3
0.48047
0.50202
0.01015
112
112
0

S-box 4
0.47852
0.50656
0.01201
112
112
0

S-box 5
0.48438
0.50105
0.00924
112
112
0

S-box 6
0.47656
0.49819
0.00784
112
112
0

S-box 7
0.48633
0.50593
0.00925
112
112
0

S-box 8
0.48828
0.50251
0.00835
112
112
0

S-box 9
0.48828
0.50258
0.00644
112
112
0

S-box 10
0.48438
0.50739
0.00981
112
112
0

S-box 11
0.47656
0.49784
0.00950
112
112
0

S-box 12
0.47070
0.49679
0.01026
112
112
0

S-box 13
0.48242
0.50021
0.01085
112
112
0

S-box 14
0.49023
0.50718
0.00901
112
112
0

S-box 15
0.48438
0.50265
0.00883
112
112
0

S-box 16
0.48633
0.50544
0.00861
112
112
0

3.4. Linear Approximation Probability

Matsui defines the extreme value of the imbalance of an event as the linear approximation probability. It is notable that the parity of the input bits that is, the mask , is equal to the parity of the output bits, i.e., the mask . The linear approximation probability of a given S-box is defined in the following equation:

where and are input and output masks, respectively, and the set represents the set of all possible inputs; is the number of elements of. The value of linear approximation indicates the strength of S-box against various linear attacks. In Table 21, the maximum count and the LP value for all proposed S-boxes is 144 and 0.0625. These values of LP of the proposed S-boxes are appropriate against linear attacks.

Table 21: Linear and differential approximation probability analysis.

S-box
Linear approximation probability
Differential approximation probability

Max count
LP
Max value
DP

S-box 1
144
0.0625
4
0.015625

S-box 1
144
0.0625
4
0.015625

S-box 2
144
0.0625
4
0.015625

S-box 3
144
0.0625
4
0.015625

S-box 4
144
0.0625
4
0.015625

S-box 5
144
0.0625
4
0.015625

S-box 6
144
0.0625
4
0.015625

S-box 7
145
0.0664
4
0.015625

S-box 8
144
0.0625
4
0.015625

S-box 9
144
0.0625
4
0.015625

S-box 10
144
0.0625
4
0.015625

S-box 11
144
0.0625
4
0.015625

S-box 12
144
0.0625
4
0.015625

S-box 13
144
0.0625
4
0.015625

S-box 14
144
0.0625
4
0.015625

S-box 15
144
0.0625
4
0.015625

S-box 16
144
0.0625
4
0.015625

3.5. Differential Approximation Probability

The degree of differential uniformity is known as differential approximation probability (DPs) of S-box. Mathematically, it can be given as

Briefly, it can be explained as follows: an input differential must be mapped to an output differential uniquely for each i. Here, represents all the possible input values and the number of its elements is given by . Table 21 depicts the results of DP, which include the maximum and DP value.

Moreover, Table 22 represents the values of proposed S-boxes along with AES, Skipjack, Xyi, APA, Gray, and residue prime S-boxes.

Table 22: Comparison of the performance indexes of the proposed S-boxes with some standard S-boxes.

S-boxes
Nonlinearity
SAC
BIC
BIC-SAC
DP
LP

AES S-box
112
0.5058
112
0.504
0.0156
0.062

APA S-box
112
0.4987
112
0.499
0.0156
0.062

Gray S-box
112
0.5058
112
0.502
0.0156
0.062

Skipjack S-box
105.7
0.4980
104.1
0.499
0.0468
0.109

Xyi S-box
105
0.5048
103.7
0.503
0.0468
0.156

Residue prime
99.5
0.5012
101.7
0.502
0.2810
0.132

Reference [27]
106
0.4978
103.92
—
—
—

Reference [28]

0.505
—
—
—
—

Reference [29]
104
0.5241
103
0.50181
0.1625
0.0486

S-box 1
112
0.496094
112
0.50119
0.015625
0.0625

S-box 2
112
0.495117
112
0.50600
0.015625
0.0625

S-box 3
112
0.494141
112
0.50202
0.015625
0.0625

S-box 4
112
0.50708
112
0.50656
0.015625
0.0625

S-box 5
112
0.503174
112
0.50105
0.015625
0.0625

S-box 6
112
0.501709
112
0.49819
0.015625
0.0625

S-box 7
112
0.502441
112
0.50593
0.015625
0.0625

S-box 8
112
0.503906
112
0.50251
0.015625
0.0664

S-box 9
112
0.485596
112
0.50258
0.015625
0.0625

S-box 10
112
0.509766
112
0.50739
0.015625
0.0625

S-box 11
112
0.50415
112
0.49784
0.015625
0.0625

S-box 12
112
0.501221
112
0.49679
0.015625
0.0625

S-box 13
112
0.500977
112
0.50021
0.015625
0.0625

S-box 14
112
0.508301
112
0.50718
0.015625
0.0625

S-box 15
112
0.498779
112
0.50265
0.015625
0.0625

S-box 16
112
0.496582
112
0.50544
0.015625
0.0625

3.6. Statistical Analyses

To evaluate the visual strength of the substitution with the help of the proposed S-boxes, various statistical analyses are made on the host and substituted images. In this proposed work, statistical analyses like homogeneity, entropy, contrast, energy, and correlation are used to evaluate the substitution ability of the 16 proposed S-boxes. These analyses are given as

where give the row and column locations of an image. The pixel value at kth row and lth column is represented by and is the probability of the image pixel. In equation (8), are mean and standard deviation, respectively.

Correlation analysis helps to find the similarity between the host and substituted image. The correlation analysis provides the range which indicates the perfect, negative, and positive correlation. This is interval for correlation and value of 1 indicates the perfect correlation.

The randomness of the digital image can be calculated with the help of entropy. The higher value of entropy from the interval represents the higher amount of randomness in a digital image. For any viewer, it is only possible with the help of contrast analysis to intensely recognize the objects in the texture of an image. With the help of contrast analyses, one can observe the maximum distinction in image pixels. The range of the contrast can be given by. For constant image, the value of contrast is zero. The goal of finding close distribution between the matrix and its diagonal is obtained in homogeneity analysis. The matrix used in this analysis is named gray level cooccurrence matrix (GLCM) and the range of homogeneity lies between 0 and 1. The range for energy analysis also lies in the interval [0, 1]. The results of Table 23 are obtained by applying these analyses on the original and encrypted images. For all the proposed 16 S-boxes, we calculated the values of the statistical analyses.

Table 23: MLC analyses for the proposed S-boxes.

S-boxes
Entropy
Correlation
Contrast
Energy
Homogeneity

S-box 1

S-box 2

S-box 3

S-box 4

S-box 5

S-box 6

S-box 7

S-box 8

0.1256
9.6635
0.0182
0.4509

S-box 9
7.2763
0.1099
8.9653
0.0179
0.4609

S-box 10
7.2763
0.0747
9.7106
0.0177
0.4453

S-box 11
7.2763
0.0927
9.7905
0.0182
0.4529

S-box 12
7.2763
0.0879
9.8720
0.0182
0.4531

S-box 13
7.2763
0.0868
8.5569
0.0188
0.4606

S-box 14
7.2763
0.0862
8.6308
0.0179
0.4510

S-box 15
7.2763
0.1245
9.3829
0.0184
0.4522

S-box 16
7.2763
0.0843
9.5898
0.0179
0.4533

A 256 × 256 JPEG image of Lena is considered for MLC analysis. Figure 2 shows the results of image encryption with 16 proposed S-boxes.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

(q)

Figure 2: Original Lena image and the encrypted images using all 16 primitive irreducible polynomials S-boxes. (a) Lena image, (b) S-box 1, (c) S-box 2, (d) S-box 3, (e) S-box 4, (f) S-box 5, (g) S-box 6, (h) S-box 7, (i) S-box 8, (j) S-box 9, (k) S-box 10, (l) S-box 11, (m) S-box 12, (n) S-box 13, (o) S-box 14, (p) S-box 15, and (q) S-box 16.

4. Balanced Boolean Function

4.1. Balance Property

The imbalance of a Boolean function weak system against linear cryptanalysis highlights the importance of balance property. The balance property indicates that the higher the magnitude of a function’s imbalance, the more the chances of a high probability linear approximation. A Boolean function is balanced. If the cardinality or Hamming weight of these two functions, that is, is the same, then it is named the balance function.

4.2. Balance Property of the Proposed S-Box

All the Boolean functions involved in proposed S-boxes are balanced just like the Boolean functions of AES, , AES and other well-known S-boxes. The nonlinearity of the proposed S-boxes is equal to 112.

5. Conclusion

In this paper, a scheme for the synthesis of S-boxes over 16 isomorphic Galois fields is presented. Here, we fixed all the parameters of affine power affine transformation, that is, for 16 S-boxes. We have 16 primitive irreducible polynomials of degree 8 and they prompt us to construct 16 Galois field extensions of order 256. By using elements of the Galois field, corresponding to each different pair of the parameters, one can construct different S-boxes. These S-boxes obtained as a result of APA transformation which is bijective, pass nonlinearity test, and out bit independent criterion (BIC) which demonstrates that the existing S-boxes have high confusion producing capability. The evaluation of constructed S-boxes is done with some algebraic and statistical analyses. The results of these analyses highlight the characteristics of all the proposed S-boxes and later these S-boxes are equated with some of the existing S-boxes. In addition to this, we also ensured that all these constructed S-boxes are balanced that guarantee the strength of our S-boxes. Hence, we have concluded that a large class of S-boxes can be obtained by varying parameters of affine power affine transformations. These S-boxes can be used for secure communication.

Data Availability

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

Conflicts of Interest

There are no conflicts of interest among the authors.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant no. R.G.P. 1/234/41.

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