Security and Communication Networks
Volume 2018 (2018), Article ID 5823230, 8 pages
https://doi.org/10.1155/2018/5823230

Research Article

To Study the Effect of the Generating Polynomial on the Quality of Nonlinear Components in Block Ciphers

Shahid Mahmood,¹ Shabieh Farwa,² Muhammad Rafiq,² Syed Muhammad Jawwad Riaz,² Tariq Shah,³ and Sajjad Shaukat Jamal³

¹Department of Mechanical Engineering, Sarhad University of Science and Information Technology, Peshawar, Pakistan
²Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan
³Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

Correspondence should be addressed to Shabieh Farwa; drsfarwa@gmail.com

Received 1 November 2017; Revised 1 March 2018; Accepted 5 March 2018; Published 10 April 2018

Academic Editor: Amir Anees

Copyright © 2018 Shahid Mahmood 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 box (-box), being the only nonlinear component, contributes to the confusion creating capability of a cryptosystem. Keeping in view the predominant role of -box, many design algorithms to synthesize cryptographically stronger -boxes have gained pivotal attention. A quick review of these algorithms shows that all these ideas mainly concentrate on the choice of bijective Boolean functions, with nonobservance to the irreducible polynomial that generates the Galois field. In this paper, we propose that the selection of irreducible polynomial has a deep influence on the highly desirable features of an -box such as nonlinearity, strict avalanche, bit independence, linear approximation probability, and differential approximation probability. We underpin our claim by investigating a detailed model, which deploys the same algorithm but different polynomials and produces unusual changes in the results regarding the performance parameters of -box.

1. Introduction

Electronic exchange of data has undoubtedly revolutionized the communication in recent years but, on the other hand, the secure transfer of confidential material over Internet has become the biggest challenge nowadays. It definitely demands seriously high level of security. The main problem is to avoid unauthorised access to the secret data. To achieve the desired level of security, many techniques such as cryptography, watermarking, and steganography have been the major focus of research for past few years [1–5]. In this paper, we deal with cryptography.

Cryptography is categorized into two types, symmetric key cryptography and asymmetric key cryptography. The symmetric key cryptography can be further split into two types: block ciphers and the stream ciphers. Advanced Encryption Standard is an example of block cipher that was officially adopted by the US government as the Federal Information Processing Standard (FIPS) in May 2002. AES algorithm [6] is based on four steps: round key addition, byte substitution, shift row, and mix column, but the most influential of all these is the byte substitution step. This step relies on a substitution box (-box), which serves as the only nonlinear component in any substitution-permutation network (SPN).

It has been established that the substitution box (-box) is a standout in different block ciphers and is a widely used mechanism in any substitution-permutation network as a source to produce nonlinearity [6]. It renders an absolutely complex, unforeseeable layout to requite various blocks of bits in output data. To extend high resistance against unexpected surveillance, -box structure is required to fulfil certain standards. The indispensable involvement of -box to induce complexity and nonlinearity motivates studying the properties and algorithms for safer and more reliable -boxes. In this regard, many advanced structural developments are witnessed in literature. Khan et al. [7] proposed a technique for -box construction based on chaotic Lorenz systems. Hussain et al. presented -box algorithms using generalized Bakers map [8] and projective general linear group [9]. Algebraic, analytical, and chaotic approaches for -box are studied in [10–13]. Özkaynak et al. [14] applied fractional-order chaotic Chen system, to develop -box. Tian and Lu [15] structured dynamic chaos-based -box in conjunction with DNA sequence operation. Some other more efficient algorithms could be reviewed in [6, 16–20]. In addition to this, applications of -boxes in digital image encryption, steganography, and watermarking have become quite popular and influential in recent years [4, 11, 13, 21].

The study of innovation in design algorithms for -boxes witnesses that the change of model and the selection of Boolean function contribute little to the performance indices of an -box. We, in this paper, propose that the performance of an -box is highly related to the background Galois field. The fact that finite fields of the same order are isomorphic is definitely of worth but the scrambling effect of a nonlinear Boolean function applied on two different fields of the same order might vary. Since in cryptography, an -box is the salient component used to produce confusion in the data, it is worth studying that the confusion creating ability is associated with the choice of the irreducible polynomial used to form the background Galois field.

In [9], Hussain et al. presented an algorithm for generating -box through the application of a linear fractional transformation on the Galois field , structured by the polynomial . We in the proposed work show that the same algorithm used for a different polynomial exhibits highly improved values of nonlinearity, strict avalanche criterion (SAC), bit independent criterion (BIC), linear approximation probability (LAP), and differential approximation probability (DAP). By comparing the numerical results of these tests, we prove that different polynomials produce significantly different results. This observation leads to revising the existing models by choosing different background polynomials as it could be more influential in improvement of ideas rather changing the whole scheme.

We organize the contents of this paper as follows. In Section 2, we discuss the properties of the background Galois field . The detailed algorithm for the design of the -box is presented in Section 3. Section 4 deals with the analyses of -boxes against several common attacks and the comparison of respective results. We further compare the cryptographic standing of both of the newly synthesized -boxes with the state-of-the-art AES -box. Conclusion is presented in Section 5.

2. Generating Polynomial and the Galois Field

For any prime , Galois field is expressed as the factor ring where is an irreducible polynomial of degree . For we choose an irreducible polynomial of degree that generates the maximal ideal of the principal ideal domain . We know that the multiplicative group of the resultant field is cyclic and hence each nonzero element of the field can be expressed as a power of the generator .

In order to support our claim regarding the effect of polynomial, we choose two irreducible primitive polynomials and of degree , to construct Galois fields and , respectively, where and , as used in [9]. We may choose other polynomials as well to compare our calculations but the selected pair beautifully serves for the purpose. Let represents the multiplicative group of the Galois field, . The exponential form of elements of the multiplicative group , along with their inverses, is represented in Table 1; however the elements of are presented in Table of [9]. In the next section, we use these calculations to develop the corresponding -boxes.

Table 1: Exponential representation and the multiplicative inverses of elements of .

1

184

208

92

159

104

134

46

173

247

139

52

48

67

117

23

252

238

83

195

204

253

16

26

181

24

180

153

121

130

73

179

232

126

141

119

220

145

248

217

175

102

188

198

108

8

172

13

53

226

84

12

49

90

148

244

223

132

203

65

224

156

68

225

60

116

14

63

157

254

143

131

31

110

154

240

150

124

186

212

196

239

19

51

227

94

193

99

163

54

149

4

158

86

192

190

235

162

88

113

123

42

189

6

135

160

169

45

199

74

155

122

100

215

237

66

15

221

36

152

29

112

101

78

187

34

140

200

171

30

72

58

202

7

105

167

176

246

11

127

35

255

71

249

38

183

243

55

91

77

241

120

28

75

111

62

69

93

5

106

168

98

89

207

231

177

136

161

107

201

129

47

9

216

41

137

166

233

32

27

25

242

146

2

209

79

125

43

103

96

234

95

87

205

20

81

213

44

109

128

170

133

59

21

194

230

164

3

185

251

228

80

197

236

114

174

40

219

218

37

118

245

57

61

64

50

85

211

250

206

165

33

178

191

97

214

115

18

82

76

151

182

147

56

222

138

10

39

144

229

210

17

22

70

142

3. Algorithm for -Box

An   -box is defined by a vector Boolean function , defined as

where and each of ’s is regarded as a component Boolean function.

For a field , the general linear group is a group formed by all invertible matrices. A projective general linear group of degree over a field is defined to be the quotient group of by its center. For this paper, we form the   -box by considering the action of the Galois field on the projective linear group ; that is, we take a function defined as follows:

In (2), is known as a linear fractional transformation (LFT) with and satisfying the nondegeneracy condition . The ease of implementation, lesser computational labour, and high algebraic complexity of an LFT are the prime features that give incentive to employ this map for byte substitution. We may choose any values for LFT parameters that satisfy the aforementioned condition but, for the presented calculations, we, in particular, choose the same values as in [9], so that a comparison could be set easily. We consider and . The images of the map , when applied on and , produce our -boxes and , respectively, as shown in Tables 2 and 3.

Table 2: -box .

3

214
37
74
126
4
18
26
219
62
45
226
50
136
104
148

154

38
200
199
185
228
170
245
177
114
137
231
139
35
8
134

158

27
223
232
17
157
217
49
83
141
171
42
47
206
64
194

106

29
30
110
14
40
72
236
105
221
202
87
241
41
11
71

186

28
253
175
67
31
23
33
66
189
117
118
94
149
135
252

235

43
124
125
90
229
204
215
218
100
101
249
243
54
173
166

138

234
244
201
167
44
250
25
16
187
207
246
107
103
161
1

183

99
179
240
129
123
188
193
20
143
155
174
7
220
213
239

108

84
113
184
5
57
208
153
75
112
223
178
180
150
65
24

224

248
102
89
70
111
59
172
95
131
198
163
93
164
55
209

86

132
6
225
51
79
53
34
97
48
197
142
182
210
91
247

195

15
10
144
85
63
168
238
196
162
98
32
251
254
203
156

80

152
237
24
127
78
165
12
52
222
122
58
211
36
140
191

216

146
109
96
147
73
116
190
128
68
56
77
115
160
19
92

69

2
192
121
145
21
76
61
60
181
133
151
159
0
88
205

13

230
22
82
39
119
9
255
120
46
81
212
227
130
176
169

Table 3: -box .

198
214
241
163
130
165
217
127
179
123
111
197
43
141
237
3

168
201
17
121
142
101
232
174
11
249
16
156
10
50
183
65

72
184
200
132
58
47
27
159
231
189
8
18
206
194
177
31

193
92
122
192
85
137
243
49
178
170
36
135
230
95
100
128

13
109
227
0
224
144
208
78
173
32
139
234
107
82
172
81

51
233
12
154
94
161
244
55
7
34
251
225
153
93
254
138

102
240
115
242
110
134
124
79
157
160
90
238
73
53
169
250

136
118
112
48
40
114
22
246
46
131
23
69
52
235
248
2

116
91
117
26
166
25
219
59
54
229
120
245
89
185
99
226

105
45
60
199
164
191
228
202
37
104
143
209
220
147
44
186

145
125
203
29
38
41
215
108
64
88
119
74
213
96
211
83

218
146
196
205
67
152
129
175
84
158
207
176
80
62
150
86

57
155
195
216
75
19
1
87
33
68
71
236
239
255
35
212

148
188
133
15
204
187
42
182
97
56
24
221
252
30
77
181

4
247
167
21
9
222
180
190
151
140
39
171
14
126
66
253

103
223
70
98
28
20
63
162
61
113
149
210
106
5
6
76

4. Performance Analysis of -Boxes

The cryptographic strength if the -boxes, generated in the foregoing section, are examined through the most widely used analysis techniques such as nonlinearity, bit independence, strict avalanche, and linear and differential approximation probabilities. In the following subsections we present all these performance indices one by one and compare the performance of and with one another, as well as, with the ever-prevailing algorithm AES.

4.1. Nonlinearity

Nonlinearity analysis measures the distance of the reference function from all of the affine functions. Nonlinearity criterion outlines the total number of bits that must be altered in the truth table of a Boolean function to get close to the nearby affine function [22].

Table 4 shows that, for , the average nonlinearity measure is , which is the highest figure attained by the AES -box. Figure 1 shows the comparison which clearly depicts outstanding performance of as compared to .

Table 4: Performance Indices for new -box.

Analysis
Max.
Min.
Average
Square deviation
DP
LP

Nonlinearity
113
111
112

SAC
0.546875
0.429688
0.498291
0.0157537

BIC

111
111.751
0.6227

DP

0.015625

LP

0.064063

Figure 1: Nonlinearity of different -boxes.

4.2. Linear Approximation Probability

The measure of unevenness of an event is determined by linear approximation probability. This analysis is used to evaluate the maximum imbalance of the outcome. Mathematically, the linear approximation probability for a given -box is defined as follows:

where represents the set of all possible inputs and and are the input and output masks, respectively. Numerical results presented in Table 5 and compared in Figure 2 show that the linear approximation probability of is much better than .

Table 5: Comparison of performance indices for different -boxes.

-box
Nonlinearity
SAC
BIC
DP
LP

AES
112
0.5058
112.0
0.0156
0.062

112
0.498291
111.751
0.015625
0.064063

105.5
0.507
106
0.0242
0.140

Figure 2: LP of different -boxes.

4.3. Differential Approximation Probability

For further analysis, we use the differential approximation probability, which determines the differential uniformity demonstrated by an -box. The mathematical expression for DP is given by the following:

In the above-mentioned expression input and output differentials are represented by and , respectively. The smaller the differential uniformity, the stronger the -box. It is evident from Table 5 and Figure 3 that in terms of the differential approximation probability is much stronger than .

Figure 3: DP of different -boxes.

4.4. Strict Avalanche Criterion

This criterion examines the changes in the output bits caused as a result of single input bit change. This is one of the most desirable features of any cryptographic design that when we change a single input bit, changes must occur in half of the output bits. In other words an -box, is said to satisfy SAC if, for a change in an input bit, the probability of change in the output bit is . The results are shown in Table 5 and Figure 4.

Figure 4: SAC of different -boxes.

4.5. Bit Independence Criterion

The independent behavior of the pair of variables and the variations of input bits are considered as important factors of bit independence criterion. In bit independence criterion, input bits are transformed exclusively, and then output results are scrutinized for their independency [23]. Bit independence has great worth in cryptographic structures. The goal of reaching the maximum complexity and perplexity in a system can be achieved through this property of increasing independence between the bits. In cryptographic systems, the increased independence between bits is an essential requirement as it makes harder to understand and forecast the design of the system.

The numerical results of BIC when applied to the proposed -box are given in Table 5 and are compared in Figure 5. It can be observed that according to these results our -box is pretty similar to the AES -box and is much better than .

Figure 5: BIC of different -boxes.

One can observe that overall performance of is much better than that of . The performance parameters for seem to be pretty close to that of AES -box. The algorithm used for both and is the same but the primitive polynomial selected to generate the Galois field is different, which really contributes to the outputs.

5. Conclusion

The kernel of the presented work lies in the fact that the choice of the background Galois field and its generating primitive polynomial matters to the function and performance of the substitution boxes. This fact leads to the fascinating idea that, rather than the development of new algorithms, the improvement of the existing algorithms is worth studying as its least laborious but most effective. We propose, on the basis of the example discussed, that the effect of the choice of generating polynomial may lead to an intensive research in future to modify the design models of -boxes. It will definitely affect the applications of -boxes in other branches of the digital communication, such as steganography, watermarking, and image encryption.

Conflicts of Interest

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

Acknowledgments

The authors are grateful to the Sarhad University of Science and Information Technology (Pakistan), for providing partial funding for this research work.

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