Construction of new substitution boxes using linear fractional transformation and enhanced chaos

Chinese Journal of Physics 60 (2019) 564–572

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Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph

Construction of new substitution boxes using linear fractional transformation and enhanced chaos

T

⁎

Sajjad Shaukat Jamala, , Attaullahb, Tariq Shahb, Ali H. AlKhaldia, Mohammad Nazim Tufailc a

Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan c Department of Mathematics, University of Management and Technology, Sialkot, Pakistan b

A R T IC LE I N F O

ABS TRA CT

Keywords: Sine-logistic map Increased chaotic range S-box Encryption

Substitution boxes are used in different security techniques and cryptosystems to ensure the secure communication of data. To enhance the randomness and perplexity of data, chaos theory has utmost importance in encryption schemes and multimedia security. In this paper, Substitution boxes are developed by using linear fractional transformation and combination of chaotic systems with the increased chaotic range as compared to their seed maps. The Substitution boxes are assessed by using various analyses which include nonlinearity, strict Avalanche criterion, bit independence criterion, linear and differential approximation probabilities. Majority logic criterion is also performed to evaluate its application in various encryption systems.

1. Introduction Since the early nineties, the use of chaos theory has been promoted in many fields like physics, engineering, biology and weather forecasting. The property of creating perplexity and confusion is the main feature of chaotic systems and this is quite valuable in the study of cryptography. The accessibility/inaccessibility of the initial values describes the certainty/uncertainty of the chaotic system. This nonlinear behaviour of the chaotic system ensures secure communication through an insecure communication path by creating randomness and perplexity in the plain text. Diffusion is generally developed by the random application of non-linear dynamical structures. Minor alteration in the initial values describes an entirely different behaviour of the chaotic structures which indicates their sensitivity to initial conditions. Initially, Shanon has introduced the idea of using chaos theory for safe communication of data which attracted scholars of a different realm of life to develop chaos-based secure communication theory [1]. The most promising feature in developing novel cryptosystems with the help of chaos theory is their sensitivity to initial conditions. Construction of strong substitution box (S-box) is the most remarkable application of chaotic cryptography in the block cipher. In the recent past, many techniques have been developed for the construction of S-box using the chaotic and algebraic structures. In view of [2], S-box of size l gives the matrix of l × l. This construction requires comprehensive search, the extra time and long processing for the selection of best suited S-box. To negate the risk of differential cryptanalysis, S-box must be evaluated with suitable algebraic and statistical analysis so that the desired confusion properties are attained for secure data communication. [3]. With a small size of S-box, results are interesting yet these results turn out to be more susceptible if the configuration comprises of huge size.

⁎

Corresponding author. E-mail address: [email protected] (S.S. Jamal).

https://doi.org/10.1016/j.cjph.2019.05.038 Received 3 February 2019; Received in revised form 6 May 2019; Accepted 14 May 2019 Available online 11 June 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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The methodology of utilizing chaotic frameworks as a part of cryptology, multiple S-boxes have been developed with suitable properties [4]. In literature, S-boxes are constructed using two-dimensional and three-dimensional chaotic maps [5,6]. Two-dimensional map (Bake map) provides exceptional encryption features whereas implantation and capability issues are identified in three-dimensional chaotic maps (Baker map). Tent map is another case of such techniques [7]. S-boxes are mainly associated with nonlinearity during encryption procedure so there is a significance of substitution procedure. In this work, we used an amalgamation of chaotic maps alongside linear fractional transformation. This will oppose any approach of cryptanalysis. These S-boxes can further be used in different established cryptosystems. The application of S-boxes in image encryption and for other security purposes is very common nowadays. These S-boxes are constructed with the help of algebraic structures. It includes S-boxes used in advanced encryption standard (AES), data encryption standard (DES) and many more. The construction of S-boxes in these standards is comparatively complex as compared to the construction of S-boxes with the help of chaotic maps. Though the existing chaotic maps have limited chaotic ranges and also have uniformity issues. These drawbacks of chaotic maps may have an impact on the strength of cryptosystems. To counter this issue, there is a need for a chaotic map which has the enhanced chaotic range and strengthens the cryptosystem against the malicious attacks. In our work, the combination of mathematics and enhanced chaotic range maps are used to construct S-box. Nowadays, many researchers have been using chaos for different security purposes which include information and multimedia security. In fact, chaos has counted as one of the most unfailing sources to provide improved security against the challenges of fast internet transmissions. Due to tremendous qualities of chaotic systems such as sensitivity to initial conditions, unpredictability, complexity strength has allowed its application in different multimedia security applications [8–14]. In 2008, Hung and Hu, have used temporal transfer entropy for chaotic communication [15]. An image encryption technique based on S-box and Lorenz chaotic system is recently presented in [16]. The 3-D chaotic systems containing six terms having three multipliers for folding trajectories is also represented in [17]. On the other hand, hyperchaotic system and chaos in two-coupled van der Pol oscillators ate thoroughly discussed in [18,19]. In this paper, construction of S-boxes is based on the concept of pure mathematics and chaotic map (having increased chaotic range). Linear fractional transformation is applied to random values obtained from the Sine-Logistic map (SLM). The confusion and diffusion properties of S-box are enhanced by the robust properties of the combined cryptographic structure of Sine and Logistic maps. The complex dynamics of SLM are thoroughly discussed in [20,21]. Recently, information entropy has been utilized for cryptanalysis. The cryptanalysis of chaotic image encryption has its importance which is depicted in [22]. This system has cryptographic properties of confusion and randomness which are assessed by the different analysis [23]. These analyses indicate the strength of proposed S-box for encryption schemes and multimedia security. The rest of the paper is organized as follows. Detailed mathematical background for the SLM is studied in Section 2. Construction of proposed S-box is discussed in Section 3. Section 4 comprises the statistical and algebraic analysis for the newly constructed S-box. In Section 5, results for the majority logic criterion are discussed. The conclusion of the proposed technique is presented in Section 6. 2. Review of different chaotic maps Chaotic maps have different applications in secure communication due to their nonlinear behaviour. The detailed background of one-dimensional (1D) chaotic maps (Sine and Logistic) is discussed in this section. Joining these two maps through XOR operation gives the chaotic map for our proposed S-boxes. 2.1. The chaotic sine map The phenomenal breakthrough in the field of secure communication is attained with the inclusion of chaos theory. The sine map has chaotic behaviour for the parameter α ∈ [3.45, 4]. Interestingly, it has a periodic motion of 2 for the values of α between 3 and 3.45. Furthermore, the sequences are non-periodic and normally distributed in the interval (0, 1). The mathematical expression for the sine map is

Yn + 1 = S (α, Yn ) =

αSin (πYn ) , 0<α≤4 4

(1)

The chaotic behaviour of Sine map can be seen by both the bifurcation and Lyapunov exponent diagrams as depicted in Fig. 1(a). Considering the weaknesses of chaotic sine map, it is evident from the bifurcation graph that this map demonstrates chaotic and nonchaotic behaviour for various interims. Although the Sine map has a chaotic range in the interval 3.57 ≤ σ ≤ 4 with the definite selection of parameters, the map shows non-chaotic behaviour even for this mentioned range. Also, the chaotic sequence has data range less than [0, 1], which represents the inconsistency in this interval. 2.2. The chaotic logistic map The logistic map has an identical behaviour with the chaotic sine map. Mathematically logistic map is described using the simple dynamical equation

Yn + 1 = L (β , Yn ) = βYn (1 − Yn ), 0 < σ ≤ 4; x n ∈ [0, 1]

(2)

Its bifurcation and Lyapunov Exponent diagrams are more or less identical with those of chaotic Sine map (Fig. 1(b)). Thus, both 565

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Fig. 1. Lyapunov and bifurcation diagrams of (a). Sine, (b). Logistic and (c). SLM.

Sine and Logistic maps have the same problems discussed in Section 2.1. The dynamical degradation of Logistic map is explained in [24,25]. The chaotic range of logistic map will not be enhanced even if we take the variant exponent of Yn. 2.3. The chaotic sine-logistic map The SLM is the combination of chaotic Sine and Logistic map. The mathematical expression is defined in Eq. (3). The parameters of both seed maps are also combined for simplicity

Yn + 1 = ΠSL (μ, Yn β ) = (S ((4 − μ), Yn β ) + L (μ, Yn β )) mod1 = ((4 − μ) Sin (πYn β )/4 + μYn β (1 − Yn β )) mod1

(3)

where 0 < μ ≤ 4. The bifurcation and Lyapunov exponent diagrams show that its chaotic sequence has a uniform distribution in the interval [0, 1] and the chaotic behaviour of the SLM is in the entire range of the parameter. Both diagrams are given in Fig. 1(c). 3. Construction of chaotic S-box using group action Block ciphers are considered as one of the vital components to building cryptosystem and they have a phenomenal dependence on the quality of S-box. In this substitution process, m binary input bits transform into n binary output bits. This m × n nonlinear S-box transformation is defined as a function

h: GF (2m) → GF (2n) The resistance against any differential and linear cryptanalysis in encryption scheme depends on the suitable selection of S-box. In this paper, SLM is used for the construction of new proposed S-boxes due to its wider chaotic range and cryptographic properties. Fig. 2 of the flow chart shows that the initial values as input for the design of proposed S-box are taken from chaotic SLM. These values are then assigned to the linear fractional transformation for the action of the projective general linear group on the finite field 28 having 256 elements. We can intricate it as

h: PGL (2, GF (28)) × GF (28) → GF (28) qm + r , 0 ≤ m ≤ 255 h (m) = sm + t

(4) 566

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Fig. 2. Algorithm for the construction of proposed S-box.

where q, r, s, t ∈ GF(28). These parameters are taken from the output values of SLM. The proposed chaotic S-box has 256 unique values. The design of the proposed S-box depicts that it takes integral values between 0 and 255 for all (q, r, s,t) along with independent variable m. The algorithm completes its loop once it is confirmed that q × t − r × s has a non-zero value. 3.1. Proposed S-boxes In this paper, S-boxes are constructed by taking different exponents of chaotic SLM. The mathematical description of the map corresponds to first S-box is given in Eq. (3) and the maps correspond to the rest of the four S-boxes are described in Eqs. (5) to (8). By keeping β as constant, we use the SLM with the exponent as a parameter.

(4 − μ) Sin (πYn 2) Yn + 1 = ΠSL (μ, Yn 2) = ⎛ + μYn 2 (1 − Yn 2) ⎞ mod1 4 ⎝ ⎠

(5)

Yn + 1 = ΠSL (μ, Yn1/2) = ((4 − μ) Sin (πYn1/2)/4 + μYn1/2 (1 − Yn1/2)) mod1

(6)

Yn + 1 = ΠSL (μ, Yn6/7) = ((4 − μ) Sin (πYn6/7)/4 + μYn6/7 (1 − Yn6/7)) mod1

(7)

⎜

⎟

Yn + 1 = ΠSL (μ, Yn10/9) = ((4 − μ) Sin (πYn10/9)/4 + μYn10/9 (1 − Yn10/9)) mod1

(8)

Moreover, Tables 1–5 represent the tabular form of proposed S-boxes. Different exponents are used in the construction are listed with tables. The 256 distinct values of S-boxes are given in 16 rows and 16 column table. The columns are represented by C1 to C16 and rows are given by R1 to R16. Table 1 S-box corresponds to the SLM given in Eq. (3), β = 1.

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

1 62 158 124 222 247 18 87 162 60 189 209 132 48 224 166

88 251 202 25 31 242 136 221 160 187 120 230 2 70 234 44

219 205 146 19 109 145 220 203 61 53 26 89 84 186 133 95

77 143 118 67 164 176 110 63 16 135 112 14 127 114 235 204

115 47 138 73 212 185 252 142 86 130 236 151 29 121 170 92

240 69 148 206 210 57 183 163 150 125 99 30 116 55 214 248

245 3 137 168 101 237 5 103 208 196 7 54 40 229 254 50

33 0 239 22 56 64 91 28 8 227 152 169 153 167 226 98

165 188 108 233 104 100 216 96 213 173 119 243 171 126 97 4

85 149 66 81 39 238 15 194 231 6 79 37 255 59 154 12

198 94 43 193 225 228 42 175 232 157 217 76 107 195 65 250

35 190 49 181 249 128 241 46 11 83 45 93 180 90 200 161

117 58 218 105 141 111 38 80 78 134 102 178 32 223 51 36

201 246 20 113 159 199 253 129 13 17 156 21 207 184 9 139

192 27 172 211 106 34 72 24 123 244 131 71 82 182 68 191

10 177 52 122 140 215 155 41 147 144 75 197 174 23 74 179

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Table 2 S-box corresponds to the chaotic map given in Eq. (5), β = 2 .

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

88 195 146 199 232 190 128 117 207 139 161 245 203 92 186 69

12 242 187 111 250 45 99 137 214 181 81 119 84 162 55 18

237 30 77 106 147 131 103 183 93 74 113 177 225 57 156 182

143 68 254 165 145 95 17 151 248 233 174 70 153 180 196 62

124 100 213 188 129 138 61 155 51 23 144 179 96 240 114 85

251 39 148 152 82 189 120 178 33 167 224 26 9 108 142 52

102 109 200 34 192 172 63 2 25 220 205 10 171 160 121 198

202 226 112 149 35 221 98 75 218 136 125 158 244 252 238 28

169 73 135 175 163 133 123 173 48 236 13 140 41 91 21 49

223 71 5 230 79 193 141 89 191 115 105 29 19 159 14 132

122 243 222 101 65 94 197 60 66 78 219 50 15 3 7 126

184 166 154 234 86 185 22 4 249 216 6 211 127 80 38 107

164 27 231 118 217 58 20 40 227 134 59 76 53 47 104 54

239 83 206 157 212 208 228 0 229 253 44 32 247 210 37 241

168 150 255 130 194 235 31 97 43 1 36 170 72 110 56 215

64 116 46 67 42 176 201 246 87 90 204 209 16 11 8 24

Table 3 S-box corresponds to the chaotic map given in Eq. (6), β = 1/2.

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

39 78 243 16 9 135 196 133 93 36 100 119 234 164 101 88

188 66 253 87 67 197 127 99 24 65 162 76 172 186 236 97

183 77 195 35 56 0 106 4 131 31 170 179 49 241 200 73

175 189 50 71 138 125 239 91 124 220 10 61 58 54 129 25

230 238 178 136 139 32 7 194 159 34 13 89 152 12 45 251

47 141 187 210 62 252 29 217 211 42 18 213 114 3 205 118

63 23 247 17 142 225 163 168 20 192 48 30 240 246 90 6

85 228 92 202 96 143 81 245 147 72 156 105 86 26 208 44

160 121 176 19 134 46 21 248 84 22 70 60 108 37 203 5

1 212 113 130 171 79 237 242 115 126 199 150 221 177 154 201

55 98 117 64 214 250 2 206 190 232 120 244 180 151 83 57

43 123 229 102 227 169 191 33 185 144 38 110 155 233 109 27

254 216 165 111 41 14 132 128 52 222 193 53 15 82 94 224

215 112 204 51 103 28 184 255 157 158 209 174 249 69 149 148

107 166 161 226 59 140 219 231 137 11 116 122 75 153 40 235

182 68 145 181 80 95 104 198 167 207 173 8 74 218 223 146

Table 4 S-box corresponds to the chaotic map given in Eq. (7), β = 6/7 .

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

232 59 249 250 188 234 235 77 48 64 233 226 191 221 230 185

142 132 78 123 190 90 201 60 103 29 164 231 23 37 184 12

245 127 212 161 177 243 253 15 49 101 109 124 120 30 112 81

1 205 224 204 18 149 65 150 193 144 4 172 238 225 180 121

66 196 35 146 210 171 154 36 152 242 157 244 98 236 227 5

216 92 0 111 32 63 116 218 170 206 183 114 138 11 26 207

28 239 166 254 200 75 240 69 73 126 128 251 24 51 215 38

88 228 8 199 148 87 50 125 55 181 158 96 16 255 252 156

79 3 208 47 219 213 9 209 13 229 187 115 41 91 82 248

40 68 203 31 160 162 20 6 137 175 54 17 159 155 194 222

44 176 178 130 74 129 33 27 140 220 39 168 84 85 45 122

237 14 241 107 71 93 139 102 43 153 61 174 105 186 167 2

182 58 97 163 19 165 53 141 46 95 57 211 202 119 198 134

56 179 117 169 10 192 189 131 104 113 217 118 133 76 83 151

108 62 22 80 106 94 135 21 214 147 197 110 143 89 34 223

7 100 70 67 247 173 136 42 72 86 99 195 52 25 246 145

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Table 5 S-box corresponds to the chaotic map given in Eq. (8), β = 10/9.

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

83 154 231 4 9 51 176 49 157 80 208 215 234 112 209 140

124 130 253 151 131 177 223 195 12 129 98 152 120 110 248 193

119 153 163 67 76 0 202 16 35 31 106 103 69 229 168 137

123 125 70 147 42 221 251 143 220 188 10 93 78 86 33 13

242 250 102 40 43 64 19 162 63 66 25 141 44 24 89 239

91 57 111 166 94 252 29 173 167 74 6 181 198 3 185 214

95 23 247 5 58 225 99 104 20 160 68 30 228 246 142 18

149 240 156 170 192 59 133 245 39 136 60 201 150 14 164 88

96 205 100 7 50 90 21 236 148 22 146 92 216 81 171 17

1 180 197 34 107 155 249 230 199 222 179 54 189 101 46 169

87 194 213 128 182 238 2 186 126 232 204 244 116 55 135 77

75 207 241 210 227 105 127 65 109 36 82 218 47 233 217 15

254 172 113 219 73 26 48 32 84 190 161 85 27 134 158 224

183 196 184 71 211 28 108 255 61 62 165 122 237 145 53 52

203 114 97 226 79 56 175 243 41 11 212 206 139 45 72 235

118 144 37 117 132 159 200 178 115 187 121 8 138 174 191 38

Fig. 3. Bifurcation diagrams for the expressions given in (a) Eq. (5), (b) Eq. (6), (c) Eq. (7), (a) Eq. (8).

As mentioned before, Fig. 1(c) give the bifurcation and Lyapunov diagram whereas the bifurcation diagrams of S-boxes with different exponents are shown in Fig. 3.

4. Analysis of substitution boxes Special features of S-boxes are being analysed by using different statistical and theoretic methodologies [26]. This assessment of the quality of S-box decides its application in various encryption procedures and for security purposes. In [27], a detailed depiction of the method which utilizes differential properties of the block cipher is described. This category of cryptanalysis is used in DES algorithm, multiple ciphers and on different S-boxes. Information theory technique can also be used to analyse the strength of the block cipher [28]. This procedure comprises variant tests which include evaluation of nonlinearity, the scheme of input and output Table 6 Comparison of nonlinearity of various S-boxes. S-boxes

0

1

2

3

4

5

6

7

Ave

S-box-1 S-box-2 S-box-3 S-box-4 S-box-5 S8 AES Jakimoski [4] Tang [5] Gray Prime Chen [6] Skipjack Wang [7] APA AES Xyi

108 104 112 112 112 112 98 100 112 94 100 104 104 112 112 106

106 108 112 112 112 112 100 103 112 100 102 108 106 112 112 104

108 106 112 112 112 112 100 104 112 104 103 108 106 112 112 106

110 102 112 112 112 112 104 104 112 104 104 108 102 112 112 106

110 106 112 112 112 112 104 105 112 102 106 108 102 112 112 104

108 108 112 112 112 112 106 105 112 100 106 104 104 112 112 106

104 106 112 112 112 112 106 106 112 98 106 104 104 112 112 104

100 108 112 112 112 112 108 109 112 94 108 106 102 112 112 106

106.75 106 112 112 112 112 103.2 104.5 112 99.5 104.3 105.75 103.7 112 112 105

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Table 7 Results of algebraic analysis for the proposed S-boxes. S-box

BIC

SAC

BIC/SAC

DP

LP

Bijective

S-box-1 S-box-2 S-box-3 S-box-4 S-box-5

106.286 103.357 112 112 112

0.507 0.509 0.504 0.504 0.504

0.504 0.505 0.504 0.504 0.504

0.0390 0.0391 0.0156 0.0156 0.0156

158/0.125 160/0.125 1440.0625 144/0.0625 144/0.0625

Yes Yes Yes Yes Yes

bits (bit independence criterion and strict avalanche criterion) that provide features and linking of input and output bits and approximation probability (linear and differential approximation probabilities) shows the probability of events along with differential uniformity to get an iterative method. Nonlinearity results of proposed S-boxes are compared with numerous well-known S-boxes in Table 6. The results of nonlinearity show that the proposed S-boxes have comparable strength to the existing S-boxes. Moreover, the results of strict avalanche criterion, bit independence criterion, linear and differential approximation probabilities are depicted in Table 7. These results indicate that our constructed S-boxes are best suited for different security purposes to create confusion in cryptosystems. 5. Majority logic criterion The S-boxes satisfy the criteria of MLC are then used for different encryption techniques and for multimedia security. Majority logic criterion (MLC) is explained in Ref. [29]. The statistical strength of the proposed S-boxes is assessed by using these analyses. As in encryption process, host image is distorted and these alterations fixed the strength of the algorithm. The proposed chaos-based Sboxes fulfil all the requirements as shown in Table 8. A 256 × 256 JPEG image of baboon is considered for MLC analysis. The outcomes of image encryption with SLM S-boxes of various rational exponents and histograms are given in Figs. 4 and 5, respectively. 6. Conclusion In this letter, an algorithm for developing S-boxes with the support of combined chaotic maps and linear fractional transformation is presented. The group action of the projective general linear group is applied to random values attained from SLM. Due to proposed S-boxes, the perplexity and confusion in the plaintext are enhanced that becomes a challenge for the cryptanalyst to decode any of the information in the encryption process. Numerous algebraic and statistical tests have been performed in order to investigate the complexity and performance of the constructed S-boxes. By seeing the outcomes of different analyses on proposed S-boxes, it is found that proposed S-boxes with increased chaotic range provide the best results for any application of secure communication. Acknowledgment 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 number R.G.P. 2/58/40. Conflict of interest There is no conflict of interest between authors. Table 8 Comparison of MLC results. Images

Contrast

Correlation

Entropy

Energy

Homogeneity

Plain Text Proposed I Proposed II Proposed III Proposed IV Proposed V AES APA Prime S8_AES Gray Xyi Skipjack

0.7179 8.6409 8.6708 8.7145 8.4508 8.5500 7.5509 8.1195 7.6236 7.4852 7.5283 8.3108 7.7058

0.6782 −0.0004 −0.0043 0.0139 −0.0054 −0.0066 0.0554 0.1473 0.0855 0.1235 0.0586 0.0417 0.1025

7.1273 7.9822 7.9834 7.9837 7.9834 7.9817 7.2531 7.2531 7.2531 7.2357 7.2531 7.2531 7.2531

0.1025 0.0173 0.0174 0.0174 0.0177 0.0176 0.0202 0.0183 0.0202 0.0208 0.0203 0.0196 0.0193

0.7669 0.4063 0.4075 0.4059 0.4100 0.4072 0.4662 0.4676 0.4640 0.4707 0.4623 0.4533 0.4689

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Fig. 4. Baboon JPG image and its encrypted images using two rounds of encryption.

Fig. 5. Histograms of images given in Fig. 4.

References [1] C.E. Shannon, Communication theory of secrecy systems, Bell Syst. Tech. J 28 (4) (1949) 656–715. [2] G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurc. Chaos 9 (07) (1999) 1465–1466. [3] A.F. Webster, S. Tavares, Chapter-3 Advances in Cryptology: Proceedings of CRYPTO_85, Chapter-3 Advances in Cryptology: Proceedings of CRYPTO_85, (1986),

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