Content uploaded by Majid Khan
Author content
All content in this area was uploaded by Majid Khan on Apr 06, 2019
Content may be subject to copyright.
Nonlinear Dyn (2012) 70:2303–2311
DOI 10.1007/s11071-012-0621-x
ORIGINAL PAPER
A novel technique for the construction of strong S-boxes
based on chaotic Lorenz systems
Majid Khan ·Tariq Shah ·Hasan Mahmood ·
Muhammad Asif Gondal ·Iqtadar Hussain
Received: 12 May 2012 / Accepted: 17 September 2012 / Published online: 11 October 2012
© Springer Science+Business Media Dordrecht 2012
Abstract In cryptographic systems, the encryption
process relies on the nonlinear mapping of original
data or plaintext to the secure data. The mapping of
data is facilitated by the application of the substitution
process embedded in the cipher. It is desirable to have
resistance against differential cryptanalysis, which as-
sists in providing clues about the composition of keys,
and linear secret system, where a simple approxima-
tion is created to emulate the original cipher charac-
teristics. In this work, we propose the use of nonlin-
ear functional chaos-based substitution process which
employs a continuous time Lorenz system. The pro-
posed substitution system eliminates the need of inde-
pendent round keys in a substitution-permutation net-
work. The performance of the new substitution box is
evaluated by nonlinearity analysis, strict avalanche cri-
terion, bit independence criterion, linear approxima-
tion probability, and differential approximation proba-
bility.
M. Khan ()·T. Shah
Department of Mathematics, Quaid-i-Azam University,
Islamabad, Pakistan
e-mail: mk.cfd1@gmail.com
H. Mahmood
Department of Electronics, Quaid-i-Azam University,
Islamabad, Pakistan
M.A. Gondal ·I. Hussain
Department of Sciences and Humanities, National
University of Computer and Emerging Sciences,
Islamabad, Pakistan
Keywords Chaos ·Lorenz system ·Substitution box
1 Introduction
The objectives of a cryptographic system are to ob-
scure information present in the plain text in order to
secure the encrypted data. The integral part of creating
confusion is the introduction of randomness in data
at the output [37]. The random behavior of chaotic
systems exhibits desirable properties suitable for non-
linear dynamic systems such as the substitution pro-
cess in a cipher without independent round keys. The
chaotic systems are highly sensitive to initial condi-
tions and exhibit random behavior, which is deter-
ministic if the initial information is available, and in
the absence of this initial information, the system ap-
pears to be random to an observer. These properties
are desirable and attractive in the design of crypto-
graphic systems. The application of chaotic sequences
to the construction of substitution boxes, used in Ad-
vanced Encryption Standard (AES), is capable of cre-
ating confusion and applying diffusion to the original
data [2–7,9,10,12–36,38–40].
The substitution process in the AES encryption pro-
cess is the only nonlinear part, which creates confu-
sion and obscures the data. The substitution process
is accomplished by the use of the substitution box (S-
box) that is an array of size n×nand is defined as
S:{0,1}n→{0,1}n.
2304 M. Khan et al.
Several methodologies for the construction of cryp-
tographically strong S-boxes have been seen in litera-
ture. In [1], a method is proposed which relies on an
exhaustive search to construct a new S-box. Although
the proposed method yields good results, the construc-
tion of new S-boxes with large values of nis computa-
tionally complex and impractical. Keeping in view the
methods used by cryptanalysis [41], an S-box of size
5×5 is presented in [11] with strong resistance to dif-
ferential cryptanalysis. In addition, results show that
only odd values of dimension nyield S-boxes with
acceptable properties. Recently, the theory of chaos
is also employed for the construction of S-boxes. In
[12,28], chaotic maps are used to generate S-boxes in
multiple steps. In another construction method based
on chaotic techniques [9], a three-dimensional chaotic
Baker map is used to generate an 8 ×8 S-box. This
method exhibited some attractive properties pertain-
ing to robustness and resistance to cryptanalysis; the
implementation aspects were not addressed in detail
[39]. This method is further improved by the use of
a continuous-time chaotic Lorenz system [32]. In or-
der to obtain discrete data from the chaotic system, the
system trajectory values are converted to digital num-
bers for selected time steps and a linear functional al-
gorithm [20] is applied to these coded discrete outputs.
This method exhibits cryptographically strong proper-
ties as compared to other algorithms, which synthe-
size S-boxes based on chaotic methods. In this paper,
we mainly relate our chaotic system with linear func-
tional transformation in order to generate a strong S-
box.
The remaining sections of this paper are organized
as follows: In Sect. 2, we present the mathematical
background for the chaotic Lorenz system. The behav-
ior of the trajectory based on the initial condition is
also presented in this section. In Sect. 3, the perfor-
mance analysis results for the new S-box. Section 4
is devoted to results and discussions. The last section
presents the conclusion.
2 Chaotic Lorenz system
The Lorenz system is used to design atmospheric
model in 1950 [32] and is the first numerical study
of chaos. The system dynamics are represented by the
following equations:
Fig. 1 The plot of Lorenz system along x–yaxis, for α=10,
β=28, γ=8/3
dx
dt =α(y −x),
dy
dt =(βx −y−xz), (1)
dz
dt =(xy −γz).
The space plots resulting from the equations in (1)are
showninFigs.1,2,3,4. The values of the parameters
are α=10, β=28 and γ=8/3. The intervals used in
the states of the system are −40 ≤x≤40, −40 ≤y≤
40, and −40 ≤z≤40. The system exhibits chaotic
behavior for the selected parameters and intervals.
2.1 Chaos based algorithm for S-box design
The algorithm of the chaos based S-box design is pre-
sented in Fig. 5. This algorithm is divided into two
parts: diffusion and substitution. The first two steps
describe the diffusion process, whereas the remaining
portion depicts the realization of the S-box.
Algorithm
A.1: System trajectories are obtained by solving the
Lorenz system with selected initial conditions
and chaotic parameter values employing the
four-step Runge–Kutta method.
A.2: Selected trajectory is sampled at every (number
of data/256) step.
A.3: Use the linear functional transformation [20].
Outputs corresponding to each sample is coded
starting from 0 to 255.
A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems 2305
Fig. 2 The plot of Lorenz
systems for xalong t-axis
for α=10, β=28,
γ=8/3
Fig. 3 Plot of Lorenz
systems for yalong t-axis
for α=10, β=28,
γ=8/3
Fig. 4 Plot of Lorenz
systems for zalong t-axis
for α=10, β=28,
γ=8/3
2306 M. Khan et al.
Fig. 5 Flow chart of proposed chaotic S-box
A.4: We select the distinct first 256 values from these
chaotic random sequences to generate chaos-
based S-box.
In the diffusion process, the system trajectories are
evaluated by the solution of the Lorenz chaotic sys-
tem. The number of orbits obtained depends on the
dimension of the system, and is selected as a design
parameter. The initial conditions of the system are se-
lected at this stage. The Runge–Kutta method is ap-
plied to generate the chaotic parameters. A trajectory
is selected and sampled at 8-bit resolution. The objec-
tive is to construct an S-box capable of substituting
8 bits of data; as a result, 256 samples are generated.
Thus, coded samples used in the S-box range from 0
to 255. The entries in the S-box are populated by using
the codes generated by the samples obtained from the
selected system trajectory. A coding table is used to
map the sampled values from the output of the Lorenz
system to an entry in S-box (see Table 1).
In this work, the system trajectory is generated for
1,000 data samples while keeping the values of initial
conditions as x=1, y=0, z=0. In order to ignore
the transients of the chaotic system, first 1,000 sam-
ples are ignored. The system trajectory along xy-axes
is shown in Fig. 1. The resulting S-box based on the
chaotic system is presented in Table 1.
3 Analysis of the proposed chaotic S-boxes
It is vital to assess the performance of the proposed
S-box in an effort to establish its usefulness in en-
cryption. Several properties are listed in the literature,
which indicate the strength of any S-box [1]. Among
some of the prevailing methods used by cryptanaly-
sis include differential analysis used for the analysis
of DES [8] and information theoretic analysis with ex-
cerpts from the original concepts presented by Shan-
non [37]. In this work, we analyze the proposed S-box
for five different properties, which include nonlinear-
ity, strict avalanche criterion (SAC), bit independence
criterion (BIC), linear approximation probability (LP),
and differential approximation probability (DP). In or-
der to determine the strength of the proposed S-box,
the results of these analyses are prudently analyzed.
In the following subsections, we present the details of
these analyses and discuss the results pertaining to the
strength the S-box under analysis.
3.1 Nonlinearity
In the nonlinearity analysis, the constituent Boolean
functions are assessed with reference to the behavior
of the input/output bit patterns. The set of all affine
functions is used to compare the distance from the
given Boolean function. Once the initial distance is
determined, the bits in the truth table of the Boolean
function are modified to approximate to the closest
affine function. The number of modifications required
to reach the closest affine functions bears useful char-
acteristics in determining the nonlinearity of the trans-
formation used in the encryption process. The measure
of nonlinearity is bounded by [23],
Ng=2m−11−2−mmaxS(g)(w).(2)
A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems 2307
Tabl e 1 Algebraic structure of S-box in the form of 16 by 16 matrix
3 167 29 139 249 80 34 165 250 251 238 110 33 38 140 17
0 41 135 164 236 71 16 209 99 143 151 70 188 184 252 242
60 120 231 105 49 66 128 121 125 218 178 196 89 154 244 192
155 82 162 185 138 97 213 50 10 113 54 237 183 22 202 194
208 191 129 136 197 137 26 152 168 103 13 65 132 39 79 61
119 160 44 207 102 175 95 72 74 235 55 63 247 144 203 20
8 177 223 92 254 90 228 118 224 219 117 240 7 6 19 147
21 186 241 48 1 216 122 93 69 73 5 15 158 114 106 187
88 130 87 68 78 98 245 47 84 234 176 141 255 51 149 53
225 214 123 35 28 166 233 220 248 211 101 45 198 115 77 52
94 193 86 133 76 85 67 200 226 14 62 4 40 146 239 126
36 230 148 150 11 75 56 153 96 215 30 145 25 100 58 174
181 172 190 57 163 64 171 124 217 111 18 131 31 243 195 253
246 182 201 104 221 27 109 107 232 157 199 83 161 42 227 112
179 159 12 210 169 127 170 189 2 206 108 204 173 23 81 116
229 91 24 37 32 43 134 222 59 142 180 205 9 46 156 212
Tabl e 2 The results of nonlinearity analysis of different S-
boxes
S-boxes Nonlinearity
Proposed 105.25
Wan g [39] 104.00
Chen [9] 100.00
Tan g [12] 100.00
Jakimoski [28]98.00
The Walsh spectrum, S(g)(w) is defined as
S(g)(w) =
w∈F2m
(−1)g(x)⊗χ.w.(3)
The results of the non-linearity analysis are shown in
Table 2.
3.2 Strict Avalanche Criterion Analytically
In strict avalanche criterion, the behavior of the out-
put bits is analyzed that results from a change at the
input bit applied to the nonlinear S-box transforma-
tion. It is desired that almost half of the output bits
change their value or simply toggle their state in re-
sponse to a single change at the input. The change in
the output bit patterns cause a series of variations in
the entire substitution–permutation network (S–P net-
work), and thus causes an avalanche effect. The extent
of these changes assists in determining the resistance
to cryptanalysis and the strength of the cipher. The re-
sults of the strict avalanche criterion is shown in Ta-
ble 3. A comparison of the SAC for different S-boxes
is listed in Table 4.
3.3 Bit Independent Criterion
The bit independence criterion (BIC) also relies on the
changes at the input bits and the properties exhibited
by the independence behavior of pairwise input/output
variables of avalanche vectors [23–25]. This criterion
is analyzed by modifying single input bit from the
plaintext.
3.4 Linear approximation probability
The imbalance of an event between input and output
bits is quantified by the linear approximation proba-
bility test [34,35]. In this method, the parity of the
input bits given by a certain mask Ωk and the parity
of the output bits Ωl are used to determine the linear
probability of bits given as
P=max
Ωk,Ωl=0
#{k/k •Ωk =S(k) •Ωl}
2m−1
2,(4)
where Ωk and Ωl are the input/output masks used in
determining the linear approximation probability. The
total number of elements is given by 2mand Kis the
set of all possible inputs.
2308 M. Khan et al.
Tabl e 3 The results of
Strict avalanche criterion
for proposed S-box
0.5156 0.4687 0.4843 0.4375 0.5468 0.5000 0.4531 0.4375
0.5468 0.5625 0.4843 0.4687 0.5156 0.5625 0.4687 0.5312
0.5156 0.4687 0.4687 0.5625 0.4062 0.5156 0.5000 0.4687
0.5156 0.5312 0.4843 0.4531 0.5156 0.5937 0.5000 0.5625
0.5781 0.5000 0.4687 0.4843 0.4375 0.4531 0.3906 0.5781
0.5156 0.5312 0.6093 0.5625 0.5312 0.4375 0.5312 0.5000
0.5468 0.5312 0.5468 0.5312 0.5312 0.6250 0.4375 0.4218
0.4531 0.4062 0.4843 0.5312 0.5156 0.5468 0.4843 0.5000
Tabl e 4 Comparison of SAC analysis of proposed chaotic S-
boxes with other S-boxes
S-boxes SAC
Proposed 0.4930
Wan g [39] 0.4850
Chen [9] 0.4999
Tan g [12] 0.4993
Jakimoski [28] 0.4972
3.5 Differential approximation probability
It is desirable that the nonlinear transformation ex-
hibits differential uniformity. In order to ensure the
uniform mapping, a differential at the input, given
as ki, uniquely maps to an output differential li
for all i. The differential approximation probability is
mathematically defined as
DPs(x→y )
=#{k∈K/S(k) ⊕S(k ⊕k) =l}
2m.(5)
The proposed chaotic S-box is evaluated by differen-
tial approximation probability test. The results show
that the performance of the new chaotic S-box is com-
parable to some of the commonly used S-boxes.
4 Results and discussions
The comparison of the strong encryption capabilities
shows that the performance of the proposed S-box is
Tabl e 5 The nonlinearity
of BIC of proposed S-box – 102 106 102 94 92 96 96
102 – 106 106 104 102 96 100
106 106 – 102 104 106 106 104
102 106 102 – 102 102 102 100
94 104 104 102 – 96 100 94
92 102 106 102 96 – 98 96
96 96 106 102 100 98 – 96
96 100 104 100 94 96 96 –
Tabl e 6 The dependent
matrix in BIC of the
proposed S-box
– 0.4765 0.5273 0.5175 0.4843 0.5117 0.5097 0.4882
0.4765 – 0.5039 0.4785 0.5078 0.4960 0.5078 0.5312
0.5273 0.5039 – 0.4960 0.4912 0.4824 0.5097 0.4863
0.5175 0.4785 0.4960 – 0.4902 0.4863 0.5078 0.5097
0.4843 0.5078 0.4921 0.4902 – 0.5117 0.4960 0.5253
0.5117 0.4960 0.4824 0.4863 0.5117 – 0.5136 0.5000
0.5097 0.5078 0.5097 0.5078 0.4960 0.5136 – 0.4804
0.4882 0.5312 0.4863 0.5097 0.5253 0.5000 0.4804 –
A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems 2309
comparable or superior to some prevailing S-boxes
used in the area of cryptography. The nonlinearity
analysis depicts that the properties are comparable to
the S-boxes use as a benchmark in this work. Table 2
presents a list of results of nonlinearity analysis. The
result of SAC is very close to 0.5 which assures the
acceptability of this S-box to encryption application
(see Table 4). In Table 7, a comparison of BIC is pre-
sented between the proposed S-box and AES, APA,
Gray, and Prime S-boxes [20–26]. The results are in
Tabl e 7 BIC of SAC analysis of S-boxes
S-boxes Average
Proposed S-box 0.476
AES 0.504
APA 0.499
Gray 0.502
Prime 0.502
Tabl e 8 Linear approximation analysis of S-boxes
S-boxes Proposed box AES APA S8AES Skipjack
Max LP 0.140 0.062 0.062 0.062 0.109
Max Value 160 144 144 144 156
agreement with the desired range. In further analysis,
the linear approximation analysis shows that the new
S-box conforms to the range of values specified for the
good nonlinear components used in encryption appli-
cations. The results are shown in Table 8. Finally, the
differential approximation probability analysis is pre-
sented in Table 9and the comparison with already ex-
isting S-boxes are shown in Table 10. In this test, it is
observed that the performance of the chaotic S-box is
comparable to the existing well-known S-boxes used
as benchmarks in this paper.
5 Conclusion
In this paper, we present a method to construct a new
S-box with the application of the Lorenz system of
differential equations. In order to evaluate the perfor-
mance of the proposed S-box, a comparison is pre-
sented by the application of strict avalanche criterion,
linear approximation probability, differential approxi-
mation probability, bit independent criterion, and non-
linearity analysis. The existing S-boxes, which are
used for the purpose of benchmarking, include AES,
APA, Gray, and Prime S-boxes. The results yield that
the new S-box have desirable properties suitable for
encryption applications for secure communications.
Tabl e 9 The differential approximation probability of proposed chaotic S-box
0123456789101112131415
0.031 0.031 0.031 0.023 0.031 0.023 0.03 0.046 0.046 0.031 0.031 0.031 0.031 0.046 0.031 0.031
0.031 0.031 0.031 0.031 0.031 0.039 0.031 0.023 0.046 0.023 0.031 0.031 0.031 0.031 0.031 0.039
0.023 0.031 0.031 0.031 0.031 0.031 0.031 0.023 0.031 0.031 0.023 0.031 0.031 0.031 0.031 0.023
0.031 0.039 0.031 0.031 0.031 0.039 0.031 0.031 0.031 0.031 0.023 0.031 0.031 0.031 0.031 0.031
0.031 0.031 0.031 0.031 0.039 0.500 0.031 0.046 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031
0.031 0.031 0.031 0.031 0.023 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.039 0.023 0.031 0.023
0.031 0.031 0.031 0.031 0.031 0.023 0.031 0.031 0.031 0.023 0.031 0.031 0.023 0.023 0.039 0.031
0.031 0.031 0.031 0.039 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.039
0.031 0.031 0.031 0.023 0.031 0.031 0.031 0.031 0.031 0.023 0.023 0.031 0.031 0.046 0.031 0.031
0.031 0.031 0.031 0.031 0.031 0.023 0.015 0.023 0.031 0.023 0.023 0.031 0.031 0.046 0.031 0.031
0.031 0.031 0.031 0.023 0.031 0.031 0.023 0.039 0.031 0.023 0.031 0.031 0.023 0.031 0.023 0.039
0.031 0.039 0.031 0.031 0.039 0.023 0.031 0.031 0.031 0.039 0.031 0.023 0.023 0.031 0.031 0.023
0.023 0.023 0.031 0.031 0.031 0.031 0.031 0.031 0.046 0.039 0.031 0.031 0.031 0.023 0.031 0.031
0.023 0.023 0.031 0.031 0.031 0.023 0.031 0.015 0.031 0.023 0.031 0.023 0.023 0.023 0.023 0.023
0.023 0.031 0.031 0.031 0.023 0.015 0.031 0.031 0.031 0.023 0.031 0.023 0.031 0.023 0.023 0.031
0.046 0.031 0.031 0.031 0.023 0.039 0.031 0.031 0.023 0.023 0.031 0.039 0.031 0.031 0.031 –
2310 M. Khan et al.
Tabl e 10 Comparison of differential approximation probability
of proposed chaotic S-box with existing S-boxes
S-boxes Proposed Box AES Gray Skipjack Xyi
Max DP 0.03 0.0156 0.0156 0.0468 0.0468
References
1. Adams, C., Tavares, S.: In: Advances in Cryptology: Pro-
ceedings of CRYPTO. Lect. Notes. Comput., vol. 89,
pp. 612–615 (1989)
2. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Crypt-
analysis of a discrete chaotic cryptosystem using external
key. Phys. Lett. A 319, 334–339 (2000)
3. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Crypt-
analysis of a chaotic secure communication system. Phys.
Lett. A 306, 200–205 (2003)
4. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Crypt-
analysis of a discrete chaotic cryptosystem using external
key. Phys. Lett. A 319, 334–339 (2003)
5. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Crypt-
analysis of dynamic look-up table based chaotic cryptosys-
tems. Phys. Lett. A 326, 211–218 (2004)
6. Alvarez, G., Montoya, F., Romera, M., Pastor, G.: Com-
ments on “Modified Baptista type chaotic cryptosystem via
matrix secret key” [Phys. Lett. A 372 (2008) 5427]. Phys.
Lett. A 373, 3398–3400 (2009)
7. Baptista, M.S.: Cryptography with chaos. Phys. Lett. A
240, 50–54 (1998)
8. Biham, E., Shamir, A.: Differential cryptanalysis of DES-
like cryptosystems. J. Cryptol. 4(1), 3–72 (1991)
9. Chen, G., Chen, Y., Liao, X.: An extended method for ob-
taining S-boxes based on three-dimensional chaotic Baker
maps. Chaos Solitons Fractals 31, 571–577 (2007)
10. Dawson, M., Tavares, S.: In: Advances in Cryptology: Pro-
ceedings of EURO-CRYPT. Lect. Notes. Comput., vol. 91,
pp. 352–367 (1991)
11. Detombe, J., Tavares, S.: In: Advances in Cryptology: Pro-
ceedings of CRYPTO. Lecture Notes in Comput. Sci.,
vol. 92, pp. 165–181 (1992)
12. Guoping, T., Xiaofeng, L., Yong, C.: A novel method for
designing S-boxes based on chaotic maps. Chaos Solitons
Fractals 23, 413–419 (2005)
13. He, D., Chen, Y., Chen, J.: Nonlinear dynamics, crypt-
analysis and improvement of an extended chaotic maps-
based key agreement protocol. Nonlinear Dyn. (2012).
doi:10.1007/s11071-012-0335-0
14. Hussain, I., Shah, T., Gondal, M.A., Mahmood, H.: A new
algorithm to construct secure keys for AES. Int. J. Con-
temp. Math. Sci. 5(26), 1263–1270 (2010)
15. Hussain, I., Shah, T., Mahmood, H., Afzal, M.: Compar-
ative analysis of S-boxes based on graphical SAC. Int. J.
Comput. Appl. 2(5), 975–8887 (2010)
16. Hussain, I., Shah, T., Gondal, M.A., Khan, W.A.: Construc-
tion of new S-box using a linear fractional transformation.
World Appl. Sci. J. 14(12), 1779–1785 (2011)
17. Hussain, I., Shah, T., Gondal, M.A., Mahmood, H.: Some
analysis of S-box based on residue of prime number. Proc.
Pak. Acad. Sci. 48(2), 111–115 (2011)
18. Hussain, I., Shah, T., Gondal, M.A., Wang, Y.: Analyses of
SKIPJACK S-box. World Appl. Sci. J. 13(11), 2385–2388
(2011)
19. Hussain, I., Shah, T., Gondal, M.A.: An efficient image en-
cryption algorithm based on S8 S-box transformation and
NCA map. Opt. Commun. (2012). doi:10.1016/j.optcom.
2012.06.011
20. Hussain, I., Shah, T., Gondal, M.A., Mahmood, H.: A pro-
jective general linear group based algorithm for the con-
struction of substitution box for block ciphers. Neural Com-
put. Appl. (2012). doi:10.1007/s00521-012-0870-0
21. Hussain, I., Shah, T., Gondal, M.A., Mahmood, H.: Anal-
ysis of S-box in image encryption using root mean square
error method. Z. Naturforsch. A 67a, 327–332 (2012)
22. Hussain, I., Shah, T., Gondal, M.A., Mahmood, H.: Gen-
eralized majority logic criterion to analyze the statisti-
cal strength of S-boxes. Z. Naturforsch. A 67a, 282–288
(2012)
23. Hussain, I., Shah, T., Gondal, M.A., Mahmood, H.: Con-
struction of S8 Lui J S-boxes and their application. Comput.
Math. Appl. (2012). doi:10.1016/j.camwa.2012.05.017
24. Hussain, I., Shah, T., Mahmood, H., Gondal, M.A.: A pro-
jective general linear group based algorithm for the con-
struction of substitution box for block ciphers. Neural Com-
put. Appl. (2012). doi:10.1007/s00521-012-0870-0
25. Hussain, I., Shah, T., Mahmood, H., Gondal, M.A.:
A group theoretic approach to construct cryptographically
strong substitution boxes. Neural Comput. Appl. (2012).
doi:10.1007/s00521-012-0914-5
26. Hussain, I., Shah, T., Mahmood, H., Gondal, M.A.: S8
affine power affine S-boxes and their application. Neural
Comput. Appl. (2012). doi:10.1007/s00521-012-1036-9
27. Ivancevic, T., Jain, L., Pattison, J., Hariz, A.: Nonlinear dy-
namics and chaos methods in neuro-dynamics and complex
data analysis. Nonlinear Dyn. 56, 23–44 (2009)
28. Jakimoski, G., Kocarev, L.: Chaos and cryptography: Block
encryption ciphers based on chaotic maps. IEEE Trans. Cir-
cuits Syst. 48(2), 163 (2001)
29. Lee, C.-C., Chen, C.-L., Wu, C.-Y., Huang, S.-Y.: An
extended chaotic maps-based key agreement protocol
with user anonymity. Nonlinear Dyn. (2011). doi:10.1007/
s11071-011-0247-4
30. Li, S., Mou, X., Ji, Z., Zhang, J., Cai, Y.: Improving security
of a chaotic encryption approach. Phys. Lett. A 290, 127–
160 (2001)
31. Maccari, A.: A perturbation method for nonlinear two-
dimensional maps. Nonlinear Dyn. 19, 295–312 (1999)
32. Özkaynak, F., Özer, A.B.: A method for designing strong S-
boxes based on chaotic Lorenz system. Phys. Lett. A 374,
3733–3738 (2010)
33. Pieprzyk, J., Finkelsten, G.: Towards effective nonlinear
cryptosystem design. IEE Proc. E 135, 325–335 (1988)
34. Shah, T., Hussain, I., Gondal, M.A., Khan, W.A.: Construc-
tion of cryptographically strong 8 ×8 S-boxes. World Appl.
Sci. J. 13(11), 2389–2395 (2011)
35. Shah, T., Hussain, I., Gondal, M.A., Mahmood, H.: Sta-
tistical analysis of S-box in image encryption applications
based on majority logic criterion. Int. J. Phys. Sci. 6(16),
4110–4127 (2011)
36. Shah, T., Hussain, I., Gondal, M.A.: Image encryp-
tion algorithm based on PGL(2,GF(2∧8)) S-boxes
A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems 2311
and TD-ERCS chaotic sequence. Nonlinear Dyn. (2012).
doi:10.1007/s11071-012-0440-0
37. Shannon, C.E.: Communication theory of secrecy systems.
Bell Syst. Tech. J. 28(656), 715 (1949)
38. Wang, G.: Chaos synchronization of discrete-time dynamic
systems with a limited capacity communication channel.
Nonlinear Dyn. 63, 277–283 (2011)
39. Wang, Y., Wong, K.W., Liao, X., Xiang, T.: A block cipher
with dynamic S-boxes based on tent map. Commun. Non-
linear Sci. Numer. Simul. 14, 3089–3099 (2009)
40. Wang, Y., Wong, K.W., Li, C., Li, Y.: A novel method to
design S-box based on chaotic map and genetic algorithm.
Phys. Lett. A 376, 827–833 (2012)
41. Webster, A.F., Tavares, S.: In: Advances in Cryptology:
Proceedings of CRYPTO. Lect. Notes. Comput. Sci.,
vol. 85, pp. 523–534 (1986)
