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IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 4, Ver. IV (July – Aug. 2015), PP 91-101
www.iosrjournals.org
DOI: 10.9790/0661-174491101 www.iosrjournals.org 91 | Page
New Dynamical Key Dependent S-Box based on chaotic maps
F. J. Luma1, H. S. Hilal1 and A. Ekhlas2
1, Department of Computer Science, University of Technology,
P.O Box 35092, Baghdad, www.uotechnology.edu.iq
2 Department of Computer Science, Mustansiriyah University, baghdad, Iraq,
Abstract: The strength and security of cryptographic algorithms is determined by substitution non-linear S-
boxes, so the construction of cryptographically strong S-boxes is important in the design of secure
cryptosystems. In In this paper, an efficient method for designing dynamical key dependent S-boxes based on 2D
logistic map and 2D cross map. S-box is a nonlinear transformation where each byte of the State is replaced by
another byte using the substitution table. Each individual byte of State is mapped into a new byte by using 2D
cat map. One S-box is used for each message. The aim of the proposed approach is to generate more secure S-
boxes. The generated S-boxes will increase the complexity and has better results in security analysis. The new S-
box is analyzed and tested for the following criteria: avalanche effect, strict avalanche effect, key sensitivity,
differential and linear cryptanalysis. All the results have shown that the proposed method is a good candidate
for designing dynamical S-boxes that can be widely used in block cipher.
Keyword: Dynamical S-Box, Block Cipher, Key-Dependence, 2D logistic map, 2D cross map, 2D cat map.
I. Introduction
The S-box is the core component of most block ciphers, such as the Data Encryption Standard (DES),
the Advanced Encryption Standard (AES) and so on. A strong block cipher should be resistant to various
attacks, such as linear and differential cryptanalysis. This is generally achieved if the S-boxes used satisfy a
number of criteria, such as the bijection, avalanche effect, the strict avalanche criteria (SAC) and so on. S-boxes
are fixed and they used in SPN cipher systems as the important nonlinear component. The S-box used in
encryption process could be chosen under the control of key, instead of being fixed.
According to the chaotic systems properties such as the ergodic, mixing and random-like behavior, it
seems to be convenient and simple to obtain "good"
S-Boxes by modifying slightly the initial conditions or system parameters. Many approaches for obtaining S-
Boxes based on chaos have been
presented:-
Jakimoski and Kocarev, 2001 were presented the first chaotic S-boxes. They have proposed a four-step
method to create Sboxes by using chaotic maps, which includes choosing a chaotic map, discretizing the
chaotic map, key Two well-known chaotic maps (exponential and logistic) are used to generate the S-boxes,
and suggested key-dependent chaotic S-boxes to enhance the security, in which the parameters and the
number of iterations of the discretized exponential map were considered as the keys [1]. Their methodology
includes N-th iteration of chaotic for each logistic map, where they choose N=1000.
In 2005, Tang et al., proposed a method based on baker map. In this method, by iterating a chaotic logistic
map, a 8-bit sequence of binary random variables is generated from a real value trajectory obtained and turn
it to a decimal integer on the range of 0-2, then an integer table can be obtained. Then a key-dependent
permuting is used to shuffle the table nonlinear by a Baker map [2].
In [3], another method proposed by Tang and Liao that is based on discretized chaotic map. This method
consists of: First, an integer sequence that can be regarded as secret key K, K= X0= {1, 2, ..., 2 }, is
obtained in an arbitrary way. Second, for a given M = 2n and A, the chaotic map iterates more than k times
with the initial value X0, one can obtain a permuted integer sequence {X}. Finally, by translating the {X} to
a 2n/2×2n/2 table, the S-box is obtained.
In 2007, Chen et al., proposed another method for designing S-boxes based on three-dimensional baker map
which has more intensive chaotic characters than the two-dimensional one [4].
Muhammad Asim et al, 2007 propose method based on the mixing property of the piecewise linear chaotic
map (PLCM). In this method, the output range [0.1, 0.9] is divided into 256 intervals of equal length. Label
each region sequentially from 0 to m, where m is equal to 255. Iterate the PLCM using the selected initial
condition.
Whenever the PLCM visits a particular region, store that number in an array S [5].
In 2009, R.Yin, propose a new key-dependent S-box based on the iteration of continuous logistic chaotic
maps. Since S-boxes operate in the discrete state space, they making a mapping between these discrete
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 92 | Page
states and the continuous states of chaotic systems [6].
J. Peng, S. Jin, L. Lei, R. Jia, 2012 propose a method for generating dynamical key-dependent S-boxes
based on hyperchaotic Chen system . The secret key first is mapped to the initial condition and control
parameter for the hyperchaotic system. Then, iterate the Chen system to generate a hyperchaotic sequence
which is subsequently used to construct the S-box [7].
Mona Dara and Kooroush Manochehri, 2013 propose a new method based on chaotic logistic map that uses
cipher key to generate initial value. They use cipher key to generate initial value of logistic map (X0) and
outputs of logistic map as the values dynamic S-box [8].
Cristian-Iulian Rîncu, Vasile-Gabriel IANA, 2014 combine simple chaotic maps, which are frequently used
in chaos based cryptography.
They use three of the chaotic maps Logistic map, Tent map and Piece-Wise Linear Chaotic Map. These
maps will be iterated starting from the initial condition x0, the output becoming the next initial condition. a
number of iterations of the three chaotic maps are performed, equal to the value of the S -box dimension, in
order to ensure the beginning of a normal chaotic operating mode. Then the iteration of the three chaotic
dynamical systems is continuing and at each iteration the real value is kept of each system. The three real values
are transformed into integer values [9]. In this paper, we propose new method based on the mixing property of
the chaotic maps (2D logistic map, 2D cross map) to design dynamical dependent key S-box. One S-box is used
to encrypt each message. Each byte is permuted using 2D cat map before substituted with the byte in the
proposed S-box.
The remaining part of the paper is organized as follows: the chaotic functions analysis is given in
Section 2. The description of the algorithm is
given in Section 3. Section 4 presents the criteria for a ‗‗good n × n bit a cryptographically S-box. Section 5
presents the experiment results of the
propose S-box.
II. Basic Theory.
In this paper we used three chaotic maps: 2D cross map, 2D logistic map and 2D cat map to construct the new
S-box.
2.1 2D cross map.
(1)
Where μ and k are the control parameters of the system, respectively. When μ=2 and k=6, this system exhibits a
great variety of dynamics behavior 10][.
Manisha Raj1, Shelly Garg [11] proposed a new image encryption algorithm based on DNA sequence addition
and five chaotic maps (Logistic Map, Cross Chaotic Map, Duffing map, Tinkerbell map, Gingerbreadman map).
The chaotic maps are compared through the simulation results, histogram analysis and correlation analysis.
They have found out that Cross Chaotic Map, showed best results than other chaotic maps. It is sensitive to the
secret keys, it has larger key space, and it gives best encrypted image.
2.2 2D logistic map.
One of the most known and widely used chaotic systems is the 1 D Logistic map, which is defined as follows
[11] :
(2)
where µ is the control parameter. The system is in chaos on condition that 3.569<μ<4.0.
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 93 | Page
The 2D logistic map is an extension of 1D logistic map. It increases the key space as well as the dependency on
control parameters. In 2D logistic map, it is bit harder to guess the secret information. It also exhibits greater
amount of chaotic behavior on the generation of sequence [12]. In overall, it increases the complexity of the
algorithm.
(3)
Where (xi,yi) is the pair-wise point at the ith iteration and μ1, μ2, μ2, γ1 and γ2 are the system parameters.
When 2.75 <μ1<=3.4, 2.75 < μ2<=3.45, 0.15 <γ1<=0.21, 0.13 <γ2<=0. 15, the system is in chaotic state and can
generate two chaotic sequences in the region (0, 1].
The 2D logistic map has a higher complexity compared to the conventional 1D logistic map. The
complexities of the 1D and 2D logistic maps can be measured by using various means such Information Entropy
and Lyapunov Exponent with respect to different pairs of initial values [13]. By this comparison they found that
the 2D logistic map has a higher information entropy scores than 1D logistic map, which implies that its
trajectory is more random-like. Meanwhile, the 2D logistic map also has a larger Lyapunov exponent than the
1D logistic map, which implies that the 2D logistic map is more dynamic.
2.3 2D Cat Map.
A 2D Cat map is first presented by V.I. Arnold in the research of ergodic theory. Let the coordinates of a
positions P = {(x, y) | x, y = 1, 2, 3. . . N}, a 2D Cat map with two control parameters [14] is as follows:
(4)
Where, a, b are positive integers which are control parameters and (x‘, y‘) is the new position of the original
pixel position (x, y) of N x N plain-image when cat map is applied once to the original. By replacing the
position of the image pixel points with new coordinate, cat map permutes/shuffles the organization of pixels of
plain-image. After several iterations, the correlation among the adjacent pixels is disturbed completely and the
image appears distorted and meaningless.
Inverse transformation for deciphering is given as follows: -
(5)
III. Design Dynamical Key Dependent S-Box.
Substitution is a nonlinear transformation which performs confusion of bits. A nonlinear transformation
is essential for every modern encryption algorithm and is proved to be a strong cryptographic primitive against
linear and differential cryptanalysis. Nonlinear transformations are implemented as lookup tables (S-boxes). The
central idea of the proposed S-box is based on the mixing property of chaotic nonlinear dynamical systems. The
proposed S-box is a table of 16 × 16 integer values (256 bytes). The S-box is created by using 2d logistic map
and cross map. The main idea of the proposed S-box consists of the following major steps:-
Step1:- the initial condition (x0 and y0) is input to the 2D cross map and 2D logistic map. These numbers are
floating point numbers where the precision is 10−16 for each of x0 and y0, considered as the keys of the proposed
S-box.
Step2:- Iterate the cross map and logistic map 100 times and ignore the results, in order to eliminate the
transient effect of chaotic map.
Step3:- Iterate the cross map and logistic map one time. The two outputs of cross map(x1 and y1 ) are Xored
with the two outputs of logistic map (x2 and y2 ) :-
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 94 | Page
(6)
Step4:- the new outputs (nx and ny) are translated to integer numbers belong [0..255] by using the following
equation:-
(7)
Step5: the two integer numbers 1 2 are inserted to the S-box table while they are not found in the S-box table in
order to avoid repeating the same numbers.
Step6:- repeat from step 2 until the S-box contains a permutation of all possible 256 byte values.
Table (1) shows an example of S-box created from the initial conditions (x0 = 0.2376589876543123 and y0=
0.9765432786554125).
S-box is a nonlinear transformation where each byte of the State is replaced by another byte using the
substitution table. Each individual byte of State is mapped into a new byte in the following way (as shown in
figure (1-a)): The leftmost 4 bits of the byte are used as a row value and the rightmost 4 bits are used as a
column value. These row and column values are permuted to anther values for row and column by using the cat
chaotic map eq(4).
The byte in the new row and column is then served as indexes into the S-box to select a unique byte output
value.
For example, if S-box in figure 1 is used to substitute the byte (190) which
is 10111110 in binary, the leftmost 4 bits (R=1011 = 11=b) is the row value and the rightmost 4
bits(C=1110=14=e) is the column value.
This row and column values are diffuses using cat map eq(4) (with the
control parameters a =41and b=68) as follows;
(8)
Finally, use the new row (9) and column (2) as index to find the substituted byte in the S-box (in table 1) which
is the byte (102).
The inverse substitute byte transformation is performed in the following way (as shown in figure (1-
b)): the byte is search in the S-box table and take its
row and column. These row and column values are return to their real value by using the inverse of cat map.
Then the new row and column is
concatenate to make the original byte.
For example, the substituted byte (102) from previous example can be return to its original byte value in the
following way: search the S-box table to find the byte (102) and take its row (9) and column (2). This row and
column values input to inverse cat map to produce the original row and column as follows:
(9)
So, the row (11) which is in binary (1011) and column (14) which is in binary (1110) are concatenating to
produce the original byte
(10111110=190).
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 95 | Page
Table(1): S-box created from the 2d logistic map and cross map using the proposed procedure.
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 96 | Page
IV. Criteria for a ‘‘good n × n bit a cryptographically S-box.
The only nonlinear components in cryptosystems are the S -boxes. The differential cryptanalysis was
introduced by Biham and Shamir for DES-like
cryptosystems [15]. After that Dawson and Tavares [16] expanded S-boxes‘ design criteria based on information
theory and revealed how S-boxes
provided immunity to the differential attack. In general, some cryptographic properties are widely accepted as
the essential properties for ―good‖ S-boxes
and can be used to evaluate the S-box [18,17,8, 9, 6, 7,4] .
There are several properties which are cryptographically desirable in an s-box. They are:-
• The Bijective Property.
• The avalanche effect.
• The strict avalanche criteria (SAC).
• Differential attacks.
• Nonlinearity.
• Key sensitivity.
4.1 The Bijective Property.
An n x n bit s-box need to be a bijection that is, that every possible input
vector x maps to a unique output vector y. The bijective property is checked by using the method introduced in
the literature [18, 19]. If the Boolean function f(x)=(f1,f2,….,fn) of an S-box such that: -
(8)
,where ai ϵ {0,1} , (a1,a2,…,an)=(0,0,…,0) and wt (⋅) is the Hamming weight (the Hamming weight is the
number of "1" bits in the binary sequence, the S-box is bijective.
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 97 | Page
4.2 The avalanche effect.
Any secure component in a cryptographic algorithm has one of the key characteristic representatives
which is the avalanche effect [9]. By presenting this characteristic, we can sure that half of the S-box output bits
will be modified when complementing a single bit in the input vector. These characteristics are tested according
to the avalanche vectors by using [9]:
(9)
Where the difference between x and xi lies only in i bit. When the
complementing of any unique input bit determined the modification of N/2 output bits on average, the avalanche
effect is fulfilled.
4.3 The strict avalanche criteria (SAC).
The strict avalanche criterion (SAC) was introduced by A.F. Webster and
Tavares in 1985.SAC is satisfied when single input bit in an S-box is changed, each output bit should change
with a probability of one half. The
SAC of an S-box is tested by constructing the dependence matrix. If each element and the mean value of the
matrix are both close to the ideal value 0.5, the S-box is considered as nearly fulfills the SAC. The dependence
matrix should be calculated According to [20] by following those steps: -
1. Let x a random vector of an n-bits and its m-bits ciphertext y, y=s(x).
2. A set of n-vectors (xl,x2, ... ,xn), such that x and xj differ only in one bit j, for all j 1≤ j ≤ n . Another set
of vectors (y1,y2, ... ,yn) is created such that yj=s(xj).
3. The avalanche vectors (v ,v ,..,v ) is calculated by:
(10)
4. The bit i of vj is added to element aij of m x n dependence matrix A.
5. Repeat these steps for k random vector x.
6. Each element of the matrix A is divided by k.
The value of each element in A (aij) varies from 0 to 1 which gives an idea about plaintext bit j and
ciphertext bit i. If aij is close to 1 then indicate that if bit j is complemented then bit i should change its value. If
all element of the matrix A have a value close to 0.5, then the S-box satisfies the SAC criterion.
4.4 Differential attacks.
Differential cryptanalysis was introduced the first time by Biham and Shamir in [15]. The S-box should
have the differential uniformity to resist the differential cryptanalysis. This mean the nonlinear transformation S-
box should ideally have differential uniformity. The differential approximation probability of a given S-box,
DPs, is a measure for differential uniformity and is defined as [17]: -
(11)
Where where X is the set of all possible input values, elements, ∆x is the input differential and ∆y is the
intruder first calculates differential pairs (∆x, ∆y) for∆y input differential ∆x and the output differential
2n is the number of its output differential. The each=s( xS)-⊕boxs(xwhere⨁△ thex),then find the result of DPs.
This equation means that the DPs is the maximum probability of the output
differential ∆y when the nonzero input differential is ∆x. Ideally, a nonlinear S-Box should have a differential
uniformity: an input differential ∆x is mapped to a unique output differential ∆y.
4.5 Nonlinearity.
Linear cryptanalysis studies linear probability approximations of the cryptosystem. The S-box must
resistant to this type of attack. The goal is to construct linear equations between input plaintext and output cipher
text and enumerate all linear approximations of the S-Box in a linear approximation table. Otherwise, the
nonlinearity is measured by calculating probability approximation (LPs) [17]: -
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 98 | Page
(12)
Where a, b ϵ {I, 2, .., 2n-1}, x.a is the parity of the binary product of x and a. To increase linear attack
complexity, the value of LPs must be decreased.
4.6 Key sensitivity.
Sensitivity analysis is the study of how the uncertainty in the output of a model can be apportioned to
different sources of uncertainty in the model input [21]. An essential factor for the S-Box is the sensitivity on
the key. In other words, a small changing in the keys should cause a large change in the S-Box. This means that
a small difference on key values, the created S-boxes should be completely uncorrelated.
To ensure the sensitivity of the key, the analysis is done using Pearson's correlation coefficients.
Consider a pair of S-boxes given by: S1 = [x1, . . . , xN] and S2 = [y1, . . . , yN]. Therefore, the corresponding
correlation coefficient is:
To ensure the sensitivity of the key, the analysis is done using Pearson's
value of CS1, S2 is to ±1, the stronger the correlation between the two S-boxes. In the case of two independent
S-boxes, the value of CS1,S2 is equal to0.
5. Experiment results.
In this section, we construct 100 dynamical key dependent S-Box from nearby or successive keys and
the Criteria for good S-box results are as
follows: -
1. The Bijective Property: -All the S-boxes are bijective. The value of all generated S-box is 128, which is
the same as the ideal value. This mean that in each S-box every possible input vector x maps to a unique
output vector y.
2. The avalanche effect: - we test the avalanche effect for the S-box in table (1). Figure (2) shows the
frequency of the number of the modified bits belonging to the output values obtained after changing each of
the input N positions for all of the S-box elements in table(1). Because the complementing of any unique
input bit determined the modification of N/2 (8/2=4) output bits on average, the avalanche effect is
considered fulfilled. We found that all the 100 created S-boxes are fulfilled the avalanche effect.
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 99 | Page
Figure (2): The results of the avalanche effect test.
3. The strict avalanche criteria (SAC):- the dependence matrix for the S-box in table (1), with 8 input bits, it
is not difficult to compute a complete dependence matrix containing the elements for all the 2N input
vectors that can appear, is calculated and shown in table (2). The mean value is 0.51342 which is close to
ideal value (0.5). All the mean values of the dependence matrixes of 100 S-boxes are located within [0.46,
0.53], which are also close to the ideal value 0.5, indicating that all the S-boxes have excellent SAC
property.
4. Differential attacks: - The DPs of the S-Box in table (1) is calculated and the result is 12/256 which is close
to the ideal value (2-6≤ DPAES-SBox≤ 2-4). All the DPs of 100 S-boxes are located within [10/256, 12/256],
which are also close to the ideal value, indicating that all the S-boxes have good immunity against differential
attacks.
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 100 | Page
5. Nonlinearity: - The linear approximation probability of the S-Box in table (1) is evaluated to LPs=0.07910.
Linear approximation probability of statistic AES S-Box varies from 2-2 to 2-3, so decreasing LPs, leads to
increasing linear attack complexity. All the LPs of 100 S-Box are calculated and the average of LPs is
0.0760089. This means that all the S-boxes can resistant to this type of attack.
6. Key sensitivity: - In order to investigate the S-box‘s sensitivity to the secret keys, the Pearson's correlation
coefficients between the S-boxes in table (1) and table (3) with the key1 and key2 are calculated. In here, key1
is set to
x0 = 0.2376589876543123 and y0= 0.9765432786554125 key2 is set to
x0 = 0.2376589876543124 and y0= 0.9765432786554126
Where the key1 is slightly changed to key2. The S-boxes generated by key1 and key2 are shown in table (1) and
table (2), respectively, and the correlation coefficient between them -0.0199495, indicating that the S-boxes are
very sensitive to the keys.
Table (2): S-box created from the 2d logistic map and cross map using the proposed procedure.
New Dynamical Key Dependent S-Box based on chaotic maps
DOI: 10.9790/0661-174491101 www.iosrjournals.org 101 | Page
V. Conclusions
The S-box is the core component of most block ciphers. In this paper, new
method for creating dynamical key-dependent S-boxes based on 2D logistic map and 2D cross map is presented.
The results show that generated S- Boxes have very low linear and differential probabilities and satisfy also the
avalanche criterion and strict avalanche criterion (SAC). In addition, the S-box‘s sensitivity to the keys is
investigated by the correlation coefficient. The algorithm used to generate S-boxes and the selected chaotic
maps ensure a simple and quick implementation using simple digital computing systems. Because the proposed
S-box is satisfied all the criteria of good S-box, we can used for designing block cipher with dynamical S-boxes.
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