Chaotic Encryption Scheme Based on A Fast

Permutation and Diffusion Structure

Jean De Dieu Nkapkop1,2, Joseph Effa1, Monica Borda2, Laurent Bitjoka3, and Mohamadou Alidou4
1
Department of Physics, University of Ngaoundéré, Cameroon
2
Department of Communications, Technical University of Cluj-Napoca, Romania
3
Department of Electrical Engineering, Energetics and Automatics, University of Ngaoundéré, Cameroon
4
Department of Physics, University of Maroua, Cameroon
Abstract: The image encryption architecture presented in this paper employs a novel permutation and diffusion strategy based
on the sorting of chaotic solutions of the Linear Diophantine Equation (LDE) which aims to reduce the computational time

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observed in Chong's permutation structure. In this scheme, firstly, the sequence generated by the combination of Piecewise

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Linear Chaotic Map (PWLCM) with solutions of LDE is used as a permutation key to shuffle the sub-image. Secondly, the
shuffled sub-image is masked by using diffusion scheme based on Chebyshev map. Finally, in order to improve the influence of
the encrypted image to the statistical attack, the recombined image is again shuffle by using the same permutation strategy

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applied in the first step. The design of the proposed algorithm is simple and efficient, and based on three phases which provide
the necessary properties for a secure image encryption algorithm. According to NIST randomness tests the image sequence
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encrypted by the proposed algorithm passes all the statistical tests with the high P-values. Extensive cryptanalysis has also
been performed and results of our analysis indicate that the scheme is satisfactory in term of the superior security and high
speed as compared to the existing algorithms.
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Keywords: Fast and secure encryption, chaotic sequence, Linear Diophantine Equation, NIST test.
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Received March 17, 2015; accepted October 7, 2015

1. Introduction phase, and they form the basis of many existing chaos-
based image encryption algorithms [2, 7].
With the fast development of image transmission Nevertheless, to assure an efficient encryption scheme,
through computer networks, especially the Internet, the
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some conditions should be fulfilled such as a large key

security of digital images has become a most important
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space, randomness of the cipher-image and a high

concern. Image encryption differs from text encryption sensitivity on the initial conditions. A large key space
due to some intrinsic features of images which include is necessary to resist brute-force attacks [9] and a
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bulk data capacities, high redundancy, strong secure encrypted image corresponds to an image that
correlations among pixels, etc. [8]. These features make cannot be statistically distinguished from a truly
conventional cipher systems such as DES, AES and random sequence. Indeed, the cipher-images should
RSA unsuitable for practical image encryption [13]. present a good level of randomness and moreover,
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In order to overcome image encryption problems, in should be very sensitive to the used of initial key or
recent years, many scientists and engineers have seeds and to the plain-image [1].
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designed image encryption algorithms based on one or Some existing image encryption algorithms were
more chaotic maps [5, 11]. Due to desirable properties designed with a fast diffusion strategy, but their
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of nonlinear dynamical systems such as high sensitive permutation is not fast enough because at this stage
dependence on initial conditions and control parameter, the discretization of chaotic sequences in finite values
ergodicity, unpredictability, mixing, etc., which are is time consuming. To achieve high security level
analogous to the confusion and diffusion properties of performance, they also need more than one round in
Shannon [12], the chaos-based encryption has their permutation-diffusion structure.
suggested a new and efficient way to deal with the The key challenge now in cryptography being to
intractable problem of fast and highly secure image consider the trade-offs between the security level and
encryption [11]. efficiency, in this paper, a chaotic cipher for gray
Fridrich [6] suggested that a suitable chaos-based images by a fast permutation-diffusion structure is
image encryption algorithm should be composed of two proposed.
phases: one phase is to permute the order of the image In the proposed scheme, image is split in n sub-
pixels using chaotic map(s) while the other phase is to images. The design of the proposed algorithm is then
alter the numerical values representing the color of each based on three phases. In phase I, image permutation
pixel, again using chaotic map(s). These two phases are is based on the sorting of the solutions of the Linear
referred to as the confusion phase and the diffusion
Diophantine Equation (LDE) [4] whose coefficients are
integers and dynamically generated from any type of
chaotic systems to enhance the speed at the permutation
stage. This method also leads to a stronger permutation
effect. In phase II, we generate diffusion template using
Figure 1. Permutation based on Chirikov Standard Map: (a) The
image diffusion based on Chebyshev map. Then the plain image of Lena with 512×512 size. (b) The test image after
image is masked by performing XOR operation on the applying the Chirikov standard map once. (c) The test image after
shuffled image and diffusion template. Finally, in phase applying the Chirikov standard map three times. (d) The test
III, the recombined image is encrypted by using image after applying the Chirikov standard map five times.
permutation key based on the sorting of the solutions of However, to get the correlation among the adjacent
the LDE. To avoid the cyclic digitization of chaotic pixels completely disturbed and the image completely
numbers in the generation of permutation key and then unrecognizable this permutation needs five rounds of
achieve high speed performance, we generate iterations. We then propose at the following sub-
permutation key at the initialization step by using section to replace this kind of permutation by a one-
ascending or descending sorting of the solution of LDE; round permutation based on the sorting of chaotic

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thereafter, this permutation key is just dynamically solutions of the Linear Diophantine Equation (LDE)

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updated for each sub-image by including inside d > 3 for permuting the pixel position in the image.
chaotic numbers and then sorting the result for
obtaining and updated permutation key. Also, diffusion

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2.2. Image permutation based on the sorting
key is updated for each sub-image.
of chaotic solutions of the LDE
The rest of the paper is organized in the following
manner: comparison between two image permutations
is described in section 2. The diffusion phase in the
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on the ascending or descending sorting of the chaotic
proposed cryptosystem is presented in section 3. The solutions of the LDE. The LDE is defined by Equation
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simulation and performance analysis are discussed in 2 below [4]:
section 4 and the conclusions are made in section 5.
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ax + by =
e (2)

2. Comparison Between Two Image where a, b and e are the constants and belongs to the
Permutations set of natural integers, x and y the general solutions.
In order to determine the solutions of LDE as a
In order to decorrelate the strong relationship between random process, coefficients a and b of the LDE can
adjacent pixels, the permutation process is usually used. be evaluated from the chaotic system in the Algorithm
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1 below:
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2.1. Image Permutation Based On Chirikov

Algorithm 1: Fragment to get a and b of the LDE
Standard Map
1. Require: T1, T2, Wr, λ0
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To demonstrate the superiority of the proposed image 2. Initialisation: Wλ0 ← 0; Wx0 ← 0;

permutation, we recall here image permutation based 3. for k=0 to 15 do
on Chirikov standard map. In order to incorporate Wλ ← 2(k/(k+1))×T1(k+1)+Wλ0
Wx ← 2(k/(k+1))×T2(k+1)+ Wx0
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Chirikov standard map into image encryption that

operated on a finite set, it has to be discretized. The end for
4. λ ← λ0+ Wλ /(10Wr)
discretized version Chirikov standard map can be
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5. x0 ← Wx /Wr
obtained by changing the range of (x, y) from the square 6. Require: x01, x02 from chaotic system
[0, 2π)×[0, 2π) to the discrete lattice N×N as follows 7. a ← 2× fix(2p x01) +1
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[3]: 8. b ← 2× fix(2p x02) +1

 xi=+1 ( xi + yi ) mod N The control parameter λ and the initial condition x0 are
 (1) deduced from the keys T1 and T2. λ0 is a constant such
  2π xi +1 
 y=i +1  yi + K sin  mod N that the behavior of Equation (3) remains chaotic for
  N 
all the range of λ and x. Wr=8160 is the greatest value
where N is the width or length of a square image, and K of Wλ0. T1(k+1) and T2(k+1) respectively correspond to
is a positive integer which can be used as the the values assigned to the ASCII symbols of key T1
permutation key. and T2. fix(.) is the integer part of function. p is the
Chirikov standard map is then employed to shuffle number of bits used for the quantization. The precision
the pixel positions of the plain image. It application to a used for the digitization of the chaotic values by using
grayscale test image with 512×512 size is shown in Matlab is about ε ≈ 10-15.
Figure 1. The ciphering key being K=512. In this work, we have considered as chaotic system,
the Piece Wise Linear Chaotic Map (PWLCM)
described by the following equation:
  1
3. Diffusion Phase in the Proposed
 x(n − 1) × λ , if 0 ≤ x(n − 1)  λ

Cryptosystem
] [ x(n − 1) − λ ] ×
1
n) F [ x(n − 1)
x(= = , if λ ≤ x(n − 1)  0.5 (3)
 0.5 −λ
 F [1 − x(n − 1) ] , if 0.5 ≤ x(n − 1)  1 In diffusion stage, the pixel values are modified
 sequentially to confuse the relationship between cipher

The PWLCM is known to be chaotic when its control image and plain image in order to increase the entropy
parameter λ is within ]0, 0.5[ and its initial condition is of the plain image by making its histogram uniform.
chosen within the interval ]0, 1[ [4]. In this paper, Chebychev map is used to generate
Then the coefficients a and b of the LDE evaluated in keystream K(n) in order to mask the pixel in first time.
the Algorithm Fragment 1 above are used to calculate The Chebychev map is described by:
( xn ) cos ( k .cos −1 xn ) , xn ∈ [ −1, 1]
the solutions of LDE which is sorted to generate the
x(n=
+ 1) Tk = (4)
permutation key for encryption. The procedure is shown
in the Algorithm 2 below: Where k ∈ [2, +∞[ is control parameter. The initial
value x(0) and parameter k are used as the key.
Algorithm 2: Fragment to get permutation key IZ
The diffusion procedure is described as follows:

t
1. Require: N, t
[G, C, D] ← gcd(a,b) 1. Randomly select a parameter k and an initial value

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2.
3. x ← mod(C+(b/G)t, N) + 1 x(0) for equation 4. Iterate equation 4 t times to
4. y ← mod(D - (a/G)t, x) + 1 avoid the harmful affect of the initial values, where
5. [I, J] ← sort(x)

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t is a preset integer and served as secret encryption
6. [I1, J1] ← sort(y) key, too.
7. Iz ← J(J1)
2. Let I denotes a gray scale permute image with size
Where t=(0, 1,…,N-1), N is the length of image.
gcd(x,y) is the greatest common divisor, mod(x,y) is the
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p(2),…p(n),…,p(N×M)}.
I and get P={p(1),
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modulo and sort(x) array elements in ascending or 3. Obtain for each iteration one key stream element
descending order. IZ is the permutation key. from the current state of the chaotic according to:
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For the security to be strengthened, it is necessary for
the permutation key IZ to be updated for each sub- =K (n) mod  floor ( ( ( x(n) + 1) / 2 ) ×1014 ) , L  (5)
images. The updating process is carried out by
replacing d>3 number of values of IZ with newly where floor(x) returns the value of x to the nearest
generated d number of chaotic numbers from the integers less than or equal to x, mod(x, y) returns
chaotic system and then sorting it for obtaining the the remainder after division and L is the gray levels
updated IZ. of plain-image
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4. Then apply the bitwise exclusive-OR to the

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In the proposed permutation scheme, the total image

frame is divided into n sub-images. For the first sub- permuted image pixels P by the following equation:
image the encryption is carried out by using Algorithm
c(=
n)
1
(
K (n) ⊕ {[ p (n) + K (n) ] mod N } ⊕ c(n − 1) ) (6)
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Fragment 2 and for the other sub-images, the 2

permutation key is only refreshed without solving LDE Where p(n), K(n), c(n) are the currently operated
equations. This helps to save computational time and at pixel, key stream element and output cipher-pixel,
the same time the length of permutation key is large
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respectively, and c(n-1) is the previous cipher-pixel.

enough to attain high security level.
5. Alter the control parameter k of the Chebychev map
The application of this method to a grayscale test
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in each round iteration as follows:

image with 512×512 size is shown in Figure 2. if x(n)>0, then
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k ← k + 10-14 ×c(n-1) (7)

else
k ← k - 10-14 ×c(n-1) (8)
6. In order to more secure the diffusion process,
Figure 2. Permutation based on the sorting of chaotic solutions of
continue to apply the bitwise exclusive-OR by
the LDE: (a) The plain image of Lena with 512×512 size. (b) The using the sequence c(n) of Equation 6 and diffusion
test image after applying the proposed permutation once. key X as follows:

As can be seen in this Figure 2, this method leads to a X ← mod(bx+ay,256) (9)

stronger permutation effect as compared to the previous c(n) ← c(n) ⊕X (10)
case and need only one round of permutation.
where x and y are obtained using Algorithm 2, a and
b are obtained using Algorithm 1.
Finally, in order to increase the randomness of the This initial condition is used for the generation of
entire 1-D ciphered image obtained in the Equation 10 two chaotic integers which are used as coefficients a
as well as the sensitivity of the cipher to small changes and b of the LDE to generate the permutation key IZt.
in the plain image, an image dependent initial condition This new permutation key of length L is thus deduced
is determined as follows: from the LDE for shuffling the 1-D image as a whole.
Notice that, before permuting the image as a whole,
Initialisation: λ0 ← 0
the n sub-images used in the permutation and diffusion
for j=1 to N×M do
stages are recombined to obtain a 1-D image.
λ0 ← λ0 + p(j)/(255×L) (11) The decryption procedure is followed in a reversed
end for order of the encryption procedure. The flowchart of
the proposed encryption and decryption algorithm is
The control parameter is the same as one used for the then described in Figure 3.
generation of IZ. L is equal to the length of the whole
image.

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Figure 3. Flowchart of the encryption and decryption algorithm.

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2×1032. The range of k should be restricted to a

4. Simulations and Performance Analysis particular interval of 2π to prevent Chebyshev map
from producing periodic orbits, then for k there will be
4.1. Key Space approximately 2π×1015 different possible values.
The key space is the total number of different keys that By considering only symbols “a-z”, “A-Z” and “0-
can be used in the encryption/decryption procedure. The 9”, the complete key space of the proposed image
key of the proposed cryptosystem is composed of two encryption scheme is 6232×4π×1047≈2347 which is large
parts: permutation key T1, T2 and diffusion key (x0, k). enough to resist brute-force attack.
In our work a 256-bit key corresponding to 32 ASCII
symbols is considered. 4.2. Histogram
In hexadecimal representation, the number of Image histogram clarifies how the pixel’s values of
different combinations of secret keys is equal to 2256. image are distributed. The histograms of the plain-
Therefore, the total number of possible values of x0 that image and the cipher-image are shown in Figure 4. As
can be used as a part of the key is approximately
shown in this figure, it is obvious that the histograms of Table 1. The results of the NIST tests.
the encrypted image are nearly uniform and Passing ratio Uniformity
Test name Result
significantly different from the histograms of the plain- of the test P-value
Frequency 0.9875 0.006355 PASSED
image. Hence it does not provide any clue to be Block frequency 0.9895 0.047478 PASSED
employed in a statistical analysis attack on the Cumulative sums 0.9870 0.170922 PASSED
encrypted image. For instance, the histogram in Figure Runs 0.9890 0.339271 PASSED
Longest run 0.9895 0.616305 PASSED
4(f) which corresponds to the ciphered Black image Rank 0.9880 0.583145 PASSED
highlights the effectiveness of the algorithm, as all the FFT 0.9860 0.096000 PASSED
256 gray-levels present the same probability. Non-overlapping
0.9895 0.999668 PASSED
template
Overlapping template 0.9870 0.412733 PASSED
Universal 0.9875 0.383827 PASSED
Approximate entropy 0.9905 0.893482 PASSED
Random excursions 0.9922 0.430809 PASSED
Random excursions
0.9875 0.892512 PASSED
variant
Serial 0.9895 0.595549 PASSED

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Linear complexity 0.9905 0.551365 PASSED

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4.4. Correlations Between Adjacent Pixels

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First, randomly select 1000 pairs of two adjacent
pixels from an image. Then, calculate their correlation
coefficient using the following formulas:
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N
N

∑ ( x − x )( y − y )
i i
(12)
rxy = i =1
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1 N
2  1
N
2

Figure 4. Example of histograms: (a)-(c) plain-images; (d)-(f)  N ∑ ( xi − x )   N ∑ ( xi − y ) 

= i 1=  i 1 
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ciphered images. From top to bottom are presented histograms of
Lena 512×512, Airplane 512×512 and Black 512×512. N
1
x=
N
∑x
i =1
i (13)
4.3. Randomness Test
N
1
∑y
The US NIST designed a set of 15 statistical tests to test
y= i (14)
randomness of binary sequences produced by N i =1
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pseudorandom number generators.

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For each test, a P-value is computed from binary where xi and yi are greyscale values of i-th pair of
sequence. In all tests, if the computed P-value is < 0.01, adjacent pixels, and N denotes the total numbers of
then conclude that the sequence is non-random. samples.
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Otherwise, conclude that the sequence is random. The results of the correlation coefficients for
In our experiment, m = 2000 different keystreams, horizontal, vertical and diagonal adjacent pixels for
each sequence having a length of n = 1000,000 bits the plain images and its cipher images are given in
in

which are generated using our scheme. The P-values for Table 2 and 3. Table 2 shows that the correlations
various tests are listed in Table 1. In this case, we coefficients evaluated with the proposed algorithm are
obtained the confidence interval [0.983, 0.996]. We better than those presented in the others references.
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notice that the results of the tests are satisfactory for the
Table 2. Comparative study of the correlation coefficients of the
whole set of tested outputs. All the sequences pass
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proposed algorithm with some existing algorithm.

successfully the NIST tests. These results show the
quality of the produced sequences with the pseudo- Proposed
Image Type algorithm [12] [2]
random number generator.
Horizontal Plain-image 0.9719 0.9404 0.9792
Cipher-image -0.0005 0.0088 0.0217
Lena Vertical Plain-image 0.9850 0.9299 0.9809
Cipher-image -0.0032 -0.0087 0.0086
Plain-image 0.9593 0.9257 0.9551
Diagonal
Cipher-image -0.0002 -0.0060 0.0118

4.5. Information Entropy Analysis

In information theory, entropy is the most significant
feature of disorder, or more precisely unpredictability.
Table 3. Correlation coefficients of two adjacent pixels in the others To evaluate the performance promotion of the
images. proposed encryption scheme, the NPCR and UACI are
plotted against the cipher cycles and compared with
Image Size Type Horizontal Vertical Diagonal
that of the existing scheme, as shown in Figures 5(a)
Plain-image 0.9663 0.9642 0.9370
Airplane 512×512 and (b), respectively. As can be seen from Figure 5,
Cipher-image -0.0018 -0.0035 0.0035
four overall encryption rounds are needed to achieve a
Plain-image 1 1 1
Black 512×512
satisfactory security level by using Lian et al.’s
Cipher-image 0.0019 0.0012 0.0000 encryption scheme [10], three overall encryption
rounds are needed to achieve a satisfactory security
It is well known that the entropy H(m) of a message level by using Zhu et al. [15]; one overall encryption
source m can be measured by: round is needed to achieve a satisfactory security level
2M by using Chong Fu et al. [3] and our algorithm.
H = −∑ p ( mi ) log 2 ( p ( mi ) ) (15)
i =1
However, compared to Chong Fu et al’s algorithm,
our method improves their NPCR and UACI only with
where M is the number of bits to represent a symbol; one round of permutation and diffusion process.

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p(mi) represents the probability of occurrence of

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symbol mi and log denotes the base 2 logarithm so that
the entropy is expressed in bits.
For a purely random source emitting 28 symbols, the

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entropy is H(m) = 8 bits. The test result on different
images for one round is defined in Table 4. It appears
that the entropy of the ciphered images is almost equal
to eight, compared to that of the plain-images. It can
also be noticed that the encrypted version of the image
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“Black” is a truly random image, thus confirming the
efficiency of the proposed cipher.
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Table 4. Information entropy of plain-images and cipher-images by Figure 5. NPCR and UACI performance of the proposed scheme
the proposed algorithm. and the others existing scheme. (a) NPCR performance. (b) UACI
performance.
Image Type Lena Airplane Black
Plain- 4.7. Key Sensitivity Analysis
7.4456 6.7043 0
image
Recall that secure cryptosystem requires not only a
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Entropy
Cipher-
large key space but also a high key sensitivity. That is,
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7.9992 7.9993 7.9978

image
a slight change in the key should cause some large
changes in the ciphered image. This property makes
4.6. Differential attack Analysis
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the cryptosystem of high security against statistical or

The diffusion performance is commonly measured by differential attacks. To evaluate the key sensitivity, the
means of two criteria, namely, the number of pixel plain Lena image is encrypted using four slightly
different test keys:
in

change rate (NPCR) and the unified average changing

intensity (UACI). NPCR is used to measure the Table 5. Slightly different keys for encryption.
percentage of different pixel numbers between two
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x0 = k= T1 = T2 =
(i)
images. The NPCR between two ciphered images A and 0.48729650284971 5.78259581295362 azertyuiopqsdfgj azertyuiopqsdfg0
B of size M×N is [14]:
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x0 = k= T1 = T2 =
(ii)
0.48729650284970 5.78259581295362 azertyuiopqsdfgj azertyuiopqsdfg0
m n

∑∑ D ( i, j )
i =1 (16) (iii)
x0 = k= T1 = T2 =
= × 100 0.48729650284971 5.78259581295361 azertyuiopqsdfgj azertyuiopqsdfg0
j
NPCRAB
m×n x0 = k= T1 = T2 =
(iv)
0.48729650284971 5.78259581295362 azertyuiopqsdfg1 azertyuiopqsdfg0
where
x0 = k= T1 = T2 =
(v)
1 A ( i, j ) ≠ B ( i, j )
 0.48729650284971 5.78259581295362 azertyuiopqsdfgj azertyuiopqsdfg2
D ( i, j ) =  (17)

0 otherwise
The corresponding cipher images are shown in Figures
The second criterion, UACI is used to measure the 6(a), (b), (d), (f) and (h), respectively. The differences
average intensity of differences between the two between any two cipher images are computed and
images. It is defined as [13]: given in Table 6. The differential images between (a)
and (b), (a) and (d), (a) and (f) and (a) and (h) are
100 m n A ( i, j ) − B ( i, j )
UACI AB = ∑∑
m×n 1 1 255
(18) shown in (c), (e), (g) and (l) of Figure 6 respectively.
 on the key has been investigated. The permutation
process was generated by sorting the chaotic solutions
of the LDE whose coefficients are integers and
dynamically generated from PWLCM. The proposed
scheme thus requires few chaotic numbers for the
generation of complex permutation and diffusion keys.
By using this technique, the permutation step and the
spreading process are significantly accelerated
contrary to that of Chong Fu et al. [3]. As a result, one
Figure 6. Key sensitivity test: (a) Ciphered image using key (i). (b) round of encryption with the proposed algorithm is
ciphered image using key (ii). (c) Differential image between (a) safe enough to resist exhaustive attack, chosen
and (b). (d) Ciphered image using key (iii). (e) Differential image
between (a) and (d). (f) Ciphered image using key (iv). (g)
plaintext attack and statistical attack. Simulations have
Differential image between (a) and (f). (h) Ciphered image using been carried out to compare its performance with that
key (v). (l) Differential image between (a) and (h). of existing methods. We have also performed an
exhaustive testing process of the randomness of the

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Table 6. Differences between cipher images produced by slightly generated binary sequences using the NIST suite in

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different keys.
order to prove the viability of the proposed method.
Test This makes it a very good candidate for real-time
Figure keys Difference (%)
image encryption applications.

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(a) (b) (d) (f) (h)
(a) (i) ---- 99.6059 99.6040 99.6014 99.6086
Acknowledgement
(b)
(d)
(f)
(ii) 99.6059 ----
(iii) 99.6040 99.6132
(iv)
99.6132

99.6014 99.6277 99.6239

----
99.6277
99.6239
----
99.6346
99.6117
99.6197
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J.D.D Nkapkop gratefully acknowledges the AUF for
their financial support.
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(h) (v) 99.6086 99.6346 99.6117 99.6197 ----
References
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Therefore, the proposed scheme is highly sensitive
[1] Alvarez G., and Li S., “Some Basic
to the key.
Cryptographic Requirements for Chaos-Based
Cryptosystems,” International Journal of
4.8. Efficiency Analysis
Bifurcation and Chaos, vol. 16, no. 8, pp. 2129-
Running speed of the algorithm is an important aspect 2151, 2006.
for a good encryption algorithm, particularly for the [2] Chen G., Mao Y., and Chui C., “A symmetric
l
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real-time internet applications. We evaluated the image encryption scheme based on 3D chaotic
performance of encryption scheme by using Matlab cat maps,” Chaos Solitons & Fractals, vol. 21,
7.10.0. Although the algorithm was not optimized, no. 3, pp. 749-761, 2004.
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performances measured on a 2.0 GHz Pentium Dual- [3] Chong F., Chen J., Zou H., Meng W., and Zhan
Core with 3GB RAM running Windows XP are Y., “A chaos-based digital image encryption
satisfactory. scheme with an improved diffusion strategy,”
The average computational time required for 256 OPTICS EXPRESS, vol. 20, no. 3, pp. 2363-
in

gray-scale images of size 512×512 is shorter than 110 2378, 2012.

ms. By comparing this result with those presented in [4] Eyebe A., Effa J., Samrat L., and Ali M., “A fast
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[14], the scheme can be said high-speed as we only chaotic block cipher for image encryption,”
used a 2.0 GHz processor and the Matlab 7.10.0 Commun. Nonlinear Sci. Numer. Simulat., vol.
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software. Indeed, the modulus and the XOR functions 19, no. 3, pp. 578-588, 2014.
are the most used basic operations in our algorithm. [5] Faraoun K., “A chaos-based key stream
The comparison between the simulations times generator based on multiple maps combinations
required at the permutation stage shows that the and its application to images ,” The International
computational time required in our experiment is three Arab Journal of Information Technology, vol. 7,
times less than that of Chong Fu et al. [3]. This means no. 3, pp. 231-1215, 2010.
that the actual computational times of our scheme could [6] Fridrich J., “Symmetric ciphers based on two
be at least smaller if implemented in the same dimensional chaotic maps,” International
conditions than the Chong Fu et al’s. algorithm. Journal of Bifurcation and Chaos, vol. 8, no. 6,
pp. 1259-1284, 1998.
5. Conclusions [7] Guan Z., Huang F., and Guan W., “Chaos-based
image encryption algorithm,” Physics Letters A,
In this paper, an encryption algorithm for the fast vol. 346, no. 1-3, pp. 153-157, 2005.
generation of large permutation and diffusion keys with
a good level of randomness and a very high sensitivity
[8] Jastrzebski K., and Kotulski Z., “On improved Jean De Dieu Nkapkop received
image encryption scheme based on chaotic map his Master’s degree in Electronics,
lattices,” Engineering Transactions, vol. 57, no. Electrical Engineering and
2, pp. 69-84, 2009. Automatic, from Department of
[9] Li C., Li S., Asim M., Nunez J., Alvarez G., and Physics of the University of
Chen G., “On the security defects of an image Ngaoundere, Cameroon in 2012.
encryption scheme,” Image and Vision Currently, he is preparing his Ph.D.
Computing, vol. 27, no. 9, pp. 1371-1381, 2009. degree in the field of chaos-based cryptography.
[10] Lian S., Sun J., and Wang Z., “A block cipher
Joseph Effa received his Ph.D. in
based on a suitable use of the chaotic standard
Electronics from the University of
map,” Chaos, Solitons & Fractals, vol. 26, no. 1,
Yaounde I, Cameroon in 2009. His
pp. 117-129, 2005.
research interests include chaos
[11] Mohammad S., and Mirzakuchaki S., “A fast
detection, synchronization of
color image encryption algorithm based on
chaotic systems, nonlinear
coupled two-dimensional piecewise chaotic map,”
transmission line and chaos-based

t
Signal processing, vol. 92, no. 5, pp. 1202-1215,

irs
image encryption.
2012.
[12] Shannon C., “Communication theory of secrecy Monica Borda received his Ph.D.
systems,” Bell Syst. Tech. J., vol. 28, no. 4, pp. in Telecommunication from the

n F
656-715, 1949. University of Bucharest, Romania in
[13] Wang J., and Chen G., “Design of a chaos-based 1987. She is Professor in
digital image encryption algorithm in time
domain,” in IEEE International Conference on
at IT Electronics Engineering
Telecommunication
and
at
Computational Intelligence & Communication Communications Department from
ic IAJ
Technology, pp. 26-29, 2015. the Technical University of Cluj-Napoca, Romania.
[14] Wang Y., Wong K., Liao X., Chen G., “A new She has more than 40 years’ experience of education
io
chaos-based fast image encryption algorithm,” and research in the topics included Information
Appl. Soft Comput., vol. 11, no. 1, pp. 514-522, Theory and Coding, Cryptography and Genomic,
2011. Signal Processing and Image Processing.
[15] Zhu Z., Zhang W., Wong K., and Yu H., “A Laurent Bitjoka received his Ph.D.
chaos-based symmetric image encryption scheme in Biomedical Engineering from the
using bit-level permutation,” Information
l

University of Tours, France in 1994.

Sciences, vol. 181, no. 6, pp. 1171-1186, 2011.
ub

His research interests include signal

processing, image processing,
automatic and biophysics.
eP

Alidou Mohamadou received his

Ph.D. in mechanics from the
in

University of Yaounde I, Cameroon

in 2007. His research interests
nl

include nonlinear dynamics and

chaos, nonlinear dynamic of DNA,
O

pattern formation, nonlinear

transmission line, modulational instability and plasma
physics.