COLOR IMAGE ENCRYPTION BASED ON MULTIPLE CHAOTIC SYSTEMS

International Journal of Network Security & Its Applications (IJNSA) Vol.8, No.5, September 2016
DOI: 10.5121/ijnsa.2016.8503 39
COLOR IMAGE ENCRYPTION BASED ON MULTIPLE
CHAOTIC SYSTEMS
Yuting Xi, Xing Zhang and Ruisong Ye
Department of Mathematics, Shantou University
Shantou, Guangdong, 515063, P. R. China
ABSTRACT
This paper proposed a novel color image encryption scheme based on multiple chaotic systems. The
ergodicity property of chaotic system is utilized to perform the permutation process; a substitution
operation is applied to achieve the diffusion effect. In permutation stage, the 3D color plain-image matrix
is converted to a 2D image matrix, then two generalized Arnold maps are employed to generate hybrid
chaotic sequences which are dependent on the plain-image’s content. The generated chaotic sequences are
then applied to perform the permutation process. The encryption’s key streams not only depend on the
cipher keys but also depend on plain-image and therefore can resist chosen-plaintext attack as well as
known-plaintext attack. In the diffusion stage, four pseudo-random gray value sequences are generated by
another generalized Arnold map. The gray value sequences are applied to perform the diffusion process by
bitxoring operation with the permuted image row-by-row or column-by-column to improve the encryption
rate. The security and performance analysis have been performed, including key space analysis, histogram
analysis, correlation analysis, information entropy analysis, key sensitivity analysis, differential analysis
etc. The experimental results show that the proposed image encryption scheme is highly secure thanks to its
large key space and efficient permutation-substitution operation, and therefore it is suitable for practical
image and video encryption.
KEYWORDS
Generalized Arnold Map, Permutation, Substitution, Chaotic System, Image Encryption
1. INTRODUCTION
Nowadays more and more images and videos are transmitted through network due to the dramatic
developments of IT era. Cryptographic approaches are therefore critical for secure image storage
and distribution over public networks. As an effective technique to protect contents from being
intercepted, tampered and destroyed illegally, encryption has attracted much attention recently.
Chaos has been extensively adopted in encryption due to its ergodicity, pseudo-randomness and
sensitivity to initial conditions and control parameters, which are in line with the fundamental
requirements like confusion and diffusion in cryptography [1]. These properties make chaotic
systems a potential candidate for construction cryptosystems and many chaos-based image
encryption algorithms are proposed [2,3,4,5,6,7,8,9,10]. Ye proposed an image encryption
scheme with an efficient permutation-diffusion mechanism, which shows good performance,
including huge key space, efficient resistance against statistical attack, differential attack, known-
plaintext attack as well as chosen-plaintext attack [6]. In both the permutation and diffusion
stages, generalized Arnold maps with real number control parameters are applied to generate
pseudo-random sequences and therefore enlarge the key space greatly. Meanwhile, a two-way
diffusion operation is executed to improve the security of the diffusion function. Wang et al. [11]
employed Logistic map in the permutation process and Gravity Model in the diffusion process to
achieve good security and performance. Wen et al. constructed a new improved chaotic system by

40
a nonlinear combination of 1D Logistic map and sine map. Wang et al. [13] pointed out that
Arnold map has short periodic and is easy to be cracked by chosen-plaintext attack, so they used
this map in a different way to overcome the short periodic issue and enhance the security. In
[14,15], Chebyshev maps and ordinary differential equations were used to generate key stream
respectively to enhance the security and performance of the proposed image encryption. In [16],
Chen et al. presented a novel image encryption scheme using Gray code based permutation
approach. The new permutation strategy takes full advantage of Gray-code achievements, and is
performed with high efficiency. A plain pixel-related image diffusion scheme is introduced to
compose a complete cryptosystem.
In this paper, we use multiple chaotic systems to generate pseudo-random sequences for color
image encryption with permutation-substitution mechanism. In the permutation stage, we use two
chaotic systems to disorder the pixels’ positions of the plain-image based on the ergodicity of
generalized Arnold maps. Firstly, the 3D color plain-image matrix is converted to 2D image
matrix, then the sum of the pixel values of the 2D image matrix will be used to be the initial gray
value seed. The key streams applied to perform the permutation are yielded by randomly
choosing one of two generalized Arnold maps according to the yielded seed and each pixel’ gray
value of the plain-image. As a result, the permutation process strongly depends on the plain-
image and therefore the image encryption scheme can resist efficiently known-plaintext attack
and chosen-plaintext attack. In the diffusion stage, four vectors are generated by another
generalized Arnold map, then the gray values of row and column pixels of 2D image matrix are
mixed with the pseudo-random number sequences via bitxoring operation. The security and
performance of the proposed image encryption scheme has been analyzed thoroughly, including
statistical analysis (histograms, correlation coefficients, information entropy), key sensitivity
analysis, key space analysis, differential analysis, encryption rate analysis, etc. All the experiment
results show that the proposed image encryption scheme is highly secure and demonstrates
excellent performance.
The remainder of this paper is organized as follow. The proposed image encryption scheme is
presented in Section 2. Section 3 shows the experimental results and performance analysis.
Finally, conclusions are drawn in the last section.
2. THE PROPOSED IMAGE ENCRYPTION SCHEME
2.1. ARNOLD MAP
There are two stages in the proposed color image encryption scheme, permutation and diffusion.
Arnold map, a kind of two-dimensional area-persevering chaotic map, will be adopted in both
processes to shuffle the positions of the plain image pixels and weaken the relationship between
adjacent pixels. The mathematical formula of classical Arnold map is given by
1
1
1 1
mod1
1 2
n n
n n
x x
y y
   
   +
   
   
   +   
 
= , 
 
(1)
where " x mod 1" represents the fractional part of a real number x . The map is area preserving
since the determinant of its linear transformation matrix is 1. The unit square is first stretched by
the linear transform matrix and then folded back to the unit square by the modulo operation,
which can be shown in Fig. 1. The 2D classical Arnold map can be generalized by introducing
two real parameters to Eq. (1):

41
1
1
1
mod1
1
n n
n n
x xp
y yq pq
   
   +
   
   
   +   
 
= , + 
(2)
where p q, are the real system control parameters. The generalized Arnold map (2) has one Lyapunov
characteristic exponent
2 2
1
1+ + +4
=1+ >1
2
ab a b ab
σ ,
so the generalized Arnold map is always chaotic for >0a , 0b > . The extension of ,a b from positive
integer numbers to positive real numbers is an essential generalization of the control parameters, enlarging
the key space significantly if it is used to design cryptosystem. Fig. 2 (a) shows an orbit of
( ) ( )0 0, = 0.5231,0.7412x y with length 1500 generated by the generalized Arnold map (2) with
=5.324, =18.2a b , the x-coordinate and the y-coordinate sequences of the orbit are plotted in Fig. 2 (b) and
Fig. 2(c) respectively. Some other good dynamical features in the generalized Arnold map, such as
desirable auto-correlation and cross-correlation features are demonstrated in Figs. 2(d)-(f). The good
chaotic nature makes it can provide excellent random sequence, which is suitable for designing
cryptosystem.
Fig.1. The Arnold map
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 50 100 150 200 250 300 350 400 450 500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k
x(k)
(a) The orbit of (0.5231, 0.7412) (b) Sequence { , 0, ,1500}kx k = L
0 50 100 150 200 250 300 350 400 450 500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k
x(k)
(c) Sequence { 0, ,1500}ky k = L, (d) Auto-correlation of { , 0, ,1500}kx k = L

42
(e) Auto-correlation of { , 0, ,1500}ky k = L (f) Cross-correlation of kx and ky sequences
Fig.2. Orbit derived from the generalized Arnold map with a=5.324, b=18.2.
2.2. PERMUTATION PROCESS
In the permutation process, the 3D color plain-image matrix A with size 3M N× × is converted
to a 2D image matrix B with size 3M N× firstly. Then we will create a zero vector C with
length 3M N× × . Now what we need to do is just put the 3M N× × pixels in B into C randomly.
In this image encryption scheme, we realize it by using two generalized Arnold maps, the detail
operation procedures are described as follows.
Step 1. Set the appropriate parameters 1 1a b, , 2 2a b, for two generalized Arnold maps and
their common initial values x y, . Another sufficiently large integer 1M is also set to be the iterate
times, 1M is usually set to be one integer number close to 3M N× × . In this paper, we set 1M to
be 3M N× × .
Step 2. Modulate x y, by iterating two generalized Arnold maps for 1N times respectively to
avoid the harmful effect of transitional procedure of the chaotic orbits. 1N can be set as a secret
key. For example, we set 1 200N = in the experiments.
Step 3. Calculate
3
1 1
( ) ( 256)
M N
i j
sum B i j p mod sum
= =
= , , = , .∑∑ (3)
We initialize the index matrix index to be zero matrix with size 3M N× and set the initial ergodic
counting number 0K = . For 1 1i M= : , we execute Steps 4-6.
Step 4. Calculate ( 2)q mod p= , , q is equal to 0 or 1, so we can use q to dynamically assign
one generalized Arnold map to generate hybrid chaotic sequences. If 0q = , we iterate the first
generalized Arnold map once to generate new x y, , otherwise, the second generalized Arnold
map will be iterated.
Step 5. The position coordinates ( )s t, can be calculated by
( ) 1 ( 3 ) 1,s floor x M t floor y N= × + , = × + (4)
where ( )floor x returns the nearest integer less than or equal to x .

43
Step 6. If ( ) 0index s t, = , then set 1K K= + , ( ) ( )C k B s t= , , ( )p B s t= , and ( ) 1index s t, = ;
otherwise, skip this step.
Step 7. Generally speaking, the traversing counting number K is often less than 3M N× ×
after the loop Steps 4-6 is finished. So we need to put the pixels which are not traversed orderly
from left to right and top to bottom into the remainder part of vector C . We finally converted the
vector C into one 2D matrix C with height M and width 3N .
2.3. SUBSTITUTION PROCESS
In the diffusion stage, four vectors are generated by another generalized Arnold map, then the
gray values of row and column pixels of 2D image matrix C are masked with the four pseudo-
random number sequences via bitwise XOR operation. The detail operation procedures are
described as follows.
Step 1. For an given generalized Arnold map with parameter a b, and initial value 1 1x y, , we
modulate 1 1x y, by iterating this generalized Arnold map for 100 times to avoid the harmful effect
of transitional process.
Step 2. Generate two chaotic sequences ( ) ( ) 1x i y i i M, , = : by the generalized Arnold map,
then two row sequence SVR and IVC with length M will be generated by
( ) ( ( ) 256)
( ) ( ( ) 256) 1, ,
SVR i floor x i
IVC i floor y i i M
= × ,
= × , = .L
(5)
Step 3. Generate two chaotic sequences 1( ) 1( ) 1 3x i y i i N, , = : by the generalized Arnold map
with initial values ( ) ( )x M y M, , then two row sequences SVC and IVR with length 3N will be
generated by
( ) ( ( ) 256)
( ) ( ( ) 256) 1, ,3
IVR i floor x i
SVC i floor y i i N
= × ,
= × , = .L
(6)
Step 4. Get the cipher image CC by masking C with the four pseudo-random number
sequences via bitwise XOR operation.
(1 ) (1 ) (1)CC C IVR SVR,: = ,: ⊕ ⊕ , ( ) ( ) ( 1 ) ( ) 2, , ,CC i C i CC i SVR i i M,: = ,: ⊕ − ,: ⊕ , = L
( 1) ( 1) (1)CC CC IVC SVC′:, = :, ⊕ ⊕ , ( ) ( ) ( 1) ( ) 2, ,3CC j CC j CC j SVC j j N:, = :, ⊕ :, − ⊕ , = .L
3. SECURITY AND PERFORMANCE ANALYSIS
According to the basic principle of cryptology [17], an ideal encryption scheme should have large
key space to make brute-force attack infeasible, it should also well resist various kinds of attacks
like statistical attack, differential attack, chosen-plaintext attack, etc. In this section, the security
analysis has been performed on this proposed encryption scheme, such as, key space analysis,
statistical analysis, correlation between plain and cipher images, key sensitivity analysis,
differential analysis, encryption rate analysis, etc. Experimental simulations and extensive
performance analysis for the proposed scheme and the comparable scheme proposed in [18] have
been carried out. The cipher keys for the comparable algorithm are the same as those in [18]. All
the simulations are performed on a computer equipped with an Intel Xeon 2.13 GHz CPU 2GB

44
memory and 300GB hard disk space running Windows 7 Professional. The compilation platform
is Matlab 7.1. The experimental results prove the superior security and high efficiency of this
scheme.
3.1. KEY SPACE ANALYSIS
Key space is composed of all the possible cipher keys in the proposed image encryption scheme.
An ideal image encryption scheme should contain sufficiently large key space for compensating
the degradation dynamics in PC and should be large enough to effectively resist brute-force attack
and prevent invaders decrypting original data even after they invest large amounts of time and
resources. It was pointed out that the key space should beat at least 100
2 in order to resist all kinds
of common attacks [9]. Regarding our proposed image encryption scheme, the key space consists
of the initial values x y, and parameters 1 1a b, , 2 2a b, of the two chaotic systems in permutation
process and the initial values 1 1x y, , parameters a b, in diffusion process. All the initial values
1 1x y x y, , , and the control parameters 1 1 2 2a b a b a b, , , , , are floating point numbers. If they are
represented as floating number with precision 14
10−
as we have used in the key sensitivity test, the
total number of cipher keys is 14 10 140
10 10×
= , which is approximately equal to 465 bits. The key
space is large enough to resist the brute-force attack.
3.2. STATISTICAL ANALYSIS
Shannon pointed out the possibility to solve many kinds of ciphers by statistical analysis [17].
Therefore, passing the statistical analysis on cipher-image is crucial for a cryptosystem. Indeed,
an ideal cryptosystem should be robust against any statistical attack. To prove the security of the
proposed encryption scheme, we perform the following statistical tests.
(i) Histogram analysis. A histogram shows the distribution of pixel values in an image by plotting
the number of pixels at each grey level. An ideal histogram of an effectively ciphered image
should be uniform and much different from that of the plain image. For an 8-bit gray image, there
are 256 different possible intensities, the histogram shows the distribution of pixels among those
256 intensity values. For a 24-bit color image, we can draw the histogram for red, green, blue
channels respectively. Encrypt the color image Lena one round with cipher keys
(0 3201 0 6317 0 4807 0 7815 100 33 4 8 62 48). , . , . , . , , , , , , . Then we plot the histograms for red, green, blue
channels of Lena and the cipher-image in Fig. 3. It is obvious that the histograms of the cipher
image are uniform and quite different from that of the plain image, which implies that the
redundancy of the plain image is successfully hidden after the encryption, so it does not provide
any useful information for statistical attacks.
(a) plain-image Lena (b) cipher-image of Lena

45
0 50 100 150 200 250
0
100
200
300
400
500
600
700
800
0 50 100 150 200 250
0
100
200
300
400
500
600
0 50 100 150 200 250
0
100
200
300
400
500
600
700
(c) (d) (e)
0 50 100 150 200 250
0
100
200
300
400
500
600
0 50 100 150 200 250
0
100
200
300
400
500
600
700
800
900
0 50 100 150 200 250
0
100
200
300
400
500
600
(f) (g) (h)
Fig. 3. The histogram analysis result. (c)-(e) Red, green, blue channel component of plain-image; (f)-(h)
Red, green, blue channel component of cipher-image.
(ii) Correlation analysis of adjacent pixels. It is of common sense that in a meaningful image
each pixel is highly correlated with its adjacent pixels either in horizontal, vertical or diagonal
direction. For an ideal encryption technique, the cipher image should get rid of the drawback of
high correlation between pixels. In order to quantify and compare the correlation between plain-
image and cipher-image, we calculate the correlation coefficients for all the pairs of horizontally,
vertically and diagonally adjacent pixels of them respectively. The correlation coefficients of the
selected pairs in horizontal, vertical and diagonal direction are calculated according to Eq. (7),
where ix and iy are the ith selected pair pixels. T is the total pixel pairs’ number of the sample.
The correlation coefficients in horizontal, vertical or diagonal direction of the selected pairs for
plain-image Lena and the cipher-image are given in Table 1. From the data in Table 1, we can see
that even though there is high correlation in plain-image, the correlation in cipher-image is
negligible. The proposed image encryption technique significantly reduces the correlation
between the adjacent pixels of the plain-image.
( )
( ) ( )
cov ,
,
x y
Cr
D x D y
= ( ) ( )( ) ( )( )
1
1
cov ,
T
i i
i
x y x E x y E y
T =
= − −∑ ,
( ) ( ) ( )( )
2
1 1
1 1
,
T T
i i
i i
E x x D x x E x
T T= =
= = −∑ ∑ . (7)
(iii) Information entropy analysis. Information entropy is one of the criteria to evaluate the
randomness and the unpredictability of an information source. The entropy ( )H m of a message
source m is defined by
2 1
2
0
( ) ( )log ( )( )
L
i i
i
H m P m P m bits
−
=
= −∑ , (8)

46
where m is the source, L is the number of bits to represent the symbol im , and ( )iP m is the
probability of the symbol im . For a truly random source consist of 2L
symbols, the entropy is L.
So for an effective encryption algorithm, the entropy of the cipher image with 256 gray levels
should be close to 8. Otherwise, the information source is not random enough and there exists a
certain degree of predictability, which makes the encryption algorithm insecure.
For a 24-bit color image, the information entropy for each color channel( Red, Green and Blue)
is given by
8
2 1
2
0
1
( ) ( ( )log
( )
R G B R G B
i R G B
i i
H m P R I
P RI
−
/ / / /
/ /
=
= −∑ .
We have calculated the information entropy for plain-image Lena and its cipher-image by the
proposed encryption scheme. The value of information entropy for the cipher-image produced by
the proposed image encryption scheme is very close to the expected value of truly random image,
i.e., 8bits. Therefore the proposed encryption scheme shows extremely robustness against entropy
attacks. For comparison, we also calculate the information entropy of cipher image of Lena by
algorithm in [18]. The results are shown in Table 2. We can see the information leakage in this
proposed encryption procedure is negligible and when faced with entropy analysis attack, this
proposed encryption show good performance.
Table 1. Correlation between adjacent pixels of plain-image and cipher-image.
Image color channel horizontal vertical diagonal
Plain-image Lena R 0.9446 0.9720 0.9212
Cipher-image of Lena R -0.0020 0.0030 0.0035
Plain-image Lena G 0.9465 0.9729 0.9360
Cipher-image of Lena G -0.0036 0.0007 0.0041
Plain-image Lena B 0.9046 0.9465 0.8677
Cipher-image of Lena B 0.0040 0.0043 0.0031
(iv) Correlation between plain-images and cipher-images. For an efficient encryption scheme, the
cipher image should be much different from plain image and has low correction with plain image.
We have already analyzed the correction between plain-image and cipher-image by computing
the two-dimensional correlation coefficients between various color channels of plain-image and
cipher-image. The two-dimensional correlation coefficients are calculated by
1
1 1
2 21 1
1 1 1 1
( )( )
( ( ) )( ( ) )
H W
i j i jH W
i j
AB
H W H W
i j i jH W H W
i j i j
A A B B
C
A A B B
, ,×
= =
, ,× ×
= = = =
− −
=
− −
∑∑
∑∑ ∑∑
,
1 1 1 1
1 1H W H W
i j i j
i j i j
A A B B
H W H W
, ,
= = = =
= , = ,
× ×
∑∑ ∑∑
where A represents one of the three channels of plain-image, B represents one of the three
channels of cipher-image. A and B are the mean value of the two-dimension matrix A and B

47
respectively. H and W are the height and width of image A or B. So we can get nine different
correlation coefficients for a pair of plain-image and cipher-image ( RR RG RB GR GG GBC C C C C C, , , , , ,BRC
BGC and )BBC . For example, RRC means the correlation between red channel of plain-image and
red channel of cipher-image. The results of correction between Lena and its cipher image are
shown in Table.3. We can see that the correlation between various channels of plain image and
cipher image are very small, hence the cipher-image owns the characteristic of a random image.
3.3. KEY SENSITIVITY ANALYSIS
Extreme key sensitivity is an essential feature of an effective cryptosystem, and key sensitivity of
a cryptosystem can be observed in two ways: ( )i completely different cipher-images should be
produced even if we use slightly different keys to encrypt the same plain-image. ( )ii the cipher
image cannot be correctly decrypted even if there is tiny difference between encryption and
decryption keys. For key sensitivity analysis, we will use the following cipher keys to perform the
simulation ( one is master cipher key, the other keys are produced by introducing a sight change
to one of the parameter of master cipher with all other parameters remain the same) . Master
cipher key: MKEY (0 3201 0 6317 0 4807 0 7815 100 33 4 8 62 48). , . , . , . , , , , , , ; Ten slightly different keys:
SKEY1 14
(0 3201 10 0 6317 0 4807 0 7815 100 33 4 8 62 48)−
. − , . , . , . , , , , , , ,
SKEY2 14
(0 3201 0 6317 10 0 4807 0 7815 100 33 4 8 62 48)−
. , . − , . , . , , , , , , ,
SKEY3 14
(0 3201 0 6317 0 4807 10 0 7815 100 33 4 8 62 48)−
. , . , . − , . , , , , , , ,
SKEY4 14
(0 3201 0 6317 0 4807 0 7815 10 100 33 4 8 62 48)−
. , . , . , . − , , , , , , ,
SKEY5 14
(0 3201 0 6317 0 4807 0 7815 100 10 33 4 8 62 48)−
. , . , . , . , − , , , , , ,
SKEY6 14
(0 3201 0 6317 0 4807 0 7815 100 33 10 4 8 62 48)−
. , . , . , . , , − , , , , ,
SKEY7 14
(0 3201 0 6317 0 4807 0 7815 100 33 4 10 8 62 48)−
. , . , . , . , , , − , , , ,
SKEY8 14
(0 3201 0 6317 0 4807 0 7815 100 33 4 8 10 62 48)−
. , . , . , . , , , , − , , ,
SKEY9 14
(0 3201 0 6317 0 4807 0 7815 100 33 4 8 62 10 48)−
. , . , . , . , , , , , − , ,
SKEY10 14
(0 3201 0 6317 0 4807 0 7815 100 33 4 8 62 48 10 )−
. , . , . , . , , , , , , − .
(i)To evaluate the key sensitivity in the first case, we encrypt plain-image Lena with MKEY and
get the first cipher image, then we encrypt Lena with SKEY1-SKEY10 and get other ten cipher
images. We have computed the correlation coefficients between the first cipher image and the
other ten cipher images. The results are given in Table 4. From the table, we can see that all the
correlation coefficients are very small which indicate that even there is only slightly difference
between the cipher keys, the cipher images are greatly different. Hence the proposed encryption
scheme is extremely sensitive to the cipher keys.
(ii)Decryption using keys with slight difference are also performed in order to evaluate the key
sensitivity of the second case. Firstly, we decrypt the cipher image using MKEY and we get the
plain-image Lena. Secondly, ten decrypted images are generated when we decrypt the cipher-
image using SKEY1-SKEY10. We have computed the correlation coefficients between Lena and
this ten decrypted images, the results have been given in Table 5. From the data, we can see even
there is only a slightly difference between the decipher keys, the deciphered images have low
correlation coefficients with the plain-image Lena. So for the second case, the proposed
encryption scheme is of highly sensitive to the cipher keys too.

48
Table 2. Information entropy analysis.
Image R G B
Plain-image Lena 7.2763 7.5834 7.0160
Cipher-image 7.9972 7.9971 7.9977
Cipher-image [18] 0.9973 7.9973 7.9967
Table 3. Correlation between plain-image Lena and its cipher-image.
Cipher-image R G B
Plain-image Lena R -0.0037 -0.0049 0.0001
G 0.0044 -0.0052 0.0014
B 0.0045 -0.0020 0.0024
3.4. DIFFERENTIAL ATTACK ANALYSIS
For an efficient encryption scheme, a slightly difference in the plain image should cause amount
of difference in the cipher image. Two indicators, which are number of pixels change rate
(NPCR) and unified average changing intensity (UACI), are used to measure the influence of one
pixel change in the plain image on the cipher image. In order to calculate NPCR and UACI,
suppose two plain images 1I and 2I with difference in only one pixel, and their cipher images are
denoted as 1C and 2C . Then we create a matrix D, when 1 2( ) ( )C i j C i j, = , , ( ) 0D i j, = ; otherwise,
( ) 1D i j, = . NPCR and UACI are calculated by
( )
100
i j
D i j
NPCR %
W H
,
,
= × ,
×
∑ 1 2( ) ( )1
( ) 100
255i j
C i j C i j
UACI %
W H ,
| , − , |
= ×
×
∑ ,
where W, H are the width and height of the images. We have performed the differential attack
analysis in two cases: ( )i Encrypt the plain-image Lena using the proposed encryption scheme
and get a cipher image. Add 1 to the last pixel value of Lena, then we get a new plain image.
Encrypt the new plain-image, then the new cipher image is compared with the old cipher image to
calculate NPCR and UACI. The results are given in Table 6. ( )ii Randomly choosing 10
pixels( one at a time ) from Lena and add their values by one unit. The average NPCR and UACI
are given in Table 7. From the two tables, we can see even though the algorithm in [18] have
better performance on NPCR in Table 6, but on the whole, the proposed encryption scheme is
more stable and has better performance on differential analysis.
Table 4. Key sensitivity analysis I.
Correlation coefficients between the encrypted images obtained using MKEY and
SKEY1 SKEY2 SKEY3 SKEY4 SKEY5 SKEY6 SKEY7 SKEY8 SKEY9 SKEY10
Crr 0.0030 -0.0089 -0.0011 -0.0012 0.0021 -0.0030 -0.0000 -0.0021 -0.0080 -0.0027
Crg 0.0027 -0.0039 0.0014 -0.0066 -0.0042 0.0024 0.0024 0.0040 -0.0044 0.0008
Crb 0.0088 -0.0016 -0.0021 0.0084 0.0033 0.0010 0.0047 0.0078 -0.0013 -0.0041
Cgr 0.0001 -0.0038 0.0063 -0.0043 -0.0009 0.0006 -0.0048 -0.0012 0.0020 0.0006
Cgg 0.0009 -0.0005 -0.0010 -0.0025 -0.0042 0.0025 0.0025 -0.0043 0.0039 -0.0018
Cgb 0.0018 0.0004 0.0015 0.0032 0.0048 -0.0003 -0.0029 -0.0036 0.0044 -0.0016
Cbr 0.0064 0.0009 0.0026 0.0009 0.0037 0.0010 0.0025 0.0070 0.0020 -0.0063
Cbg 0.0048 -0.0006 -0.0009 0.0021 0.0005 0.0090 -0.0046 -0.0008 0.0032 -0.0003
Cbb 0.0033 -0.0048 -0.0007 0.0038 0.0033 -0.0032 -0.0054 -0.0006 0.0060 0.0027

49
Table 5. Key sensitivity analysis II.
Correlation coefficients between the decrypted images obtained using MKEY and
SKEY1 SKEY2 SKEY3 SKEY4 SKEY5 SKEY6 SKEY7 SKEY8 SKEY9 SKEY10
Crr 0.0086 0.0038 -0.0065 0.0004 0.0158 0.0015 0.0014 0.0119 0.0097 0.0064
Crg 0.0067 0.0081 -0.0061 0.0044 0.0011 -0.0054 0.0050 0.0046 0.0147 0.0065
Crb 0.0046 0.0014 -0.0005 -0.0051 -0.0023 -0.0078 0.0116 0.0027 0.0090 -0.0044
Cgr 0.0097 0.0054 -0.0044 -0.0014 0.0106 -0.0026 0.0035 0.0091 0.0092 0.0035
Cgg 0.0052 0.0086 -0.0063 0.0015 0.0012 -0.0061 0.0068 0.0090 0.0137 0.0058
Cgb 0.0053 0.0030 -0.0003 -0.0056 -0.0026 -0.0124 0.0103 0.0041 0.0043 -0.0028
Cbr 0.0099 0.0046 -0.0043 -0.0024 0.0099 -0.0029 0.0044 0.0059 0.0088 0.0007
Cbg 0.0050 0.0047 -0.0048 -0.0006 0.0003 -0.0050 0.0057 0.0086 0.0095 0.0046
Cbb 0.0042 0.0027 0.0016 -0.0063 -0.0021 -0.0132 0.0071 0.0021 0.0011 -0.0008
Table 6. Differential analysis I.
NPCR(%) UACI(%)
Red Green Blue Red Green Blue
The proposed scheme 99.61 99.59 99.59 33.55 33.60 33.55
The scheme proposed in [18] 99.57 99.64 99.65 33.42 33.19 33.38
Table 7. Differential analysis II.
Average NPCR(%) Average UACI(%)
Red Green Blue Red Green Blue
The proposed scheme 99.64 99.60 99.60 33.44 33.52 33.53
The scheme proposed in [18] 79.68 87.69 79.66 26.75 29.31 26.69
4. CONCLUSIONS
In this paper, we proposed a color image encryption scheme based on multiple chaotic systems. In
this encryption algorithm, a parameter depending on the plain-image is applied to dynamically
assign two generalized Arnold maps to generate chaotic sequences, so the proposed encryption
scheme can well resist chosen/known plain-image attack and the chaotic sequences show good
chaotic properties. In the diffusion stage, four vectors are generated by another generalized
Arnold map, then the gray values of row and column pixels of 2D image matrix are mixed with
the pseudo-random number sequences via bitxoring operation, which greatly weaken the
correction between plain-image and cipher-image. Simulation results show that the proposed
encryption scheme has good performance to resist all kinds of attacks.
ACKNOWLEDGEMENTS
This research is supported by National Natural Science Foundation of China (No. 11271238).
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50
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AUTHORS
Yuting Xi, master degree candidate at department of mathematics in Shantou University.
Xing Zhang, master degree candidate at department of mathematics in Shantou University.
Ruisong Ye was born in 1968 and received the B.S. degree in Computational Mathematics in 1990 from
Shanghai University of Science and Technology, Shanghai, China and the Ph. D. degree in Computational
Mathematics in 1995 from Shanghai University, Shanghai, China. He is a professor at Department of
Mathematics in Shantou University, Shantou, Guangdong, China since 2003. His research interest includes
bifurcation theory and its numerical computation, fractal geometry and its application in computer science,
chaotic dynamical system and its application in computer science, specifically the applications of fractal
chaotic dynamical systems in information security, such as, digital image encryption, digital image hiding,
digital image watermarking, digital image sharing.

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