A CHAOTIC CONFUSION-DIFFUSION IMAGE ENCRYPTION BASED ON HENON MAP

International Journal of Network Security & Applications (IJNSA) Vol. 11, No. 4, July 2019
DOI: 10.5121/ijnsa.2019.11402 19
A CHAOTIC CONFUSION-DIFFUSION IMAGE
ENCRYPTION BASED ON HENON MAP
Ashraf Afifi
Department of Computer Engineering, | Computers and Information Technology College,
Taif University, Al-Hawiya 21974, Kingdom of Saudi Arabia
ABSTRACT
This paper suggests chaotic confusion-diffusion image encryption based on the Henon map. The proposed
chaotic confusion-diffusion image encryption utilizes image confusion and pixel diffusion in two levels. In
the first level, the plainimage is scrambled by a modified Henon map for n rounds. In the second level, the
scrambled image is diffused using Henon chaotic map. Comparison between the logistic map and modified
Henon map is established to investigate the effectiveness of the suggested chaotic confusion-diffusion
image encryption scheme. Experimental results showed that the suggested chaotic confusion-diffusion
image encryption scheme can successfully encrypt/decrypt images using the same secret keys. Simulation
results confirmed that the ciphered images have good entropy information and low correlation between
coefficients. Besides the distribution of the gray values in the ciphered image has random-like behavior.
.
KEYWORDS
Chaos, Encryptions, Henon map, Shuffle, Confusion, Diffusion.
1. INTRODUCTION
This document describes and is written to conform to, author guidelines for the journals of
AIRCC series. The rapid development of digital technology and network communications during
the past decade has made a great change in people’s work and life. Images have special features is
used how the bulk data capacity and the high level of data redundancy. These features make
image encryption harder than texts. To overcome this limitation, several methods of image
encryption have been suggested to secure the digital image contents. The chaotic systems
properties have been reported to be suitable for images ciphering. The main properties are the
sensitivity to initial conditions, the ergodicity and mixing and the determinism [1-9]. Therefore,
the secure transmission of confidential and sensitive information on networks has become a major
research problem.
Image chaotic ciphering in previous works relied on either image confusion or diffusion process.
Confusion image is generated by permuting positions of pixels. The confusion functions are
mainly the Standard map [10], the Arnold cat map [2], the Baker map [11], and the logistic map.
Diffusion is the process of changing pixels gray values of the image. The main diffusion
functions are Chen’s map, logistic map, and the Henon map.
In [2], Peterson ciphered “cat image” by scrambling it based on Arnold cat map. However, after
iterating enough times the ciphering algorithm, the plain image reappeared. So, we concluded that
image ciphering by confusing or diffusing alone is not enough to resist to an eventual attack. In
Gao et al.’s image cryptosystem [3], the plain image pixels are masked by a pseudo-random
chaotic sequence generated by power and tangent function. In Kwok scheme [4], the keystream
for masking is obtained by the cascading of the skewed Tent map and a high-dimensional Cat
map. Nien at [5] suggested encrypting the RGB components of a digital color image separately

20
using three variables of a third-order RLC chaotic circuit. Pareek at [6] used an external key and a
logistic map output to select one of the eight possible operations is selected to encrypt image
pixel.
In [7], Fridrich reported that chaos-based image encryption includes two iterative stages namely
chaotic confusion and pixel diffusion. Zhang at [12] used a discrete exponential chaotic map for
pixel permutation and simple XOR logic function for diffusion.
Our paper suggests a Henon map based confusion-diffusion, then compares it with the logistic
map, and then uses the suggested modified Henon map for confusion-diffusion mechanism in the
suggested chaotic image encryption system.
The rest of the paper is structured as follows: The Henon map based confusion and diffusion is
explored in Section 2. Section 3 introduces the suggested image encryption technique with
confusion and diffusion. The test results discussions are presented in section 4. Finally, section 5
concludes this work.
2. FUNDAMENTALS KNOWLEDGE
2.1. Henon Chaotic System
A Henon chaotic system is a 2-D dynamic system as suggested in [13] to simplify the Lorenz
map [14] defined by properties of eq.1 .
.....,.........2,1,0
1
1
2
1





i
bxy
yaxx
ii
iii
(1)
The initial parameters are a, b and the initial point is. ),( 00
yx Each point ),( nn
yx is mapped
to a new point ),( 11  ii
yx through the Henon map. For a=0.3 and b=1.4 as suggested in [15].
The parameter a, the parameter b, initial value x0
and the initial value 1x may represent the key
and make it is hard to predict the secret information.
2.1.1. Pixel Permutation
In the permutation level, the new location of each bit is computed by eq.2. The pair of
),( ''
yx is the new position of ),( yx . At this phase, the n x n input image is permuted r times
with different parameters P and Q, where ri ,.........1 .
Let’s consider
n
y
x
QPQ
P
y
x
iii
i
mod
1
1
'
'



















(2)
By setting P and Q as:

21
,.......2,1,0),36.0,2.1
4.1
3.01
00
2



iQP
PQ
QPP
ii
iii
.
2.1.2. Pixel diffusion
In the diffusion phase, each pixel diffusion is calculated by eq. 3. The pair of
))1(),1((  iyix is the new value of ),( ''
yx . At this phase, the scrambled image is
diffused by n iterations based on the image size.
.....,.........2,1,0
)(4.1
)()(3.01
1(
)1( 2






 








i
ix
iyix
iy
ix
, (3)
2.2. The Chaotic 2-D Logistic Map
The chaotic 2-D logistic map [16] is another ciphering scheme. The definition of the 2-D logistic
map ),( yxF can be described by the following eq. 4.















)yx(β)y(1yy
β)x(1xx
),(
ii
2
2ii21i
2
1ii11i
i
i
x
N
yxF


(4)
Where 1 , 1β , 2 , 2β are the control system parameters, and i varies as 0, 1, 2… and so on,
0x and 0y are initial conditions values. For 4.375.2 1  , 21.0β15.0 1  ,
45.37.2 2  , 15.0β13.0 1  and 1,a0 ii  b .
3. THE PROPOSED IMAGE ENCRYPTION TECHNIQUE
In the suggested modified Henon map will be defined in terms of two basic processes namely
ciphering and deciphering. As shown in Fig.1, it has two stages for ciphering /deciphering,
respectively. The ciphering stage starts by reading the plain image. Then, fed to the chaotic Henon
map confusion and apply the chaotic Henon map diffusion ciphering is applied according to eq. 3
and eq. 4.
The ciphering steps can be summarized as follows:
 Read the plainimage.
 Apply a chaotic Henon map confusion on the plainimage.
 Apply a chaotic Henon map diffusion on the previously confused plainimage.

22
Figure 1: The Proposed Image Encryption
The receiver starts by applying the deciphering stage in the same steps but reverse order used in the
ciphering stage.
The steps of deciphering stage are summarized as follows:
 Read the cipher image.
 Apply the inverse of the chaotic diffusion Henon map on the ciphered image.
 Apply the inverse of the chaotic confusion Henon map on the resulted image produced in
step 2 to obtain the find decrypted image.
4. SIMULATION RESULTS
In this section, some security analysis results on the suggested ciphering technique are described,
including some important tests like statistical and differential analysis [17-20] and comparing its
performance with a chaotic 2-D logistic map ciphering. The three images of the Bata, Girl, and
peppers of 256x256-sized are shown in Fig.2 have been used in the tests.
a) Bata b) Girl c) Peppers
Figure 2: Test Images - Bata, Girl and Peppers Images
The encryption outcomes of plainimages employing the proposed chaotic Henon map encryption
technique and conventional 2-D logistic map are illustrated in Fig. 3 for Bata, Girl and Peppers
images, respectively. It is obvious that the encryption with the proposed chaotic Henon map
outperforms in the concealment of all images details.
Decrypted ImageOriginal Plain image
Apply the inverse a chaotic
diffusion Hénon map
Apply a chaotic Hénon map confusion
Apply the inverse a chaotic
confusion Hénon map
Apply a chaotic Henon map diffusion
Send encrypted image to the receiver Read encrypted image

23
Image
Encrypted image with Henon and Logistic
2-D Logistic map Henon map
Bata
Girl
Peppers
Figure 3: Encrypted images using proposed a chaotic Henon map and 2-D Logistic map.
4.1. Abstract Correlation Analysis
The Abstract section begins with the word, The correlation rxy is calculated as [15-16, 21]:
1
1
( )
N
iN
i
E x x

  (5)
21
1
( ) ( ( )) ,
N
iN
i
D x x E x

  (6)
1
1
cov( , ) ( ( ))( ( )),
N
i iN
i
x y x E x y E y

   (7)
cov( , )
,
( ) ( )xy
x y
D x D y
r  (8)
where ; x and y are gray values of pixels of the two images x, y.
After measuring the correlation between the plainimage and encrypted image, if it does not equals
1, the plainimage and its encrypted image are the same, if it equals 0, the encrypted image is
completely different from the original and if it equals -1, the encrypted image is the negative of
the plainimage.
4.2. Entropy Analysis
Shannon entropy calculates the information involved within the image. It is calculated in bits. It
can be calculated as [16]:
2 1
2
0
1
( ) ( )log
( )
N
i
i i
H m P m bits
P m


  (9)
Where ; mi is a symbol and 2N-1 is the occurrence number of mi in the image. Consequently,
large entropy value indicates good encryption.

24
4.3. The Histogram Analysis
The histogram deviation (HD) asses the encryption quality by examining how it enlarges between
the difference between the plain and the encrypted images according to [22]. The H
D can be
estimated as follows:
254
1
(0) (255)
( )
2 i
H
d d
d i
D
W H

 
 
 


(10)
Where d(i) is the absolute value of the difference between the histograms of the plain image and
encrypted images at pixel level i. the variables W and H are the dimensions of the image. The
objective is to obtain as high DH
as possible.
The irregular deviation (DI ) asses the encryption accuracy in terms of how much the irregularity
is the deviation caused by encryption. The irregular deviation DI
is calculated as [22]:
255
0
( )Di
I
h i
D
W H




(11)
 ( )D Hh i h i M 
(12)
where )(ih is the histogram of the encrypted image at level i, and MH is the uniform histogram
mean value for an ideal encrypted image. Therefore, achieving low values for I
D means high
encryption quality.
4.4. Unified Average Changing Intensity (UACI)
The UACI asses the mean intensity of the difference among the plain image C1 and the encrypted
image C2 according to [16, 21]. UACI can be calculated as follows:
1 2
,
( , ) ( , )1
100%,
255i j
C i j C i j
UACI
W H
 
    
 (13)
To obtain more security, the output of such encrypted image outperforms symbols with UACI
near to 33%.
4.5. The NPCR Measure
The NPCR asses the percentage of dissimilar pixels number to the entire pixels number among
the two encrypted images E1(xi,yj )andE2 (xi,yj ). When a single pixel changes.
The D(xi,yj )is calculated by E1(xi,yj )andE2 (xi,yj )as follows:
D(xi,yj )=
0ifE1(xi,yj )=E2 (xi,yj )
1Otherwise
ì
í
ï
îï

25
According to [16,21] [21], the NPCR can be calculated as follows:
NPCR(E1,E2 )=
D(xi,yj )
i,j
å
WxH
x100% (14)
achieving greater NPCR value, the algorithm is more secure.
we will compare the chaotic 2-Logistic map to the suggested a chaotic Henon map numerically.
Table 1 shows entropy, corr., HI, HD, NPCR , and UACI of the three images compared with
their ciphered ones when being ciphered with a chaotic 2-D Logistic map and the suggested a
chaotic Henon map.
Table 1: The quality matrices results of the estimated correlation, entropy, HD, HI, NCPR, and UACI for
encrypted images using a chaotic 2-D Logistic map and by the suggested a chaotic Henon map
From this table, The entropy in the suggested a chaotic Henon map is greater than the chaotic 2-D
Logistic map in “Bata”,” Girl” and in ”Peppers” images and it is near 8. it is noticed that the corr.,
HD, and HI in all encrypted images by the suggested a chaotic Henon map are better than those
of encrypted by a chaotic 2-D Logistic map.
From Figure 4, the HD and HI in the suggested a chaotic Hanon map is better than the chaotic 2-
D Logistic map.
Image Original image
Encrypted image with Henon and 2-D Logistic
2-D Logistic Henon
Bata
Girl
Peppers
Figure 4: Histogram results of the plainimages and encrypted using the proposed Henon and 2-D Logistic
Image Bata Girl Peppers
Logistic Henon Logistic Henon Logistic Henon
Entropy 7.9993 7.9999 7.9929 7.9990 7.9990 7.9993
Corr. 0.00061 0.00113 0.0182 0.00034 0.000764 0.0018
HI 0.4616 0.4703 0.5709 0.5750 0.6349 0.6388
HD 0.8677 1.4986 0.8399 1.4838 0.5528 1.4999
NCPR % 99.9634 % 99.6125 % 97.9763 % 99.5754 % 99.6082 % 99.5964
UACI %29.82 %33.21 %32.58 %31.94 %29.63 %29.56

26
It is also noticed that the values of UACI are very near to 33%. So, the suggested encryption
technique is secure. The NPCR values are higher than 99% and therefore, changing a single pixel
in the plainimage to be ciphered will lead to a ciphered image that is quite different from the
outcoming encrypted image of the plainimage.
4.6. The PSNR measure
The PSNR is a calculation that quantifies the difference between the plainimage and the
decrypted image. It is computed as [15,17-18]:
PSNR(I,D)=10
10
log
(255)2
I(xm.yn( )-D xm,yn( )
2
n=1
N
å
m=1
M
å
æ
è
ç
ç
ç
ç
ö
ø
÷
÷
÷
÷
(15)
Where I(xm,yn ) and D(xm,yn ) are the pixel values of the original and decrypted image at
position (xm,yn ), respectively. High PSNR values indicate good immunity against noise. The
noise immunity results are given in Tables 2, 3, 4, 5 and 6. The results demonstrated that the
proposed chaotic Henon map image encryption has a better immunity against AWGN and speckle
noise than 2-D logistic which verifies the best choice for ideal optical telecommunication
applications that can cancel the noise effects.
Table 2: The PSNR for decrypted images using a chaotic Henon map and 2-D logistic map.in the presence
of AWGN noise for zero mean and different variances (0.05, 0.1, 0.15, and 0.2).
Image
PSNR
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata 9.8060 8.7381 8.4292 8.2652 14.2682 11.8853 10.6056 9.7804
Girl 9.5594 8.9466 8.6681 8.5212 13.9399 11.5771 10.3719 9.6379
Peppers 11.6611 10.3500 9.8276 9.4888 13.7447 11.4455 10.2872 9.5879
Table 3: The decrypted images using a chaotic Henon map and 2-D logistic map.in the presence of AWGN
noise for zero mean and different variances (0.05,0.1,0.15, and 0.2).
Image
PSNR
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata
Girl
Peppers

27
Table 4: The PSNR for decrypted images using a chaotic Henon map and 2-D logistic map.in the presence
of Speckle noise for different variances (0.05,0.1,0.15, and 0.2).
Image
PSNR
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata 12.2715 11.2794 9.5478 8.8939 22.1912 19.4066 17.7472 16.5825
Girl 10.3008 9.9735 9.9786 9.5904 21.3148 18.3272 16.6205 15.4219
Peppers 14.8926 12.9859 12.1722 11.4799 18.8753 16.0799 14.5075 13.4376
Table 5: The decrypted images using a chaotic Henon map and 2-D logistic map.in the presence of Speckle
noise for different variances (0.05,0.1,0.15, and 0.2).
Image
PSNR
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata
Girl
Peppers
4.7. The Structure Similarity Index (FSIM)
The FSIM evaluate the deciphered image and can be defined as [23]:
(16)
Where Ω is the spatial domain of image, denotes overall similarity among two images and
denotes a value of phase congruency. High value for FSIM means noise immunity is
better. The obtain FSIM results are given in Table 6, and 7 for decrypted images. These results
confirm that the suggested a chaotic Henon map image encryption technique is noise resistant.
Table 6: The FSIM for decrypted images using a chaotic Henon map and 2-D logistic map.in the presence
of AWGN noise for zero mean and different variances (0.05,0.1,0.15, and 0.2).
Image
FSIM
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata 0.5846 0.5524 0.5439 0.5425 0.7223 0.6482 0.6060 0.5776
Girl 0.4942 0.4460 0.4329 0.4264 0.6178 0.5355 0.4922 0.4640
Peppers 0.5658 0.5051 0.4792 0.4669 0.6534 0.5763 0.5361 0.5099

28
Table 7: The FSIM for decrypted images using a chaotic Henon map and 2-D logistic map.in the presence
Image
FSIM
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata 0.6683 0.6245 0.5749 0.5837 0.9330 0.8945 0.8675 0.87463
Girl 0.5277 0.4991 0.5015 0.4752 0.8637 0.8082 0.7712 0.77440
Peppers 0.7104 0.6257 0.5829 0.5506 0.8201 0.7551 0.7131 0.6840
4.8. The Structure Similarity Index (SSIM)
The SSIM can be computed as [23]:
(17)
Where, C1, C2 denote minor constants, and denote the average of and regions,
respectively. denotes the variance of region and denotes covariance among two
regions . High value for SSIM means noise immunity is better. The obtain SSIM
results are given in Table 8 and 7. These results confirm that the suggested a chaotic Henon map
image encryption technique has better resistance to noise.
Table 8: The SSIM for decrypted images using a chaotic Henon map and 2-D logistic map.in the presence
of AWGN noise for zero mean and different variances (0.05,0.1,0.15, and 0.2).
Image
SSIM
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2
Bata 0.1399 0.0853 0.0684 0.0588 0.4055 0.2942 0.2390 0.2040
Girl 0.1249 0.0786 0.0613 0.0531 0.2709 0.1911 0.1553 0.1328
Peppers 0.2007 0.1245 0.0956 0.0748 0.3245 0.2360 0.1952 0.1708
Table 9: The SSIM for decrypted images using a chaotic Henon map and 2-D logistic map.in the presence
Image
SSIM
2-D Logistic Henon
0.05 0.1 0.15 0.2 0.05 0.1 0,15 0.2
Bata 0.3005 0.2322 0.1336 0.1508 0.8378 0.7467 0.6798 0.6313
Girl 0.1763 0.1421 0.1447 0.1215 0.6675 0.5765 0.5246 0.4905
Peppers 0.4108 0.2794 0.2278 0.1875 0.5918 0.4787 0.4170 0.3758
5. CONCLUSIONS
Aiming to improve the encryption security, we purpose in this paper a chaotic confusion-
diffusion image encryption scheme based on Henon map. The security measures are performed
on both the 2-D logistic map and the proposed chaotic confusion-diffusion Hénon map to
compare their performance.

29
The test results have proved that the proposed image encryption technique offers high-security
level, and outperforms with its excellent potential of encryption. Compared to other methods of
encryptions from the state of arts, the suggested technique offers high-security level and can
immune many types of attacks such as the known-plaintext, cipher text-only attack, differential
attack, and statistical attack.
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S. Faragallah, (2016) “Wavelet Fusion for Encrypting Images with Few Details”, Computers and
Electrical Engineering, Vol. 60, pp. 450-470.
AUTHOR
Ashraf Afifi received the B.Sc.(Hons.), M.Sc., and Ph.D. degrees in Electronic and
Communication Engineering from Zagazig University, Egypt, in 1987, 1995, and
2002, respectively. He is currently Associate Professor with the Department of
Computer Engineering, Faculty of Computers and Information Technology, Taif
University, Saudi Arabia. He is a coauthor of about 30 papers in international journals
and conference proceedings. His research interests cover communication security,
image processing, and image encryption. Email: a.afifi@tu.edu.sa, Mobile: 00966507275265.

A CHAOTIC CONFUSION-DIFFUSION IMAGE ENCRYPTION BASED ON HENON MAP

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