International Journal of scientific research and management (IJSRM)
||Volume||3||Issue||11||Pages|| 3754-3761||2015||
Website: www.ijsrm.in ISSN (e): 2321-3418
Analysis of Chaotic Image Encryption Using 1D Logistic Map, 2D
Arnold Cat Map and 3D Arnold Cat Map
Jenny Joseph1, Josy Elsa Varghese2
1
Pursuing M.Tech, Dept. of Computer Science and Engineering, Caarmel Engineering College, MG University, Kerala
[email protected]
2
Assistant professor, Dept. of Computer Science and Engineering,Caarmel Engineering College, MG University, Kerala
[email protected]
Abstract:
Analyzing the randomness, complexity and efficiency of chaotic image encryption using 1D Logistic map,
2D Arnold cat map or 3D Arnold cat map is proposed here. The keys for the encryption process are
generated from the biometric image of the sender. Initially the biometric fingerprint image of the sender is
captured using a scanner, which is preprocessed to the grey level image. By using an efficient technique
on the grey level image, the initial condition for the 1D Logistic map and the keys for the encryption are
generated. The keys are Feature Vector, transform orders and angles for Dual Parameter Fractional
Fourier Transform. The key generation process is followed by the encryption process, which includes
randomization using chaotic map, transformation using DPFrFT and decomposition using Hessenberg
Decomposition. The resulting image after the three processes undergoes inverse Dual Parameter
Fractional Fourier Transform to get the encrypted image. The decryption process is the reverse of
encryption process. The analyses is conducted using Histogram Analysis, Correlation Analysis,
Information Entropy, NPCR and UACI
Keywords: 1D Logistic map, 2D Arnold cat map, 3D Arnold cat map, DPFrFT.
complexity. They are considered to be more
complex and unpredictable. This random behaviour
INTRODUCTION
Due to the rapidly changing technologies in the of chaotic systems makes it more suitable to use in
digital world, the multimedia data such as image, communication to increase security and privacy of
video etc can be distributed all over the world within data.
no time. But unfortunately along with the Motivated by the chaotic properties, recently many
developing technologies, new and efficient ways for researchers have been proposed and analysed many
attacking is also emerging. So the protection of chaos-based multimedia encryption. For example,
multimedia
data
becomes
an
important H.Gao et al [1] proposed a new Nonlinear Chaotic
factor.Cryptography is a commonly used technique, Algorithm which uses power function and tangent
which protects the contents of the messages that is function for image encryption. The process encrypts
transmitted over insecure channels and gives the image using the chaotic sequence generated by
emphasis to the privacy of information. Nowadays NCA map. In [2], Chaudhary et al proposed a
many new and efficient ways of cryptographic chaotic image encryption based on Complex
techniques has been developed. But most of the RandomNumber Generators. These random numbers
algorithms faced disadvantages such as small key are generated by the iteration of 1D Logistic map.
space, low level of security etc. To meet these Here an extra parameter is added in the 1D Logistic
challenges, many chaos-based algorithms have been equation which increases security.
suggested. The chaotic cryptography is the The security of the encryption process can be
application of chaos theory in a system for increased more, by the incorporation of Biometrics.
performing different cryptographic tasks. The Biometrics can be used to uniquely identify a person
chaotic systems are nonlinear systems which cannot using his physical or behavioural characteristics. In
be broken up into small pieces and solved most of the biometric based cryptographic systems,
separately. They have to be dealt with their full the keys for the encryption are generated from the
Jenny Joseph1, IJSRM volume 3 issue 11 Nov 2015 [www.ijsrm.in]
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DOI: 10.18535/ijsrm/v3i11.17
biometrics. One such example is the method
reported earlier by Hao et al [3] which integrates iris
biometric with cryptography. The system randomly
generates a biometric key which is encoded using
Reed Solomon code and Hadamard code to produce
pseudo iris code. It is then XORed with user’s iris
code. The locked data along with hash value of key
is then stored in a smart card. However, the
biometric data is not very secret. If someone can
capture the iris image of the target using a hidden
camera he can generate the key and reveal the data.
This paper analyzes the randomness, complexity and
efficiency of chaotic image encryption using 1D
Logistic map, 2D Arnold cat map or 3D Arnold cat
map. The biometric fingerprint image of the sender
is captured using a scanner, which is pre-processed
to the grey level image. The initial condition for
iterating 1D Logistic map, Feature Vector, transform
orders and angles for Dual Parameter Fractional
Fourier Transform (DPFrFT) are generated from the
grey level image. The encryption process
randomizes the original image using chaotic map to
get randomized image. The randomized image
undergoes transformation using DPFrFT and
decomposition using Hessenberg Decomposition to
get randomized encrypted image. Then randomized
encrypted image undergoes inverse DPFrFT to get
encrypted image. The decryption process is the
reverse of encryption.
RELATED WORKS
Cryptography is considered as a best method for data
protection against frauds. The two important
techniques in the Cryptography are Encryption and
Decryption. Encryption involves encrypting the data
with a secret key so that only an authorised person
with the key can decrypt it. Thus to make the
encryption more secure the key used should be
managed securely. To make the key management
more secure and to increase the security of the
system, there are a lot of previous works in which
Cryptography is combined with chaos and
Biometrics. A brief explanation of some of the
works is given below.
Zhenjun et al [4] proposed a secure image
encryption using Arnold Transform and Random
Strategies. Arnold transform is a widely used
technique for picture scrambling. In this paper, three
random strategies are used. First random division,
second random iterative numbers and third random
encryption order. The random division is performed
on the image using different squares controlled by a
key, to get series of square blocks. To get the
iterative number of the Arnold transform in a
particular block, produce pseudo random numbers
using a random number generator. For processing
the square blocks, they use a random order.
Experiments are conducted on images sized
512*512.The simulation shows the effectiveness of
the proposed scheme and the usefulness of random
division, random generator etc.
Pawan et al [5] proposed a chaotic image encryption
using 3D Logistic map, 3D Chebyshev map, 3D and
2D Arnold cat map. This technique uses 2D Arnold
cat map to scramble the image. Then using the 3D
Chebyshev map three keys are generated separately
for Red, Green and Blue component from the
scrambled image. These three keys are given as the
initial condition for the 3D Logistic map to produce
X, Y and Z components. X component is XORed
with Red component, Y component is XORed with
Green component and Z Component is XORed with
Blue component. Then all the components are
combined as a single image. The additional
scrambling and substitution is done using 3D Arnold
cat map. The simulation shows that the additional
scrambling and substitution using 3D Arnold cat
map increases security.
Gaurav et al [6] proposed a multimedia encryption
using Dual Parameter Fractional Fourier Transform,
Piecewise Nonlinear Chaotic map and the iris image
of the sender. The keys for the encryption process
are generated from the iris image of the sender. The
original image is scrambled with the help of the
chaotic sequence generated by the iteration of
Piecewise Nonlinear Chaotic map. The scrambled
mage is then transformed using DPFrFT.
Decompose the feature vector into orthogonal
matrices using Hessenberg Decomposition. Then the
image is encrypted using these orthogonal matrices
and inverse transfromation.Results shows that the
Piecewise Nonlinear Chaotic map and the DPFrFT
increase the efficiency of encryption.
In [7], Sunil et al proposed a cancellable key
generation from the fingerprint image. It includes
three stages extracting minutia points from
fingerprint, Secured Feature Matrix generation and
key generation from Secured Feature Matrix. The
fingerprint undergoes histogram equalization to
increase the local contrast of the image. Then Gabor
filter is applied on the resulting image so that the
filer has high response at ridges. Then the image
undergoes binarization so that the image is
converted to binary image. This increases contrast
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DOI: 10.18535/ijsrm/v3i11.17
between ridges and valleys. Then minutia points are
extracted using Ridge Thinning method. The
Secured Feature matrix is generated using AES
encryption method. Then the key is generated by
decrypting Secured Feature matrix using AES
decryption. The approach is tested on five
fingerprint images and the 256-bit keys generated
are also shown in simulation.
.The parameter and x0 may
represent the key.
The 2D chaotic map used in this paper is 2D
Arnold’s cat map. The 2D Arnold’s cat map is given
by equation (5):
3. Preliminaries
Before going to the paper, we will give a brief
introduction about Dual Parameter Fractional
Fourier
Transform
(DPFrFT),
Hessenberg
Decomposition, 1D Logistic map, 2D Arnold Cat
map and 3D Arnold Cat map.
3.1DualParameter Fractional Fourier Transform
DPFrFT is a type of Fractional Fourier Transform
which transforms the image according to the
transform orders. It is more time efficient than
Fractional Fourier Transform. Any application that
needs more randomness can use DPFrFT. It is
defined mathematically by equation (1):
dimension of image. Here
(1)
where α is the transform order, θ is the angle,
is
the Rotational Matrix,
s the transpose of the
Rotational Matrix, x is the original function and
s
the transformed function.
The original function x can be regenerated using
inverse DPFrFT. It is defined by the following
equation (2):
transform. The third parameter inserted is z which is
given by z = (a*x + b* y + z) mod n.
(2)
3.2 Hessenberg Decomposition
Hessenberg decomposition is the decomposition of a
matrix B into an upper Hessenberg matrix H by a
sequence of similarity transformations. It is defined
by equation (3):
B=PHPT
(3)
T
where P is an orthogonal matrix, P is the transpose
of orthogonal matrix P and H is a square matrix in
which all the entries below the first diagonal is 0.
3.3 Chaotic Map
Chaos is a kind of characteristic of a nonlinear
system. A Nonlinear chaotic map is used to generate
complex random number sequence and can be used
to randomize the image. In this paper the basic
encryption technique is carried out using 1D, 2D and
3D chaotic maps.
The 1D chaotic map used in this paper is the simple
logistic map. The logistic map is defined in equation
(4) :
(4)
=
mod n
(5)
b is the position variable (0 < b < n) and a is the
momentum variable, where at = bt – bt-1and n is
pixels before transform and
represents location of
represents location
of pixels after transform.
The 3D chaotic map that is used in this paper is 3D
Arnold’s cat map. The 3D Arnold’s cat map is
defined as equation (6):
=
Here
mod n
(6)
represents location of pixels before
transform and
represents location of pixels after
4. Proposed System
The proposed system analyses the randomness,
complexity and efficiency of image encryption using
different chaotic maps such as 1D Logistic map, 2D
Arnold Cat map and 3D Arnold Cat map. The block
diagram of the proposed system is depicted below in
Figure 1 and the methodology is described below:
Figure 1:Block Diagram of Proposed Encryption
System
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The methodology of the system is described as
follows
4.1 Key Generation from Biometrics
Let A be the biometric image having size
m1*n1.Initially, the image A is pre-processed to the
grey level image. The keys are generated from the
grey level image using the following key generation
process.
Randomly select two pixel values say px and py from
the biometric image A.
Take the difference between the two pixels px and py
and divide it with total number of grey levels in the
biometric image to get the threshold(T) as shown in
the following equation (7):
T=px-py/(2L-1)
(7)
Using the threshold T, the key for the 1D Logistic
map is calculated as per the equation (8):
K= (2L*T)mod1
(8)
Normalize the biometric image A using the equation
(9):
(9)
Generate the Feature Matrix FB from the normalized
image using the equation (10):
FB (i, j) =
1,if B(i,j) ≥T
0, if B (i, j) < T.
(10)
Arrange the two dimensional Feature Matrix FB into
the one dimensional Feature Vector Fv.
To get the transform orders and angles for DPFrFT,
partition Feature Vector into two equal segments Fv1
and Fv2 of order (m*n)/2.The transform orders and
angles can be calculated as per equations (11),(12),
(13) and (14):
αx=
(11)
αy=
(12)
θx=
θy=
(13)
(14)
4.2 Image Encryption
The encryption includes Randomization using
chaotic map Transformation using DPFrFT and
decomposition using Hessenberg Decomposition.
The process is described in the following steps:
Randomization using chaotic map.
If the chaotic map is Logistic map(equation 4), then
iterate it by using the key K generated from the Key
Generation process and store the chaotic sequence
generated to get the random matrix (Rk),which
randomizes the original image of size m1*n1 into
randomized image using the following equation
(15):
Ir(i,j)= ln Rk (i, j)/ ln I (i, j)
(15)
If the chaotic map is 2D Arnold Cat map (equation
5), the iteration will shuffle the pixels of the image
and will give the randomized image.
If the chaotic map is 3D Arnold Cat map (equation
6), the iteration will shuffle the pixels and the
intensity values of the image to give the shuffled and
substituted randomized image.
2) Transformation using DPFrFT
Apply transformation on the randomized image
using DPFrFT using equation (16):
=
(16)
3) Decomposition using Hessenberg Decomposition
Distribute Feature Vector Fv repeatedly into two
arrays A1 and A2 of size m1*m1 and n1*n1.
Apply Hessenberg Decomposition on two arrays A1
and A2 to get two orthogonal matrices Q1 and Q2
using the following equation (17):
A1=Q1HQ1TandA2=Q2HQ2T
(17)
4) Final encryption using randomization and inverse
transformation
Using Q1 and Q2,randomized encrypted image Ire is
generated as follows in equation (18):
Ire = Q1 Ir Q2T
(18)
Perform inverse
transformation using inverse
DPFrFT on Ire to get the encrypted image J as in
equation (19):
=
(19)
4.3 Image Decryption
The decryption process is as follows:
1) Transformation of encrypted image J, using
DPFrFT
The encrypted image undergo transformation using
DPFrFT as shown in the equation (20):
=
(20)
2) Decomposition of transformed image J
Using the two orthogonal matrices Q1 and Q2 which
is generated by Hessenberg Decomposition, the
partially decrypted image Jd is generated as in
equation (21):
Jd=Q1TJQ2
(21)
3) Inverse transformation of partially decrypted
image Jd
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The partially decrypted image Jd undergoes inverse
transformation using inverse DPFrFT to get
randomized decrypted image Jrd as in equation (22):
=
(22)
4) Undo Randomization of randomized decrypted
image Jrd
If the chaotic map is Logistic map (equation (4)),
then using
the random matrix(Rk) undo the
randomization and decrypt the image using the
following equation (23):
=
(23)
If the chaotic map is 2D Arnold Cat map or 3D
Arnold Cat map, multiplying the inverse of the map
with the image will give the decrypted image as in
equation (24):
(24)
Analysis, Information Entropy, Histogram Analysis,
NPCR and UACI.
5.1 Correlation Analysis
Correlation Coefficient is an inevitable factor to
evaluate the encryption process. The correlation
between two adjacent pixels in an image shows the
degree of relationship between them. It expresses a
measure of security. For a good encryption, the
adjacent pixels in the horizontal/vertical /diagonal
direction should be uncorrelated. It can be visually
tested by plotting the correlation distribution
between adjacent pixels in the encrypted image. It is
measured to ensure that the redundant information
available in the encrypted image is as low as
possible.
For calculating the correlation coefficient, the
following equation (25), (26), (27) an (28) are used:
(25)
(a)
(b)
(c)
(26)
(27)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 2: Images of original image, encrypted image
and decrypted image.(a) original image (b)
encrypted image using 1D Logistic map (c)
decrypted image using 1D Logistic map (d) original
image (e) encrypted image using 2D Arnold Cat
map (f) decrypted image using 2D Arnold Cat map
(g) original image (h) encrypted image using 3D
Arnold Cat map (f) decrypted image using 3D
Arnold Cat map
(28)
where x and y are grey values of two adjacent pixels
in the encrypted image. If the correlation coefficient
is equal to zero or near to zero, then the adjacent
pixels are not correlated. If it is equal to -1, then the
encrypted image is a negative of original image.
For simulation purpose, here same image is
encrypted using 1D Logistic map, 2D Arnold Cat
map and 3D Arnold Cat map. Then the correlation
coefficients between horizontally adjacent pixels of
three encrypted images are calculated. Also the
correlation distribution of three encrypted images are
plotted in Figure 3 and the Correlation Coefficient of
different images are shown in Table 1
5. Experimental Results and Discussion
A good encryption should be highly complex and
resistant against all types of attacks. The analysis
performed here to check the efficiency, randomness
and complexity of encryption are, Correlation
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(a)
The Information Entropy is a commonly used
technique to measure the randomness. It expresses
the degree of uncertainties of the system. A good
encryption decreases the correlation between the
pixels of an encrypted image and thus it increases
the value of entropy of the image. The Information
Entropy is calculated using the equation (29):
(29)
For conducting experiment, the same image is
encrypted using 1D Logistic map, 2DArnold Cat
map and 3D Arnold Cat map. Then the entropy for
the three images is calculated.
(b)
(c)
Figure 3: Correlation Distribution of encrypted
images using (a) 1D Logistic map (b) 2D Arnold Cat
map and (c) 3D Arnold Cat map
Figure 3(a) shows the Correlation Distribution of
encrypted mage using 1D Logistic map. From the
figure we can see that the adjacent pixels of the
encrypted mage are highly correlated. Figure 3(b)
shows the Correlation Distribution of encrypted
mage using 2D Arnold Cat map. The figure shows
that the horizontally adjacent pixels are uncorrelated
than 1D map. Figure 3(c) shows the Correlation
Distribution of encrypted mage using 3D Arnold Cat
map. It is obvious that the horizontally adjacent
pixels are uncorrelated than 1D and 2D map.
Table 1: Correlation Coefficient of two encrypted
images and the corresponding chaotic map
Image
Rose
Rose
Rose
Cat
Cat
Cat
Chaotic Map
1D Logistic map
2D Arnold Cat
map Cat
3D Arnold
map map
1D Logistic
2D Arnold Cat
map Cat
3D Arnold
map
5.2 Information Entropy
Correlation
Coefficient
0.9958
0.8228
0.2354
0.9487
0.6776
0.2519
Figure 4: Information Entropy of encrypted images
using 1D Logistic map, 2D Arnold Cat map and 3D
Arnold Cat map
Figure 4 shows the Information Entropy of three
encrypted images. Red line shows the entropy of
three images using 1D Logistic map Blue line shows
the entropy of three images using 2D Arnold Cat
map and Green line shows the entropy of three
images using 3D Arnold Cat map. If entropy is high,
then randomness of encrypted image is also high.
From the figure, we can see that entropy using 1D
Logistic map is lesser than 2D Arnold Cat map
which is lesser than 3D Arnold Cat map.
5.3 Histogram Analysis
Histogram is a commonly used analysis technique in
Image Processing. An image histogram shows the
distribution of pixels in an image by plotting the
number of pixels at each color intensity level. It
shows the efficiency of the encryption process. For
an efficient encryption, the histogram of the
encrypted image will be nearly uniform.
For analyzing the histogram of image encrypted
using 1D,2D or 3D chaotic map, the same image is
encrypted using 1D Logistic map,2D Arnold Cat
map and 3D Arnold Cat map. The Figures 5(a), 5(b)
and 5(c) shows the histogram of three encrypted
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images which is encrypted using 1D Logistic map,
2D Arnold Cat map and 3D Arnold Cat map
(a)
measures the percentage average differences of
intensities between two ciphered images.
For calculating NPCR and UACI, the same image is
encrypted using 1D Logistic map, 2D Arnold Cat
map and 3D Arnold Cat map. Then some random
numbers of pixels are selected from the original
image and make some slight change. Then the
changed image is encrypted using 1D Logistic map,
2D Arnold Cat map and 3D Arnold Cat map. Then
by using the two encrypted images of one chaotic
map, it can be calculated using equations (30), (31)
and (32):
(30)
(31)
where D is a matrix of same size as the original
image(m1*n1) and P and C is the encrypted images.
UACI (Unified Average Changing Intensity):
(b)
(c)
Figure 5: Histogram Analysis of encrypted mages
using (a) 1D Logistic map (b) 2D Arnold Cat map
and (c) 3D Arnold Cat map
(32)
Ideally, NPCR should be close to 100% and UACI
should be between 30 and 40%
.
Figure 6: UACI of encrypted images using 1D
Logistic map, 2D Arnold Cat map and 3D Arnold
Cat map
The Histogram Analysis shows that histogram of
encrypted image which is encrypted using 3D
Arnold Cat map is more uniform than others.
5.4 NPCR and UACI
NPCR and UACI is used to test the influence of
how a minor change in the plain image reflected in
the ciphered image.NPCR stands for Number of
Pixels Change Rate. It measures the difference of
pixel numbers between two ciphered images.UACI
stands for Unified Average Changing Intensity .It
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Figure 7: NPCR of encrypted images using 1D
Logistic map, 2D Arnold Cat map and 3D Arnold
Cat map
6. Conclusion
The randomness, complexity and efficiency of
multimedia encryption using 1D Logistic map, 2D
Arnold Cat map and 3D Arnold Cat map is analyzed
here. The application of DPFrFT and Hessenberg
Decomposition add more randomness to the system.
Also the use of biometrics for the key generation
increases the security of the system. For simulation,
different analyses are conducted with different
images. From the analysis we can see that encryption
using 3D Arnold Cat map has more randomness,
complexity and efficiency than others. But it has
certain disadvantages such as it is periodic, so that
the original image will be regenerated after number
of iteratons.Also it requires image to have image
height equal to image width. So an image encryption
using image compression technique and more
chaoticity which overcomes the weakness of Arnold
transform can be considered as a future work.
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