A Robust Chaos-Based Medical Image Cryptosystem
SAMIRA DIB1, ASMA BENCHIHEB2, FADILA BENMEDDOUR3
1Department of Electronics
University of Jijel
BP 98 O. Aissa, Jijel 18000
ALGERIA
2Faculty of Medicine
University of Constantine 3
Constantine 25000
ALGERIA
3Department of Electronic Engineering
University of M’Sila
M’Sila 28000
ALGERIA
Abstract: - In In this paper, we propose an efficient cryptosystem for medical images. While the confusion
stage is ensured by an Arnold's cat map allowing the permutation of pixels; the diffusion stage is alleviated by
an improved logistic map used by the chaotic key-based algorithm (CKBA).
The simulation results attest that the proposed algorithm has superior security and enables efficient
encryption/decryption of medical images. Performances were evaluated by several security analyses: the NPCR
and UACI are improved over 99.60% and 33.46% respectively, and entropy is reported close to 7.8. What
makes this new cipher much stronger security.
Key-Words: - CKBA, Modified logistic map, Chaos, Cryptography, Medical image.
Received: September 18, 2021. Revised: May 7, 2022. Accepted: June 8, 2022. Published: July 2, 2022.
1 Introduction
In recent years, telemedicine systems have been
introduced in hospitals and medical centers with the
aim of establishing communication bridges offering
good medical services between specialists and
patients. For reasons of patient incapacity or the
remoteness or scarcity of certain specialties,
physicians may interact among themselves or with
their patients to diagnose a disease or prescribe a
treatment. They use clinical data such as images and
medical signals.
Various patient medical information and
specialist diagnoses are confidential and must be
secure during transfer over different communication
networks in a highly computerized and
interconnected world. In order to reinforce the
security of this information, several approaches are
used, the most important of which are cryptographic
techniques. The goal of any image encryption
method is to obtain a higher quality image in order
to keep the information secret and safe.
Because of their sensitivity to changes in initial
conditions, chaotic systems are one of the best
approaches to protect the information in secure
communications. It represents one of the most
powerful classes of cryptography by their resistance
to external attacks. Much research has been
conducted in this area over the past two decades [3–
20] and references therein.
The chaotic key-based algorithm (CKBA) for
image encryption is originally proposed by Yen and
Guo [3] and claiming strong security for image
encryption. This statement is challenged showing
that the complexity of a ciphertext attack against the
CKBA is much lower than originally announced,
and chosen/known text attacks can be applied
effectively. Since then, research work has been
carried out to improve the safety of the CKBA.
Another technique, presented in [4] and based on
the chaotic logistic map, provides high security for
gray images. The method proposed in [5] is secure
against brute force attacks, sensitive to the key and
is efficient even in terms of speed. However, the
success of this method may be limited due to the
common operations it uses. This makes external
attacks vulnerable. Despite these developments,
even faster and more secure cryptosystems are
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needed. Further, many efforts have been focused on
the use of the chaotic cryptography for medical data
encryption [19–27].
In this paper, a medical image encryption scheme
using a modified logistic (ML) map is presented.
The initial condition x0 of the chaotic ML system is
considered as the secret key of our proposed
encryption algorithm. Furthermore, the parameters
of cat map, p and q, are used as confusion key
parameters.
To test the feasibility of the proposed algorithm,
a variety of standard reference images were used
including X-ray, ultrasound, MRI and CT scanner
images. Additionally, to accurately verify the
performance of this method, several parameters
were calculated, including histogram analysis,
correlation, differential attack, PSNR, SSIM and
information entropy. The results reveal that the
proposed technique offers good results and can
therefore be adopted in a telemedicine system.
The organization of the paper is as follows:
Section 2 is devoted to the definitions of the logistic
map and the modified map of the chaotic system.
The proposed encryption method is presented in
Section 3. Section 4 is devoted to the discussion of
the obtained results. Finally, conclusions are
presented in section 5.
2 Logistic Map and Modified Map
The logistic map is typically used in most chaos-
based cryptosystems, especially in secure
communication systems.
2.1 Logistic Map
Logistic map is one-dimensional map which is
explained by a recursive function as follows:
(1)
Where r is its parameter that lies in interval [0, 4]
and .
2.2 Modified Logistic Map (ML)
In order to expand chaotic region of Logistic map
and make it suitable for cryptography, we will
approach a modification to the Logistic map and use
that one proposed in [11]:
(2)
Where n is a time index, is the initial value,
and the control parameters are α
and s used as key parameters for the proposed
cryptosystem. They are related by the following
equation: (3)
Here, the parameter s must always be greater than α,
for the values to appear inside the interval (0,1).
Moreover, the closer the value α is to s, the wider
the interval of .
In order to explain the performance of Equations
(1) and (2), Bifurcation diagram and Lyapunov
exponents are calculated and plotted with respect to
respective control parameters ‘r’ and ‘s’ in Figure
1, for the Logistic map and the Modified Logistic
Map (ML).
Regarding Fig.1, Logistic map is chaotic when
parameter “r” lies in interval [3.6, 4] and ML map is
chaotic when parameter ‘s’ lies in interval [0.1, 20].
Therefore, the chaotic range of ML is more than the
others. As shown in the figure, there are no free
white spaces and the entire area is almost covered.
More importantly, all values of s can be used to
build the key space. This property extends the key
space of the proposed cryptosystem.
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Fig. 1. (a) Bifurcation diagram of logistic map ;
(b) Bifurcation diagram of ML map ; (c) Lyapunov
Exponent of Logistic map ; (d) Lyapunov exponent
of ML map.
3 Proposed Cryptosystem
In general, chaos-based image encryption systems
are applied in two steps: replacing pixels called
confusion and changing pixel values called
diffusion.
It is known that an ordinary image has a strong
correlation of adjacent pixels which must be broken
before any encryption procedure. Several ways of
ensuring this decorrelation are used in the literature.
Within the framework of this paper, we propose the
application of Arnold Cat Map (ACM). It is a
discrete invertible system which allows a good
reorganization of the pixels’ positions of the original
image. It is represented by
(4)
where (x,y) are the pixel position of the plain image
with a size of N×N and (x’, y’) is the corresponding
pixel position. Control parameters of the ACM are p
and q, which are positive integers and will be used
as confusion key parameters:
(5)
ACM effectively changes all pixel positions as
only linear transformation with simple mod function
need to be performed. Furthermore, it has a
characteristic of area-preserving which means that if
it is iterated enough times, original image reappears.
Shortly, ACM can be considered as a permutation
method that focuses on the pixel position not the
pixel value of the plain image. Hence, the
cryptosystem requires a diffusion process to
enhance the security.
This stage is ensured by the CKBA algorithm
using the modified logistic map which offers a large
key space.
Modified CKBA algorithm
In order to protect medical images transferred on
public roads and in order to fortify their security, we
proposed in this paper a modified algorithm based
on chaotic keys (CKBA) by the use of the
traditional architecture given in [3]. The
modification is made using the modified map (ML)
given in eqt(2) rather than the logistic map eqt(1).
The encryption process is described as follows:
Step 1 : Assume that the size of the original image
(Img) is M×N.
Step 2 : Select different parameters keys namely :
- p and q and a number of iteration, to apply
Arnold’s cat map to change the pixels position.
- Two bytes key1 and key2 (8 bits) and the initial
condition x0 of the ML chaotic system as the
secret keys of the encryption system.
Not all secret keys can make well disorderly cipher-
images, the basic criterion to select key1 and
key2 should be satisfied:
, where
and
Step 3 : Run the chaotic system to make a chaotic
sequence
Step 4 : Generate a pseudo-random binary sequence
(PRBS)
from the 16-bit binary
representation of
Once {b(i)} is generated, the
encryption can start.
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- For the plain-pixel f(x,y) (0 ≤ x ≤ M – 1 , 0
≤ y ≤ N−1), the corresponding cipher-pixel f′(x,y)
is determined by the following rule :
(6)
Where and .
Step 5 : The decryption procedure is just like the
encryption since XOR and XNOR are both
involutive operations.
4 Performance analysis
In order to evaluate the proposed cryptographic
method, multiple medical images have been
adopted, including X-Ray, ultrasound, MRI and CT
scanner images which are retrieved from open
databases [29, 30].
4.1. Histogram Analysis
In image processing, histogram is the graphical
representation of the pixel values’ distribution in an
image by plotting the number of pixels at each gray
level. The randomness of the proposed algorithm
can be validated by a flat and uniformly distributed
histogram of the encrypted images.
A good cipher image has a uniform frequency
distribution of the pixel values. Fig. 2 shows
histograms of the plain and cipher images with the
studied algorithm. As can be seen, the cipher images
are so boisterous in a way that any data from them
cannot be obtained, so the proposed scheme is
powerful against histogram attacks.
4.2. Correlation Analysis
Correlation distributions and correlation coefficients
play a crucial role in the analysis of encrypted
images. It is well known that the correlation
between adjacent pixels of an informative image is
high in any direction. Whereas for the encrypted
image, the correlation must be very weak in order to
be able to withstand the various statistical attacks.
In this section, the correlations of the adjacent
pixels in the original image and encrypted image are
analysed and compared by choosing 20 000 pairs of
adjacent pixels in horizontal, vertical, and diagonal
directions from the original image and its encrypted
image.To calculate the correlation coefficient of
adjacent pixels, Eq. 7 can be adopted [28]:
(7)
Where a and b are values of 2 adjoining pixels, E(a)
and E(b) is the mean of a and b respectively, and M
is the number of adjoining pixels of the image.
If the correlation coefficient is 1, it means that
the original image and its encrypted image are
highly dependent. However, if this coefficient is 0,
then the encrypted image and the original image are
not correlated. Table 1 demonstrates the results
obtained. Indeed, it is clear that two adjacent pixels
of the original image, in the three directions, are
strongly correlated. However, for adjacent pixels of
the encrypted image, the correlation is is very small
or negligible. Therefore, the proposed algorithm has
good permutation and substitution properties.
On the other hand, the graphical representation
of the correlation is a visual inspection of the pixel
dependence of the image, where the horizontal axis
represents the intensity value of the pixel and the
vertical axis represents the value of the
neighbouring pixel, horizontal, vertical or diagonal.
For an informative image, the graph is expected to
show a strong pattern on a 45 degree line; the denser
this line, the more the tested image is correlated. In
the encrypted image, the expected graph should
have points in the whole plane since most
neighbours of any pixel have different intensity
values with respect to that pixel.
The graphical representation of the correlation
analysis is performed by plotting the pixel gray
values between adjacent pixels in the different
directions for 1024 randomly selected adjacent
pixels pairs for both the original and the encrypted
image. It is clearly shown from Fig. 3 (a)–(d) that
the strong correlation between adjacent pixels is
completely broken in all directions after applying
the proposed encryption process.
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(a) (b) (c) (d)
Fig. 2. Simulation results: (a) Original images. (b) Histograms of the original images. (c) Encrypted images.
(d) Histograms of the encrypted images.
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Table 1. Correlation between adjacent pixels of original and encrypted images.
Directions of adjacent pixels
Horizontal
Vertical
Diagonal
Image
Original
Encrypted
Original
Encrypted
Original
Encrypted
0.9956
0.0231
0.9971
-0.0034
0.9933
-0.0005
0.9836
-0.00706
0.9744
0.000201
0.9677
0.0093
0.9880
0.00005
0.9931
-0.0013
0.9828
-0.0087
0.9805
0.0422
0.9832
-0.0013
0.9664
-0.0015
0.9981
-0.0002
0.9899
-0.0034
0.9883
0.0081
0.9812
0.0250
0.9511
0.0134
0.9418
0.0003
0.9904
0.0051
0.9943
0.0044
0.9870
-0.0006
0.9737
0.0477
0.9905
0.00351
0.9714
0.00006
Fig. 3. Correlation distribution of adjacent pixels in different directions: (a) Original and encrypted images.
(b) Diagonal direction. (c) Vertical direction. (b) Horizontal direction.
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4.3. Information Entropy analysis
The pixel values of a cipher image should be
randomly distributed that they do not hold any
relation with the plain image. Entropy denotes the
degree of randomness and it is calculated using
logarithm over a probability distribution.
(8)
where, is the probability mass function.
Generally, we expect from a good image
encryption scheme that the entropy of the encrypted
image is close to the ideal case. For an image data
given in the form of a byte, the maximum entropy
value is 8. Thus, the entropy value close to 8
denotes higher randomness; else if this entropy
value is lesser then there is a possibility of attack
which can reduce the security of image
transmission. In Table 2, the entropy after
encryption is compared to that before encryption for
various images. According to the obtained values,
the average entropy of all encrypted images is very
close to the theoretical value (≈ 7.7). This means
that the proposed algorithm is much efficient and is
highly secure. So we can conclude that the main
objectives of image encryption are ensured, namely,
illegibility and indeterminacy.
Table 2 The information entropy of different images.
Images
Plain images
7.516
7.574
6.097
0.642
6.701
6.017
6.361
7.530
Cipher
7.977
7.9819
7.214
7.879
7.404
7.429
7.446
7.8412
4.4. Differential Attack Analysis
4.4.1. NPCR and UACI tests
Number of Pixel Change Rate (NPCR) and Unified
Average Change in Intensity (UACI) are the two
most important parameters to measure the resistance
of developed algorithm against differential attacks.
This analysis can be made by observing the
relationship between two cipher images, one
obtained from the normal image and another from
single pixel changed plain image. NPCR measures
the minimum number of pixels altered and UACI
measures the average difference between the two
cipher images. The NPCR and UACI are evaluated
by Eqs. (9) and (10) [28] :
(9)
(10)
Subject to :
where E1 and E2 are the encrypted images.
Evaluation of encrypted images by NPCR and
UACI parameters is depicted in Table 3. The results
reveal that a swift change in the original images
leads to a change in the cipher ones. This signifies
that the proposed scheme has a high ability to resist
differential attack and the image encryption schema
has a high sensitivity to a minor change in the
original images.
The average values of NPCR and UACI for
images are estimated as 99.60 % and 33.47%
respectively.
4.4.2. SSIM and PSNR
An encryption method achieves successful
performance when the encrypted image has a low
PSNR and SSIM values. Indeed, two same images
have PSNR value as 1 and perfectly similar images
have SSIM value as 1(one).
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To determinate the change in the pixel values of
encrypted image from original image and accuracy
of decrypted image from original image, peak
signal-to-noise ratio (PSNR) and structural
similarity (SSIM) value are calculated using the
following equations [28]:
(11)
(12)
(13)
Where :
MAX: Maximum supported pixel value, MSE:
Mean squared error, : Size of the image
: Original image pixel value at location (i, j),
and : Received image pixel value at location
(i, j).
In medical image, it is important that the
decrypted image is exactly the same as the original
one. Table 4 shows the PSNR and SSIM values for
both crypted and decrypted image with respect to
the original image. These results indicate that the
PSNR is less than 10 dB and the SSIM is close to 0
for all encrypted images. In addition, all the images
have been well reconstructed. Indeed, the SSIM is
close to 1 and the PSNR is very high and tends to ∞.
Table 3. NPCR and UCAI images
NPCR (%)
99.55
99.61
99.61
99.57
99.61
99.56
99.61
99.61
UACI (%)
33.48
33.46
33.46
33.48
33.46
33.48
33.46
33.46
Table 4. The SSIM and PSNR values for encrypted/decrypted images
PSNR (dB)
Encrypted
8.77
7.86
6.81
8.14
6.91
6.81
8.46
6.58
Decrypted
∞
43.26
∞
28.42
∞
41.34
∞
∞
SSIM
Encrypted
0.02
0.04
0.03
0.00
0.04
0.02
0.02
0.06
Decrypted
1
1
1
0.97
1
0.96
1
1
4.5. Key Sensitivity Analysis
For the encryption of medical images, the sensitivity
of the keys is a very important index for building
any cryptosystem and evaluating the quality of an
encryption algorithm. An image cannot be properly
decrypted if the key is changed slightly, indicating
that the encryption method has high performance.
On the contrary, if the key changes significantly, the
image may still be partially decrypted, indicating
that the encryption performance of the encryption
method is poor.
Key sensitivity analysis can be observed in two
aspects: (i) if slightly different keys are applied to
encrypt the same images, then completely different
cipher images should be produced; (ii) if a small
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difference exists in decryption key, then the cipher
image could not be decrypted correctly.
For the first key sensitivity analysis, a test plain
of X-Ray image is encrypted with a randomly
chosen key of s=97.5678910, x0 =0.1234567. Then
a slight change 10−7 is applied to the one of the
parameters with the other remains same, and repeats
the encryption. The corresponding cipher images
and the differential images are shown in Fig. 4. The
correlation coefficients between the cipher images
are calculated and given in Table 5. Small key
changes will lead to incorrect decryption results.
This also shows that the security performance of this
method is very high. Fig .4 illustrates that the
decrypted images with incorrect secret key are quite
different from the original image, which
demonstrates that the proposed scheme is highly
sensitive to the slightest changes in secret keys.
The proposed cryptosystem should also be
sensitive to Cat map stage p, q and n parameters.
Cipher images produced by slightly different keys
are shown in Fig. 5 and the correlation coefficients
for the corresponding cipher images are given in
Table 6. It is clear from these results that a very
small difference for all encryption keys results in
completely different encryption images.
In the second case, the encrypted image could
not be properly decrypted, using a slightly different
key than the one used in the decryption. As an
example, the scanner CT image of Fig. 2(c) was
used and the result is shown in Fig. 6.
We conclude that the proposed cryptosystem is
quite sensitive to all keys and can resist differential
attacks effectively.
Fig. 4. Key sensitivity in the first case: (a) Plain image; (a’) Cipher image with s=97.5678910, x0 = 0.1234567;
(b) Cipher image with s=97.5678911, x0 = 0.1234567 ; (c) Cipher image with s = 97.5678910 ; x0 = 0.1234568;
(d) Differential image between (a’) and (b); (c’) Differential image between (a’) and (c).
Table 5. Correlation coefficients between cipher images produced by slightly different keys.
Figure 4
Key
Correlation coefficient
s
x0
(a’) - (b)
97.5678911
0.1234567
-0.0019
(a’) - (c)
97.5678910
0.1234568
0.0054
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Fig. 5. Key sensitivity in the first case: (a’) Cipher image with p = 7 , q = 5 , n = 10 ; (b) Cipher image with
p=6, q = 3 , n = 10 ; (c) Cipher image with p = 5 , q = 3 , n = 7.
Table 6. Correlation coefficients between cipher X-Ray images produced by slightly different keys.
Figures
Keys
Differential
Figures
Correlation
coefficient
p
q
n
3(b)
6
4
10
5(a’)
7
5
10
3(b)-5(a’)
0.0097
5(b)
6
3
10
3(b)-5(b)
0.0076
5(c)
5
3
7
3(b)-5(c)
0.0016
Fig. 6. Key sensitivity in the second case: (a) Wrong decrypted image with s = 97.5678911, x0= 0.1234567 ;
(b) Wrong decrypted image with s = 97.5678910, x0 =0.1234568 ; (c) Correct decrypted image with
s=97.5678910, x0 = 0.1234567
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4.6. Data Loss and Noise Attack Analysis
During transmission, the image is subject to data
loss. A powerful image cryptosystem must
withstand this loss. Cropping attack and noise attack
analysis is performed to find how the algorithm is
stronger against a loss of data when an image is sent
over the public channel.
Fig. 7 shows examples simulation of the data
loss attack. A CT scan and ultrasound test images
are first encrypted using the proposed algorithm.
Next, a 60x60 data slice of the encrypted images is
performed. According to the obtained result, the
decrypted images contain most of the original
information. Even after cropping, the intended
receiver will be able to retrieve the plain images to
some extent, hence against this cropping its
robustness is been proved.
Moreover, the ability of defending the noise
attacks is measured by adding different types of
noise. In this test, the salt and pepper noise are
added in cipher image of ultrasound and MRI with
the density of 1% and 2%, which are given in Fig. 8.
The Gaussian noise influences the cipher image
with the intensity of 0.0001 and 0.0002 are shown in
Fig.9.
From these Figures, it is concluded that the
proposed method has good robustness to defend the
noise attacks. It is demonstrates that the proposed
method can still succeeds in recovering the image
when the cipher image subjects to different noise
attacks.
Fig. 7. Data loss analysis: (a) Original Ultrasound and CT scanner images;
(b) Cipher image with 60×60 data cut; (c) Decrypted image of (b).
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Fig. 8. Noise attack analysis: (a) Original Ultrasound and MRI images; (b)-(c) Cipher images added with
1% “salt & pepper” and 2% “salt & pepper” noise ; (d)-(e) Decrypted images.
Fig.9. Noise attack analysis: (a) Original CT and Ultrasound images; (b)-(d) Cipher images under Gaussian
noise with the degree of Mean=0, Variance=0.0001, 0.0002 and 0.0003; (e)-(g) Decrypted images.
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5. Conclusion
To contribute to the security of medical image
encryption, this paper proposes an efficient chaos-
based cryptosystem is proposed for medical images.
The system uses the CKBA with a modified
logistics (ML) map that provides a wide range of the
key space due to the unlimited value of the
command parameter. Various criteria were used to
analyze and validate the security and performance
of the developed algorithm.
A small change in the simple image or in any
cryptosystem setting will provide totally different
keys even if the same encryption key is used. The
experimental results proved the robustness of this
algorithm achieving desirable protection for real-
time medical image security applications. Further
key sensitivity evaluation proves that the method in
this paper is very sensitive to the small changes of
the keys, and the encryption security is high. The
results of plaintext sensitivity evaluation show that
the NPCR average value of all encrypted images is
as 99.60%.
For future work, we suggest implementing the
presented method for other file types such as video
and color 3D images very often encountered in
ultrasound and scanner techniques.
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