Image Encryption Algorithm based on Elliptic Curves
SARA CHILLALI*, LAHCEN OUGHDIR
Engineering Sciences Laboratory – LSI,
Higher Normal School of Fez (ENSF),
Sidi Mohamed Ben Abdellah University,
Fez,
MOROCCO
*Corresponding Author
Abstract: - Several cryptography schemes on the images use chaos theory, these schemes are very effective but
have weaknesses such as the distribution of keys between the two entities that want to exchange an image
confidentially. In this work, we propose an efficient encryption scheme using elliptic curves, we show that the
proposed scheme is tamper-proof against any attack. It also proves security because of the problem of discrete
logarithm on this elliptic curve whose calculation is difficult. This scheme achieves the required objectives such
as non-repudiation, confidentiality, and integrity. During the security analysis of the proposed scheme, we
verified that many security attributes have been satisfied.
Key-Words: - Cryptography, Elliptic curve, Image processing, Analysis of data, Embedded systems, Matlab.
Received: July 16, 2022. Revised: October 22, 2023. Accepted: November 24, 2023. Published: December 31, 2023.
1 Introduction
The embedded systems are open and free domains.
That is to say, when you send information from one
point to another, it is redirected via several stations
where you can read this data.
The analysis of this data via these systems is in
fact as open as sending a postcard.
The protection of transmitted data is an open
domain which can only be ensured by effective
encryption.
This work consists of implementing an encryption
algorithm applied to standard image files (BMP,
JPEG, TIFF), based on simulated attacks.
Cryptology is a mathematical science that has
two branches; cryptography and cryptanalysis.
On the one hand, traditional cryptography is the
study of methods of transmitting data in a
confidential way.
Cryptanalysis, on the other hand, is the study of
cryptographic processes in order to find weaknesses
and particularly to be able to decrypt encrypted
messages.
Image processing is certainly the most
innovative process that man has known, motivated
by his needs in various fields, including imaging,
the later, generates sensitive problems solved by the
various image processing techniques.
The digital image is represented by a matrix of
points called pixels (Picture Element), each one has
a characteristic of a gray level or a color coming
from the corresponding location in the real image or
calculated from an internal description of the scene
to represent.
We can represent an image by a matrix ;
The real-time image processing system is a
visualization and interpretation system. The
functions performed by each module of the system
are as follows:
1- The video camera transforms the optical image
into an analog signal.
2- The conversion of the signal into a set of digital
data by performing coordinated sampling
operations, and the quantification of the amplitudes
of the light intensity measured by the digitization
module.
3- The storage and visualization of digital data is
ensured by the visualization module and the result
will be saved in a file.
In [1], the authors invented a key distribution
method based on a very difficult mathematical
problem to solve; this problem is the discrete
logarithm on a cyclic group of finite order.
In [2], the authors gave an encryption method
using matrices built in [3], as points out in his article
of image cryptography based on chaos and cubic. In
other cryptography problems we find discrete
logarithm problem on a Montgomery Curves in [4]
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and the closest vector problem in, [5]. In this work,
we expose a new method of image encryption,
based on the last problem, using a Diffie-Hellman
key exchange on an elliptic curve. The security of
this type of encryption is proven because the
resolution of the problem of discrete logarithm is
almost impossible on these curves.
In the following, denotes a finite field where
p is a prime number greater than or equal to 5, [6].
In this context, an elliptic curve over is a
plane curve defined by an equation of the form
: ; where a and b are in and
is not equal to zero.
We denote this curve by : , we can write :
The law ‘+’ defined on by :
For each three points and
of the elliptic curve defined by such
that . Then is given by:
, for ,;
; and
, with:
t=
and
s=
A point multiplication can be considered as a
series of consecutive point additions:
is an abelian group of neutral element
[0 :1 :0], called the point at infinity.
The principle of public key cryptography is
based on two keys, one is public and the other is
private. Finding the private key from the public key
is equivalent to solving a difficult problem.
On elliptic curves, it is the problem of the
discrete logarithm that must be solved to find the
secret key. That is to say, find knowing
with and are known and .
2 Proposed Encryption Scheme
Let be an elliptic curve on the field
defined by , where is a large prime number
such as the discrete logarithm problem in is
difficult.
2.1 Key Exchange between Entities A and B
1) The two entities A and B agree on a prime
number , a generator of a cyclic subgroup of
known order of the elliptic curve .
2) A chooses a random integer secret ,
and sends to B.
3) B also chooses a random integer secret
, and sends to A.
4) A computes .
5) B computes .
The secret key between A and B is : .
2.2 Secret Key Calculation Algorithm
Each for himself, A and B build the secret
encryption key ; by the following steps:
Step1 :
Construction of matrix A of size [T,3,3] :
Let , the matrix
; is
constructed as following :
For and ;
Step2 :
Build an encryption matrix B, whose size is the size
of the image:
Let the size of the image to be encrypted,
then
Step3 :
Convert matrix B to a column matrix C of size
, this matrix is constructed as following :
Finally convert the decimal numbers of the
matrix C into binaries of size t=size(T) , so the
obtained matrix
is the secret key
built between the two entities A and B.
Each can be written as , where
equals to 0 or 1.
2.3 Encryption Algorithm
The need to encrypt, store and transmit a digital
image or any other graphic form comprising
thousands of bytes and at the rate of more than 5000
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images per day, for a traffic for example, we
question the amount of data produced for this last.
This problem had to be solved by cryptography and
coding in order to guarantee confidentiality,
integrity, authentication and non-repudiation while
maintaining a faithful replica of the original data.
If we look at the image on the physical plane, we
find that the image signal is very redundant in
nature. Neigh boring pixels generally have similar
gray levels, which shows a significant spatial
correlation. By exploiting dissociation, it is possible
to reduce the disturbance of this redundancy.
Let
a secret key between A and B.
B wants to send an image confidentially « img » to
A, it follows the structure of the proposed algorithm
which consists of five encryption steps, as described
by:
Step1 :
Turn the image, « img » in to a matrix . The
public size of is .
Step2 :
Convert matrix M to a column matrix S of size ,
such that :
Step3 :
Convert the decimal numbers of the matrix S into
binaries of size t ; .
Step4 :
Calculate,
where
and
Step5 :
Convert the binary numbers from matrix
to
decimals modulo 256 to get a matrix column , then
the matrix of the decrypted image is the matrix
such that:
The transformation of into image,
"encryption(img)" is the encryption of the image
origin "img".
2.4 Decryption Algorithm
A receives "encryption(img)" sent by B, he has the
private key which allows him to reciprocally
calculate the matrix M, which we can transform to
the image "img". Figure 1 illustrates such a
situation:
Fig. 1: Encrypt and decrypt Algorithm
2.5 Security
Cryptography on elliptic curves (ECC) is a public
key encryption technique based on the theory of
elliptic curves that can be used to create faster
cryptographic keys, smaller and more efficient.
According to some researchers, ECC can
achieve a level of security with a 164-bit key than
other systems require a 1024-bit key. ECC was
developed by a mobile e-commerce security
provider. The security of our contribution is based
on the problem of the discrete logarithm on an
elliptic curve, until now, there is no specific
algorithm which solves such a problem on special
elliptic curves.
3 Example and Illustration
To check the performance of the proposed scheme,
the results of simulation experiments were evaluated
by several criteria in this section. Our experimental
environment was a desktop PC with 64-bitWindows
10 OS, Intel i7-2600 CPU, and 8GB RAM.
The programming language was Matlab, the
test images were chosen from image database.
In this part all illustrations are done by
MATLAB, we assume that ;
and .
Let the elliptical curve defined on by an
equation:
We have :
Let , the secret
key between A and B, we count ,
; .
3.1 Encryption and Decryption Simulation
The images where we have chosen for the
encryption and decryption test are downloaded from
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a standard image processing library, we have
selected 20 black and white images and 15 colors
with different extensions (bmp, jpg, tif) and
different sizes, [7].
The intrinsic features of bit distribution in
digital images were revealed. Higher bits of pixels
hold higher weight of an image’s information, and
there are strong correlations among the higher bit
planes.
In the instance of Figures, the 8th bit plane and
the 7th bit plane tend to have opposite values.
These features shall not be neglected in a secure
cryptosystem.
To build our way through bit maps, the strategy
has been extended to color, grayscale and black and
white images with different extensions in this
article.
By these means, an ordinary image of size [m,
n, p] is extended to a matrix M of the same size. All
the p bit planes of the pixels of the ordinary image
have been placed in the p th matrix M (:,:, p) ;
p=1,2,3.
After generating a key matrix of size [n, m, p],
the p bit planes were restored by XOR operations
which were performed on pixels. As illustrated in
the following figures, the entire cryptosystem is
managed by Matlab. The inputs to the algorithm are
the ordinary image "img" of size [n, m, p] and the
parameters (Bmp, Jpg, Tif). The output is the
encrypted image.
We will include five examples of figures, one in
black and white and the other in color, the set is
detailed in Figure 2, Figure 3, Figure 4, Figure 5 and
Figure 6.
Fig. 2: Image encryption and decryption
experimental result using Mandrill
Fig. 3: Image encryption and decryption
experimental result using Mandrill
Fig. 4: Image encryption and decryption
experimental result using Lena
Fig. 5: Image encryption and decryption
experimental result using Peppers
Fig. 6: Image encryption and decryption
experimental result using House
3.2 Analysis by Histogram
The histogram is the foundation of various spatial
image processing techniques, e.g., image
enhancement. Moreover, the inherent information of
histograms is useful in image compression and
segmentation.
Fig. 7: Histogram of Mandrill image Red and Blue
color component
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Fig. 8: Histogram of Mandrill image Red, Green
and Blue color component
Fig. 9: Histogram of Mandrill image Blue color
component
In Figure 8, the histograms of the three channels
R, G and B of the encrypted image are uniformly
disturbed compared to the other channels of the
original image, Thus the encryption algorithm used
shows the dependence of the statistical properties of
the images encrypted and original images is almost
random, hence cryptanalysis is very difficult.
This is also checked for white and black images
in Figure 7.
According to Figure 9, it should be noted that
the original image and the decrypted image have the
same histogram, which can be explained by the
performance of our approach.
3.3 Analysis by Correlation
Plain images usually are redundant in the spatial
domain, which means that adjacent pixels are highly
correlated. Whereas, in cipher images, such a
correlation should be broken. To measure the
correlation between adjacent pixels, we calculated
correlation coefficients as below:
(3)
The x and y are pixel vectors of the same length.
The and are their arithmetic mean
values, and the and are their standard
deviations. The range of correlation coefficients is [-
1, 1]. If x and y are not correlated, shall be
close to 0. See the next table, Table 1, of correlation
coefficients:
Table 1. Correlation coefficients
I
D
Si
ze
(k
o)
Original Image
Encryption Image
Decryption Image
Correlation
Correlation
Correlation
H-V
H-D
V-D
H-V
H-D
V-D
H-V
H-D
V-D
1
76
9
0.8
786
0.6
764
0.9
106
-
0.0
013
0.0
007
0.0
009
0.8
786
0.6
764
0.9
106
2
76
8
0.3
565
0.1
237
0.8
074
-
0.0
026
0.0
012
0.0
019
0.3
565
0.1
237
0.8
074
3
76
8
0.2
752
0.3
952
0.8
379
0.0
002
0.0
018
0.0
046
0.2
752
0.3
952
0.8
379
4
15
.2
0.8
830
0.7
013
0.9
189
0.0
067
-
0.0
002
-
0.0
005
0.8
830
0.7
013
0.9
189
5
36
.9
0.8
863
0.7
028
0.9
180
-
0.0
007
0.0
018
0.0
015
0.8
863
0.7
028
0.9
180
6
32
.1
0.8
843
0.6
952
0.9
173
-
0.0
002
-
0.0
018
0.0
013
0.8
843
0.6
952
0.9
173
7
19
2
0.6
378
0.4
823
0.9
418
-
0.0
053
0.0
012
0.0
045
0.6
378
0.4
823
0.9
418
8
76
8
0.4
753
0.2
939
0.4
201
0.0
009
0.0
032
0.0
001
0.4
753
0.2
939
0.4
201
9
19
1
0.8
775
0.6
791
0.9
169
0.0
041
-
0.0
001
-
0.0
014
0.8
775
0.6
791
0.9
169
1
0
61
0
0.5
759
0.3
751
0.5
282
-
0.0
023
0.0
013
0.0
022
0.5
759
0.3
751
0.5
282
1
1
76
8
0.8
788
0.6
755
0.9
119
-
0.0
016
0.0
002
0.0
011
0.8
788
0.6
755
0.9
119
1
2
12
1
0.5
387
0.3
421
0.9
242
-
0.0
094
0.0
044
0.0
035
0.5
387
0.3
421
0.9
242
1
3
76
8
0.3
565
0.1
237
0.8
074
-
0.0
026
0.0
012
0.0
019
0.3
565
0.1
237
0.8
074
1
4
38
9
0.1
661
0.2
121
0.5
228
-
0.0
027
-
0.0
011
-
0.0
005
0.1
661
0.2
121
0.5
228
1
5
76
8
0.2
752
0.3
952
0.8
379
0.0
002
0.0
018
0.0
046
0.2
752
0.3
952
0.8
379
According to the Table 1, the pixels of the
original image and those of the decrypted image are
strongly correlated, while the pixels of the encrypted
image are uncorrelated because their correlation
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coefficients are almost zero. We can say that the
proposed encryption algorithm makes plus
cryptanalysis more difficult.
3.4 Information Entropy
Information entropy was proposed by C. E.
Shannon, which is a measure of the randomness of
information. For a digital image, it is difficult to
predict the content if its information entropy is high.
The formula for calculating the entropy of
information is given in equation (4). The ideal value
of a cipher image’s information entropy is its bit-
depth d. In Table 2, the information entropy of plain
images cipher images and decrypt images are listed:
Table 2. Entopy
ID
Size
(ko)
Original
Image
Encryption
Image
Decryption
Image
Entropy
Entropy
Entropy
1
121
7.0907
7.9958
7.0907
2
768
6.8657
7.9974
6.8657
3
691
7.5459
7.9993
7.5459
4
1009
7.2176
7.9993
7.2176
5
432
7.5148
7.9994
7.5148
6
429
7.6336
7.9991
7.6336
7
8.02
0.8829
7.9859
0.8829
8
64.8
7.5785
7.9955
7.5785
9
192
7.4444
7.9985
7.4444
10
192
6.9047
7.9970
6.9047
11
192
7.4444
7.9985
7.4444
12
13.1
3.5335
7.9791
3.5335
13
64.8
7.5902
7.9956
7.5902
14
132
7.3946
7.9996
7.3946
15
256
7.0480
7.9983
7.0480
16
512
3.8764
7.7954
3.8764
17
64.2
7.4429
7.9953
7.4429
18
256
7.4451
7.9988
7.4451
19
256
7.2925
7.9991
7.2925
20
512
4.3812
7.7949
4.3812
The values of the entropy of the encrypted
images are almost equal to 8, it is indeed the
theoretical value, which proves the uniformity of the
histograms of the encrypted images, hence the
resistance of our approach to attacks by entropy.
In addition the entropy of the original image is equal
to that of the decrypted image so we do not lose
information on the original image.
3.5 Analysis of Embedded Capacity and
PSNR
In this subsection, we will select 19 8-bit grayscale
images with different sizes to display the
experimental results under a table. At the same time,
an objective assessment is carried out through:
analysis of the on-board capacity; the peak signal
noise ratio (PSNR) and safety analysis.
In order to measure the processed image quality,
we usually refer to PSNR value for objective
evaluation, MSE is the expected value of the square
of the difference between the original image and the
processed image, as is shown in the Eq. (5) and Eq.
(6).
Where the size of the image, m(i, j) pixel value
of the original image and img(i, j) pixel value of
the crypt image. The results obtained are collated in
Table 3:
Table 3. Crypt image MSE and PSNR
ID
Size
(ko)
MSE
PSNR
1
121
1.7835e+004
5.6521
2
768
1.2360e+004
7.2446
3
691
1.7683e+004
5.6892
4
1009
1.5032e+004
6.3947
5
432
9.7515e+003
8.2741
6
429
1.8634e+004
5.4619
7
64.8
1.2367e+004
7.2421
8
192
1.7170e+004
5.8171
9
192
3.2404e+004
3.0588
10
192
1.7170e+004
5.8170
11
13.1
1.2368e+004
7.2419
12
64.8
1.4267e+004
6.6214
13
132
1.1206e+004
7.6704
14
256
1.7725e+004
5.6790
15
512
4.3299e+004
1.8000
16
64.2
1.7542e+004
5.7239
17
256
1.7555e+004
5.7208
18
256
1.7972e+004
5.6189
19
512
4.0593e+004
2.0803
According to the results obtained for the values
in dB of the PSNR, these values are lower than
13dB, which shows that one obtained a high quality
of encryption and that one cannot make a
cryptanalysis by a small change in the clear image.
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3.6 Efficiency
In the proposed scheme, the bit level permutation is
performed in a linear time complexity. During this
time, the diffusion phase is also linear. If the
encrypted image is of size [n, m, p], the time
complexity of the algorithm is O(nm). The time
complexity of the algorithm in, [8] is also O(mn).
However, our bit-level pattern is slower than the
pixel-level pattern of, [8]. We can accompany the
permutation by a sorting operation. Thus, the
efficiency of the scheme was linked to the sorting
algorithm adopted.
The efficiency is represented in Figure 10 and
the comparison between the encryption time and the
decryption time is in Figure 12, this efficiency
depends on the size of the image in ko.
To know the histogram of time, we can see the
representation Figure 11.
Fig. 10: The polygon of time
Fig. 11: The histogram of time
Fig. 12: Comparison of time
Experience in an elliptic curve cryptographic
environment, we use an 80-bit security level,
processing on an Intel i7 3.07 GHz machine. The
time required for a task is as follows: 3.21ms.
In the chaotic environment, the time spent by
the same task is as follows: 4.5ms.
We prove that the proposed scheme is much
more profitable than the other existing schemes in,
[9], [10], [11] and [12], while these schemes are also
signature schemes.
The execution time of the encryption is equal to
that of the decryption by adding the construction
time of the secret encryption matrix to it.
4 Conclusion
In this article, the secret data is integrated into the
encrypted image by building a secret encryption
matrix. This scheme not only makes it possible to
realize that the secret data can be extracted in clear
text and in encrypted text, but also that the capacity
of integration is increased on the basis of the
guarantee that the original image is completely
restored. At the same time, the algorithm is used for
self-integration, which improves the PSNR value.
The main contributions of this document are
summarized as follows:
1. Improved integration capacity.
2. By homomorphic property, secret data can be
extracted in the field of plain text and encrypted
text.
3. At the same time, the diagram uses a block
strategy and the correlation within the block.
Acknowledgement:
The authors gratefully acknowledge the helpful
comments and suggestions of the reviewers.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Sara Chillali carried out the simulation and the
cosimulation and has implemented the Algorithm
in Matlab Simulink.
- Lahcen Oughdir was also responsible for this
Recherch.
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on SIGNAL PROCESSING
DOI: 10.37394/232014.2023.19.19
Sara Chillali, Lahcen Oughdir
E-ISSN: 2224-3488
191
Volume 19, 2023
