Digital Image Encryption using Laplace Transform and LFSR
Srushti Gandhi ^{1} , Ravi Gor ^{2},
^{1}^{ }Research
Scholar, Department of Mathematics, Gujarat University, Gujarat, India
^{2}^{ }Department
of Mathematics, Gujarat University, Gujarat, India

ABSTRACT 

In the world
of rapid evolution of exchanging digital data, data security is essential to
protect data from the unauthorized parities. With the broad use of digital
images of various fields, it is important to preserve the confidentiality of
image’s data from any unauthorized access. Cryptography is a technique that
assists in the development of such algorithms for security purpose. In this
paper, key is generated using a random number generator based on Linear Feedback
Shift Register (LFSR) and Laplace Transformation. 

Received 24 July 2022 Accepted 25 August 2022 Published 13 September 2022 Corresponding Author Srushti Gandhi, srushtigandhi@gujaratuniversity.ac.in DOI 10.29121/IJOEST.v6.i5.2022.390 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Laplace
Transform, LFSR, Image EncryptionDecryption 
1. INTRODUCTION
In the present world, technologies have been progressing rapidly. Primarily, most people prefer to use internet to transfer data from sender to receiver. There are several ways to transmit data such as email, message, whatsapp, and many more. In the present communicating world, images are used widely. Yet security and authenticity are the main issue for sending data through internet. Data security means protection of data from attackers and unauthorised parties/users. Encryption is one of the best techniques to secure data. Image encryption is a technique that converts original image into cipher image which is difficult to identify. No one can access image without knowing the decryption key. Image encryption process has applications in various fields like in corporate world, health care, military, multimedia etc.
Cryptography plays key role in the field of security. It is the battlefield for mathematicians and scientists. Cryptography consists of encryption and decryption process. Encryption is the process of converting plain text into cipher text. Decryption is the reverse process of encryption i.e., the process of converting cipher text into plain text. Several cryptographic algorithms have been proposed till date such as AES, DES, RSA, IDEA, etc.
Image encryption techniques are different from data encryption techniques. There are several security problems for digital image processing and transmission. So, it is necessary to maintain integrity and confidentiality of an image. Any single change in the pixel value does not change the entire image. Thus, digital images are less sensitive than data. A small manipulation or modification in digital image is acceptable as compared to text message. But it is more susceptible to decrypt by an attacker.
Various images are transmitted and stored in large amount over wireless network and internet. Thus, there is rapid development in multimedia and network technology. Digital image plays significant role in multimedia technology. Thus, it is important for the user to maintain privacy as well as security. To provide privacy and security to the users, digital image encryption and decryption process is important to protect from any unauthorised parties/user. Image, audio, and video encryption have applications in various fields like internet communications, multimedia, medical imaging, military etc.
2. LITERATURE REVIEW
Anandkumar (2015) proposed an image encryption algorithm that worked efficiently. It is very secure, with a prominent level of security and low computational requirements. The simulation results that the method had advantages based on their imageprocessing approaches. As a result, the algorithms are found to be effective for image encryption. It can provide security in open networks.^{}
Kapur et al. (2015) planned a simple and secure procedure to secure images. The image encryption procedure made use of two Pseudo Random Number generators. In the first step, Linear Feedback Shift Register algorithm was used to swap the rows of the original image. This was followed by the swapping of the columns. This produced an intermediary cipher image. In the second step, Blum Blum Shub algorithm was used to substitute the intensity of each pixel of the intermediary cipher image. This produced the final encrypted image.^{}
Mondal et al. (2016) proposed a highly secure encryption algorithm. They used permutationsubstitution architecture for encryption and decryption of an image. In the permutation step, image pixels of the plain image are shuffled using Linear Feedback Shift Register (LFSR). The output of this step is an intermediary cipher image which is of the same size as that of the plain image. In the substitution step, sequence of random numbers was generated using the RC4 key stream generator. It was XORed with the pixel value of the intermediary cipher image to produce the final cipher image. Experimental results and security analysis of the proposed scheme show that the proposed scheme is efficient and secure.^{}
Chepuri (2017) projected Laplace Transform algorithm for encrypting images. The Laplace Transform algorithm has been updated to work with RGB images. The results of the experiments showed that they were able to successfully encrypt and decrypt a variety of images, and that the technique has a decent encryption effect. When compared to the original image, the cipher image created by their technology was completely different. This method offers enhanced security and is appropriate for secure image transmission over the Internet.^{}
Jumaa (2018) solved the problem of secret key exchanging with the communicated parities by using a random number generator based on Linear Feedback Shift Register (LFSR). The random key generator was used to encrypt and decrypt the data using the Advance Encryption Standard (AES). They also encrypted and decrypted grayscale and colour RGB images. Three elements were important to the functionality of their proposed system in their paper: The first feature dealt with the obstetrics of creating a random and safe encryption key, the second with encrypting the plain or secret image using the AES technique, and the third with recovering the original image by decrypting the encrypted or cipher one.^{}
Devi et al. (2018) proposed a new medical image encryption algorithm. For image confusion, they applied a Henon map, and for diffusion, they used a Linear Feedback Shift Register (LFSR). The researchers looked at metric values and claimed that their system could withstand differential attacks. They show that their approach can ensure the security of DICOM images.^{}
Jain and Sharma (2019) presented a technique for digital image encryption which is enhancing its security by using the Laplace Transform Algorithm.^{}
3. TERMINOLOGIES
3.1. LAPLACE TRANSFORMATION^{ }Anandkumar (2015)
Laplace Transformation is a technique for solving differential equations. Here, differential equation of time domain formed is first transformed to algebraic equation of frequency domain. After solving the algebraic equation in frequency domain, the result is then finally transformed to time domain to achieve the ultimate solution of the differential equation. In other words, it can be said that the Laplace Transformation is nothing but a shortcut method of solving differential equation.
The Laplace Transforms is usually used to simplify a differential equation into a simple and solvable algebra problem. Even when the algebra becomes a little complex, it is still easier to solve than solving a differential equation.
Let be the function of time, for all ; then the Laplace Transform of can be defined as:
Provides that the integral exists. Where the Laplace Operator, will be real or complex .
Some of the Laplace Transformation
properties are:
1) Linearity:
Let ,
be constants. ,
be
the functions of time, ,
then
2) Change
of scale property:
If ,
then
;
frequency scaling
;
time scaling
3) Differentiation:
And many more.
3.2. LINEAR FEEDBACK SHIFT REGISTER (LFSR) Devi et al. (2018), ^{ }Srushti and Gor (2022)^{}
LFSR is a shift register whose input is liner function of its previous state. LFSR is built from simple shift register with a small number of XOR gates. Shift register is a type of digital circuit using a cascade of flipflops where the output of one flipflop is connected to the input of the next.
Initial value of LFSR is called seed. LFSR consists of clocked
storage elements (flipflops) and a feedback path. The number of storage
elements gives degree of LFSR. LFSR with flipflops is said to be of degree. As operation of register is
deterministic, the stream of values produced by register is determined by its current state. As register has finite number,
if possible, states it must eventually enter a repeating cycle. LFSR with
wellchosen feedback function can produce a sequence of bits that appears
random and has exceptionally long cycle. LFSRs are bits counter exhibiting pseudorandom
behaviour.
An mstage linear feedback shift register (LFSR) is characterized by feedback polynomial of degreem over , if the feedback polynomial is primitive the sequence of states generated is periodic and is of period .Here, . There are 255 possible initial states. Each initial state generates a periodical sequence of states of periodic the sequence of states of the period.The sequence generated with different initial states are shifted versions of each other.
Figure 1^{}
Figure 1 LFSR 
LFSRs are used in many key stream generators because:
·
LFSRs are well suited to hardware
implementation.
·
They can produce sequences with good
statistical properties.
·
They can produce sequences of large
period.
·
Because of their structure, they can be
rapidly analysed using algebraic techniques.
In the purposed scheme, sequences of 8 bit are used for generating key sequences. We denote this sequence as .
4. PROPOSED WORK
4.1. ENCRYPTION ALGORITHM
Step 1: An 8bit image of size pixels converted into a onedimensional array of pixel , where and Next, convert each unsigned pixel value ranging from 0 to 255 into a block of 8bit.
Step 2: Bit by bit XOR operation is employed between generated sequence which is generated by Laplace Transform and which is generated by LFSR using by to obtain final key sequence .
_{ }
Step 3: The binary image pixels are XORed with key sequence {Ki} to obtain encrypted pixel . This block of 8bit is intern converted into decimal digit 0 to 255.
_{}
Step 4: Repeat step 3 to encrypt all image pixels.
Transform all encrypted digits into
an array of size to
obtain the encrypted image.
4.2. DECRYPTION ALGORITHM
Step 1: Encrypted image of size pixels is transformed into one dimensional array of pixels , where and . Then convert each unsigned pixel values into a block of 8 bit.
Step 2: The key sequence is obtained by and as defined in (3) is used to decrypt the image. The obtained block of decrypted 8bit is converted into decimal digits and called as .
Step 3: One dimensional array of decrypted pixels is converted into an array of size to obtain the decrypted image.
5. EXAMPLE
Figure 2
Figure 2 Original Image 
Using PYTHON.
Image array: <PIL.JpegImagePlugin. JpegImageFile image mode=RGB size=576x720 at 0x182A11243D0>
Numpy array: <class 'numpy. ndarray'>
Image shape: (720, 576, 3)
Pillow image: <class 'PIL.Image.Image'>
Image mode: RGB
Image size: (576, 720)
Image array/in matrix form:
Total pixel value: 284832
1) Key generation using Laplace Transformation:
Using ,
the output is .
On considering the coefficient of the given output, binary of is .
This binary is used to generate key for LFSR.
2) Key generation using LFSR:
Using Laplace Transform key and a primitive polynomial ;
we have.
1.0011E+11 
0 
1.0011E+10 
1 
1.001E+12 
0 
1.001E+11 
1 
1.01E+12 
0 
1.01E+11 
1 
………. 

1.1101E+10 
1 
1.0011E+12 
(Repeating) 
Here, from clock, period will be repeating and will give pseudorandom sequence. So, 0011101100001 is considered as key .
5.1. FOR XOR OPERATION
5.1.1. ENCRYPTION ALGORITHM
Step 1: An 8bit grayscale image of size pixels converted into a onedimensional array of pixel . where and Next, convert each unsigned pixel value ranging from 0 to 255 into a block of 8bit.
A onedimensional array of pixel :
Step 2: Bit by bit XOR operation is employed between generated sequence which is generated by Laplace Transform and which is generated by LFSR using by to obtain final key sequence .
Input a binary number: .
The decimal value of the number is .
Step 3: The binary image pixels are XORed with key sequence {Ki} to obtain
encrypted pixel . This block of 16bit is intern converted into
decimal digit 0 to 255.
Step 4: Repeat step 3 to encrypt all image pixels.
Transform all encrypted digits into an array of size to
obtain the encrypted image.
Figure 3
Unit 16
Figure 3 Encrypted XOR Image 
For encrypted image:
Image array: <PIL.PngImagePlugin.
PngImageFile image mode=I size=1728x720 at
0x1829FEEFE50>
Numpy
array: <class 'numpy.ndarray'>
Image shape: (720, 1728)
Pillow image: <class 'PIL.Image.Image'>
Image mode: I
Image size: (1728, 720)
Image array/in matrix form:
Total pixel value: 3732480
5.1.2. DECRYPTION ALGORITHM
Step 1: Encrypted grayscale image of size pixels is transformed into one dimensional
array of pixels ,where and .
Then convert each unsignd pixel values into a block of 16 bit.
Step 2: The key sequence is obtained by and as defined in (3) is used to decrypt the
image. The obtained block of decrypted 8bit is converted into decimal digits and called as
.
Step 3: One dimensional array of decrypted pixels is converted into an array of size to obtain the decrypted image.
Figure 4
Figure 4 Decrypted XOR image 
5.2. XNOR OPERATION
5.2.1. ENCRYPTION ALGORITHM
Step 1: An 8bit grayscale image of size pixels converted into a onedimensional array of pixel . where and Next, convert each unsigned pixel value ranging from 0 to 255 into a block of 8bit.
A onedimensional array of pixel :
Step 2: Bit by bit XOR operation is employed between generated sequence which is generated by Laplace Transform and which is generated by LFSR using by to obtain final key sequence .
Input a binary number: .
The decimal value of the number is .
Step 3: The binary image pixels are XORed with key sequence {Ki} to obtain
encrypted pixel . This
block of 16bit is intern converted into decimal digit 0 to 255.
Step 4: Repeat step 3 to encrypt all image pixels.
Transform all encrypted digits into an array of size to
obtain the encrypted image. Unit 8
Figure 5
Figure 5 Encrypted XNOR Image 
For encrypted image:
Image array: <PIL.PngImagePlugin.
PngImageFile image mode=I size=1728x720 at
0x1829FEEFE50>
Numpy
array: <class 'numpy.ndarray'>
Image shape: (720, 1728)
Pillow image: <class 'PIL.Image.Image'>
Image mode: I
Image size: (1728, 720)
Image array/in matrix form:
Total pixel value: 3732480
5.2.2. DECRYPTION ALGORITHM
Step 1: Encrypted grayscale image of size pixels is transformed into one dimensional
array of pixels ,where and .
Then convert each unsigned pixel values into a block of 16 bit.
Step 2: The key sequence is obtained by and as defined in (3) is used to decrypt the
image. The obtained block of decrypted 8bit is converted into decimal digits and called as
.
Step 3: One dimensional array of decrypted pixels is converted into an array of size to obtain the decrypted image.
Figure 6
Figure 6 Decrypted XNOR image 
6. RESULTS AND DISCUSSION
6.1. Visual Testing
The visual testing is done online on https://www.textcompare.org/image/.
Figure 7
Figure 7 Visual Testing 
Comparing original and encrypted image, similarity between
them is shown in the below table. White dotes show the similar pixel values of
original image and encrypted image.
By comparing original image with encrypted image and
decrypted image, the difference is
shown in below table. The white dot indicates the similar pixel value of
original image and encrypted image. The difference is found with full
transparency when less ignored along with original size and
movement with different intensity.
Figure
8
Figure 8 Difference between original and encrypted XOR image 
Figure
9
Figure 9 Difference between original and decrypted XOR image 
Image 
Difference 
Original
and Encrypted XOR image 
98.12
% 
Original
and Decrypted XOR image 
0.18
% 
Figure 10
Figure 10 Difference between original and encrypted XNOR image 
Figure11
Figure 11 Difference between original and decrypted XNOR image 
Image 
Difference 
Original and Encrypted XNOR
image 
98.59 % 
Original
and Decrypted XNOR image 
0.18 % 
Image 
Difference 
Original
and Encrypted image 
94.35
% 
Original
and Decrypted image 
0.18
% 
By comparing original image with encrypted image and
decrypted image, the difference is shown
in below table. The white dot indicates the similar pixel value of original
image and encrypted image. The difference is found with full transparency when
less ignored along with scale to same size and movement with
different intensity.
Figure 12
Figure 12 Difference between original and encrypted image 
Figure 13
Figure 13 Difference between original and decrypted image 
Figure 14
Figure 14 Difference between original and encrypted XNOR image 
Figure 15
Figure 15 Difference between original and decrypted XNOR image 
Image 
Difference 
Original and Encrypted XNOR image 
91.36 % 
Original and Decrypted
XNOR image 
0.18 % 
6.2. SENSITIVITY ANALYSIS^{ }Srushti and Gor (2022)^{}
Image quality and vision outcomes were generated because of the experiment. All the analysis are performed with the use of PYTHON. The following parameters are used to evaluate image quality:
6.2.1. NUMBER OF PIXELS CHANGE RATE (NPCR)
When the difference between two encrypted images is negligible, NPCRs are used to verify the number of changing pixels between them. The NPCR can be mathematically defined as follows:
NPCR=
Where
is the weight and height of the encrypted
interferogram,
is the interferogram encrypted before pixel
change,
is the interferogram encrypted after pixel
change,
is the bipolar network
The optimal NPCR value is:
Image 
NPCR 
Encrypted XOR
image 
100% 
Decrypted XOR image 
0% 
Encrypted XNOR
image 
100% 
Decrypted XNOR image 
0
% 
6.2.2. MEAN SQUARED ERROR (MSE) AND PEAK SIGNAL TO NOISE RATIO (PSNR)
The PSNR block analyses the peak signaltonoise ratio between two images in decibels. The PSNR ratio is used to compare the quality of the original and encrypted images. The better the quality of the compressed or reconstructed image, the higher the PSNR.
To compare image compression quality, the mean square error (MSE) and peak signaltonoise ratio (PSNR) are evaluated. The MSE is a measure of the peak error between the encrypted and original image, whereas the PSNR is a measure of the cumulative squared error.
The smaller the MSE value, the smaller the error.
The PSNR is calculated by first calculating the meansquared error (MSE) using the equation:
MSE
PSNR can be calculated as:
PSNR
The optimal MSE and PSNE values are:
Image 
MSE 
PSNR 
Encrypted XOR
image 
4096 
4096 dB 
Decrypted XOR image 
4096 
4096 dB 
Encrypted XNOR
image 
0 
0 0 dB 
Decrypted XNOR image 
0 
00 0 dB 
6.2.3. UNIFIED AVERAGE CHANGING INTENSITY (UACI)
UACI is used to calculate the average intensity of the difference between the two encrypted images ( and ). It is used to determine the strength of an encryption scheme. Its quality is determined by the format and size of the image. The average intensity variation between the ciphered and original images is measured using UACI. The highest UACI suggests that the recommended technique is resistant enough to a variety of attacks.
For an image of size , UACI is calculated as follows:
UACI
On comparing the original image with encrypted image and decrypted image, the optimum UACI value is:
Image 
UACI 
Encrypted XOR
image 
40.96% 
Decrypted XOR image 
40.96% 
Encrypted XNOR
image 
0% 
Decrypted XNOR image 
0
% 
6.2.4.
ENTROPY ANALYSIS
Information entropy is the degree of the uncertainty associated with a random event. It tells us the amount of information present in the event. It increases with uncertainty or randomness. It finds its application in various fields such as statistical inference, lossless data compression and cryptography. The entropy of m can be calculated as:
where is the total number of symbols.
and is the probability of symbol .
In case of the original digital image, should theoretically be equal to as there are 256 values of the information
source in red, green, blue, and green colours of the image with the same
probability.
Entropy values are:
Image 
Entropy 
Original image 
8 
Encrypted XOR image 
7.99 
Decrypted XOR image 
7.99 
Encrypted XNOR image 
7.55 
Decrypted
XNOR image 
7.55 
6.2.5. TIME TAKEN FOR ENCRYPTION AND DECRYPTION OF AN IMAGE
With XOR operation: The time taken to encrypt the image is 0.0 sec and time taken to decrypt the image is 3.9375 sec.
With XNOR operation: The time taken to encrypt the image is 0.0 sec and time taken to decrypt the image is 2.828125
sec.
6.2.6. COMPARISON^{ }Srushti and Gor (2022)^{}^{}
The table shows the comparison of the results of encryption process done using Laplace transformation with LFSR and RSA with LFSR:
Key
Generation 
Laplace
Transform & LFSR 
RSA
& LFSR 

Key Operation 
With XOR operation 
With XNOR operation 
With XOR operation 
Image type 
Input image: .jpg Encrypted image: .png Decrypted Image: .png 
Input image: .jpg Encrypted image: .png Decrypted Image: .png 
Input image: .jpg Encrypted image: .png Decrypted Image:.png 
Visual Testing 
98.59 % 
91.36% 
99.78% 
NPCR 
100 % 
100 % 
100 % 
MSE 
4096 
0 

PSNR 
40.96 dB 
0 dB 

UACI 
40.96 % 
0 % 
78.12 % 
Entropy 
7.99 
7.55 

Time taken for encryption and decryption process 
0.0 sec & 3.9375 sec 
0.0 sec & 2.828125sec 
0.0 sec & 3.796875 sec 
Strength 
Laplace Transformation produces a disordered key. LFSR produces random and secure key. 
RSA algorithm produces a strong public key. LFSR produces random and secure key. 

Limitation 
Randomness of the key depends on which is to be chosen for Laplace Transformation. Key generation by LFSR depends on the primitive polynomial. 
Strength of algorithm depends on the selection of the prime numbers. Key generation by LFSR depends on the primitive polynomial. 
Here, NPCR, MSE, PSNR and entropy results gives better result for encryption and decryption using XOR between Laplace Transform & LFSR.
NPCR and time taken for encryption and decryption process gives better results for encryption and decryption using XNOR between Laplace Transform & LFSR.
Visual testing, NPCR and UACI gives better results for encryption and decryption using XOR between RSA & LFSR.
Thus, using XOR between Laplace Transform & LFSR gives better results for encryption and decryption process while XNOR between Laplace Transform & LFSR gives the worst results.
7. CONCLUSION
The key utilized for image encryption and
decryption in this paper is produced using a Laplace transform and Linear
Feedback Shift Register (LFSR). The suggested approach is extremely sensitive
to the LFSR's initial state. A Laplace transform generates the first key, then
LFSR uses the first key to generate the second key. Then both keys are XORed and
XNORed together to produce a strong key, which is the
final key. As a result, the hacker will have a tough time guessing that key. As
a result, if the wrong key is used, the image will be radically different.
The results show original and encrypted image is
highly uncorrelated and perceptually different. Laplace transform uses more
controlled parameters as compared to another algorithms, which enhances the
data security. The result has been proven using PYTHON that it is efficient and
secure.
Compression between RSA with LFSR is done that shows that using XOR between Laplace Transform & LFSR gives better results for encryption and decryption process.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
REFERENCES
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Jain, A., & Sharma, S. (2019). A Novel Digital Image Encryption Method Based on Laplace Transform Algorithm. International Journal of Electronics Engineering.
Jumaa, N. K. (2018). Digital Image Encryption Using AES and Random Number Generator. Iraqi Journal of Electrical and Electronic Engineering, 14(1).
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Mondal, B., Sinha, N., & Mandal, T. (2016). A Secure Image Encryption Algorithm Using lfsr and rc4 Key Stream Generator. In Proceedings of 3rd International Conference of Advanced Computing, Networking and Informatics. Springer, New Delhi, 227237. https://doi.org/10.1007/9788132225386_24.
Srushti, G., and Gor, R. (2022). Digital Image Encryption using RSA and LFSR. International Journal of Engineering Science Technologies, 6(4), 116. https://doi.org/10.29121/IJOEST.v6.i4.2022.351.
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