CRYPTOGRAPHIC METHOD TO ENHANCE
DATA SECURITY USING RSA ALGORITHM AND MELLIN TRANSFORM
Akash Thakkar ^{1}, Ravi Gor ^{2}
^{1}^{ }Research
scholar, Department of Applied Mathematical Science, Actuarial Science and
Analytics, Gujarat University, India
^{2}^{ }Department
of Applied Mathematical Science, Actuarial Science and Analytics, Gujarat
University, India

ABSTRACT 

Cryptography
is the technique of using mathematical algorithms to encrypt and decrypt the information. The process of converting plaintext to
ciphertext is known as encryption, whereas the process of converting
ciphertext to plaintext is known as decryption. Encryption and decryption
methods based on Mellin Transform are unable to
provide more security while transmitting the information. RSA algorithm is an
Asymmetric key cryptography algorithm. The purpose of this study is to
present a cryptographic method that uses the RSA algorithm and Mellin Transform to improve communication security. 

Received 11 March 2023 Accepted 10 April 2023 Published 26 April 2023 Corresponding Author Akash
Thakkar, akashthakkar@gujaratuniversity.ac.in DOI 10.29121/IJOEST.v7.i2.2023.490 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2023 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: Cryptography,
Encryption, Decryption, RSA Algorithm, Mellin
Transform 
1. INTRODUCTION Thakkar and Gor (2022), Thakkar and Gor (2022)
Cryptography is one of the most utilized techniques for data security. The techniques used to protect data in cryptography are based on mathematical concepts and a set of rulebased calculations known as algorithms. Two crucial cryptographic functions are encryption and decryption. Normal data is transformed into an unreadable form through encryption, and the reverse is accomplished through decryption. There are three main categories of cryptography:
· Symmetric key cryptography (secret key cryptography)
· Asymmetric key cryptography (public key cryptography)
· Hash Function
Symmetric key cryptography is a type of encryption where the same
key is used to both encrypt and decrypt the data. Symmetric key cryptography is
quick and easy, but it has the drawback that the sender and receiver must
securely exchange keys. DES, AES, IDEA, RC4, Blowfish, Twofish
are some Symmetric key algorithms.
Asymmetric key cryptography is also known as publickey
cryptography. A pair of keys is used to encrypt and decrypt the data in
Asymmetric key cryptography. Data is encrypted with the public key and
decrypted with the corresponding private key. RSA, DSA, ElGamal,
Rabin, ECC are some Asymmetric key algorithms.
1) RSA
ALGORITHM Rivest (1978)^{},^{
} Thakkar and
Gor (2021)
RSA is public key cryptosystem developed by Rivest R.,
Shamir A., Adleman L. in 1978. RSA algorithm is widely used for secure data
transmission. There
are mainly three phases in RSA algorithm.
1)
Key Generation
2)
Encryption algorithm
3)
Decryption algorithm
1)
key Generation
RSA involves two keys: public key and private key. Public key is used
for encryption and private key is used for decryption of data.
1)
Choose two prime numbers and
2)
Find such that
3)
Find the Phi of,
4)
Choose ansuch that and such that and share no divisors other than 1.
5)
Determine such that
Public Key: and Private Key:
2)
Encryption algorithm
The process of converting Plain Text into Cipher Text is called as
Encryption process.
3)
Decryption algorithm
The process of converting Cipher Text into Plain Text is called as
Decryption process.
Some integral transforms contribute to the process of
cryptography. The features of integral transforms are used to create encryption
and decryption methods.
2) MELLIN
TRANSFORM (MT) Johar (2019),
Santana (2014)
The Mellin Transform is an integral
transform named after mathematician Hjalmar Mellin
(18541933). The Mellin Transform is extremely useful
for certain applications including solving Laplace equations.
Let a function defined for all positive values of, then the Mellin Transform of is defined by
Properties
of Mellin Transform:
1) Scaling:
2) Inverse of independent variable:
3) Multiplication by Power of .
4) Derivative:
5) Where:
6) Convolution:
2. LITERATURE REVIEW
Rivest
et al. (1978) introduced a method
namely RSA to encrypt and decrypt the data. The RSA algorithm is the most widely
used public key cryptography algorithm. One of the reason RSA has become most
widely used is because it has two keys, one is for encryption and other one is
for decryption. Thus, it is promising confidentiality, integrity, authenticity and nonrepudiation of data.
Milanov (2009) concluded that RSA is a strong encryption algorithm that has stood a partial test of time. RSA is a public key cryptosystem that enables secure communications and digital signatures. Its security is based in part on the difficulty of factoring large numbers.
Malhotra & Singh (2013) studied various cryptographic algorithms. They provided a study of the research work done in cryptography field and various cryptographic algorithms being used. It is recapitulated that RSA is being used widely. This paper presented the current scenario and can provide a direction to naive users.
Santana
(2014) developed a scheme in cryptography whose construction is based on the
application of Mellin Transform.
Lone and Uddin (2016) studied common attacks on RSA and its variants with possible countermeasures.^{}
Nisha and Farik (2017) reviewed RSA public key cryptography algorithm. They examined its strengths and weaknesses and propose novel solutions to overcome the weakness.
Tayal
et al. (2017) provided an overview of network security and various techniques for
improving network security. They demonstrated various schemes used in
cryptography for network security purposes.
Mohammadi et al. (2018) compared two public key
cryptosystems. They focused on the efficient implementation and analysis of the
two most popular algorithms for key generation, encryption, and decryption
schemes of RSA and ElGamal. RSA is based on the
difficulty of prime factorization of a very large number and the ElGamal algorithms hardness is essentially equivalent to
the difficulty of finding discrete logarithm modulo a large prime number. These
two systems are compared in terms of various parameters such as performance, security,
and speed. They concluded that RSA is more efficient for encryption than ElGamal and RSA is less efficient for decryption than ElGamal.
Johar (2019) presented basic
introduction of Mellin Transform and its examples.
Mok
and Chuah (2019) studied brute force attack on RSA cryptosystem. They concluded that
prime factorization attack is the most efficient way on RSA cryptanalysis.
Nagalakshmi et al. (2019) provided the conditions for the RSA Cryptosystem based on the Laplace
transform techniques. The proposed algorithm was implemented using a highlevel
programme, and its time complexity was tested using RSA cryptosystem
algorithms. The comparison shows that the proposed algorithm improves data
security when compared to RSA cryptosystem algorithms and the use of the
Laplace transform in cryptosystem schemes.
Thakkar and Gor
(2021) represented a review of literature concerned
with cryptographic algorithms and mathematical transformations. The review of
RSA and ElGamal algorithms aids readers in better
understanding the differences between the two asymmetric key cryptographic
algorithms and how they work and review of
mathematical transformations helps the reader to understand how mathematical
transformations are used in cryptography.
Thakkar and Gor
(2022) developed a cryptographic method using RSA algorithm and Kamal
Transform to enhance communication security. This paper provided frequency test
and statistical analysis on the proposed method.
Thakkar and Gor
(2022) presented a cryptographic method using ElGamal
algorithm and Kamal Transform to improve security of communication. The
frequency test and statistical analysis on the proposed method are provided in
this work.
Thakkar and Gor (2022) provided a
cryptographic method using the ElGamal algorithm and Mellin Transform to improve security of communication. The
frequency test and statistical analysis on the proposed method are provided in
this paper.
3. PROPOSED ALGORITHM OF THE MATHEMATICAL MODEL
The proposed method is RSA algorithm with application of Mellin Transform (RSAMT). The proposed work is to improve
security of communication. When two people want to transfer the data, they will
follow the given steps for encryption and decryption. The following method
provides an overview of the proposed cryptographic scheme.
1) Method
of Key Generation
Following are the steps involved in Key Generation.
Step 1: Generate four large random prime numbers
Step 2:
Calculate and
Step 3: Select
the public exponent , such that
Step 4: Find
the secret exponent , such that
Step 5: Generate
polynomial using public exponent . i.e.,
2) Method
of Encryption
Following are the steps involved in Encryption.
Step 1: Select the plain text , convert into ASCII code integer
Step 2: Calculate
Step 3: Generating
function from that we get
Step 4: Take
Mellin Transform of a function. i.e.,
Step 5: Choose
and get
Step 6: Find
such that
Step 7: Find
such that and (value of )
Step 8: Calculate
cipher text then get integer of cipher text
Step 9: Each
integer of cipher text is converted to its construct by ASCII
character are stored as the cipher text
Following are the steps involved in Decryption.
Step 1: Consider the Cipher text and key
received from the sender
Step 2: Cipher text converted to ASCII values of
Step 3: Each
integer of is converted into and get
Step 4: Calculate
and get
Step 5: Find
the polynomial assuming as a coefficient
Step 6: Apply
inverse Mellin Transform. i.e.,
and get integer
Step 7: Each integer are converted to their corresponding ASCII
code values and hence get the original plain text
Public
key:
Private
key:
4. NUMERICAL EXAMPLE
This section contains an example of an encryption and decryption method. Note that, the parameters are chosen to make computation easier, however they are not in the useable range for secure transmission.
If Alice (sender) wants to send an encrypted message to Bob (receiver).
Bob first computes his parameters using steps as given in method of Key Generation.
Step 1: Primes
Step
2: and
Step
3: , such that
Step
4: , such that
Step
5: Polynomial
using public exponent
i.e.,
Bob then sends his public key to Alice.
Alice computes his parameters to encrypt the message using steps as given in method of Encryption.
Step
1: Plain text =
“ M@th ”, ,
convert into ASCII code integer
Step
2:
Step
3:
we get,
Step
4: Take
Mellin Transform of a function. i.e.,
Step
5: Choose and
we get,
Step
6: Find such that ,
we get,
Step
7: Find
such that and (value of )
we get,
Step
8: Calculate
cipher text ,
we get,
Step
9: Each
integer of cipher text is converted to its construct by ASCII character 㬊 ᒌ ɬ ⠑ and stored as the cipher text “ 㬊ᒌɬ⠑ ”
Alice
then sends (, cipher text ) to Bob.
Bob decrypts the cipher text using steps as given in method
of Decryption.
Step 1: Consider
the Cipher text and key received from the sender
Step
2: Cipher text “ 㬊ᒌɬ⠑ ”
converted to ASCII values of
Step
3: Each
integer of is converted into
,
we get,
Step
4: Calculate
we have,
we get,
Step
5: The
polynomial assuming
as a coefficient
Step
6: Apply
inverse Mellin Transform
and get and we have (as value of )
From get integer
Step
7: Each integer are converted to them
corresponding ASCII code values and hence get the
original plain text = “ M@th ”
5. TESTING AND ANALYSIS Thakkar and Gor (2022)^{},^{ }^{ }Thakkar and Gor (2022)^{}
The statistical analysis and frequency testing for this proposed method are presented. The graph of RSA algorithm and proposed method RSAMT is shown here and also compared with each other. In statistical analysis, correlation coefficients are obtained for RSA, MT, and the proposed RSAMT.
1) Frequency
Test
Figure 1 show that the frequency of the same character in plaintext after encryption with RSA algorithm is the same, where the xaxis and yaxis represent plaintext and frequency level of ciphertext, respectively.
Figure 1
Figure 1 RSA Algorithm Ciphertext Frequency Distribution 
Figure 2 demonstrate that the frequency of each character in a plaintext change after encryption with the proposed method RSAMT, where the xaxis and yaxis represent plaintext and frequency level of ciphertext, respectively.
Figure 2
Figure 2 The Proposed Algorithm Ciphertext Frequency Distribution 
Figure 3 show that graphical representation of the frequency distribution shown in Figure 1 and Figure 2 for each algorithm.
Figure 3
Figure 3 Ciphertext Frequency Distribution of RSA And RSAMT 
According to the frequency test, the proposed method RSAMT has a different frequency for each repeated character in a plaintext after encryption.
2) Statistical
Analysis
In statistics, correlation coefficients are used to assess how closely two variables are related. The aim of the proposed method of research is to examine and create an algorithm that strongly resists cryptographic attacks. The correlation shows the relationship between two values. The correlation coefficient between plaintext and ciphertext are examined. Plaintext and ciphertext are identical if the correlation coefficient is one. Plaintext and ciphertext are completely different if the correlation coefficient is near to zero. If the correlation coefficient is less than one, ciphertext is the inverse of plaintext. As a result, encryption success is associated with lower correlation coefficient values. Table shows the experimental finding and the correlation coefficient value of the proposed encryption method.
Table 1
Table 1 The Correlation Test from Plaintext to Ciphertext 

Message 
Algorithm 
Correlation 
Applied 
RSA 
0.19820314 

MT 
0.67223518 

RSAMT 
0.05420039 
CryPto 
RSA 
0.67704516 

MT 
0.61845471 

RSAMT 
0.29986904 
M@th 
RSA 
0.91830612 

MT 
0.09195426 

RSAMT 
0.40369100 
According to the correlation test, proposed method RSAMT is more effective than RSA and MT. Correlation coefficient values are closer to zero with the proposed method RSAMT. However, for specific types of data (messages), RSA or MT may outperform RSAMT. Such circumstances and conditions under which performance can be generalized are a research path to pursue.
6. CONCLUSION
One of the most crucial foundational technologies for ensuring the security of data communication is cryptography. An application of Mellin Transform for cryptography is a weak strategy since encrypted data can be deciphered using simple modular arithmetic. RSA is most widely used technique for keeping data secret. The primary method of breaking RSA still depends on how quickly huge prime numbers can be factored. The proposed research based on innovative approach that combines RSA algorithm with Mellin Transform of function to generate four huge prime numbers. Without having access to the private key, it is challenging to crack this method. As a result, the proposed method combining RSA algorithm and Mellin Transform has the potential to improve communication security.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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