Cryptography is the science of using mathematics to encrypt and decrypt data. Cryptography includes two phases: Encryption and Decryption. Encryption is the process of transforming plaintext to ciphertext, whereas decryption is the reverse procedure. Encryption and decryption schemes based on Sumudu Transform are unable to give more security while communicating the information. ElGamal is a public key algorithm that is based on the discrete logarithm problem. The purpose of this study is to introduce a cryptographic method that uses the ElGamal algorithm and Sumudu Transform to improve communication security.
Cryptography is a method of protecting data. In cryptography, the procedures used to protect information are based on mathematical principles and a set of rule-based calculations known as algorithms. Cryptography consists of two components: encryption and decryption. Encryption is the process of converting normal data into an unreadable format and decryption is the act of recovering the unreadable data. Cryptography is classified into three types:
In symmetric key cryptography, the same key is used for encryption and decryption. It is fast and efficient but the drawback is that the sender and receiver must exchange the keys in secure manner. DES, AES, IDEA, RC4, Blowfish, Twofish are some Symmetric key algorithms.
Asymmetric key cryptography also known as public key cryptography that uses two different keys: a public key for encryption and private key for decryption. RSA, DSA, ElGamal, Rabin, ECC are some Asymmetric key algorithms.
II. LITERATURE REVIEW
ElGamal  (1985) introduced a method of public key cryptosystem and signature scheme based on discrete logarithms. The security of both systems relies on the difficulty of computing discrete logarithms over finite fields.
Watugala  (1993) introduced Sumudu Transform to show interesting properties which makes it easy to visualize. Thus, it is an ideal transform for control engineers and applied mathematicians.
Asiru  (2002) discussed the general properties of the Sumudu Transform and some special functions that occur frequently in physical and engineering applications.
Allen  (2008) discussed the implementation of several attacks on plain ElGamal encryption and discussed attacks which rely on the underlying mathematics.
Bodkhe and Panchal  (2015) introduced a new cryptographic application using Sumudu transform and private key. It is very difficult to find the private key by any other attack. After producing key, they use this key for encryption and decryption that algorithm based on Sumudu transformation and modular arithmetic.
Grewal  (2015) discussed ElGamal System which is a public key cryptosystem based on the discrete logarithm problem. He examined its security, advantages, disadvantages and its applications.
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.
Tuncay  (2017) analyzed security based on Sumudu Transform in cryptography and concluded that without knowing the key, the encrypted text can be decrypted.
Dissanayake  (2018) studied an improvement of the basic ElGamal public key cryptosystem. The public key of the ElGamal system is not changed in this method. But, the sending structure of message and the decryption process are changed. The ElGamal cryptosystem is not secure under adaptive Chosen Ciphertext Attack (CCA). This improved cryptosystem is immune against Chosen Plaintext Attack (CPA) and Chosen Ciphertext Attack (CCA) attacks. Therefore, this improved system is very suitable for small messages or key exchanges.
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.
Ranasinghe and Athukorala  (2020) discussed generalization of the ElGamal public key cryptosystem. They presented a generalization to the original ElGamal system which also relies on the discrete logarithm problem. The encryption process of the scheme is improved such that it depends on the prime factorization of the plaintext. If the plaintext consists of only one distinct prime factor the new method is similar to that of the basic ElGamal algorithm. The proposed system preserves the immunity against the Chosen Plaintext Attack (CPA).
Nagalakshmi et. al.  (2020) proposed an implementation of ElGamal scheme for Laplace transform cryptosystem. The time analysis is compared with existing algorithms and comparison reveals that the proposed cryptosystem enhances the data security.
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 improve security of communication. This paper provided frequency test and statistical analysis on the proposed method.
Thakkar and Gor  (2022) developed 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) developed 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 work.
Thakkar and Gor  (2023) developed a cryptographic method using RSA algorithm and Mellin Transform to improve security of communication. This paper provided frequency test and statistical analysis on the proposed method.
Thakkar and Gor  (2023) developed a cryptographic method using the RSA algorithm and Sumudu Transform to improve security of communication. The frequency test and statistical analysis on the proposed method are provided in this work.
III. PROPOSED ALGORITHM OF THE MATHEMATICAL MODEL
The proposed method is ElGamal algorithm with application of Sumudu Transform (ElGamal-ST). 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.
Cryptography is one of the most important fundamental tools to provide security to data communication. An application of Sumudu Transform for cryptographic process is a weak scheme because encrypted data can be decrypted by elementary modular arithmetic. ElGamal is a widely used public key cryptosystem that is based on the difficulty of computing discrete logarithms over finite fields. The proposed work expands on innovative method using ElGamal algorithm with application of Sumudu Transform. It is impossible to break this method without knowing the private key. Therefore, this proposed method ElGamal-ST can provide more security of communication.
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