Secure transaction in e-commerce refers to the safe and secure exchange of information and money between buyers and sellers in an online marketplace. E-commerce has revolutionized the way people buy and sell goods and services, making it easy for customers to shop from anywhere in the world, at any time of the day. The convenience of online shopping has also led to the need for secure transactions to protect both buyers and sellers from online threats and fraud. However, with the growth of e-commerce, there have also been concerns about the security of online transactions. Here are some of the most common security issues of e-commerce:

• Payment Security: One of the biggest concerns for consumers when shopping online is the security of their payment information. Cybercriminals may intercept and steal sensitive data such as credit card numbers, names, and addresses. To prevent this, it’s important for e-commerce websites to have strong encryption protocols to protect customer data.
• Data Privacy: Customers share a lot of personal information when they make an online purchase. This data may include names, addresses, phone numbers, and email addresses. If this data falls into the wrong hands, it can be used for identity theft or other criminal activities. Businesses must ensure that they are handling this data securely, with proper encryption, storage, and access controls.
• Phishing and Malware Attacks: Cybercriminals often use phishing and malware attacks to steal sensitive information from customers. Phishing attacks involve sending fake emails or websites that appear to be legitimate to trick customers into sharing their personal information. Malware attacks involve installing malicious software on a customer’s computer to steal data. E-commerce businesses should be vigilant in monitoring for these attacks and should have strong anti-malware and anti-phishing measures in place.
• Website Security: The security of e-commerce websites is also critical to protect against hacking and data breaches. Businesses should ensure that their websites are secure with SSL/TLS encryption, firewalls, and other security measures. They should also monitor for suspicious activity, such as multiple failed login attempts.

Secure transactions in e-commerce are crucial to maintaining the trust of customers and ensuring the safety and security of online transactions. E-commerce platforms must employ various security measures to protect the sensitive information of buyers and sellers and prevent fraudulent activities. They are encryption, authentication, and secure payment systems.

Encryption ensures that sensitive information such as credit card details, passwords, and personal data are securely transmitted over the internet, making it difficult for hackers to intercept or steal such information. Authentication involves verifying the identity of users, ensuring that only authorized individuals have access to sensitive information. Secure payment systems ensure that the payment information is transmitted securely, preventing unauthorized access and fraudulent activities. This involves the use of secure payment gateways, which encrypt and process the payment information, ensuring that the transaction is secure and protected [1].

Encryption is the process of converting plaintext into a coded form, making it unreadable to unauthorized users. Public key cryptography, such as the RSA (Rivest, Shamir, and Adleman) and Rabin cryptosystems, are widely used encryption techniques that ensure electronic commerce transactions’ confidentiality, integrity, and authenticity.

RSA cryptography is the oldest, most used, and most efficient of the various public-key cryptosystems, developed by Rivest et al. [2] in 1978. Rivest et al. [3] first proposed the problem of factorization in the year 1978. However, RSA’s security [4] cannot be guaranteed theoretically; it is a slow algorithm that can only encrypt a small amount of data simultaneously. There have been several attempts to overcome the limitations of the RSA algorithm. Michael O. Rabin [5] made one such attempt in 1979. To increase the speed of the encryption of RSA, he proposed a variant of RSA, later known as Rabin’s cryptography. Rabin is essentially RSA with the optimal choice of public key exponent (e)public key exponent (e), where encryption uses integer two as the public key exponent, which takes a shorter computation time for encryption. This feature makes the Rabin cryptosystem relatively faster in encryption than Standard RSA. Rabin algorithm makes use of two keys like RSA. Here, the public key is a common modulus (n), and the private keys are the prime factors used to compute n. Hence, the security of the Rabin algorithm entirely depends on n. In cryptanalysis, determining the factors of common modulus n plays a vital role. If someone breaks the factor, obtaining the message becomes an easy task. Using the Rabin cryptosystem, getting plaintext back from the cipher text is considered as hard as factoring. Because of this feature, Rabin cryptosystem is used in numerous research applications [6,7,8,9]. Rabin cryptography can secure e-commerce transactions by encrypting sensitive information using the public key and decrypting it using the private key.

Here is an example of a secure transaction using Rabin cryptography:

• Alice wants to purchase a book from an online store.
• The online store has a publicly available public key.
• Alice uses Rabin encryption to encrypt her credit card information and other personal data using the online store’s public key. This generates the ciphertext.
• Alice sends the ciphertext to the online store.
• The online store receives the ciphertext and uses its private key to decrypt the message.
• The online store processes the transaction and sends a confirmation message to Alice.
• The confirmation message is encrypted using Alice’s public key.
• Alice receives the encrypted confirmation message and uses her private key to decrypt it.

In this example, Rabin cryptography ensures that Alice’s credit card information and personal data are secure during the transaction. The online store’s public key encrypts Alice’s information, and only the online store’s private key can decrypt the ciphertext. Similarly, Alice’s public key encrypts the confirmation message, and only Alice’s private key can decrypt the message. This provides a secure way for Alice and the online store to exchange information without the risk of unauthorized access or interception. It’s important to note that Rabin cryptography are susceptible to brute force attacks and side-channel attacks. Therefore, using a secure implementation of these algorithms and keeping the private keys secure is essential. To improve the security of the Rabin cryptosystem, researchers contributed several ideas to make the Rabin cryptosystem strong. 

Williams [5] uses unique prime numbers to make the system more efficient using the quadratic residue theory and the Jacobi symbol in the decryption. This leads to obtaining a proper message back out of four decrypted values. However, this technique results in Poor performance due to the involvement of the Jacobi symbol computation in the encryption and decryption process, causing increased computational complexity and the need for extra bits, which increases cipher text overhead. The work proposed in [10] optimized the Rabin cryptosystem by using reciprocal numbers to solve Rabin’s 4-to-1 situation in decryption in 1999. In this method, Encryptor calculates and sends two additional bits of information with its ciphertext to indicate the proper square root. However, it still requires more computational costs since it uses the Jacobian symbol for encryption and decryption. Lynn Margaret Batten and Hugh Cowie Williams [11] introduced a unique scheme known as the ‘R-W signature scheme,’ which is considered the most efficient decryption method compared to existing methods. This scheme uses the concept of the Chinese Reminder Theorem (CRT) to obtain the correct plaintext back out of 4 outcomes of the decryption algorithm using private keys 𝛼 and 𝛽. In 1997, authors in [12] proposed an RSA-type system using n-adic expansions and permutation functions, showing that the proposed method is faster. The authors introduced a new concept [13] built on the hardness of factoring and pointed similarity of the trapdoor permutation of the proposed scheme with the Rabin cryptosystem. He also suggested that the proposed method is best suited for practical application by developing a hybrid encryption scheme using a new trapdoor one-way permutation. The work in [14] deals with deterministic aspects and identification problems of the Rabin cryptosystem during decryption. The paper [15] proposed a fault attack against the Rabin cryptosystem using a one-byte permutation on public key n. However, the above-discussed methods are either too complex or easy to crack.

Some researchers turn to the modulus process to improve the Rabin cryptosystem. In [16], the authors analyzed and compared three types of algebraic analysis on AAβ cryptosystem. The study includes congruence relation, which is used to solve the Aaβ equation. Continued fractions and Coppersmith’s theorem are used to retrieve the factors from the equation. The authors developed an asymmetric scheme based on the integer factorization problem [IFP], including the square root scenario in [17], which is analytically proved to have 1 to 1 decryption. Mahad et al. [17] introduced a new optimized solution to correct the Rabin cryptosystem decryption failure of 4 to 1 by reducing the plaintext phase space from 𝑥 𝜖 ℤ_𝛼𝛽, to 𝑥 𝜖 2^2𝑛−2,2^2𝑛−1 ⊂ℤ_𝛼𝛽, where 𝛼𝛽 is a composite of 2 strong primes 𝛼𝛽𝜖 22^𝑛,2^2𝑛+2. Also, the specified proposed [18] method makes the encryption process fast, and computation is not included much. In [19], the authors proposed two methods using common modulus n=𝛼²𝛽. In the first method, the authors restricted the message range to 𝑀 ∈ 𝑍_𝛼𝛽. In the second method, the range of plaintext is restricted between 0 to 2^2𝑛−2. In both schemes, the authors introduced a mathematical notation to obtain actual plaintext 𝑥_𝑖 among four possible candidates 𝑥₁,𝑥₂,𝑥₃,𝑥₄  which is calculated using, where 𝑊_𝑖 is an integer, 𝐶_𝑖 is the cipher.

All Rabin encryption variant techniques stated in the literature above, a one-time execution of modulo nn squaring is registered with complexity 𝑂(n²). This feature of Rabin makes the system the quickest and most efficient compared to RSA. In the Rabin cryptosystem, encryption can be done using Equation (2).

However, most of the researchers majorly concentrated on the decryption side of the Rabin cryptosystem. The decryption side of the Rabin cryptosystem proposed in [11,14,15,19] uses two prime factors as the key and uses the Chinese Reminder Theorem (CRT) to obtain the plaintext. In these approaches, the decryption procedure produces four possible plaintexts, of which only one will always be correct. In addition to the correct plaintext, decryption has three false plaintext results to judge the actual answer. This is the main issue and significant disadvantage of Rabin-type algorithms. If the algorithm is used to encrypt a text message, then obtaining back in the decryption is not a difficult task. If the plaintexts are numerical values, this algorithm becomes challenging in decryption.

This limitation has been resolved in the paper [20] with a new Rabin-like cryptosystem without using the Jacobi symbol. In this approach, the decryption function needs a single prime 𝑝 as the key by computing a single mod function and giving the required plaintext without any failure. In [21] work of Rabin P is assessed on the microprocessor platform in terms of runtime and energy consumption. The following points summarize the limitations of all existing Rabin cryptosystems.

• Case I: In the case of the existing works, it is easy to recover the plaintext if the intruder can efficiently factor in the public key n.
• Case II: Not all the plaintexts can be used for encryption/decryption.
• Case III: It requires plaintext padding systems or sending extra bits to improve encryption and decryption.
• Case IV: Insufficient expansion of the plaintext-ciphertext ratio.