A Novel and Secure FakeModulus Based RabinӠ Cryptosystem
Abstract
:1. Introduction
 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 ecommerce 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. Ecommerce businesses should be vigilant in monitoring for these attacks and should have strong antimalware and antiphishing measures in place.
 Website Security: The security of ecommerce 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.
2. Related Work
 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.
 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 $\mathrm{n}$.
 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 plaintextciphertext ratio.
3. Mathematical Preliminaries
3.1. Range of Plaintext
3.2. FakeModulus Principle
4. Methodology Proposed
4.1. Key Generation
Algorithm 1: Key Generation 
Input: 2 large prime numbers $\alpha $ and $\beta $ by satisfying $\left(\alpha +1\right)\text{}mod\text{}4==0$ and $\left(\beta +1\right)\text{}mod\text{}4==0$. 
Output: Fakemodulus $\u04e1$. 
Steps: 

4.2. Encryption
Algorithm 2: Encryption 
Input: Plaintext ${x}_{i}$ and fakemodulus $\u04e0$. 
Output: Cipher text ${C}_{i}$. 
Steps: 
Encrypt the plaintext ${x}_{i}$, where the range of ${x}_{i}$ is 0 < ${x}_{i}<\frac{{\alpha}^{2}}{2}$ using 
${C}_{i}\equiv {x}^{2}\left(mod\u04e1\right)$ 
4.3. Decryption
Algorithm 3: Decryption 
Input: Cipher text ${C}_{i}$ and secret key $\alpha $ 
Output: Plaintext ${x}_{i}$. 
Steps: 

4.4. Example
4.4.1. Key Generation
4.4.2. Encryption
4.4.3. Decryption
5. Cryptanalysis
 By factoring the prime numbers using Fermat’s Factorization method [24]
 Breaking the plaintext using cipher value and shared public key by brute force.
5.1. Obtaining Private Keys from Fermat’s Factorization Method
5.2. Obtaining Plaintext from Cipher Text and Modulus in Rabin Cryptosystem Using Brute Force Method
5.3. Case Study
 It is observed that RabinP, with the fakemodulus approach, denoted as fake RabinP, requires a higher number of steps to crack the plaintext from the given ciphertext.
 The time consumption for RabinP and RabinP with the fakemodulus is approximately equivalent for prime numbers with lower bit lengths (e.g., 8, 10, and 12 bits). However, as the bit length increases beyond 16 bits, the gap between the time curves widens significantly.
 Based on the statistical comparison, it is evident that breaking the code using the proposed fakemodulus approach, demands more time and steps compared to the traditional RabinP algorithm.
6. Results and Analysis
6.1. Visual Analysis
6.2. Histogram Analysis
6.3. Entropy Analysis
6.4. Differential Analysis
6.5. Complexityl Analysis
6.6. Randomness Analysis
7. Discussions
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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I  ${\mathbf{\u01b3}}_{\mathit{i}}$  ${\mathbf{\u01b3}}_{\mathit{i}}{}^{2}$  ${\mathbf{\u01b3}}_{\mathit{i}}{}^{2}\mathit{n}$  $\U0001d4e3=\sqrt{{\mathbf{\u01b3}}_{\mathit{i}}{}^{2}\mathit{n}}$ 

1  147  21,609  136  11.661903789690601 
2  148  21,904  431  20.760539492026695 
3  149  22,201  728  26.981475126464083 
4  150  22,500  1027  32.046840717924134 
5  151  22,801  1328  36.4417343165772 
6  152  23,104  1631  40.38564101261734 
7  153  23,409  1936  44 
Key Size  $\mathbf{\u1d87}$ $={\mathit{\alpha}}^{2}\mathit{\beta}$  Steps k  $\mathbf{Factoring}\text{}\mathbf{Time}\text{}\mathbf{in}\text{}\mathbf{\mu}\mathbf{s}$  Factors Obtained 

8  4,307,411  6631  6.2408447265625  17,161, 251 
10  278,726,051  120,579  62.55626678466797  273,529, 1019 
12  17,411,169,179  1,998,079  1064.565896987915  4,255,969, 4091 
14  1,105,352,737,843  32,732,805  17,681.19716644287  67,551,961, 16,363 
16  70,363,372,715,879  528,613,693  3,430,048.5668182373  1,073,938,441, 65,519 
Key Size 
$$\mathbf{\u1d87}={\mathit{\alpha}}^{2}\mathit{\beta}$$
 Steps k 
$$\mathbf{Factoring}\text{}\mathbf{Time}\text{}\mathbf{in}\text{}\mathbf{\mu}\mathbf{s}$$
 Factors Obtained 

8  4,307,411  5899  12.034177780151367  17,161, 501 
10  278,726,051  7253  11.652231216430664  50,731, 10,983 
12  17,411,169,179  1137  0.8997917175292969  208,363, 167,103 
14  1,105,352,737,843  818  0.7925033569335938  1,536,953, 1,438,325 
16  70,363,372,715,879  12,748,236  144,303.49683761597  46,174,339, 3,047,703 
$\mathit{k}$  ${\mathit{M}}_{\mathit{i}}=\sqrt{{\mathit{C}}_{\mathit{i}}+\mathit{k}\times \mathbf{n}}$ 

0  3397.7158503912597 
1  5578.467531500027 
2  7119.980828625875 
3  8382.657931706386 
4  9478.59594032787 
5  10,460.334985075764 
6  11,357.52767991344 
7  12,188.858108945235 
8  12,967 
Methods Proposed  Entropy of RGB Components  

Red  Green  Blue  
Ref. [11]  7.59  7.68  7.71  
Ref. [14]  7.65  7.70  7.68  
Ref. [15]  7.73  7.76  7.72  
Ref. [16]  7.71  7.73  7.70  
RabinP algorithm [19]  (Lena)  7.63  7.71  7.75 
(Baboon)  7.68  7.74  7.69  
Rabinӡ with fakemodulus  (Lena)  7.93  7.95  7.94 
(Baboon)  7.92  7.97  7.94 
NPCR  UACI  

RED  GREEN  BLUE  RED  GREEN  BLUE  
Ref. [11]  99.600  99.423  99.364  32.379  32.278  33.178 
Ref. [14]  99.591  99.490  99.564  32.619  32.311  33.214 
Ref. [15]  99.593  99.538  99.614  32.714  32.274  33.287 
Ref. [16]  99.614  99.532  99.632  32.770  32.297  33.258 
RabinP [19] (Lena)  99.619  99.629  99.636  32.799  32.382  33.284 
RabinP [19] (Baboon)  99.608  99.587  99.478  32.798  32.492  33.01 
Rabinӡ with fakemodulus (Lena)  99.641  99.638  99.646  32.957  32.300  33.310 
Rabinӡ with fakemodulus (Baboon)  99.624  99.574  99.547  33.047  32.981  32.865 
Process  Equation Used  RabinP  Rabinӡ Using FakeModulus 

Key Generation  $\u0272={\alpha}^{2}\beta $  $O\left({n}^{3}\right)$  $O\left({n}^{3}\right)$ 
$\u04e1=\u0272+\left({\alpha}^{2}\times \tau \right)$    $O\left({n}^{3}\right)$  
Encryption  ${C}_{i}\equiv {x}^{2}\left(mod\u04e1\right)$  $O({n}^{2}{log}_{2}n)$  $O({n}^{2}{log}_{2}n)$ 
Decryption  $w={C}_{i}\left(mod\alpha \right)$  $O\left({n}^{2}\right)$  $O\left({n}^{2}\right)$ 
${x}_{\alpha}={C}_{i}{}^{\frac{\alpha +1}{4}}\left(mod\alpha \right)$  $O({n}^{2}{log}_{2}n)$  $O({n}^{2}{log}_{2}n)$  
$i=\frac{{C}_{i}{x}_{\alpha}^{2}}{\alpha}mod\alpha $  $O\left(3{n}^{2}\right)$  $O\left(3{n}^{2}\right)$  
$\left(2{x}_{\alpha}\ast v\right)mod\alpha =1$  $O\left(M\ast 2{n}^{2}\right)$  $O\left(M\ast 2{n}^{2}\right)$  
$j=\left(i\ast v\right)\left(mod\alpha \right)$  $O\left(2{n}^{2}\right)$  $O\left(2{n}^{2}\right)$  
${x}_{1}={x}_{\alpha}+j\alpha $  $O\left(logn\right)O\left(n\right)$  $O\left(logn\right)O\left(n\right)$ 
Test Name  Proposed Encryption Algorithm (Lena)  Proposed Encryption Algorithm (Baboon)  Result 

Frequency  0.03427581  0.02989546  ✓ 
Block Frequency  0.02543914  0.02734212  ✓ 
Approximate Entropy  0.104512041  0.09128766  ✓ 
Linear Complexity  0.1382546  0.1087234  ✓ 
Random Excursions  0.16248531  0.10237231  ✓ 
Random Excursions Variant  0.09214753  0.10118763  ✓ 
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Ramesh, R.K.; Dodmane, R.; Shetty, S.; Aithal, G.; Sahu, M.; Sahu, A.K. A Novel and Secure FakeModulus Based RabinӠ Cryptosystem. Cryptography 2023, 7, 44. https://doi.org/10.3390/cryptography7030044
Ramesh RK, Dodmane R, Shetty S, Aithal G, Sahu M, Sahu AK. A Novel and Secure FakeModulus Based RabinӠ Cryptosystem. Cryptography. 2023; 7(3):44. https://doi.org/10.3390/cryptography7030044
Chicago/Turabian StyleRamesh, Raghunandan Kemmannu, Radhakrishna Dodmane, Surendra Shetty, Ganesh Aithal, Monalisa Sahu, and Aditya Kumar Sahu. 2023. "A Novel and Secure FakeModulus Based RabinӠ Cryptosystem" Cryptography 7, no. 3: 44. https://doi.org/10.3390/cryptography7030044