A Novel and Secure Fake-Modulus Based Rabin-Ӡ 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 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.
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 .
- 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.
3. Mathematical Preliminaries
3.1. Range of Plaintext
3.2. Fake-Modulus Principle
4. Methodology Proposed
4.1. Key Generation
Algorithm 1: Key Generation |
Input: 2 large prime numbers and by satisfying and . |
Output: Fake-modulus . |
Steps: |
|
4.2. Encryption
Algorithm 2: Encryption |
Input: Plaintext and fake-modulus . |
Output: Cipher text . |
Steps: |
Encrypt the plaintext , where the range of is 0 < using |
4.3. Decryption
Algorithm 3: Decryption |
Input: Cipher text and secret key |
Output: Plaintext . |
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 Rabin-P, with the fake-modulus approach, denoted as fake Rabin-P, requires a higher number of steps to crack the plaintext from the given ciphertext.
- The time consumption for Rabin-P and Rabin-P with the fake-modulus 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 fake-modulus approach, demands more time and steps compared to the traditional Rabin-P 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
References
- Cebeci, S.E.; Nari, K.; Ozdemir, E. Secure E-Commerce Scheme. IEEE Access 2022, 10, 10359–10370. [Google Scholar] [CrossRef]
- Rivest, R.L.; Shamir, A.; Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 1978, 21, 120–126. [Google Scholar] [CrossRef]
- Rabin, M.O. Digitalized Signatures and Public-Key Functions as Intractable as Factorization; Tech. Report MIT/LCS/TR-212; MIT Laboratory for Computer Science: Cambridge, MA, USA, 1979. [Google Scholar]
- Imam, R.; Areeb, Q.M.; Alturki, A.; Anwer, F. Systematic and Critical Review of RSA Based Public Key Cryptographic Schemes: Past and Present Status. IEEE Access 2021, 9, 155949–155976. [Google Scholar] [CrossRef]
- Williams, H. A modification of the RSA public-key encryption procedure (Corresp.). IEEE Trans. Inf. Theory 1980, 26, 726–729. [Google Scholar] [CrossRef]
- Singh, D.; Kumar, B.; Singh, S.; Chand, S.; Singh, P.K. RCBE-AS: Rabin cryptosystem–based efficient authentication scheme for wireless sensor networks. Pers. Ubiquitous Comput. 2021. [Google Scholar] [CrossRef]
- Jain, M.; Lenka, S.K. Diagonal queue medical image steganography with Rabin cryptosystem. Brain Inf. 2016, 3, 39–51. [Google Scholar] [CrossRef]
- Jain, M.; Kumar, A.; Choudhary, R.C. Improved diagonal queue medical image steganography using Chaos theory, LFSR, and Rabin cryptosystem. Brain Inf. 2017, 4, 95–106. [Google Scholar] [CrossRef]
- Rachmawati, D.; Budiman, M.A. An implementation of the H-rabin algorithm in the shamir three-pass protocol. In Proceedings of the 2017 2nd International Conference on Automation, Cognitive Science, Optics, Micro Electro—Mechanical System, and Information Technology (ICACOMIT), Jakarta, Indonesia, 23–24 October 2017; pp. 28–33. [Google Scholar] [CrossRef]
- Kurosawa, K.; Ogata, W. Efficient Rabin-type digital signature scheme. Des. Codes Cryptogr. 1999, 16, 53–64. [Google Scholar] [CrossRef]
- Batten, L.M.; Williams, H.C. Unique Rabin-Williams Signature Scheme Decryption; Report 2019/915; Cryptology ePrint Archive: 2019. Available online: https://eprint.iacr.org/2019/915 (accessed on 30 July 2023).
- Takagi, T. Fast RSA-type cryptosystems using n-adic expansion. In Advances in Cryptology—CRYPTO ‘97; CRYPTO 1997; Lecture Notes in Computer Science; Kaliski, B.S., Ed.; Springer: Berlin/Heidelberg, Germany, 1997; Volume 1294. [Google Scholar]
- Schmidt-Samoa, K. A New Rabin-Type Trapdoor Permutation Equivalent To Factoring. Electron. Notes Theor. Comput. Sci. 2006, 157, 79–94. [Google Scholar] [CrossRef]
- Elia, M.; Piva, M.; Schipani, D. The Rabin Cryptosystem Revisited. Appl. Algebra Eng. Commun. Comput. 2015, 26, 251–275. [Google Scholar] [CrossRef]
- Kaminaga, M.; Yoshikawa, H.; Shikoda, A.; Suzuki, T. Crashing Modulus Attack on Modular Squaring for Rabin Cryptosystem. IEEE Trans. Dependable Secur. Comput. 2018, 15, 723–728. [Google Scholar] [CrossRef]
- Asbullah, M.A.; Ariffin, M.R.K. Analysis on the AAβ cryptosystem. In Proceedings of the 5th International Cryptology and Information Security Conference 2016, CRYPTOLOGY 2016, Aksaray, Turkey, 21–22 September 2016; pp. 41–48. [Google Scholar]
- Ariffin, M.R.K.; Asbullah, M.A.; Abu, N.A.; Mahad, Z. A New Efficient Asymmetric Cryptosystem Based on the Integer Factorization Problem. Malays. J. Math. Sci. 2013, 7, 19–37. [Google Scholar]
- Zahari, M.; Ariffin, K.; Rezal, M. Rabin-RZ: A new efficient method to overcome Rabin cryptosystem decryption failure problem. Int. J. Cryptol. Res. 2015, 5, 11–20. [Google Scholar]
- Zahari, M.; Muhammad Asyraf, A.; Ariffin, M.R.K. Efficient methods to overcome Rabin cryptosystem decryption failure. Malays. J. Math. Sci. 2017, 11, 9–20. [Google Scholar]
- Asyraf, A.M.; Ariffin, K.; Rezal, M. Design of Rabin-like cryptosystem without decryption failure. Malays. J. Math. Sci. 2016, 10, 1–18. [Google Scholar]
- Mazlisham, M.H.; Adnan, S.F.S.; Isa, M.A.M.; Mahad, Z.; Asbullah, M.A. Analysis of Rabin-P and RSA-OAEP Encryption Scheme on Microprocessor Platform. In Proceedings of the 2020 IEEE 10th Symposium on Computer Applications & Industrial Electronics (ISCAIE), Penang, Malaysia, 18–19 April 2020; pp. 292–296. [Google Scholar] [CrossRef]
- Tutueva, A.V.; Nepomuceno, E.G.; Karimov, A.I.; Andreev, V.S.; Butusov, D.N. Adaptive chaotic maps and their application to pseudo-random numbers generation. Chaos Solitons Fractals 2020, 133, 109615. [Google Scholar] [CrossRef]
- Bhattacharjee, K.; Das, S. A search for good pseudo-random number generators: Survey and empirical studies. Comput. Sci. Rev. 2022, 45, 100471. [Google Scholar] [CrossRef]
- Kaur, M.; Kumar, V. A Comprehensive Review on Image Encryption Techniques. Arch. Computat. Methods Eng. 2020, 27, 15–43. [Google Scholar] [CrossRef]
- Ruzai, W.N.A.; Ariffin, M.R.K.; Asbullah, M.A.; Mahad, Z.; Nawawi, A. On the Improvement Attack Upon Some Variants of RSA Cryptosystem via the Continued Fractions Method. IEEE Access 2020, 8, 80997–81006. [Google Scholar] [CrossRef]
- Raghunandan, K.R.; Shetty, R.; Aithal, G. Key generation and security analysis of text cryptography using cubic power of Pell’s equation. In Proceedings of the 2017 International Conference on Intelligent Computing, Instrumentation and Control Technologies (ICICICT), Kerala, India, 6–7 July 2017; pp. 1496–1500. [Google Scholar] [CrossRef]
- Raghunandan, K.R.; Dodmane, R.; Bhavya, K.; Rao, N.S.K.; Sahu, A.K. Chaotic-Map Based Encryption for 3D Point and 3D Mesh Fog Data in Edge Computing. IEEE Access 2023, 11, 3545–3554. [Google Scholar] [CrossRef]
- Dodmane, R.; Rao, R.K.; Krishnaraj Rao, N.S.; Kallapu, B.; Shetty, S.; Aslam, M.; Jilani, S.F. Blockchain-Based Automated Market Makers for a Decentralized Stock Exchange. Information 2023, 14, 280. [Google Scholar] [CrossRef]
- Zhou, N.-R.; Tong, L.-J.; Zou, W.-P. Multi-image encryption scheme with quaternion discrete fractional Tchebyshev moment transform and cross-coupling operation. Signal Process. 2023, 211, 109107, ISSN 0165-1684. [Google Scholar] [CrossRef]
- Afolabi, A.O.; Oshinubi, K.I. Derivation of a Numerical Scheme to find any Root of any Real Number k using Newton Raphson Iterative Method. In Proceedings of the 13th iSTEAMS Multidisciplinary Conference, Accra, Ghana, 11 August 2018; pp. 107–112. [Google Scholar]
- Sahu, A.K.; Sahu, M. Digital image steganography techniques in spatial domain: A study. Int. J. Pharm. Technol. 2016, 8, 5205–5217. [Google Scholar]
- Hemalatha, J.; Sekar, M.; Kumar, C.; Gutub, A.; Sahu, A.K. Towards improving the performance of blind image steganalyzer using third-order SPAM features and ensemble classifier. J. Inf. Secur. Appl. 2023, 76, 103541. [Google Scholar] [CrossRef]
- Sahu, A.K. A logistic map based blind and fragile watermarking for tamper detection and localization in images. J. Ambient. Intell. Humaniz. Comput. 2022, 13, 3869–3881. [Google Scholar] [CrossRef]
- Rukhin, A.; Soto, J.; Nechvatal, J.; Smid, M.; Barker, E. Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; U.S. Department of Commerce: Washington, DC, USA, 2010.
- Puneeth, B.R.; Raghunandan, K.R.; Bhavya, K.; Shetty, S.; Krishnaraj Rao, N.S.; Dodmane, R.; Ramya; Sarda, M.N.I. Preserving Confidentiality against Factorization Attacks using Fake-modulus (ζ) Approach in RSA and its Security Analysis. In Proceedings of the 2022 IEEE 9th Uttar Pradesh Section International Conference on Electrical, Electronics and Computer Engineering (UPCON), Prayagraj, India, 2–4 December 2022; pp. 1–6. [Google Scholar] [CrossRef]
- Wang, X.; Liu, P. A New Full Chaos Coupled Mapping Lattice and Its Application in Privacy Image Encryption. IEEE Trans. Circuits Syst. I Regul. Pap. 2022, 69, 1291–1301. [Google Scholar] [CrossRef]
I | ||||
---|---|---|---|---|
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 | Steps k | 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 | Steps k | 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 |
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 | |
Rabin-P algorithm [19] | (Lena) | 7.63 | 7.71 | 7.75 |
(Baboon) | 7.68 | 7.74 | 7.69 | |
Rabin-ӡ with fake-modulus | (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 |
Rabin-P [19] (Lena) | 99.619 | 99.629 | 99.636 | 32.799 | 32.382 | 33.284 |
Rabin-P [19] (Baboon) | 99.608 | 99.587 | 99.478 | 32.798 | 32.492 | 33.01 |
Rabin-ӡ with fake-modulus (Lena) | 99.641 | 99.638 | 99.646 | 32.957 | 32.300 | 33.310 |
Rabin-ӡ with fake-modulus (Baboon) | 99.624 | 99.574 | 99.547 | 33.047 | 32.981 | 32.865 |
Process | Equation Used | Rabin-P | Rabin-ӡ Using Fake-Modulus |
---|---|---|---|
Key Generation | |||
- | |||
Encryption | |||
Decryption | |||
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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Share and Cite
Ramesh, R.K.; Dodmane, R.; Shetty, S.; Aithal, G.; Sahu, M.; Sahu, A.K. A Novel and Secure Fake-Modulus Based Rabin-Ӡ 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 Fake-Modulus Based Rabin-Ӡ 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 Fake-Modulus Based Rabin-Ӡ Cryptosystem" Cryptography 7, no. 3: 44. https://doi.org/10.3390/cryptography7030044