An Efficient and Secure Cryptographic Algorithm Using Elliptic Curves and Max-Plus Algebra-Based Wavelet Transform
Abstract
:1. Introduction
2. Preliminaries
2.1. Elliptic Curve (EC)
2.2. Max-Plus Algebra-Based Wavelet Transforms (MP Wavelets)
3. Proposed Scheme
3.1. Encryption Algorithm
3.2. Decryption Algorithm
4. Simulation Experiments: Security Analysis and Comparison
4.1. Correlation Analysis
4.2. Computation Time
4.3. Encryption Quality Analysis
4.4. Entropy Analysis
4.5. NIST Tests Analysis
4.6. Key Sensitivity Analysis
4.7. Plaintext Sensitivity Analysis
4.8. Key Space Analysis
4.9. Ciphertext-Only Attack
4.10. Known-Plaintext Attack
4.11. Chosen-Plaintext Attack
4.12. Chosen-Ciphertext Attack
4.13. Comparative Analysis of Cryptographic Attacks
5. Conclusions
- Our results indicate that the designed scheme offers a high level of security, while maintaining reasonable computational efficiency, making it well suited for secure text communication in various contexts, including sensitive government communications, financial transactions, and personal messaging.
- The designed encryption scheme is currently limited to text data, but it could be advantageous to broaden its application to include other types of data, such as images, audios, and videos.
- Moreover, the results obtained by the proposed encryption scheme are not fully optimized. It would be useful to apply an optimization algorithm to attain an optimal level of security.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Number of Characters in P | R | (%) | Plaintext Sensitivity | ||
---|---|---|---|---|---|
30 | 0.0515 | 90.00 | 3.5232 | 4.2817 | 100.000 |
300 | 0.0774 | 77.33 | 4.8759 | 5.6367 | 99.3333 |
999 | 0.0244 | 76.18 | 4.4759 | 5.4658 | 98.6987 |
3000 | 0.0012 | 81.20 | 4.5578 | 5.5356 | 98.5333 |
5995 | 0.0021 | 82.60 | 4.5573 | 5.5114 | 98.8490 |
1 Plaintext | Scheme | R | (%) | Plaintext Sensitivity | ||
---|---|---|---|---|---|---|
Proposed | 0.0697 | 100.000 | 2.2516 | 2.5850 | 100.000 | |
Ref. [14] | −0.3189 | 83.3333 | 2.2516 | 2.2516 | 16.6667 | |
Ref. [18] | 0.3904 | 66.6667 | 2.2516 | 2.5850 | 16.6667 | |
Ref. [17] | 0.3904 | 66.6667 | 2.2516 | 2.5850 | 16.6667 | |
Ref. [16] | 0.3904 | 66.6667 | 2.2516 | 2.5850 | 16.6667 | |
Ref. [15] | −0.5354 | 66.6667 | 2.2516 | 2.2516 | 16.6667 | |
Ref. [13] | −0.2580 | 83.3333 | 2.2516 | 2.2516 | 16.6667 | |
Ref. [12] | 0.3904 | 66.6667 | 2.2516 | 2.5850 | 16.6667 | |
Ref. [11] | 0.3904 | 66.6667 | 2.2516 | 2.5850 | 16.6667 | |
Ref. [10] | −0.3907 | 100.000 | 2.2516 | 2.2516 | 16.6667 | |
Proposed | −0.0299 | 100.000 | 3.0850 | 3.4183 | 91.666 | |
Ref. [14] | −0.4538 | 91.6667 | 3.0850 | 2.9183 | 8.3333 | |
Ref. [18] | 0.0095 | 66.6667 | 3.0850 | 2.7516 | 8.3333 | |
Ref. [17] | 0.0095 | 66.6667 | 3.0850 | 2.7516 | 8.3333 | |
Ref. [16] | 0.0095 | 66.6667 | 3.0850 | 2.7516 | 8.3333 | |
Ref. [15] | 0.1844 | 66.6667 | 3.0850 | 2.9183 | 8.3333 | |
Ref. [13] | −0.0996 | 91.6667 | 3.0850 | 2.8554 | 8.3333 | |
Ref. [12] | 0.0095 | 66.6667 | 3.0850 | 2.7516 | 8.3333 | |
Ref. [11] | 0.0095 | 66.6667 | 3.0850 | 2.7516 | 8.3333 | |
Ref. [10] | 0.2418 | 66.6667 | 3.0850 | 3.0221 | 8.3333 | |
Proposed | −0.2979 | 93.3333 | 3.9484 | 4.2817 | 96.666 | |
Ref. [14] | −0.0588 | 86.6667 | 3.9484 | 3.8314 | 6.6667 | |
Ref. [18] | 0.3411 | 63.3333 | 3.9484 | 3.0159 | 3.3333 | |
Ref. [17] | 0.3411 | 63.3333 | 3.9484 | 3.0159 | 3.3333 | |
Ref. [16] | 0.3411 | 63.3333 | 3.9484 | 3.0159 | 3.3333 | |
Ref. [15] | 0.6242 | 43.3333 | 3.9484 | 3.7069 | 3.3333 | |
Ref. [13] | 0.0223 | 66.6667 | 3.9484 | 3.0799 | 3.3333 | |
Ref. [12] | 0.3411 | 63.3333 | 3.9484 | 3.0159 | 3.3333 | |
Ref. [11] | 0.3411 | 63.3333 | 3.9484 | 3.0159 | 3.3333 | |
Ref. [10] | 0.3588 | 60.0000 | 3.9484 | 2.8115 | 3.3333 |
Number of Characters in P | Time (Seconds) | |
---|---|---|
Encryption | Decryption | |
30 | 0.0254 | 0.0256 |
300 | 0.0294 | 0.0432 |
999 | 0.0320 | 0.0956 |
3000 | 0.0372 | 0.2050 |
5995 | 0.0472 | 0.3860 |
Scheme | Number of Characters in P | Time (Seconds) | |
---|---|---|---|
Encryption | Decryption | ||
Proposed | 1126 | 0.0272 | 0.0926 |
Ref. [14] | 999 | 0.2650 | 0.3590 |
Ref. [41] | 409 | 0.0930 | 0.1400 |
Ref. [42] | 1126 | 0.1400 | 0.1090 |
Ref. [43] | 409 | 1.9500 | 0.8300 |
Test Name | Proportion | Results |
---|---|---|
Frequency (monobit) | Pass | |
Frequency (block) | Pass | |
Longest run | Pass | |
Rank | Pass | |
Non-overlapping template | Pass | |
Overlapping template | Pass | |
Universal | Pass | |
Linear complexity | Pass | |
Serial | Fail | |
Cumulative sums 1 | Fail | |
Cumulative sums 2 | Pass |
Key Sensitivity | |||
---|---|---|---|
Plaintext → Applied Sciences | |||
Original key → , | |||
Original ciphertext C → $FOU-O5aJB3TI3Q2 | |||
Modified key | Modified ciphertext | ||
p | G | ||
227 | (193, 60) | dd*F>*TG‘caj4a6, | |
239 | (206, 48) | A@BHhBlO!%9GG9$F | |
227 | (203, 22) | !!Q“wQ$QbZI$6IAt |
Scheme | Key Space | Key Sensitivity | Ciphertext Only Attack | Known-Plaintext Attack |
---|---|---|---|---|
Proposed | Pass | Pass | Pass | Pass |
Ref. [38] | Fail | Fail | Fail | Fail |
Ref. [14] | Pass | Pass | Pass | Fail |
Ref. [44] | Fail | Pass | Fail | Fail |
Ref. [39] | Fail | Pass | Fail | Fail |
Ref. [40] | Fail | Fail | Fail | Fail |
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Sattar, K.A.; Haider, T.; Hayat, U.; Bustamante, M.D. An Efficient and Secure Cryptographic Algorithm Using Elliptic Curves and Max-Plus Algebra-Based Wavelet Transform. Appl. Sci. 2023, 13, 8385. https://doi.org/10.3390/app13148385
Sattar KA, Haider T, Hayat U, Bustamante MD. An Efficient and Secure Cryptographic Algorithm Using Elliptic Curves and Max-Plus Algebra-Based Wavelet Transform. Applied Sciences. 2023; 13(14):8385. https://doi.org/10.3390/app13148385
Chicago/Turabian StyleSattar, Kanza Abdul, Takreem Haider, Umar Hayat, and Miguel D. Bustamante. 2023. "An Efficient and Secure Cryptographic Algorithm Using Elliptic Curves and Max-Plus Algebra-Based Wavelet Transform" Applied Sciences 13, no. 14: 8385. https://doi.org/10.3390/app13148385