Random Number Generators in Image Encryption: A Comparative Analysis of RANLUX, MIXMAX, and Lehmer Methods

Abstract

This study offers a comparative evaluation of three prominent random number generators frequently employed in image encryption: RANLUX, MIXMAX, and Lehmer. The level of randomness in encryption directly affects its performance and is closely linked to the structure of the generator used. RANLUX is a variant of the linear congruential generator (LCG) that enhances randomness through a decimation mechanism. MIXMAX offers strong chaotic behavior based on its Lyapunov exponent, while Lehmer stands out for its low computational cost. The mentioned algorithms are evaluated using metrics such as correlation coefficient, NPCR, PSNR, entropy, and histogram analysis. RANLUX effectively eliminates inter-pixel correlation with a correlation value of –0.0025. MIXMAX demonstrates high resistance against differential attacks with an NPCR value of 99.63%. PSNR results show that RANLUX causes the highest distortion. Entropy analysis reveals that all algorithms provide high randomness; however, RANLUX achieves the best performance with a value of 7.9992, which is closest to the theoretical maximum. The study also includes a discussion and conclusions based on scientific findings.

Keywords

• Image Encryption
• Random Number Generator
• Chaotic Systems

Görüntü Şifrelemede Rastgele Sayı Üreteçleri: RANLUX, MIXMAX ve Lehmer Yöntemlerinin Karşılaştırmalı Bir Analizi

Abstract

Bu çalışma, görüntü şifreleme alanında sıkça kullanılan üç farklı rastgele sayı üreteci olan RANLUX, MIXMAX ve Lehmer algoritmalarını karşılaştırmalı olarak incelemektedir. Şifrelemedeki rastgelelik düzeyi kullanılan üretecin yapısı şifreleme başarımını doğrudan etkilemektedir. RANLUX, azaltma mekanizmasıyla rastgeleliği artıran bir doğrusal artan rastgele sayı üreteci türevidir. MIXMAX, Lyapunov üs katsayısı ile güçlü kaotik özellikler sunarken, Lehmer algoritması düşük hesaplama maliyetiyle öne çıkmaktadır. Bahsi geçen algoritmalar korelasyon katsayısı, NPCR, PSNR, entropi ve histogram analizi gibi ölçütlerle değerlendirilmiştir. RANLUX, -0,0025’lik korelasyon değeriyle pikseller arası ilişkiyi en etkin şekilde ortadan kaldırmıştır. MIXMAX, %99,63’lük NPCR değeriyle diferansiyel saldırılara karşı yüksek direnç göstermiştir. PSNR sonuçları da RANLUX’un en yüksek bozulmayı sağladığını göstermektedir. Entropi sonuçları, tüm algoritmaların yüksek rastgelelik sunduğunu, ancak RANLUX’un 7.9992 değeriyle teorik maksimuma en yakın sonucu vererek en güçlü şifreleme performansını sergilediğini göstermektedir. Çalışmamızda bilimsel bulgulara dayanan tartışma ve sonuçlara da yer verilmektedir.

Keywords

• Görüntü Şifreleme
• Rastgele Sayı Üretici
• Kaotik Sistemler

References

1. X. Li, "Image encryption scheme based on multiple chaotic maps," in Proc. 2013 Fourth Int. Conf. Emerging Intelligent Data and Web Technologies, Xi'an, China, Sept. 2013, 261-266.
2. S. Lian, J. Sun, and Z. Wang, "A block cipher based on a suitable use of the chaotic standard map," Chaos, Solitons & Fractals, 26(1), 117–129, 2005. https://doi.org/10.1016/j.chaos.2005.11.070
3. B. Bhargava, C. Shi, and S.-Y. Wang, "MPEG video encryption algorithms," Multimedia Tools and Applications, 24(1), 57–79, 2004. https://doi.org/10.1023/B:MTAP.0000016782.67566.79
4. Y. Zhou, L. Bao, and C. L. P. Chen, "Image encryption using a new parametric switching chaotic system," Signal Processing, 93(11), 3039–3052, Nov. 2013. [Online]. Available: https://doi.org/10.1016/j.sigpro.2013.01.031
5. C. Li, T. Xie, Q. Liu, and G. Cheng, "Cryptanalyzing image encryption using chaotic logistic map," Nonlinear Dynamics, 78(3), 1545–1551, 2014. [Online]. Available: https://doi.org/10.1007/s11071-014-1533-8
6. P. Sridevi and J. Suguna, "Image encryption using Arnold map," Asian Journal of Computer Science and Technology, 8(S1), 70–73, 2019.
7. B. Zhang and L. Liu, "Chaos-Based Image Encryption: Review, Application, and Challenges," Mathematics, 11(11), 2585, Jun. 2023, doi: 10.3390/math11112585.
8. T. Haider, S. A. Blanco, and U. Hayat, "A novel pseudo-random number generator based on multivariable optimization for image-cryptographic applications," Expert Systems with Applications, vol. 240, 2024, 122446, doi: 10.1016/j.eswa.2023.122446.

9. O. Gökbulut and R. Demirci, "XorShiftAnd Random Number Generator For Image Encryption," in Proc. 2022 Int. Symp. on Multidisciplinary Studies and Innovative Technologies (ISMSIT), Ankara, Türkiye, Oct. 2022, 1–5, doi: 10.1109/ISMSIT56059.2022.9932726.
10. K. H. S. Ranjan, S. S. P. Fathimath, S. Shetty, and G. Aithal, "Image encryption based on pixel transposition and Lehmer Pseudo random number generation," in Proc. 2017 2nd IEEE Int. Conf. on Recent Trends in Electronics, Information & Communication Technology (RTEICT), Bangalore, India, May 2017, 1–6, doi: 10.1109/RTEICT.2017.8256786.
11. A. P. Hana, H. S. Naima, and A.-P. Adda, "Image encryption by using a specific adaptation of Lehmer’s algorithm," J. Discrete Math. Sci. Cryptogr., 23(5), 949–971, 2020, doi: 10.1080/09720529.2019.1652402.
12. Y. Yang, X. Xiong, Z. Liu, S. Jin, and J. Wang, "High-Performance Encryption Algorithms for Dynamic Images Transmission," Electronics, 12(24), 3131, Dec. 2023, doi: 10.3390/electronics12243131.
13. D. E. Knuth, The Art of Computer Programming, Volume 2: Semi-Numerical Algorithms, 3rd ed. Reading, MA, USA: Addison-Wesley, 1998.
14. M. Luescher, "A Portable High-Quality Random Number Generator for Lattice Field Theory Simulations," Comput. Phys. Commun., 79, 100–110, 1994, doi: 10.1016/0010-4655(94)90232-1.
15. F. James and L. Moneta, "Review of High-Quality Random Number Generators," Computing and Software for Big Science, 4(2), 1–12, Jan. 2020, doi: 10.1007/s41781-019-0034-3.
16. G. Marsaglia, "Random Numbers Fall Mainly in the Planes," Proc. Natl. Acad. Sci. U.S.A., 61(1), 25–28, Jul. 1968, doi: 10.1073/pnas.61.1.25.
17. A. Sibidanov, "A revision of the subtract-with-borrow random number generators," arXiv preprint, arXiv:1705.03123, May 2017. https://arxiv.org/abs/1705.03123
18. G. Savvidy, "Anosov C-systems and random number generators," arXiv preprint, arXiv:1507.06348, Jul. 2015. https://arxiv.org/abs/1507.06348
19. K. G. Savvidy, "The MIXMAX random number generator," Computer Physics Communications, 196, 161–165, 2015. https://doi.org/10.1016/j.cpc.2015.06.003
20. D. H. Lehmer, "Mathematical methods in large-scale computing units," in Proc. 2nd Symp. Large-Scale Digital Calculating Machinery, Cambridge, MA, USA: Harvard Univ. Press, 1951, pp. 141–146.
21. S. K. Park and K. W. Miller, "Random number generators: good ones are hard to find," Commun. ACM, 31(10), 1192–1201, Oct. 1988. https://doi.org/10.1145/63039.63042
22. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 4th ed. Hoboken, NJ, USA: Pearson, 2018, ch. 3.3.
23. B. Schneier, Applied Cryptography, 2nd ed. New York, NY, USA: Wiley, 1996.
24. C. E. Shannon, "Communication theory of secrecy systems," Bell Syst. Tech. J., 28(4), 656–715, Oct. 1949. https://doi.org/10.1002/j.1538-7305.1949.tb00928.x
25. S. Xu, Y. Wang, Y. Guo, and C. Wang, "A novel chaos-based image encryption scheme," in Proc. 2009 Int. Conf. on Information Engineering and Computer Science (ICIECS), Wuhan, China, Dec. 2009, pp. 1–4, doi: 10.1109/ICIECS.2009.5365275.
26. Y. Wu, J. P. Noonan, and S. Agaian, "NPCR and UACI randomness tests for image encryption," Journal of Selected Areas in Telecommunications (JSAT), Apr. 2011.
27. G. A. Sathishkumar, K. Bhoopathy Bagan, and N. Sriraam, "Image encryption based on diffusion and multiple chaotic maps," Int. J. Netw. Secur. Its Appl. (IJNSA), 3(2), 181–194, Mar. 2011.
28. G. A. Sathishkumar, R. Srinivas, and K. Bhoopathy Bagan, "Image encryption using random pixel permutation by chaotic mapping," in Proc. 2012 IEEE Symposium on Computers & Informatics (ISCI), Penang, Malaysia, Mar. 2012, 1–5, doi: 10.1109/ISCI.2012.6222703.
29. C. E. Shannon, "A mathematical theory of communication," The Bell System Technical Journal, vol. 27(3), 379–423, Jul. 1948, doi: 10.1002/j.1538-7305.1948.tb01338.x.
30. Y. Mao, G. Chen, and S. Lian, "A novel fast image encryption scheme based on 3D chaotic Baker maps," Int. J. Bifurcation Chaos, 14(10), 3613–3624, 2004. doi: 10.1142/S0218127404011132
31. L. Y. Zhang, X. Hu, Y. Liu, K.-W. Wong, and J. Gan, "A chaotic image encryption scheme owning temp-value feedback," Commun. Nonlinear Sci. Numer. Simul., 19(8), 3106–3118, Aug. 2014. doi: 10.1016/j.cnsns.2014.03.006
32. S. Hoşmeyve, A. C. Bilecan, B. Karasulu and İ. Ünlü, “Secure Message Transmission Via Object Detection Verification Using Images Encoded With Lehmer Algorithm Based Random Key,” Gazi Journal of Engineering Sciences, 9(1), 108-127, 2023. doi:10.30855/gmbd.0705057