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Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

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Open Access Research Article

Design and analysis of a pseudo random number generator based on the logistic-sine system and a nonlinear feedback shift register : Application to image encryption 

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* Corresponding author · click or hover a name for details

pp. 2645–2670Vol. 28Issue 7October 2025DOI: 10.47974/JDMSC-1847 Crossmark XML
Received:
15 Mar 2023
Published Online:
09 Jun 2025
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-1847
Pages:
2645–2670

Abstract

This study aimed to improve the output quality and statistical characteristics of random output sequences by combining the output of an 8-bit Fibonacci nonlinear feedback shift register (NLFSR) of degree three with a sequence of keys generated by a coupled logistic-sine system (LSS). Subsequently, the proposed pseudorandom number generator PRNG-LSS-NLFSR was investigated using a simple image-encryption application. Theoretical randomness evaluation through the National Institute of Standards and Technology (NIST) and other statistical test of its application by means of the histogram, entropy analysis, correlation test, and other tests confirm that the proposed PRNG-LSS-NLFSR is simple, has random and nonlinear features, and displays a strong cryptographic security aspect.

Keywords

Subject Classifications

11T7168P2594A6068R01

References

[1] V. Himthani, V. S. Dhaka, M. Kaur, and P. Hemrajani, “V-Net architecture based advanced visually meaningful image encryption technique,” J. Discrete Math. Sci. Cryptogr., vol. 25, no. 7, pp. 2183–2194 (Oct. 2022), doi: 10.1080/09720529.2022.2133255.
[2] R. Ge, G. Yang, J. Wu, Y. Chen, G. Coatrieux, and L. Luo, “A Novel Chaos-Based Symmetric Image Encryption Using Bit-Pair Level Process,” IEEE Access, vol. 7, pp. 99470–99480 (2019), doi: 10.1109/ACCESS.2019.2927415.
[3] Z. Qiao, I. Taralova, and S. El Assad, “Efficient Pseudo-chaotic Number Generator for Cryptographic Applications,” Int. J. Intell. Comput. Res., vol. 11, no. 1, pp. 1041–1048 (Dec. 2020), doi: 10.20533/ijicr.2042.4655.2020.0126.
[4] H.-C. Tang and C. S. Chen, “Spectral test of DX and DL multiple recursive random number generators,” J. Discrete Math. Sci. Cryptogr., vol. 21, no. 1, pp. 171–178 (Jan. 2018), doi: 10.1080/09720529.2017.1338602.
[5] M. K. Khairullah, A. A. Alkahtani, M. Z. Bin Baharuddin, and A. M. Al-Jubari, “Designing 1D Chaotic Maps for Fast Chaotic Image Encryption,” Electronics, vol. 10, no. 17, Art. no. 17 (Jan. 2021), doi: 10.3390/electronics10172116.
[6] S. Kanamaru, Y. Shimada, K. Fujiwara, and T. Ikeguchi, “Performance evaluation of chaotic random numbers generated from responses of integer logistic maps,” Nonlinear Theory Its Appl. IEICE, vol. 12, no. 3, pp. 489–499 (2021), doi: 10.1587/nolta.12.489.
[7] V. Kanth, T. Martinsen, and P. Stanica, “The Self-Shrinking Conflation Generator: A Proposed Improvement to the Self-Shrinking Generator,” Eur. J. Pure Appl. Math., vol. 15, no. 4, Art. no. 4 (Oct. 2022), doi: 10.29020/nybg.ejpam.v15i4.4504.
[8] J. Walczak and R. Stępień, “Discrete models of the NLFSR generators,” Comput. Appl. Electr. Eng., vol. Vol. 9, 2011, Accessed: Jan. 15 (2023). [Online]. Available: http://yadda.icm.edu.pl/baztech/element/bwmeta1.element.baztech-58e76496-f009-450f-b2b1-d0121a67f8f7.
[9] A. A. Kuznetsov, O. V. Potii, N. A. Poluyanenko, Y. I. Gorbenko, and N. Kryvinska, “Research of Second-Order Properties of NLFSR. Comparative Analysis of M-NLFSR and M-LFSR,” in Stream Ciphers in Modern Real-time IT Systems: Analysis, Design and Comparative Studies, A. A. Kuznetsov, O. V. Potii, N. A. Poluyanenko, Y. I. Gorbenko, and N. Kryvinska, Eds., in Studies in Systems, Decision and Control. , Cham: Springer International Publishing, pp. 419–465 (2022). doi: 10.1007/978-3-030-79770-6_15.
[10] M. Garland, S. Le Grand, J. Nickolls, J. Anderson, J. Hardwick, S. Morton, E. Phillips, Y. Zhang, and V. Volkov, “Parallel computing experiences with CUDA,” IEEE Micro, vol. 28, no. 4, pp. 13–27 (Jul. 2008), doi: 10.1109/MM.2008.57.
[11] J. C. Cerda, C. D. Martinez, J. M. Comer, and D. H. K. Hoe, “An efficient FPGA random number generator using LFSRs and cellular automata,” in 2012 IEEE 55th International Midwest Symposium on Circuits and Systems (MWSCAS), pp. 912–915 (Aug. 2012). doi: 10.1109/MWSCAS.2012.6292169.
[12] E. Dubrova, “A List of Maximum Period NLFSRs.” 2012. Accessed: May 28 (2023). [Online]. Available: https://eprint.iacr.org/2012/166
[13] D. Zhao, H. Peng, L. Li, S. Hui, and Y. Yang, “Novel way to research nonlinear feedback shift register,” Sci. China Inf. Sci., vol. 57, no. 9, pp. 1–14 (Sep. 2014), doi: 10.1007/s11432-013-5058-4.
[14] M. A. Murillo-Escobar, C. Cruz-Hernández, L. Cardoza-Avendaño, and R. Méndez-Ramírez, “A novel pseudorandom number generator based on pseudorandomly enhanced logistic map,” Nonlinear Dyn., vol. 1, no. 87, pp. 407–425 (Sep. 2016), doi: 10.1007/s11071-016-3051-3.
[15] M. Sharma, R. K. Ranjan, and V. Bharti, “A pseudo-random bit generator based on chaotic maps enhanced with a bit-XOR operation,” J. Inf. Secur. Appl., vol. 69, no. C (Sep. 2022), doi: 10.1016/j.jisa.2022.103299.
[16] “A Modified Discretized Chaotic Map and Its Generated Pseudo Binary Random Number,” Mater. Today Proc., vol. 65, pp. 3806–3813 (Jan. 2022), doi: 10.1016/j.matpr.2022.06.577.
[17] M. Hemattil, A. Ahmadi, S. V. Makkil, and M. Ahmadi, “Hardware Design of Chaotic Pseudo-Random Number Generator Based on Nonlinear Feedback Shift Register,” 2018 IEEE 61st Int. Midwest Symp. Circuits Syst. MWSCAS, pp. 980–983 (Aug. 2018), doi: 10.1109/MWSCAS.2018.8624001.
[18] H. S. Alhadawi, M. F. Zolkipli, S. M. Ismail, and D. Lambić, “Designing a pseudorandom bit generator based on LFSRs and a discrete chaotic map,” Cryptologia, vol. 43, no. 3, pp. 190–211 (May 2019), doi: 10.1080/01611194.2018.1548390.
[19] F. Dridi, S. El Assad, W. El Hadj Youssef, and M. Machhout, “Design, Hardware Implementation on FPGA and Performance Analysis of Three Chaos-Based Stream Ciphers,” Fractal Fract., vol. 7, no. 2, Art. no. 2 (Feb. 2023), doi: 10.3390/fractalfract7020197.
[20] H. Bourekouche, S. Belkacem, and N. Messaoudi, “Efficient image encryption scheme using a nonlinear shift register and chaos,” Journal of Information and Optimization Sciences, vol. 45, no. 1, pp. 157–180 (2024).
[21] G. Yao and U. Parampalli, “Improved transformation algorithms for generalized Galois NLFSRs,” Cryptogr. Commun., vol. 14, no. 2, pp. 229–258 (Mar. 2022), doi: 10.1007/s12095-021-00500-3.
[22] P. K. Kumar and B. Mondal, “Lightweight Stream Cipher for Health Care IoT,” in 2023 IEEE 2nd International Conference on Industrial Electronics: Developments & Applications (ICIDeA), pp. 444–449 (Sep. 2023). doi: 10.1109/ICIDeA59866.2023.10295196.
[23] G. Yao, “Transformation and Security Analysis of NLFSR-based Stream Ciphers,” 2020, Accessed: Sep. 27 (2023). [Online]. Available: http://hdl.handle.net/11343/267826.
[24] S. Deb, B. Biswas, and N. Kar, “Study of NLFSR and Reasonable Security Improvement on Trivium Cipher,” in Information Systems Design and Intelligent Applications, J. K. Mandal, S. C. Satapathy, M. Kumar Sanyal, P. P. Sarkar, and A. Mukhopadhyay, Eds., in Advances in Intelligent Systems and Computing. New Delhi: Springer India, pp. 731–739 (2015). doi: 10.1007/978-81-322-2250-7_73.
[25] H. Bourekouche, Contribution to chaotic encryption methods for digital data, Ph.D. dissertation, Ministry of Higher Education (2024).
[26] I. Al-Hejri and S. Almuhammadi, “Constructing New NLFSR Functions with Optimal Periods,” Int. J. Interdiscip. Telecommun. Netw. IJITN, vol. 12, no. 2, pp. 71–80 (2020), doi: 10.4018/IJITN.2020040106.
[27] T. Hu, Y. Liu, L.-H. Gong, S.-F. Guo, and H.-M. Yuan, “Chaotic image cryptosystem using DNA deletion and DNA insertion,” Signal Process., vol. 134, pp. 234–243 (May 2017), doi: 10.1016/j.sigpro.2016.12.008.
[28] Y. Zhou, L. Bao, and C. L. P. Chen, “A new 1D chaotic system for image encryption,” Signal Process., vol. 97, pp. 172–182 (Apr. 2014), doi: 10.1016/j.sigpro.2013.10.034.
[29] V. M. Padmapriya, B. Sowmya, M. Sumanjali, and A. Jayapalan, “Chaotic Encryption based secure Transmission,” in 2019 International Conference on Vision Towards Emerging Trends in Communication and Networking (ViTECoN), pp. 1–5 (Mar. 2019). doi: 10.1109/ViTECoN.2019.8899588.
[30] Joan. S. Muthu, A. J. Paul, and P. Murali, “An Efficient Analyses of the Behavior of One Dimensional Chaotic Maps using 0–1 Test and Three State Test,” in 2020 IEEE Recent Advances in Intelligent Computational Systems (RAICS), pp. 125–130 (Dec. 2020). doi: 10.1109/RAICS51191.2020.9332470.
[31] C. Pak, K. An, P. Jang, J. Kim, and S. Kim, “A novel bit-level color image encryption using improved 1D chaotic map,” Multimed. Tools Appl., vol. 78, no. 9, pp. 12027–12042 (May 2019), doi: 10.1007/s11042-018-6739-1.
[32] L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, N. Heckert, J. Dray, and S. Vo, “A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications,” NIST, Sep. 2010, Accessed: Feb. 07 (2023). [Online]. Available: https://www.nist.gov/publications/statistical-test-suite-random-and-pseudorandom-number-generators-cryptographic
[33] M. ParsiMehr, K. Shayesteh, and K. Godini, “The modeling and prediction of the quality of the groundwater resources in Tuyserkan plain using the optimized artificial neural network,” J. Adv. Environ. Health Res., vol. 8, no. 2, pp. 100–110 (Apr. 2020), doi: 10.22102/jaehr.2020.210891.1150.
[34] M. M. Al-Mhadawi, E. A. Albahrani, and S. H. Lafta, “Efficient and secure chaotic PRNG for color image encryption,” Microprocess. Microsyst., vol. 101, pp. 104911 (Sep. 2023), doi: 10.1016/j.micpro.2023.104911.
[35] M. M. Al-Mhadawi and A. A. Albahrani, “Hybrid Method as Pseudo-Random Bits Generator,” in 2019 First International Conference of Computer and Applied Sciences (CAS), pp. 250–255 (Dec. 2019). doi: 10.1109/CAS47993.2019.9075715.
[36] M. Ťažký, L. Bodnárová, L. Ťažká, R. Hela, M. Meruňka, and P. Hlaváček, “The Effect of the Composition of a Concrete Mixture on Its Volume Changes,” Materials, vol. 14, no. 4, Art. no. 4 (Jan. 2021), doi: 10.3390/ma14040828.
[37] J. Arif et al., “A Novel Chaotic Permutation-Substitution Image Encryption Scheme Based on Logistic Map and Random Substitution,” IEEE Access, vol. 10, pp. 12966–12982 (2022), doi: 10.1109/ACCESS.2022.3146792.
[38] H. T. Elshoush, B. M. Al-Tayeb, and K. T. Obeid, “Enhanced Serpent algorithm using Lorenz 96 Chaos-based block key generation and parallel computing for RGB image encryption,” PeerJ Comput. Sci., vol. 7, pp. e812 (Dec. 2021), doi: 10.7717/peerj-cs.812.
[39] B. Ge, X. Chen, G. Chen, and Z. Shen, “Secure and Fast Image Encryption Algorithm Using Hyper-Chaos-Based Key Generator and Vector Operation,” IEEE Access, vol. 9, pp. 137635–137654 (2021), doi: 10.1109/ACCESS.2021.3118377.
[40] M. Singh, N. Baranwal, K. N. Singh, and A. K. Singh, “Using GAN-Based Encryption to Secure Digital Images with Reconstruction through Customized Super Resolution Network,” IEEE Trans. Consum. Electron., pp. 1–1 (2023), doi: 10.1109/TCE.2023.3285626.
[41] S. Vishwakarma and S. Qureshi, “Secure Transmission of Video using (2,2) Visual Cryptography Scheme and Share Encryption using Logistic Chaos Method,” Int. J. Sci. Res. Comput. Sci. Eng. Inf. Technol., vol. 3, no. 1, pp. 1502–1514 (Feb. 2018), doi: 10.32628/CSEIT1831357.
[42] B. Norouzi, S. Mirzakuchaki, S. M. Seyedzadeh, and M. R. Mosavi, “A simple, sensitive and secure image encryption algorithm based on hyper-chaotic system with only one round diffusion process,” Multimed. Tools Appl., vol. 71, no. 3, pp. 1469–1497 (Aug. 2014), doi: 10.1007/s11042-012-1292-9.
[43] A. Souyah and K. M. Faraoun, “Fast and efficient randomized encryption scheme for digital images based on Quadtree decomposition and reversible memory cellular automata,” Nonlinear Dyn., vol. 84, no. 2, pp. 715–732 (Apr. 2016), doi: 10.1007/s11071-015-2521-3.
[44] Z. Man, J. Li, X. Di, and O. Bai, “An Image Segmentation Encryption Algorithm Based on Hybrid Chaotic System,” IEEE Access, vol. 7, pp. 103047–103058 (2019), doi: 10.1109/ACCESS.2019.2931732.
[45] H. Bourekouche, S. Belkacem, and N. Messaoudi, “Lightweight medical image encrypting and decrypting algorithm based on the 3D intertwining logistic map,” International Journal of Informatics and Applied Mathematics, vol. 6, no. 2, pp. 46–62 (2024).
[46] P. Ramasamy, V. Ranganathan, S. Kadry, R. Damaševičius, and T. Blažauskas, “An Image Encryption Scheme Based on Block Scrambling, Modified Zigzag Transformation and Key Generation Using Enhanced Logistic—Tent Map,” Entropy, vol. 21, no. 7, Art. no. 7 (Jul. 2019), doi: 10.3390/e21070656.
[47] M. Kumari and S. Gupta, “Performance comparison between Chaos and quantum-chaos based image encryption techniques,” Multimed. Tools Appl., vol. 80, no. 24, pp. 33213–33255 (Oct. 2021), doi: 10.1007/s11042-021-11178-3.
[48] X. Chai, “An image encryption algorithm based on bit level Brownian motion and new chaotic systems,” Multimed. Tools Appl., vol. 76, no. 1, pp. 1159–1175 (Jan. 2017), doi: 10.1007/s11042-015-3088-1.
[49] M. Kumari and S. Gupta, “Performance comparison between Chaos and quantum-chaos based image encryption techniques,” Multimed. Tools Appl., vol. 80, no. 24, pp. 33213–33255 (2021), doi: 10.1007/s11042-021-11178-3.
[50] L. Liu, Y. Zhang, and H. Zhang, “A color image encryption algorithm based on DNA computation and Chen system,” J. Phys. Conf. Ser., vol. 1074, no. 1, pp. 012096 (Sep. 2018), doi: 10.1088/1742-6596/1074/1/012096.
[51] Bourekouche H, Belkacem S, Messaoudi N. Impact of Confusion-Diffusion Complexity on Maintaining Data Security. In2024 2nd International Conference on Electrical Engineering and Automatic Control (ICEEAC) 2024 May 12 (pp. 1-6). IEEE.
[52] S. Cai, L. Huang, X. Chen, and X. Xiong, “A Symmetric Plaintext-Related Color Image Encryption System Based on Bit Permutation,” Entropy, vol. 20, no. 4, Art. no. 4 (Apr. 2018), doi: 10.3390/e20040282.
[53] P. Li, Z. Li, W. A. Halang, and G. Chen, “A multiple pseudorandom-bit generator based on a spatiotemporal chaotic map,” Phys. Lett. A, vol. 349, no. 6, pp. 467–473 (Jan. 2006), doi: 10.1016/j.physleta.2005.09.060.
[54] P. Li, Z. Li, W. A. Halang, and G. Chen, “A multiple pseudorandom-bit generator based on a spatiotemporal chaotic map,” Phys. Lett. A, vol. 349, no. 6, pp. 467–473 (Jan. 2006), doi: 10.1016/j.physleta.2005.09.060.
[55] V. Patidar, K. K. Sud, and N. K. Pareek, “A Pseudo Random Bit Generator Based on Chaotic Logistic Map and its Statistical Testing,” Informatica, vol. 33, no. 4, Art. no. 4 (2009), Accessed: Mar. 05, 2023. [Online]. Available: https://www.informatica.si/index.php/informatica/article/view/261.
[56] M. Elkandoz, W. Alexan, and H. Hussein, “Logistic Sine Map Based Image Encryption,” (Sep. 2019). doi: 10.23919/SPA.2019.8936718.
[57] D. A. Trujillo-Toledo et al., “Real-time RGB image encryption for IoT applications using enhanced sequences from chaotic maps,” Chaos Solitons Fractals, vol. 153, pp. 111506 (Dec. 2021), doi: 10.1016/j.chaos.2021.111506.
[58] A. H. Brahim, A. A. Pacha, and N. H. Said, “A new image encryption scheme based on a hyperchaotic system & multi specific S-boxes,” Inf. Secur. J. Glob. Perspect., vol. 32, no. 2, pp. 59–75 (Mar. 2023), doi: 10.1080/19393555.2021.1943572.
[59] K. Kumar, S. Roy, U. Rawat, and S. Malhotra, “IEHC: An efficient image encryption technique using hybrid chaotic map,” Chaos Solitons Fractals, vol. 158, pp. 111994 (May 2022), doi: 10.1016/j.chaos.2022.111994.
[60] L.-H. Gong, H.-X. Luo, R.-Q. Wu, and N.-R. Zhou, “New 4D chaotic system with hidden attractors and self-excited attractors and its application in image encryption based on RNG,” Phys. Stat. Mech. Its Appl., vol. 591, pp. 126793 (Apr. 2022), doi: 10.1016/j.physa.2021.126793.

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