FPGA Implementation of a Chaotic Quadratic Map for Cryptographic Applications

Abstract

A hardware implementation of a quadratic map through FPGA platform is proposed in this paper. Firstly, a
chaotic quadratic map is modeled by using Matlab/Simulink programming and then implemented into the FPGA
(Field Programmable Gate Array) to be used for key generation for cryptographic applications. When the
quadratic map is in chaotic mode, its output is unpredictable and aperiodic. Besides this, the map has a uniform
output distribution and sufficient randomness. These characteristics make the chaotic quadratic map a suitable
key generator for cryptography. This paper also reveals the successful real-time implementation of the quadratic
map using FPGA for practical applications. Experimental results confirm that the feasibility of the quadratic map
is verified under a digital hardware environment.

Keywords

• Chaos,Quadratic Map,Implementation,FPGA

References

1. 1. Kang, Z., Sun, J., Ma, L., Qi, Y. and Jian, S., (2014). Multimode synchronization of chaotic semiconductor ring laser and its potential in chaos communication. IEEE journal of Quantum Electronics, vol. 50, pp. 148-157.
2. 2. Yang, J., Chen, Y. and Zhu, F., (2015) .Associated observer-based synchronization for uncertain chaotic systems subject to channel noise and chaos-based secure communication. Neurocomputing, vol. 167, pp. 587-595.
3. 3. Eisencraft, M., at al., (2012). Chaos-based communication systems in non-ideal channels. Communications in Nonlinear Science and Numerical Simulation, vol. 17, pp. 4707-4718.
4. 4. Kaddoum, G., Coulon, M., Roviras, D. and Charge, P., (2010). Theoritical performance for asynchronous multi-user chaos-based communication systems on fading channels. Signal Processing, vol. 90, pp. 2923- 2933.
5. 5. Zaher, A. A. and Abu-Rezq, A., (2011). On the design of chaos-based secure communication systems, Communications in Nonlinear Science and Numerical Simulation, vol. 16, pp. 3721-3737.
6. 6. Turk, M. and Ogras, H., (2011). Classification of chaos-based digital modulation techniques using wavelet neural networks and performance comparison of wavelet families. Expert Systems with Applications, vol. 38, pp. 2557-2565.
7. 7. Zhu, Z. L., Zhang, W., Wong, K. W., Yu, H., (2011). A Chaos-based symmetric image encryption scheme using a bit-level permutation. Information Sciences, vol. 181, pp. 1171-1186.
8. 8. Patidar, V., Pareek, N. K., Purohit, G. and Sud, K. K., (2011). A Robust and secure chaotic standard map based pseudorandom permutation-substitution scheme for image encryption. Optics Communications, vol. 284, pp. 4331-4339.

9. 9. Murillo-Escobar, M. A. et al., (2015). A RGB image encryption algorithm based on total plain image characteristics and chaos. Signal Processing, vol. 109, pp. 119-131.
10. 10. Ye, R., and Guo, W., (2014). An image encryption scheme Multimode synchronization of chaotic semicon based on chaotic systems with changeable parameters,” I. J. Computer Network and Information Security, vol. 4, pp. 37-45.
11. 11. Zhu, H., Zhao, C. and Zhang, X., (2013). A Novel image encryption-compression scheme using hyperchaos and Chinese remainder theorem. Signal Processing: Image Communication, vol. 28, pp. 670- 680.
12. 12. Ogras, H. and Turk, M., (2017). A Robust chaosbased image cryptosystem with an improved key generator and plain image sensitivity mechanism. Journal of Information Security, vol. 8, pp. 23-41.
13. 13. Yibei, W., Man, L., Yanting, X. and Hougui, C., (2011). Research on chaos phenomena in power systems. Power engineering and automation conference, vol. 2, pp. 453-456.
14. 14. Yau, H. T., Wang, M. H., Wang, T. Y. and Chen, G., (2015). Signal clustering of power disturbance by using chaos synchronization. Int. J. Electr. Power Energy System, vol. 64, pp. 112-120.
15. 15. Ghasemi, M., Ghavidel, S., Aghaei, J., Gitizadeh, M. and Falah, H., (2014). Application of chaos-based chaotic invasive weed optimization techniques for environmental OPF problems in the power systems. Chaos, Solitons Fract., vol. 69, pp. 271-284.
16. 16. Chen, Q., Ren, X. and Na, J., (2015). Robust finite-time chaos synchronization of uncertain permament magnet synchronous motors. ISA Trans., vol. 58, pp. 262-269.
17. 17. Merah, L., Ali-Pacha, A., Said, N. H. and Mamat, M., (2013). Design and FPGA implementation of Lorenz chaotic system for information security issues. Applied Mathematical Sciences, vol. 7, pp. 237-246.
18. 18. Xue, H., Wang S. and Meng, X., (2013). Study on one modified chaotic system based on Logistic map. Res. J. Appl. Sci. Eng. Technol., vol. 5, pp. 898-904.
19. 19. Aseeri, M. A. and Sobhy,M. I., (2002). A New approach to implement Chaotic generators based on Field Programmable Gate Array (FPGA). Proc. 3rd. Int. Conf. Discrete Chaotic Dynam. Nature Soc., September.
20. 20. Mao, Y., Cao, L. and Liu, W., (2006). Design and FPGA implementation of a pseudo-random bit sequence generator using spatiotemporal chaos. IEEE Proceedings of International Conference on Communications, Circuits and Systems, pp. 2114- 2118.
21. 21. Lian, S., Sun, J. and Wang, Z., (2005). Security analysis of a chaos-based image encryption algorithm. Physica A: Statistical Mechanics and its Applications, vol. 351, pp. 645-661.
22. 22. Ramadan, N., Ahmed, H. E., Elkhamy H S. E., and Abd El-Samie, F. E., (2016). Chaos-based image encryption using an improved quadratic chaotic map. American Journal of Signal Processing, vol. 6, pp. 1- 13.
23. 23. Hathal, H. M., Abdulhussein, R. A. and Ibrahim, S. K., (2014). Lyapunov exponent testing for AWGN generator system. Communications and Network, vol. 6, pp. 201-208.
24. 24. Marton, K., Suciu, A., Sacarea, C. and Cret, O., (2012). Generation and testing of random numbers for cryptographic applications. Proceedings of the Romanian Academy, vol. 13, pp. 368-377.
25. 25. Rukhin, A., Soto, J., Nechvatal, J. and Smid, M., (2010). A Statistical Test for random and psudorandom number generators for cryptographic applications. NIST Special Publication 800-22 rev1, pp. 2-40.
26. 26. Chen, J. X., Zhu, Z. L., Fu, C., Yu, H. and Zhang, L.B (2015). A Fast chaos-based image encryption scheme with a dynamic state variables selection mechanism. Communications in Nonlinear Science and Numerical Simulation, vol. 20, pp. 846-860.