Abstract
This paper proposes a new chaotic cryptosystem for the encryption of very high-resolution digital images based on the design of a digital chaos generator by using arbitrary precision arithmetic. This can be taken as an alternative to reduce the dynamic degradation that chaotic models present when they are implemented in digital devices and to increase the security of the cryptosystems. The obtained results show that when using high-precision arithmetic, the generated sequences provide good randomness and security during a greater number of iterations of the implemented chaotic maps in comparison with the generated sequences by using the standard of simple precision or double precision according to the IEEE 754 standard for floating-point arithmetic. The proposed method does not require high-cost hardware for increasing the numerical accuracy and security. As an advantage versus other recent works, using high precision, in relation to the methods that use simple precision or double precision, it awards an exponential increase in the key space. In this manner, it is demonstrated that using multiple-precision arithmetic, a key space of \(2^{33,268}\) or higher can be obtained, depending on the level of high precision configured. The security analysis confirms that the proposed chaotic cryptosystem is secure and robust against several known attacks, as well as statistical tests of NIST and TestU01, proving that high-precision arithmetic helps to enhance the security of the cryptosystems.
Similar content being viewed by others
References
Morabito, R., Petrolo, R., Loscri, V., Mitton, N.: LEGIoT: a lightweight edge gateway for the internet of things. Fut. Gen. Comput. Syst. 81, 1–15 (2018)
Al-Fuqaha, A., Guizani, M., Mohammadi, M., Aledhari, M., Ayyash, M.: Internet of things: a survey on enabling technologies, protocols, and applications. IEEE Commun. Surv. Tutor. 17(4), 2347–2376 (2015)
Ng, I.C.L., Wakenshaw, S.Y.L.: The internet-of-things: review and research directions. Int. J. Res. Mark. 34(1), 3–21 (2017)
Kocamaz, U.E., Çiçek, S., Uyaroğlu, Y.: Secure communication with chaos and electronic circuit design using passivity-based synchronization. J. Circuits Syst. Comput. 27(04), 1850057 (2018)
Inzunza-González, E., Cruz-Hernández, C.: Double hyperchaotic encryption for security in biometric systems. Nonlinear Dyn. Syst. Theory 13(1), 55–68 (2013)
Ferreira, H.G.C., de Sousa Junior, R.T.: Security analysis of a proposed internet of things middleware. Cluster Comput. 20(1), 651–660 (2017)
Murillo-Escobar, M.A., Cruz-Hernández, C., Abundiz-Pérez, F., López-Gutiérrez, R.M.: Implementation of an improved chaotic encryption algorithm for real-time embedded systems by using a 32-bit microcontroller. Microprocess. Microsyst. 45, 297–309 (2016)
Li, S., Mou, X., Cai, Y., Ji, Z., Zhang, J.: On the security of a chaotic encryption scheme: problems with computerized chaos in finite computing precision. Comput. Phys. Commun. 153(1), 52–58 (2003)
Li, C., Lin, D., Lü, J., Hao, F.: Cryptanalyzing an image encryption algorithm based on autoblocking and electrocardiography. IEEE Multimed. 25(4), 46–56 (2018)
Zuras, D., Cowlishaw, M., Aiken, A., Applegate, M., Bailey, D., Bass, S., Bhandarkar, D., Bhat, M., Bindel, D., Boldo, S., et al.: IEEE standard for floating-point arithmetic. IEEE Std. 754–2008, 1–70 (2008)
Azzaz, M.S., Tanougast, C., Sadoudi, S., Bouridane, A.: Synchronized hybrid chaotic generators: application to real-time wireless speech encryption. Commun. Nonlinear Sci. Numer. Simul. 18(8), 2035–2047 (2013)
Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos-based cryptosystems. Int. J. Bifurc. Chaos 16(08), 2129–2151 (2006)
Li, S., Chen, G., Mou, X.: On the dynamical degradation of digital piecewise linear chaotic maps. Int. J. Bifurc. Chaos 15(10), 3119–3151 (2005)
Deng, Y., Hu, H., Xiong, W., Xiong, N.N., Liu, L.: Analysis and design of digital chaotic systems with desirable performance via feedback control. IEEE Trans. Syst. Man Cybern. Syst. 45(8), 1187–1200 (2015)
Murillo-Escobar, M.A., Cruz-Hernández, C., Abundiz-Pérez, F., López-Gutiérrez, R.M., Del Campo, OR A.: A RGB image encryption algorithm based on total plain image characteristics and chaos. Signal Process. 109, 119–131 (2015)
Li, C., Lin, D., Feng, B., Lü, J., Hao, F.: Cryptanalysis of a chaotic image encryption algorithm based on information entropy. IEEE Access 6, 75834–75842 (2018)
García-Martínez, M., Campos-Cantón, E.: Pseudo-random bit generator based on multi-modal maps. Nonlinear Dyn. 82(4), 2119–2131 (2015)
Wang, Y., Liu, Z., Ma, J., He, H.: A pseudorandom number generator based on piecewise logistic map. Nonlinear Dyn. 83(4), 2373–2391 (2016)
Dragan, L., Mladen, N.: Pseudo-random number generator based on discrete-space chaotic map. Nonlinear Dyn. 90(1), 223–232 (2017)
Murillo-Escobar, M.A., Cruz-Hernández, C., Cardoza-Avendaño, L., Méndez-Ramírez, R.: A novel pseudorandom number generator based on pseudorandomly enhanced logistic map. Nonlinear Dyn. 87(1), 407–425 (2017)
Palacios-Luengas, L., Pichardo-Méndez, J.L., Díaz-Méndez, J.A., Rodríguez-Santos, F., Vázquez-Medina, R.: PRNG based on skew tent map. Arab. J. Sci. Eng. 1–14 (2018). https://doi.org/10.1007/s13369-018-3688-y
Sahari, M.L., Boukemara, I.: A pseudo-random numbers generator based on a novel 3D chaotic map with an application to color image encryption. Nonlinear Dyn. 94(1), 723–744 (2018)
National Institute of Standards and Technology: Security requirements for cryptographic modules. US Department of Commerce, National Institute of Standards and Technology (2017)
Bassham III, L.E., Rukhin, A.L., Soto, J., Nechvatal, J.R., Smid, M.E., Barker, E.B., Leigh, S.D., Levenson, M., Vangel, M., Banks, D.L. et al.: SP 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications. National Institute of Standards & Technology (2010)
Deng, Y., Hanping, H., Xiong, N., Xiong, W., Liu, L.: A general hybrid model for chaos robust synchronization and degradation reduction. Inf. Sci. 305, 146–164 (2015)
Liu, L., Liu, B., Hanping, H., Miao, S.: Reducing the dynamical degradation by bi-coupling digital chaotic maps. Int. J. Bifurc. Chaos 28(05), 1850059 (2018)
Wang, Q., Yu, S., Li, C., Lü, J., Fang, X., Guyeux, C., Bahi, J.M.: Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems. IEEE Trans. Circuits Syst. I: Reg. Pap. 63(3), 401–412 (2016)
Yu-Ming, X., Qiang, X., Bao, B.-C.: Grid-scroll hyperchaotic system based on microcontroller digital hardware implementation. Acta Physica Sinica 59(9), 5959–5965 (2010)
Tlelo-Cuautle, E., Rangel-Magdaleno, J.J., Pano-Azucena, A.D., Obeso-Rodelo, P.J., Nuñez-Perez, J.C.: FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear Sci. Numer. Simul. 27(1–3), 66–80 (2015)
François, M., Grosges, T., Barchiesi, D., Erra, R.: Pseudo-random number generator based on mixing of three chaotic maps. Commun. Nonlinear Sci. Numer. Simul. 19(4), 887–895 (2014)
François, M., Grosges, T., Barchiesi, D., Erra, R.: A new pseudo-random number generator based on two chaotic maps. Informatica 24(2), 181–197 (2013)
Heidari-Bateni, G., McGillem, C.D.: A chaotic direct-sequence spread-spectrum communication system. IEEE Trans. Commun. 42(234), 1524–1527 (1994)
Rodríguez-Orozco, E., García-Guerrero, E.E., Inzunza-Gonzalez, E., López-Bonilla, O.R., Flores-Vergara, A., Cárdenas-Valdez, J.R., Tlelo-Cuautle, E.: FPGA-based chaotic cryptosystem by using voice recognition as access key. Electronics 7(12), 414 (2018)
Tlelo-Cuautle, E., Carbajal-Gomez, V.H., Obeso-Rodelo, P.J., Rangel-Magdaleno, J.J., Cruz Nuñez-Perez, J.: FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn. 82(4), 1879–1892 (2015)
Sadoudi, S., Tanougast, C., Azzaz, M.S., Dandache, A.: Design and FPGA implementation of a wireless hyperchaotic communication system for secure real-time image transmission. EURASIP J. Image Video Process. 2013(1), 43 (2013)
Azzaz, M.S., Tanougast, C., Sadoudi, S., Fellah, R., Dandache, A.: A new auto-switched chaotic system and its FPGA implementation. Commun. Nonlinear Sci. Numer. Simul. 18(7), 1792–1804 (2013)
Li, C., Xie, T., Liu, Q., Cheng, G.: Cryptanalyzing image encryption using chaotic logistic map. Nonlinear Dyn. 78(2), 1545–1551 (2014)
IEEE Design Automation Standards Committee et al.: Std 1076–2008. IEEE standard VHDL language reference manual. IEEE, New York, NY, USA (2008)
Li, C., Li, S., Asim, M., Nunez, J., Alvarez, G., Chen, G.: On the security defects of an image encryption scheme. Image Vis. Comput. 27(9), 1371–1381 (2009)
Ping, P., Jinjie, W., Mao, Y., Feng, X., Fan, J.: Design of image cipher using life-like cellular automata and chaotic map. Signal Process. 150, 233–247 (2018)
Özkaynak, F.: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. 92(2), 305–313 (2018)
Lu, X., Li, Z., Li, J., Hua, W.: A novel bit-level image encryption algorithm based on chaotic maps. Opt. Lasers Eng. 78, 17–25 (2016)
Cao, C., Sun, K., Liu, W.: A novel bit-level image encryption algorithm based on 2D-LICM hyperchaotic map. Signal Process. 143, 122–133 (2018)
Pak, C., Huang, L.: A new color image encryption using combination of the 1D chaotic map. Signal Process. 138, 129–137 (2017)
Kwok, H.S., Tang, W.K.S.: A fast image encryption system based on chaotic maps with finite precision representation. Chaos Solitons Fractals 32(4), 1518–1529 (2007)
Wei, H., Guo, H., Geng, H., Zhang, K., Liu, J., Liu, X.: A novel design of software system on chip for embedded system. J. Signal Process. Syst. 86(2–3), 135–147 (2017)
Larsen, A.H., Mortensen, J.J., Blomqvist, J., Castelli, I.E., Christensen, R., Dułak, M., Friis, J., Groves, M.N., Hammer, B., Hargus, C. et al.: The atomic simulation environment—a python library for working with atoms. J. Phys. Condens. Matter. 29(27), 273002 (2017)
Smith, D.M.: Using multiple-precision arithmetic. Comput. Sci. Eng. 5(4), 88–93 (2003)
Wu, Y., Noonan, J.P., Agaian, S.: NPCR and UACI randomness tests for image encryption. Cyber J. Multidiscip. J. Sci. Technol. J. Sel. Areas Telecommun. (JSAT) 1(2), 31–38 (2011)
Marinescu, D.C.: Classical and Quantum Information. Academic Press, Cambridge (2011)
Junod, P., Canteaut, A.: Advanced Linear Cryptanalysis of Block and Stream Ciphers (Cryptology and Information Security). IOS Press, Amsterdam (2011)
Siddavaatam, P., Sedaghat, R.: A novel architecture with scalable security having expandable computational complexity for stream ciphers. Facta Universitatis, Series: Electronics and Energetics 30(4), 459–475 (2017)
Shannon, C.E.: Communication theory of secrecy systems. Bell Syst. Tech. J. 28(4), 656–715 (1949)
Yuan, S., Jiang, T., Jing, Z.: Bifurcation and chaos in the tinkerbell map. Int. J. Bifurc. Chaos 21(11), 3137–3156 (2011)
Chen, L.-Q.: An open-plus-closed-loop control for discrete chaos and hyperchaos. Phys. Lett. A 281(5–6), 327–333 (2001)
Itoh, M., Yang, T., Chua, L.O.: Conditions for impulsive synchronization of chaotic and hyperchaotic systems. Int. J. Bifurc. Chaos 11(02), 551–560 (2001)
Verhulst, P.-F.: Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux mémoires de l’académie royale des sciences et belles-lettres de Bruxelles 18, 14–54 (1845)
Hénon, M.: A two-dimensional mapping with a strange attractor. In: The Theory of Chaotic Attractors, pp. 94–102. Springer (1976)
Moon, F.C., Linsay, P.S., Mallinckrodt, A.J., McKay, S.: Chaotic and fractal dynamics: an introduction for applied scientists and engineers. Comput. Phys. 8(1), 69 (1994)
Katz, J., Menezes, A.J., Van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)
L’Ecuyer, P., Simard, R.: TestU01: A C library for empirical testing of random number generators. ACM Trans. Math. Softw. (TOMS) 33(4), 22 (2007)
Li, J., Zheng, J., Whitlock, P.: Efficient deterministic and non-deterministic pseudorandom number generation. Math. Comput. Simul. 143, 114–124 (2018)
Yang, L., Xiao-Jun, T.: A new pseudorandom number generator based on complex number chaotic equation. Chin. Phys. B 21(9), 090506 (2012)
Stoyanov, B., Kordov, K.: Novel secure pseudo-random number generation scheme based on two tinkerbell maps. Adv. Stud. Theor. Phys. 9(9), 411–421 (2015)
Adlam, E., Kent, A.: Deterministic relativistic quantum bit commitment. Int. J. Quantum Inf. 13(05), 1550029 (2015)
Acknowledgements
This paper was supported by the research project approved at the 18th Internal Call for Research Projects by UABC, with number 485. The researchers A.F.V. and E.R.O. were supported for his postgraduate studies at PhD level by CONACyT. Thanks to PRODEP (Professional Development Program for Professors) for supporting the new generations and for innovating the application of knowledge with the Number 402/377/E. The authors would like to express their gratitude to TECNM for financial support under project 6578.18-P.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Flores-Vergara, A., García-Guerrero, E.E., Inzunza-González, E. et al. Implementing a chaotic cryptosystem in a 64-bit embedded system by using multiple-precision arithmetic. Nonlinear Dyn 96, 497–516 (2019). https://doi.org/10.1007/s11071-019-04802-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-019-04802-3