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A pseudo-random numbers generator based on a novel 3D chaotic map with an application to color image encryption

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Abstract

In this work, we propose a novel 3D chaotic map obtained by coupling the piecewise and logistic maps. Showing excellent properties, like a high randomness, a high complexity and a very long period, this map has enabled us to implement and investigate a new chaotic pseudo-random number generator (CPRNG). The produced pseudo-random numbers exhibit a uniform distribution and successfully pass the NIST SP 800-22 randomness tests suite. In addition, an application in the field of color image encryption is proposed where the encryption key is strongly correlated with the plain image and is then used to perform the confusion and diffusion stages. Furthermore, the ability to expand the size of our map has an impact on the complexity of the system and increases the size of the key space, making our cryptosystems more efficient and safer. We also give some statistical tests and computer simulations which confirm that the proposed algorithm has a high level of security.

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References

  1. Abbas, N.A.: Image encryption based on independent component analysis and Arnold’s cat map. Egypt. Inf. J. 17(1), 139–146 (2016)

    Article  Google Scholar 

  2. Ahmad, J., Hwang, S.O.: Chaos-based diffusion for highly autocorrelated data in encryption algorithms. Nonlinear Dyn. 82(4), 1839–1850 (2015)

    Article  MathSciNet  Google Scholar 

  3. Alvarez, G., Li, S.: Some basic cryptographic requirements for chaos-based cryptosystems. Int. J. Bifurc. Chaos 16(08), 2129–2151 (2006)

    Article  MathSciNet  Google Scholar 

  4. Atee, H.A., Ahmad, R., Noor, N.M., Rahma, A.M.S., Sallam, M.S.: A novel extreme learning machine-based cryptography system. Secur. Commun. Netw. 9(18), 5472–5489 (2016)

    Article  Google Scholar 

  5. Atteya, A.M., Madian, A.H.: A hybrid chaos-AES encryption algorithm and its implementation based on FPGA. In: New Circuits and Systems Conference (NEWCAS), 2014 IEEE 12th International, pp. 217–220. IEEE (2014)

  6. Baptista, M.: Cryptography with chaos. Phys. Lett. A 240(1–2), 50–54 (1998)

    Article  MathSciNet  Google Scholar 

  7. Bhatnagar, G., Wu, Q.J., Raman, B.: Image and video encryption based on dual space-filling curves. Comput. J. 55(6), 667–685 (2012)

    Article  Google Scholar 

  8. Boccaletti, S., Farini, A., Arecchi, F.: Adaptive synchronization of chaos for secure communication. Phys. Rev. E 55(5), 4979 (1997)

    Article  Google Scholar 

  9. Boccaletti, S., Grebogi, C., Lai, Y.C., Mancini, H., Maza, D.: The control of chaos: theory and applications. Phys. Rep. 329(3), 103–197 (2000)

    Article  MathSciNet  Google Scholar 

  10. Boyarsky, A., Lou, Y.: Approximating measures invariant under higher-dimensional chaotic transformations. J. Approx. Theory 65(2), 231–244 (1991)

    Article  MathSciNet  Google Scholar 

  11. Collet, P., Eckmann, J.P.: Iterated maps on the interval as dynamical systems. Springer, Berlin (2009)

    Book  Google Scholar 

  12. Dellago, C., Hoover, W.G.: Finite-precision stationary states at and away from equilibrium. Phys. Rev. E 62(5), 6275 (2000)

    Article  Google Scholar 

  13. Farajallah, M., El Assad, S., Deforges, O.: Fast and secure chaos-based cryptosystem for images. Int. J. Bifurc. Chaos 26(02), 021–1650 (2016)

    Article  Google Scholar 

  14. Guan, Z.H., Huang, F., Guan, W.: Chaos-based image encryption algorithm. Phys. Lett. A 346(1), 153–157 (2005)

    Article  Google Scholar 

  15. Haroun, M.F., Gulliver, T.A.: A new 3D chaotic cipher for encrypting two data streams simultaneously. Nonlinear Dyn. 81(3), 1053–1066 (2015)

    Article  Google Scholar 

  16. Hilborn, R.C.: Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford University Press on Demand, Oxford (2000)

    Book  Google Scholar 

  17. Hotoleanu, D., Cret, O., Suciu, A., Gyorfi, T., Vacariu, L.: Real-time testing of true random number generators through dynamic reconfiguration. In: Digital System Design: Architectures, Methods and Tools (DSD), 2010 13th Euromicro Conference on, pp. 247–250. IEEE (2010)

  18. Huang, C., Nien, H.: Multi chaotic systems based pixel shuffle for image encryption. Opt. Commun. 282(11), 2123–2127 (2009)

    Article  Google Scholar 

  19. Jakimoski, G., Kocarev, L.: Analysis of some recently proposed chaos-based encryption algorithms. Phys. Lett. A 291(6), 381–384 (2001)

    Article  MathSciNet  Google Scholar 

  20. Kanso, A., Ghebleh, M.: A novel image encryption algorithm based on a 3d chaotic map. Commun. Nonlinear Sci. Numer. Simul. 17(7), 2943–2959 (2012)

    Article  MathSciNet  Google Scholar 

  21. Khan, M., Shah, T.: An efficient chaotic image encryption scheme. Neural Comput. Appl. 26(5), 1137–1148 (2015)

    Article  Google Scholar 

  22. Kun, Y., Han, Z., Zhaohui, L.: An improved AES algorithm based on chaos. In: Multimedia Information Networking and Security, 2009. MINES’09. International Conference on, vol. 2, pp. 326–329. IEEE (2009)

  23. Kwok, H., Tang, W.K.: A fast image encryption system based on chaotic maps with finite precision representation. Chaos, Solitons Fractals 32(4), 1518–1529 (2007)

    Article  MathSciNet  Google Scholar 

  24. Lanford III, O.E.: Informal remarks on the orbit structure of discrete approximations to chaotic maps. Exp. Math. 7(4), 317–324 (1998)

    Article  MathSciNet  Google Scholar 

  25. Li, C., Liu, Y., Xie, T., Chen, M.Z.: Breaking a novel image encryption scheme based on improved hyperchaotic sequences. Nonlinear Dyn. 73(3), 2083–2089 (2013)

    Article  MathSciNet  Google Scholar 

  26. Li, C., Zhang, L.Y., Ou, R., Wong, K.W., Shu, S.: Breaking a novel colour image encryption algorithm based on chaos. Nonlinear Dyn. 70(4), 2383–2388 (2012)

    Article  MathSciNet  Google Scholar 

  27. Li, S., Mou, X., Cai, Y.: Improving security of a chaotic encryption approach. Phys. Lett. A 290(3), 127–133 (2001)

    Article  MathSciNet  Google Scholar 

  28. Lian, S., Sun, J., Wang, Z.: Security analysis of a chaos-based image encryption algorithm. Physica A 351(2), 645–661 (2005)

    Article  Google Scholar 

  29. Liu, S., Sun, J., Xu, Z.: An improved image encryption algorithm based on chaotic system. JCP 4(11), 1091–1100 (2009)

    Google Scholar 

  30. Lozi, R.: Chaotic pseudo random number generators via ultra weak coupling of chaotic maps and double threshold sampling sequences. In: ICCSA 2009, 3rd Conference on Complex Systems and Applications., pp. 20–24 (2009)

  31. Lozi, R.: Emergence of randomness from chaos. Int. J. Bifurc. Chaos 22(02), 021–1250 (2012)

    Article  MathSciNet  Google Scholar 

  32. Matthews, R.: On the derivation of a chaotic encryption algorithm. Cryptologia 13(1), 29–42 (1989)

    Article  MathSciNet  Google Scholar 

  33. Norouzi, B., Mirzakuchaki, S.: A fast color image encryption algorithm based on hyper-chaotic systems. Nonlinear Dyn. 78(2), 995–1015 (2014)

    Article  Google Scholar 

  34. Norouzi, B., Seyedzadeh, S.M., Mirzakuchaki, S., Mosavi, M.R.: A novel image encryption based on hash function with only two-round diffusion process. Multimedia Syst. 20(1), 45–64 (2014)

    Article  Google Scholar 

  35. Overton, M.L.: Numerical computing with IEEE floating point arithmetic. SIAM (2001)

  36. Pareschi, F., Rovatti, R., Setti, G.: On statistical tests for randomness included in the NIST SP800-22 test suite and based on the binomial distribution. IEEE Trans. Inf. Forensics Secur. 7(2), 491–505 (2012)

    Article  Google Scholar 

  37. Patidar, V., Pareek, N., Sud, K.: A new substitution-diffusion based image cipher using chaotic standard and logistic maps. Commun. Nonlinear Sci. Numer. Simul. 14(7), 3056–3075 (2009)

    Article  Google Scholar 

  38. Rhouma, R., Meherzi, S., Belghith, S.: OCML-based colour image encryption. Chaos, Solitons Fractals 40(1), 309–318 (2009)

    Article  Google Scholar 

  39. Rhouma, R., Solak, E., Belghith, S.: Cryptanalysis of a new substitution-diffusion based image cipher. Commun. Nonlinear Sci. Numer. Simul. 15(7), 1887–1892 (2010)

    Article  MathSciNet  Google Scholar 

  40. Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E.: A statistical test suite for random and pseudorandom number generators for cryptographic applications. Tech. rep, DTIC Document (2001)

  41. Seyedzadeh, S.M., Norouzi, B., Mosavi, M.R., Mirzakuchaki, S.: A novel color image encryption algorithm based on spatial permutation and quantum chaotic map. Nonlinear Dyn. 81(1–2), 511–529 (2015)

    Article  MathSciNet  Google Scholar 

  42. Shannon, C.E.: Communication theory of secrecy systems. Bell Labs Tech. J. 28(4), 656–715 (1949)

    Article  MathSciNet  Google Scholar 

  43. Soltanalian, M., Stoica, P.: Computational design of sequences with good correlation properties. IEEE Trans. Signal Process. 60(5), 2180–2193 (2012)

    Article  MathSciNet  Google Scholar 

  44. Som, S., Dutta, S., Singha, R., Kotal, A., Palit, S.: Confusion and diffusion of color images with multiple chaotic maps and chaos-based pseudorandom binary number generator. Nonlinear Dyn. 80(1–2), 615–627 (2015)

    Article  Google Scholar 

  45. Sornette, D., Arneodo, A.: Chaos, pseudo-random number generators and the random walk problem. J. de Physique 45(12), 1843–1857 (1984)

    Article  MathSciNet  Google Scholar 

  46. Cormen, Thomas: H and Leiserson. Introduction to Algorithms. 2nd Ed. McGraw-Hill, Charles E and Rivest, Ronald L and Stein, Clifford (2001)

  47. Hanrot, Guillaume, Lefevre, Vincent, Pélissier, Patrick, Théveny, Philippe, Zimmermann, Paul : The GNU MPFR library (2005)

  48. Li, Shujun, Chen, Guanrong, Mou, Xuanqin: On the dynamical degradation of digital piecewise linear chaotic maps. Int. J. Bifurc. Chaos 15(10), 3119–3151 (2005)

    Article  MathSciNet  Google Scholar 

  49. Li, Shujun, Li, Qi, Li, Wenmin, Mou, Xuanqin, Cai, Yuanlong: Statistical properties of digital piecewise linear chaotic maps and their roles in cryptography and pseudo-random coding. In: IMA International Conference on Cryptography and Coding, pp. 205–221. Springer (2001)

  50. Shujun, Li, Xuanqin, Mou, Yang, Boliya L., Zhen, Ji, Jihong, Zhang: Problems with a probabilistic encryption scheme based on chaotic systems. Int. J. Bifurc. Chaos 13(10), 3063–3077 (2003)

    Article  MathSciNet  Google Scholar 

  51. Luo, Yuling, Cao, Lvchen, Qiu, Senhui, Lin, Hui, Harkin, Jim, Liu, Junxiu: A chaotic map-control-based and the plain image-related cryptosystem. Nonlinear Dyn. 83(4), 2293–2310 (2016)

    Article  MathSciNet  Google Scholar 

  52. Özkaynak, Fatih: Brief review on application of nonlinear dynamics in image encryption. Nonlinear Dyn. pp. 1–9 (2018)

    Article  Google Scholar 

  53. Persohn, K.J., Povinelli, Richard J.: Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation. Chaos, Solitons Fractals 45(3), 238–245 (2012)

    Article  Google Scholar 

  54. Tong, X., Cui, M.: Image encryption scheme based on 3d baker with dynamical compound chaotic sequence cipher generator. Signal Process. 89(4), 480–491 (2009)

    Article  Google Scholar 

  55. Tong, X.J., Wang, Z., Zhang, M., Liu, Y., Xu, H., Ma, J.: An image encryption algorithm based on the perturbed high-dimensional chaotic map. Nonlinear Dyn. 80(3), 1493–1508 (2015)

    Article  MathSciNet  Google Scholar 

  56. Vaish, A., Kumar, M.: Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain. Optik-Int. J. Light Electron Opt. 145, 273–283 (2017)

    Article  Google Scholar 

  57. Wang, S., Kuang, J., Li, J., Luo, Y., Lu, H., Hu, G.: Chaos-based secure communications in a large community. Phys. Rev. E 66(6), 065–202 (2002)

    Google Scholar 

  58. Wang, X., Guo, K.: A new image alternate encryption algorithm based on chaotic map. Nonlinear Dyn. 76(4), 1943–1950 (2014)

    Article  Google Scholar 

  59. Wang, X.Y., Zhang, Y.Q., Zhao, Y.Y.: A novel image encryption scheme based on 2-d logistic map and DNA sequence operations. Nonlinear Dyn. 82(3), 1269–1280 (2015)

    Article  MathSciNet  Google Scholar 

  60. Wang, Y., Liu, Z., Ma, J., He, H.: A pseudorandom number generator based on piecewise logistic map. Nonlinear Dyn. 83(4), 2373–2391 (2016)

    Article  MathSciNet  Google Scholar 

  61. Wang, Y., Wong, K.W., Liao, X., Xiang, T., Chen, G.: A chaos-based image encryption algorithm with variable control parameters. Chaos, Solitons Fractals 41(4), 1773–1783 (2009)

    Article  Google Scholar 

  62. Wei, X., Guo, L., Zhang, Q., Zhang, J., Lian, S.: A novel color image encryption algorithm based on DNA sequence operation and hyper-chaotic system. J. Syst. Softw. 85(2), 290–299 (2012)

    Article  Google Scholar 

  63. Wong, K.W., Kwok, B.S.H., Law, W.S.: A fast image encryption scheme based on chaotic standard map. Phys. Lett. A 372(15), 2645–2652 (2008)

    Article  Google Scholar 

  64. Wu, X., Li, Y., Kurths, J.: A new color image encryption scheme using CML and a fractional-order chaotic system. PLoS ONE 10(3), e0119–660 (2015)

    Google Scholar 

  65. Xiao, D., Liao, X., Wei, P.: Analysis and improvement of a chaos-based image encryption algorithm. Chaos, Solitons Fractals 40(5), 2191–2199 (2009)

    Article  MathSciNet  Google Scholar 

  66. Xie, E.Y., Li, C., Yu, S., Lü, J.: On the cryptanalysis of Fridrich’s chaotic image encryption scheme. Signal Process. 132, 150–154 (2017)

    Article  Google Scholar 

  67. Yao, W., Wu, F., Zhang, X., Zheng, Z., Wang, Z., Wang, W., Qiu, W.: A fast color image encryption algorithm using 4-pixel Feistel structure. PLoS ONE 11(11), e0165–937 (2016)

    Google Scholar 

  68. Yao, W., Zhang, X., Zheng, Z., Qiu, W.: A colour image encryption algorithm using 4-pixel Feistel structure and multiple chaotic systems. Nonlinear Dyn. 81(1–2), 151–168 (2015)

    Article  MathSciNet  Google Scholar 

  69. Yuan, G., Yorke, J.A.: Collapsing of chaos in one dimensional maps. Physica D 136(1), 18–30 (2000)

    Article  MathSciNet  Google Scholar 

  70. Zaher, A.A., Abu-Rezq, A.: On the design of chaos-based secure communication systems. Commun. Nonlinear Sci. Numer. Simul. 16(9), 3721–3737 (2011)

    Article  MathSciNet  Google Scholar 

  71. Zhang, Q., Wei, X.: RGB color image encryption method based on Lorenz chaotic system and DNA computation. IETE Tech. Rev. 30(5), 404–409 (2013)

    Article  MathSciNet  Google Scholar 

  72. Zhu, C.: A novel image encryption scheme based on improved hyperchaotic sequences. Opt. Commun. 285(1), 29–37 (2012)

    Article  Google Scholar 

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Acknowledgements

The authors would like thank the anonymous reviewers for their helpful comments and Mr H. Sissaoui and Mr M-K. Blackmore for their help in the preparation of this paper.

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  1. LANOS Laboratory, Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

    Mohamed Lamine Sahari

  2. MDM Laboratory, Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000, Annaba, Algeria

    Ibtissem Boukemara

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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, 723–744 (2018). https://doi.org/10.1007/s11071-018-4390-z

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