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Construction of a novel nth-order polynomial chaotic map and its application in the pseudorandom number generator

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Abstract

Chaotic maps with good chaotic performance have been extensively designed in cryptography recently. This paper gives an nth-order polynomial chaotic map by using topological conjugation with piecewise linear chaos map. The range of chaotic parameters of this nth-order polynomial chaotic map is large and continuous. And the larger n is, the greater the Lyapunov exponent is and the more complex the dynamic characteristic of the nth-order polynomial chaotic map. The above characteristics of the nth-order polynomial chaotic map avoid the disadvantages of one-dimensional chaotic systems in secure application to some extent. Furthermore, the nth-order polynomial chaotic map is proved to be an extension of the Chebyshev polynomial map, which enriches chaotic map. The numerical simulation of dynamic behaviors for an 8th-order polynomial map satisfying the chaotic condition is carried out, and the numerical simulation results show the correctness of the related conclusion. This paper proposed the pseudorandom number generator according to the 8th-order polynomial chaotic map constructed in this paper. Using the performance analysis of the proposed pseudorandom number generator, the analysis result shows that the pseudorandom number generator according to the 8th-order polynomial chaotic map can efficiently generate pseudorandom sequences with higher performance through the randomness analysis with NIST SP800-22 and TestU01, security analysis and efficiency analysis. Compared with the other pseudorandom number generators based on chaotic systems in recent references, this paper performs a comprehensive performance analysis of the pseudorandom number generator according to the 8th-order polynomial chaotic map, which indicates the potential of its application in cryptography.

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References

  1. Naskar, P.K., Bhattacharyya, S., Nandy, D., Chaudhuri, A.: A robust image encryption scheme using chaotic Tent map and cellular automata. Nonlinear Dyn. 100(3), 2877–2898 (2020)

    Article  Google Scholar 

  2. Ben Farah, M.A., Farah, A., Farah, T.: An image encryption scheme based on a new hybrid chaotic map and optimized substitution box. Nonlinear Dyn. 99(4), 3041–3064 (2020)

    Article  Google Scholar 

  3. Zhang, Z.Q., Wang, Y., Zhang, L.Y., Zhu, H.: A novel chaotic map constructed by geometric operations and its application. Nonlinear Dyn. 102(4), 2843–2858 (2020)

    Article  Google Scholar 

  4. Zhao, Y., Gao, C.Y., Liu, J., Dong, S.Z.: A self-perturbed pseudo-random sequence generator based on hyperchaos. Chaos Solitons Fract. 4, 100023 (2019)

    Article  Google Scholar 

  5. Zang, H.Y., Meng, D.T., Wei, X.Y.: Image encryption schemes based on a class of uniformly distributed chaotic systems. Mathematics 10(7), 1027 (2022)

    Article  Google Scholar 

  6. Wang, Y., Liu, Z., Zhang, L.Y., Pareschi, F., Setti, G., Chen, G.R.: From chaos to pseudorandomness: a case study on the 2-D coupled map lattice. IEEE Trans Cybern (2021). https://doi.org/10.1109/TCYB.2021.3129808

    Article  Google Scholar 

  7. Zang, H.Y., Liu, J.Y., Li, J.: Construction of a class of high-dimensional discrete chaotic systems. Mathematics 9(4), 365 (2021)

    Article  Google Scholar 

  8. Yang, X.P., Min, L.Q., Wang, X.: A cubic map chaos criterion theorem with applications in generalized synchronization based pseudo-random number generator and image encryption. Chaos 25(5), 053104 (2015)

    Article  MathSciNet  Google Scholar 

  9. Luca, A., Ilyas, A., Vlad, A.: Generating random binary sequences using Tent map. In: International Symposium on Signals, Circuits and Systems (ISSCS), pp. 81–84. IEEE (2011)

  10. Zhang, X., Shi, Y.M., Chen, G.R.: Constructing chaotic polynomial maps. Int. J. Bifurc. Chaos. 19(2), 531–543 (2009)

    Article  MathSciNet  Google Scholar 

  11. Zhang, X.: Chaotic polynomial maps. Int. J. Bifurc. Chaos. 26(8), 1650131 (2016)

    Article  MathSciNet  Google Scholar 

  12. Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)

    Article  MathSciNet  Google Scholar 

  13. Zhou, H.L., Song, E.B.: Discrimination of the 3-periodic points of a quadratic polynomial. J. Sichuan Univ. 46(3), 561–564 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Theodore, J.R.: Chebyshev polynomials. Wiley, New York (1990)

    MATH  Google Scholar 

  15. Xu, Z.G., Tian, Q., Tian, L.: A class of topologically conjugated chaotic maps of Tent map to generate independently and uniformly distributed chaotic key stream. Acta Phys. Sin. 62(13), 120501 (2013)

    Google Scholar 

  16. Zang, H.Y., Huang, H.F., Chai, H.Y.: Homogenization method for the quadratic polynomial chaotic system. J. Electron. Inf. Technol. (2019). https://doi.org/10.11999/JEIT180735

  17. Zang, H.Y., Wei, X.Y., Yuan, Y.: Determination and properties analysis of a cubic polynomial chaotic map. J. Electron. Inf. Technol. (2021). https://doi.org/10.11999/JEIT190875

  18. Chu, J.X., Min, L.Q.: Chaos criterion theorems on specific 2n order and 2n+1 order polynomial discrete maps with application. Sciencepaper Online. http://www.paper.edu.cn/releasepaper/conTent/202001-28. Accessed 20 July 2021

  19. Hamza, R.: A novel pseudo random sequence generator for image-cryptographic applications. J. Inf. Secur. Appl. 35, 119–127 (2017)

    Google Scholar 

  20. García-Martínez, M., Campos-Cantón, E.: Pseudo-random bit generator based on multi-modal maps. Nonlinear Dyn. 82, 2119–2131 (2015)

    Article  MathSciNet  Google Scholar 

  21. Elmanfaloty, R.A., Abou-Bakr, E.: Random property enhancement of a 1D chaotic PRNG with finite precision implementation. Chaos Solitons Fract. 118, 134–144 (2019)

    Article  MathSciNet  Google Scholar 

  22. Meranza-Castillón, M.O., Murillo-Escobar, M.A., López-Gutiérrez, R.M., Cruz-Hernández, C.: Pseudorandom number generator based on enhanced Hénon map and its implementation. AEU-Int. J. Electron. Commun. 107, 239–251 (2019)

    Article  Google Scholar 

  23. Dastgheib, M.A., Farhang, M.: A digital pseudo-random number generator based on sawtooth chaotic map with a guaranteed enhanced period. Nonlinear Dyn. 89(4), 2957–2966 (2017)

    Article  MathSciNet  Google Scholar 

  24. Pierre, C., Jean-pierre, E.: Iterated Maps on the Interval as Dynamical Systems. Birkhäuser, Basel (2009)

    MATH  Google Scholar 

  25. Duan, P.A.: The general term formula for the polynomial of cosx used to express cosnx. Sciencepaper Online. http://www.paper.edu.cn/releasepaper/conTent/201004-960. Accessed 10 Dec 2021

  26. Hao, B.L.: Starting with Parabola: An Introduction to Chaotic Dynamics. Peking University Press, Beijing (2013)

    Google Scholar 

  27. Min, L.Q., Zhang, L.J., Zhang, Y.Q.: A novel chaotic system and design of pseudorandom number generator. In: International Conference on Intelligent Control and Information Processing (ICICIP), pp. 545–550. IEEE (2013)

  28. Bassham, L., Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E., Leigh, S., Levenson, M., Vangel, M., Banks, D., Hecke-rt, N., Dray, J.: A statistical test suite for random and pseudo-random number generator for cryptographic applications. National Institute of Standards and Technology. https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication800-22r1a.pdf. Accessed 20 July 2021

  29. Ecuyer, L., Simard, R.: TestU01: a C library for empirical testing of random number generators. ACM Trans. Math. Softw. 33(22), 1–40 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. IEEE Computer Society: IEEE Standard Binary Floating-Point Arithmetic, ANSI/IEEE std (1985)

  32. 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, 407–425 (2017)

    Article  MathSciNet  Google Scholar 

  33. Stoyanov, B., Kordov, K.: Novel secure pseudo-random number generation scheme based on two tinkerbell maps. Adv. Stud. Theor. Phys. 9, 411–421 (2015)

    Article  Google Scholar 

  34. Lambić, D., Nikolić, M.: Pseudo-random number generator based on discrete-space chaotic map. Nonlinear Dyn. 90, 223–232 (2017)

    Article  MathSciNet  Google Scholar 

  35. Zang, H.Y., Zhao, X.X., Wei, X.Y.: Construction and application of new high-order polynomial chaotic maps. Nonlinear Dyn. 107(1), 1247–1261 (2022)

    Article  Google Scholar 

  36. Gerardo de la Fraga, L.G., Torres-Pérez, E., Tlelo-Cuautle, E., Mancillas-López, C.: Hardware implementation of pseudo-random number generators based on chaotic maps. Nonlinear Dyn. 90, 1661–1670 (2017)

    Article  Google Scholar 

  37. Rezk, A.A., Madian, A.H., Radwan, A.G., Soliman, A.M.: Reconfigurable chaotic pseudo random number generator based on FPGA. AEU-Int. J. Electron. Commun. 98, 174–180 (2019)

    Article  Google Scholar 

  38. Thane, A., Chaudhari, R.: Hardware design and implementation of pseudorandom number generator using piecewise linear chaotic map. In: International Conference on Advances in Computing, Communications and Informatics (ICACCI), pp. 456–459. IEEE (2018)

  39. Zang, H.Y., Yuan, Y., Wei, X.Y.: Research on pseudorandom number generator based on several new types of piecewise chaotic maps. Math. Probl. Eng. 2021, 1375346 (2021)

    Google Scholar 

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  1. Mathematics and Physics School, University of Science and Technology Beijing, Beijing, 100083, China

    Xinxin Zhao, Hongyan Zang & Xinyuan Wei

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  1. Xinxin Zhao
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  2. Hongyan Zang
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  3. Xinyuan Wei
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Correspondence to Hongyan Zang.

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Zhao, X., Zang, H. & Wei, X. Construction of a novel nth-order polynomial chaotic map and its application in the pseudorandom number generator. Nonlinear Dyn 110, 821–839 (2022). https://doi.org/10.1007/s11071-022-07641-x

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