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Image compression and encryption algorithm based on uniform non-degeneracy chaotic system and fractal coding

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Abstract

This paper focuses on the design of chaotic image compression encryption algorithms. Firstly, we design a uniform non-degenerate chaotic system based on nonlinear filters and the feed-forward and feed-back structure. Theoretical and experimental analyses indicate that the system can avoid the drawbacks of the existing chaotic systems, such as chaos degradation, uneven trajectory distribution, and weak chaotic behavior. In addition, our chaotic system can produce chaotic sequences with good pseudo-random characteristics. Then, we propose a fractal image compression algorithm based on adaptive horizontal or vertical (HV) partition by improving the baseline HV partition and the time-consuming global matching algorithm. The algorithm does not need to implement time-consuming global matching operations. In addition, analysis results demonstrate that our fractal image compression algorithm can reconstruct the original image with high quality under ultra-high compression ratios. Finally, to protect the confidentiality of images, we propose a chaotic fractal image compression and encryption algorithm by using our chaotic system and fractal image compression algorithm. The algorithm achieves excellent diffusion and confusion abilities without using the hash value of plain images. Therefore, it avoids the failure of decryption caused by the tampering of hash value during the transmission process, and can well resist differential attacks and chosen-ciphertext attacks. In addition, simulation results show the algorithm is efficient and robust.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Ghadirli, H.M., Nodehi, A., Enayatifar, R.: An overview of encryption algorithms in color images. Signal Process. 164, 163–185 (2019)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Li, C., Luo, G., Qin, K., et al.: An image encryption scheme based on chaotic tent map. Nonlinear Dyn. 87(1), 127–133 (2017)

    Article  Google Scholar 

  4. Hussain, I., Shah, T., Gondal, M.A., et al.: A novel image encryption algorithm based on chaotic maps and GF (2\(^8\)) exponent transformation. Nonlinear Dyn. 72(1), 399–406 (2013)

    Article  MathSciNet  Google Scholar 

  5. Hua, Z., Zhou, Y.: Image encryption using 2D logistic-adjusted-sine map. Inf. Sci. 339, 237–253 (2016)

    Article  Google Scholar 

  6. Hua, Z., Zhu, Z., Yi, S., et al.: Cross-plane colour image encryption using a two-dimensional logistic tent modular map. Inf. Sci. 546, 1063–1083 (2021)

    Article  MathSciNet  Google Scholar 

  7. Wang, X., Zhang, H.L.: A novel image encryption algorithm based on genetic recombination and hyper-chaotic systems. Nonlinear Dyn. 83(1), 333–346 (2016)

    Article  MathSciNet  Google Scholar 

  8. Liu, H., Zhang, Y., Kadir, A., et al.: Image encryption using complex hyper chaotic system by injecting impulse into parameters. Appl. Math. Comput. 360, 83–93 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, X., Tong, X., Wang, Z., et al.: A new n-dimensional conservative chaos based on generalized Hamiltonian system and its’ applications in image encryption. Chaos Solitons Fractals 154, 111693 (2022)

    Article  MathSciNet  Google Scholar 

  10. Wang, X., Yang, J.: A privacy image encryption algorithm based on piecewise coupled map lattice with multi dynamic coupling coefficient. Inf. Sci. 569, 217–240 (2021)

    Article  MathSciNet  Google Scholar 

  11. Lu, G., Smidtaite, R., Howard, D., et al.: An image hiding scheme in a 2-dimensional coupled map lattice of matrices. Chaos Solitons Fractals 124, 78–85 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Song, Y., Zhu, Z., Zhang, W., et al.: Joint image compression–encryption scheme using entropy coding and compressive sensing. Nonlinear Dyn. 95(3), 2235–2261 (2019)

    Article  MATH  Google Scholar 

  13. Donoho, D.L.: Compressed sensing. IEEE Trans. Inform. Theory. 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52(2), 489–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhang, B., Xiao, D., Xiang, Y.: Robust coding of encrypted images via 2D compressed sensing. IEEE Trans. Multimed. 23, 2656–2671 (2020)

    Article  Google Scholar 

  16. Chai, X., Wu, H., Gan, Z., et al.: An efficient approach for encrypting double color images into a visually meaningful cipher image using 2D compressive sensing. Inf. Sci. 556, 305–340 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Barnsley, M.F., Sloan, A.D.: Fractal image compression. In: NASA. Goddard Space Flight Center, Proceedings of the Scientific Data Compression Workshop (1989)

  18. Jacquin, A.E., et al.: Image coding based on a fractal theory of iterated contractive image transformations. IEEE Trans. Image process. 1(1), 18–30 (1992)

    Article  Google Scholar 

  19. Truong, T., Kung, C., Jeng, J., et al.: Fast fractal image compression using spatial correlation. Chaos Solitons Fractals 22(5), 1071–1076 (2004)

    Article  MATH  Google Scholar 

  20. Wang, X.Y., Wang, Y.X., Yun, J.J.: An improved fast fractal image compression using spatial texture correlation. Chin. Phys. B. 20(10), 104202 (2011)

    Article  Google Scholar 

  21. Nandi, U.: Fractal image compression with adaptive quadtree partitioning and non-linear affine map. Multimed. Tools Appl. 79(35), 26345–26368 (2020)

    Article  Google Scholar 

  22. Wang, X.Y., Zhang, D.D.: Discrete wavelet transform-based simple range classification strategies for fractal image coding. Nonlinear Dyn. 75(3), 439–448 (2014)

    Article  Google Scholar 

  23. Gupta, R., Mehrotra, D., Tyagi, R.: Adaptive searchless fractal image compression in DCT domain. Imaging Sci. J. 64(7), 374–380 (2016)

    Article  Google Scholar 

  24. Al-Bundi, S.S., Al-Saidi, N.M.: Time optimizing of fractal image compression using the scatter search algorithm. Ital. J. Pure Appl. Mat. 47, 365–381 (2020)

    Google Scholar 

  25. Al-Saidi, N.M.G., Al-Bundi, S.S.: An improved harmony search algorithm for reducing computational time of fractal image coding. J. Theor. Appl. Inf. Technol. 95(8), 1669–1679 (2017)

    Google Scholar 

  26. Cheng, M.H.M., Bakhoum, E.G.: Fractal compression and adaptive sampling with HV partitioning: accelerating the scanning process in scanning probe microscopy. In: ASME International Mechanical Engineering Congress and Exposition, vol. 43857, pp. 195–201 (2009)

  27. Nandi, U.: Fractal image compression using a fast affine transform and hierarchical classification scheme. Vis. Comput. 38(11), 3867–3880 (2022)

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Wu, Y., Hua, Z., Zhou, Y.: \(n\)-dimensional discrete cat map generation using Laplace expansions. IEEE Trans. Cybern. 46(11), 2622–2633 (2015)

    Article  Google Scholar 

  30. Hua, Z., Yi, S., Zhou, Y., et al.: Designing hyperchaotic cat maps with any desired number of positive Lyapunov exponents. IEEE Trans. Cybern. 48(2), 463–473 (2017)

    Article  Google Scholar 

  31. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos. 9(07), 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu, X., Tong, X., Wang, Z., et al.: Uniform non-degeneracy discrete chaotic system and its application in image encryption. Nonlinear Dyn. 108(1), 653–682 (2022)

    Article  Google Scholar 

  34. Ye, G., Pan, C., Dong, Y., et al.: Image encryption and hiding algorithm based on compressive sensing and random numbers insertion. Signal Process. 172, 107563 (2020)

    Article  Google Scholar 

  35. Liu, J., Zhang, M., Tong, X., et al.: Image compression and encryption algorithm based on compressive sensing and nonlinear diffusion. Multimed. Tools Appl. 80(17), 25433–25452 (2021)

  36. Xu, Q., Sun, K., Cao, C., et al.: A fast image encryption algorithm based on compressive sensing and hyperchaotic map. Opt. Lasers Eng. 121, 203–214 (2019)

    Article  Google Scholar 

  37. Wang, J., Zhang, M., Tong, X., et al.: A chaos-based image compression and encryption scheme using fractal coding and adaptive-thresholding sparsification. Phys. Scripta 97(10), 105201 (2022)

    Article  Google Scholar 

  38. Daemen, J., Rijmen, V.: The Rijndael block cipher: AES proposal. In: First candidate conference (AES1), 343–348 (1999)

  39. Wang, X., Feng, L., Zhao, H.: Fast image encryption algorithm based on parallel computing system. Inf. Sci. 486, 340–358 (2019)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, Y., Noonan, J.P., Agaian, S., et al.: NPCR and UACI randomness tests for image encryption. Cyber J. Multidiscip. J. Sci. Technol. IEEE J. Sel. Area Commun 1(2), 31–38 (2011)

    Google Scholar 

  42. Wu, X., Wang, K., Wang, X., et al.: Color image DNA encryption

  43. Zhou, Y., Hua, Z., Pun, C.M., et al.: Cascade chaotic system with applications. IEEE Trans. Cybern. 45(9), 2001–2012 (2014)

    Article  Google Scholar 

  44. Wu, Y., Zhou, Y., Saveriades, G., et al.: Local Shannon entropy measure with statistical tests for image randomness. Inf. Sci. 222, 323–342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, X., Teng, L., Qin, X.: A novel colour image encryption algorithm based on chaos. Signal Process. 92(4), 1101–1108 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the following projects and foundations: National Natural Science Foundation of China (Grant No. 61902091), Project ZR2019MF054 supported by Shandong Provincial Natural Science Foundation.

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Authors and Affiliations

  1. School of Computer Science and Technology, Harbin Institute of Technology, Weihai, 264209, China

    Xudong Liu, Xiaojun Tong & Miao Zhang

  2. School of Information and Electrical Engineering, Harbin Institute of Technology, Weihai, 264209, China

    Zhu Wang

  3. Information and Communication Branch of Inner Mongolia Power (Group) Co., Ltd, Hohhot, 010000, China

    Yunhua Fan

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  1. Xudong Liu
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  2. Xiaojun Tong
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  3. Miao Zhang
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  4. Zhu Wang
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  5. Yunhua Fan
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Correspondence to Xiaojun Tong.

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Liu, X., Tong, X., Zhang, M. et al. Image compression and encryption algorithm based on uniform non-degeneracy chaotic system and fractal coding. Nonlinear Dyn 111, 8771–8798 (2023). https://doi.org/10.1007/s11071-023-08281-5

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  • DOI: https://doi.org/10.1007/s11071-023-08281-5

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