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Neural Computing and Applications

, Volume 31, Issue 8, pp 3317–3326 | Cite as

Construction of robust substitution boxes based on chaotic systems

  • Fatih ÖzkaynakEmail author
Original Article

Abstract

The construction of substitution boxes (s-boxes) is an important research area in cryptography. S-box is an important mathematical object. The aim of this study is to construct s-box designs with the best performance criteria for all chaotic system classes. The proposed method achieves the best s-box designs for all chaotic systems classes. The method is independent of the chosen chaotic system. The analyses show that maximum value of nonlinearity criterion is 106.75 and minimum value of equiprobable input/output XOR distribution table is 10. The importance of the best generated s-boxes based on chaotic systems is that cryptologic properties of the best generated s-box structures are the upper bound for chaos-based s-box literature.

Keywords

Cryptography S-box Chaos 

Notes

Acknowledgements

F. Özkaynak is deeply grateful to the editors for smooth and fast handling of the manuscript. The author would also like to thank the anonymous referees for their valuable suggestions to improve the quality of this paper.

References

  1. 1.
    Zhang H, Ma T, Huang G, Wang Z (2010) Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans Syst Man Cybern Part B Cybern 40(3):831–844CrossRefGoogle Scholar
  2. 2.
    Zhang H, Huang W, Wang Z, Chai T (2006) Adaptive synchronization between two different chaotic systems with unknown parameters. Phys Lett A 350(5–6):363–366zbMATHCrossRefGoogle Scholar
  3. 3.
    Zhang H, Liu D, Wang Z (2009) Controlling chaos: suppression, synchronization and chaotification. Springer, LondonzbMATHCrossRefGoogle Scholar
  4. 4.
    Liu H, Wang X (2010) Color image encryption based on one-time keys and robust chaotic maps. Comput Math Appl 59(10):3320–3327MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Liu H, Wang X (2011) Color image encryption using spatial bit-level permutation and high-dimension chaotic system. Opt Commun 284(16–17):3895–3903CrossRefGoogle Scholar
  6. 6.
    Liu H, Wang X, Kadir A (2012) Image encryption using DNA complementary rule and chaotic maps. Appl Soft Comput 12(5):1457–1466CrossRefGoogle Scholar
  7. 7.
    Wang X, Yang L, Liu R, Kadir A (2010) A chaotic image encryption algorithm based on perceptron model. Nonlinear Dyn 62(3):615–621MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Wang X, Wang Q (2014) A novel image encryption algorithm based on dynamic S-boxes constructed by chaos. Nonlinear Dyn 75(3):567–576CrossRefGoogle Scholar
  9. 9.
    Wang X, Teng L, Qin X (2012) A novel colour image encryption algorithm based on chaos. Sig Process 92(4):1101–1108MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhang Y, Wang X (2014) A symmetric image encryption algorithm based on mixed linear–nonlinear coupled map lattice. Inf Sci 273:329–351CrossRefGoogle Scholar
  11. 11.
    Zhang Y, Wang X (2015) A new image encryption algorithm based on non-adjacent coupled map lattices. Appl Soft Comput 26:10–20CrossRefGoogle Scholar
  12. 12.
    Wang X, Liu L, Zhang Y (2015) A novel chaotic block image encryption algorithm based on dynamic random growth technique. Opt Lasers Eng 66:10–18CrossRefGoogle Scholar
  13. 13.
    Wang X, Zhang Y, Bao X (2015) A novel chaotic image encryption scheme using DNA sequence operations. Opt Lasers Eng 73:53–61CrossRefGoogle Scholar
  14. 14.
    Zhang Y, Wang X (2014) Analysis and improvement of a chaos-based symmetric image encryption scheme using a bit-level permutation. Nonlinear Dyn 77(3):687–698CrossRefGoogle Scholar
  15. 15.
    Özkaynak F, Yavuz S (2013) Security problems of pseudorandom sequence generator based on Chen chaotic system. Comput Phys Commun 184(9):2178–2181zbMATHCrossRefGoogle Scholar
  16. 16.
    Özkaynak F, Özer A (2016) Cryptanalysis of a new image encryption algorithm based on chaos. Optik 127:5190–5192CrossRefGoogle Scholar
  17. 17.
    Wu Y, Noonan J, Agaian S (2011) NPCR and UACI randomness tests for image encryption. Cyber J Multidiscipl J Sci Technol J Sel Areas Telecommun 2:31–38Google Scholar
  18. 18.
    Cusick T, Stanica P (2009) Cryptographic boolean functions and applications. Elsevier, AmsterdamzbMATHGoogle Scholar
  19. 19.
    Matsui M (1994) Linear cryptanalysis method for DES cipher, advances in cryptology—Eurocrypt’93. Lect Notes Comput Sci 765:386–397zbMATHCrossRefGoogle Scholar
  20. 20.
    Biham E, Shamir A (1991) differential cryptanalysis of DES-like cryptosystems. J Cryptol 4:3–72MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Daemen J, Rijmen V (1998) AES proposal: Rijndael. In: First advanced encryption conference, CaliforniaGoogle Scholar
  22. 22.
    Bard G (2009) Algebraic cryptanalysis. Springer, BerlinzbMATHCrossRefGoogle Scholar
  23. 23.
    Kocarev L, Lian S (2011) Chaos based cryptography theory algorithms and applications. Springer, BerlinzbMATHCrossRefGoogle Scholar
  24. 24.
    Jakimoski G, Kocarev L (2011) Chaos and cryptography: block encryption ciphers. IEEE Trans Circ Syst I 48(2):163–169zbMATHCrossRefGoogle Scholar
  25. 25.
    Tang G, Liao X, Chen Y (2005) A novel method for designing S-boxes based on chaotic maps. Chaos Solitons Fractals 23:413–419zbMATHCrossRefGoogle Scholar
  26. 26.
    Tang G, Liao X (2005) A method for designing dynamical S-boxes based on discretized chaotic map. Chaos Solitons Fractals 23(5):1901–1909MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Chen G, Chen Y, Liao X (2007) An extended method for obtaining S-boxes based on 3-dimensional chaotic baker maps. Chaos Solitons Fractals 31:571–579MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Chen G (2008) A novel heuristic method for obtaining S-boxes. Chaos Solitons Fractals 36:1028–1036MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Özkaynak F, Özer A (2010) A method for designing strong S-boxes based on chaotic Lorenz system. Phys Lett A 374:3733–3738zbMATHCrossRefGoogle Scholar
  30. 30.
    Wang Y, Wong K, Li C, Li Y (2012) A novel method to design S-box based on chaotic map and genetic algorithm. Phys Lett A 376(6–7):827–833zbMATHCrossRefGoogle Scholar
  31. 31.
    Khan M, Shah T, Mahmood H, Gondal M, Hussain I (2012) A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems. Nonlinear Dyn 70(3):2303–2311MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hussain I, Shah T, Mahmood H, Gondal M (2012) Construction of S8 Liu J S-boxes and their applications. Comput Math Appl 64(8):2450–2458MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Hussain I, Shah T, Gondal M (2012) A novel approach for designing substitution-boxes based on nonlinear chaotic algorithm. Nonlinear Dyn 70(3):1791–1794MathSciNetCrossRefGoogle Scholar
  34. 34.
    Khan M, Shah T, Mahmood H, Gondal M (2013) An efficient method for the construction of block cipher with multi-chaotic systems. Nonlinear Dyn 71(3):489–492MathSciNetCrossRefGoogle Scholar
  35. 35.
    Özkaynak F, Yavuz S (2013) Designing chaotic S-boxes based on time-delay chaotic system. Nonlinear Dyn 74(3):551–557MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Khan M, Shah T, Gondal M (2013) An efficient technique for the construction of substitution box with chaotic partial differential equation. Nonlinear Dyn 73(3):1795–1801MathSciNetCrossRefGoogle Scholar
  37. 37.
    Hussain I, Shah T, Mahmood H, Gondal M (2013) A projective general linear group based algorithm for the construction of substitution box for block ciphers. Neural Comput Appl 22(6):1085–1093CrossRefGoogle Scholar
  38. 38.
    Hussain I, Shah T, Gondal M, Khan W, Mahmood H (2013) A group theoretic approach to construct cryptographically strong substitution boxes. Neural Comput Appl 23(1):97–104CrossRefGoogle Scholar
  39. 39.
    Hussain I, Shah T, Gondal M, Mahmood H (2013) An efficient approach for the construction of LFT S-boxes using chaotic logistic map. Nonlinear Dyn 71(1):133–140MathSciNetCrossRefGoogle Scholar
  40. 40.
    Hussain I, Shah T, Gondal M (2013) Efficient method for designing chaotic S-boxes based on generalized Baker’s map and TDERC chaotic sequence. Nonlinear Dyn 74(1):271–275MathSciNetCrossRefGoogle Scholar
  41. 41.
    Hussain I, Shah T, Gondal M, Mahmood H (2013) A novel method for designing nonlinear component for block cipher based on TD-ERCS chaotic sequence. Nonlinear Dyn 73(1):633–637MathSciNetCrossRefGoogle Scholar
  42. 42.
    Khan M, Shah T (2014) A construction of novel chaos base nonlinear component of block cipher. Nonlinear Dyn 76(1):377–382MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Khan M, Shah T (2014) A novel image encryption technique based on Hénon chaotic map and S8 symmetric group. Neural Comput Appl 25(7–8):1717–1722CrossRefGoogle Scholar
  44. 44.
    Lambić D (2014) A novel method of S-box design based on chaotic map and composition method. Chaos Solitons Fractals 58:16–21zbMATHCrossRefGoogle Scholar
  45. 45.
    Zaibi G, Peyrard F, Kachouri A, Prunaret D, Samet M (2014) Efficient and secure chaotic S-box for wireless sensor network. Secur Commun Netw 7:279–292CrossRefGoogle Scholar
  46. 46.
    Liu H, Kadir A, Niu Y (2014) Chaos-based color image block encryption scheme using S-box. AEU Int J Electron Commun 68(7):676–686CrossRefGoogle Scholar
  47. 47.
    Zhang X, Zhao Z, Wang J (2014) Chaotic image encryption based on circular substitution box and key stream buffer. Sig Process Image Commun 29(8):902–913CrossRefGoogle Scholar
  48. 48.
    Liu G, Yang W, Liu W, Dai Y (2015) Designing S-boxes based on 3-D four-wing autonomous chaotic system. Nonlinear Dyn 82(4):1867–1877MathSciNetCrossRefGoogle Scholar
  49. 49.
    Ahmad M, Bhatia D, Hassan Y (2015) A novel ant colony optimization based scheme for substitution box design. Proc Comput Sci 57:572–580CrossRefGoogle Scholar
  50. 50.
    Khan M (2015) A novel image encryption scheme based on multiple chaotic S-boxes. Nonlinear Dyn 82(1):527–533MathSciNetCrossRefGoogle Scholar
  51. 51.
    Khan M, Shah T (2015) An efficient construction of substitution box with fractional chaotic system. SIViP 9(6):1335–1338MathSciNetCrossRefGoogle Scholar
  52. 52.
    Jamal S, Khan M, Shah T (2016) A watermarking technique with chaotic fractional S-box transformation. Wirel Pers Commun 90(4):2033–2049CrossRefGoogle Scholar
  53. 53.
    Khan M, Shah T, Batool S (2016) Construction of S-box based on chaotic Boolean functions and its application in image encryption. Neural Comput Appl 27(3):677–685CrossRefGoogle Scholar
  54. 54.
    Khan M, Shah T, Batool S (2016) A new implementation of chaotic S-boxes in CAPTCHA. SIViP 10(2):293–300CrossRefGoogle Scholar
  55. 55.
    Khan M, Asghar Z (2016) A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and S8 permutation. Neural Comput Appl.  https://doi.org/10.1007/s00521-016-2511-5 CrossRefGoogle Scholar
  56. 56.
    Lambić D (2017) A novel method of S-box design based on discrete chaotic map. Nonlinear Dyn 87(4):2407–2413MathSciNetCrossRefGoogle Scholar
  57. 57.
    Farah T, Rhouma R, Belghith S (2017) A novel method for designing S-box based on chaotic map and teaching–learning-based optimization. Nonlinear Dyn 88(2):1059–1074CrossRefGoogle Scholar
  58. 58.
    Özkaynak F, Çelik V, Özer A (2017) A new S-box construction method based on the fractional-order chaotic Chen system. SIViP 11(4):659–664CrossRefGoogle Scholar
  59. 59.
    Belazi A, Latif A (2017) A simple yet efficient S-box method based on chaotic sine map. Opt Int J Light Electron Opt 130:1438–1444CrossRefGoogle Scholar
  60. 60.
    Belazi A, Latif A, Diaconu A, Rhouma R, Belghith S (2017) Chaos-based partial image encryption scheme based on linear fractional and lifting wavelet transforms. Opt Lasers Eng 88:37–50CrossRefGoogle Scholar
  61. 61.
    Belazi A, Khan M, Latif A, Belghith S (2017) Efficient cryptosystem approaches: S-boxes and permutation–substitution-based encryption. Nonlinear Dyn 87(1):337–361CrossRefGoogle Scholar
  62. 62.
    Çavuşoğlu Ü, Zengin A, Pehlivan İ, Kaçar S (2017) A novel approach for strong S-box generation algorithm design based on chaotic scaled Zhongtang system. Nonlinear Dyn 87(2):1081–1094zbMATHCrossRefGoogle Scholar
  63. 63.
    Islam F, Liu G (2017) Designing S-box based on 4D-4Wing hyperchaotic system. 3D Res 8:9CrossRefGoogle Scholar
  64. 64.
    Özkaynak F (2015) A novel method to improve the performance of chaos based evolutionary algorithms. Opt Int J Light Electron Opt 126(24):5434–5438CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2017

Authors and Affiliations

  1. 1.Department of Software Engineering, Faculty of TechnologyFırat UniversityElazigTurkey

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