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Cited by 113Chaos-based Image/Signal EncryptionCryptographic Implementations and SecurityCoding theory and cryptography

A New Hyperchaotic System-Based Design for Efficient Bijective Substitution-Boxes

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Al Solami, Eesa, Ahmad, Musheer, Volos, Christos, Doja, Mohammad Najam, Beg, Mirza Mohd Sufyan·2018·Entropy
DOI 10.3390/e20070525PMC PMC7513051
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Abstract

In this paper, we present a novel method to construct cryptographically strong bijective substitution-boxes based on the complicated dynamics of a new hyperchaotic system. The new hyperchaotic system was found to have good characteristics when compared with other systems utilized for S-box construction. The performance assessment of the proposed S-box method was carried out based on criteria, such as high nonlinearity, a good avalanche effect, bit-independent criteria, and low differential uniformity. The proposed method was also analyzed for the batch-generation of 8 × 8 S-boxes. The analyses found that through a proposed purely chaos-based method, an 8 × 8 S-box with a maximum average high nonlinearity of 108.5, or S-boxes with differential uniformity as low as 8, can be retrieved. Moreover, small-sized S-boxes with high nonlinearity and low differential uniformity are also obtainable. A performance comparison of the anticipated method with recent S-box proposals proved its dominance and effectiveness for a strong bijective S-box construction.

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  • The designs of variable-sized key-dependent S-boxes have not been considered significantly in literature, with only few proposals investigating variable-sized n x n S-boxes restricted to sizes 5 to 8.

    “The designs of variable sized key-dependent S-boxes have not been considered signi cantly in literature. There are few proposals that have investigated the design and construction of variable sized n n S-boxes restricted to sizes for 5 n 8 in [21].”

    From Bijective S-Boxes Method Using Improved Chaotic Map-Based Heuristic Search and Algebraic Group Structures (2020) · cites this paper as [21]

    Section 4.8 says methods for synthesizing S-boxes of varied small sizes are rarely investigated and identifies existing optimization-based methods for 5 ≤ n ≤ 8. It does not characterize those earlier methods as key-dependent, so that qualification is not supported by the cited passage. Page numbers are unavailable in our copy. (LLM, conf=0.85)

    Experimental — accuracy not yet measured

  • The proposed four bijective S-boxes offer considerably better resistance to linear attacks and higher nonlinear transformation of input to ciphertext than the S-boxes of references [21], [39], [59], and [60].

    “It is clear from the comparison Table that the proposed four bijective S-boxes offers considerably better resistance to linear attacks and offer high nonlinear transformation of input data to ciphertext data as compared to other S-boxes investigated in [21], [39], [59], [60].”

    From Bijective S-Boxes Method Using Improved Chaotic Map-Based Heuristic Search and Algebraic Group Structures (2020) · cites this paper as [21]

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Research works printed but not linked: 51

  1. [1]Knudsen L.R. Robshaw M.J. The Block Cipher Companion Information Security and Cryptography Springer Berlin/Heidelberg, Germany 2011
  2. [3]Ahmad M. Doja M.N. Beg M.M.S. ABC Optimization Based Construction of Strong Substitution-Boxes Wirel. Pers. Commun. 2018 101 1715 1729 10.1007/s11277-018-5787-1 · DOI
  3. [4]Makarim R.H. Tezcan C. Relating Undisturbed Bits to Other Properties of Substitution Boxes Lecture Notes in Computer Science Lightweight Cryptography for Security and Privacy Springer Cham, Switzerland 2015 109 125
  4. [5]Picek S. Jakobovic D. Miller J.F. Marchiori E. Batina L. Evolutionary Methods for the Construction of Cryptographic Boolean Functions Lecture Notes in Computer Science Genetic Programming Springer Cham, Switzerland 2015 192 204
  5. [6]Crilly A.J. Earnshaw R. Jones H. Applications of Fractals and Chaos: The Shape of Things Springer Science & Business Media New York, NY, USA 2013
  6. [7]Kocarev L. Galias Z. Lian S. Intelligent Computing Based on Chaos Springer Berlin/Heidelberg, Germany 2009
  7. [8]Kocarev L. Lian S. Chaos-Based Cryptography: Theory, Algorithms and Applications Springer Berlin/Heidelberg, Germany 2011
  8. [9]Sprott J.C. Elegant Chaos World Scientific Singapore Singapore 2010
  9. [10]Yang Q. Bai M. A new 5D hyperchaotic system based on modified generalized Lorenz system Nonlinear Dyn. 2016 88 189 221 10.1007/s11071-016-3238-7 · DOI
  10. [11]Ojoniyi O.S. Njah A.N. A 5D hyperchaotic Sprott B system with coexisting hidden attractors Chaos Solitons Fractals 2016 87 172 181 10.1016/j.chaos.2016.04.004 · DOI
  11. [12]Vaidyanathan S. Pham V.-T. Volos C.K. A 5-D hyperchaotic Rikitake dynamo system with hidden attractors Eur. Phys. J. Spec. Top. 2015 224 1575 1592 10.1140/epjst/e2015-02481-0 · DOI
  12. [13]Vaidyanathan S. Volos C. Pham V.-T. Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation Arch. Control Sci. 2014 24 409 446 10.2478/acsc-2014-0023 · DOI
  13. [14]Yang Q. Chen C. A 5D Hyperchaotic System with Three Positive Lyapunov Exponents Coined Int. J. Bifurc. Chaos 2013 23 1350109 10.1142/S0218127413501095 · DOI
  14. [15]Zarei A. Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors Nonlinear Dyn. 2015 81 585 605 10.1007/s11071-015-2013-5 · DOI
  15. [16]Ahmad M. Solami E.A. Wang X.-Y. Doja M.N. Beg M.M.S. Alzaidi A. Cryptanalysis of an Image Encryption Algorithm Based on Combined Chaos for a BAN System, and Improved Scheme Using SHA-512 and Hyperchaos Symmetry 2018 10 266 10.3390/sym10070266 · DOI · Paper page
  16. [17]Jia R. Huang Q. Peng J. Study of the Hyperchaos-based Hash Function in E-commerce Applications Proceedings of the 2010 International Conference on Intelligent Computation Technology and Automation Changsha, China 11–12 May 2010
  17. [18]Smaoui N. Karouma A. Zribi M. Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems Commun. Nonlinear Sci. Numer. Simul. 2011 16 3279 3293 10.1016/j.cnsns.2010.10.023 · DOI
  18. [19]Islam F.U. Liu G. Designing S-Box Based on 4D-4Wing Hyperchaotic System 3D Res. 2017 8 9 10.1007/s13319-017-0119-x · DOI
  19. [20]Özkaynak F. Özer A.B. A method for designing strong S-Boxes based on chaotic Lorenz system Phys. Lett. A 2010 374 3733 3738 10.1016/j.physleta.2010.07.019 · DOI
  20. [21]Khan M. Shah T. Mahmood H. Gondal M.A. An efficient method for the construction of block cipher with multi-chaotic systems Nonlinear Dyn. 2012 71 489 492 10.1007/s11071-012-0675-9 · DOI
  21. [22]Khan M. Shah T. An efficient construction of substitution box with fractional chaotic system Signal Image Video Process. 2013 9 1335 1338 10.1007/s11760-013-0577-4 · DOI
  22. [23]Liu G. Yang W. Liu W. Dai Y. Designing S-boxes based on 3-D four-wing autonomous chaotic system Nonlinear Dyn. 2015 82 1867 1877 10.1007/s11071-015-2283-y · DOI
  23. [24]Özkaynak F. Çelik V. Özer A.B. A new S-box construction method based on the fractional-order chaotic Chen system Signal Image Video Process. 2016 11 659 664 10.1007/s11760-016-1007-1 · DOI
  24. [25]Çavuşoğlu Ü. Zengin A. Pehlivan I. Kaçar S. A novel approach for strong S-Box generation algorithm design based on chaotic scaled Zhongtang system Nonlinear Dyn. 2016 87 1081 1094 10.1007/s11071-016-3099-0 · DOI
  25. [26]Wolf A. Swift J.B. Swinney H.L. Vastano J.A. Determining Lyapunov exponents from a time series Physica D Nonlinear Phenom. 1985 16 285 317 10.1016/0167-2789(85)90011-9 · DOI · Paper page
  26. [27]Wang L. Liu J. Lu Z.-R. Incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems Nonlinear Dyn. 2017 89 153 167 10.1007/s11071-017-3442-0 · DOI
  27. [28]Kaplan J.L. Yorke J.A. Chaotic Behavior of Multidimensional Difference Equations Functional Differential Equations and Approximation of Fixed Points Lecture Notes in Mathematics Springer Berlin/Heidelberg, Germany 1979 204 227
  28. [29]Belazi A. Rhouma R. Belghith S. A novel approach to construct S-box based on Rossler system Proceedings of the 2015 International Wireless Communications and Mobile Computing Conference (IWCMC) Dubrovnik, Croatia 24–28 August 2015
  29. [30]Picek S. Cupic M. Rotim L. A New Cost Function for Evolution of S-Boxes Evolut. Comput. 2016 24 695 718 10.1162/EVCO_a_00191 27482748 · DOI
  30. [31]Braeken A. Cryptographic Properties of Boolean Functions and S-Boxes Ph.D Thesis Katholieke Universiteit Leuven Leuven, Belgium 2006
  31. [32]Cusick T.W. Stanica P. Cryptographic Boolean Functions and Applications Elsevier Amsterdam, The Netherlands 2009
  32. [34]Ding C. Xiao G. Shan W. The Stability Theory of Stream Ciphers LNCS 561 Springer Science & Business Media New York, NY, USA 1991
  33. [35]Matsui M. Linear Cryptanalysis Method for DES Cipher Advances in Cryptology—EUROCRYPT ’93 Lecture Notes in Computer Science Springer Heidelberg/Berlin, Germany 1993 386 397
  34. [36]Webster A.F. Tavares S.E. On the Design of S-Boxes Lecture Notes in Computer Science Advances in Cryptology—CRYPTO ’85 Proceedings Springer Heidelberg/Berlin, Germany 1985 523 534
  35. [37]Adams C. Tavares S. The structured design of cryptographically good s-boxes J. Cryptol. 1990 3 27 41 10.1007/BF00203967 · DOI
  36. [39]Belazi A. Khan M. El-Latif A.A.A. Belghith S. Efficient cryptosystem approaches: S-boxes and permutation–substitution-based encryption Nonlinear Dyn. 2016 87 337 361 10.1007/s11071-016-3046-0 · DOI
  37. [40]Özkaynak F. Construction of robust substitution boxes based on chaotic systems Neural Comput. Appl. 2017 1 10 10.1007/s00521-017-3287-y · DOI
  38. [41]Ullah A. Jamal S.S. Shah T. A novel scheme for image encryption using substitution box and chaotic system Nonlinear Dyn. 2017 91 359 370 10.1007/s11071-017-3874-6 · DOI
  39. [42]Farwa S. Muhammad N. Shah T. Ahmad S. A Novel Image Encryption Based on Algebraic S-box and Arnold Transform 3D Res. 2017 8 26 10.1007/s13319-017-0135-x · DOI
  40. [44]Razaq A. Yousaf A. Shuaib U. Siddiqui N. Ullah A. Waheed A. A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map Secur. Commun. Netw. 2017 2017 1 16 10.1155/2017/5101934 · DOI
  41. [45]Khan M. Asghar Z. A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and S8 permutation Neural Comput. Appl. 2016 29 993 999 10.1007/s00521-016-2511-5 · DOI
  42. [46]Jamal S.S. Khan M.U. Shah T. A Watermarking Technique with Chaotic Fractional S-Box Transformation Wirel. Pers. Commun. 2016 90 2033 2049 10.1007/s11277-016-3436-0 · DOI
  43. [47]Khan M. Shah T. Batool S.I. A new implementation of chaotic S-boxes in CAPTCHA Signal Image Video Process. 2015 10 293 300 10.1007/s11760-014-0741-5 · DOI
  44. [48]Khan M. Shah T. Batool S.I. Construction of S-box based on chaotic Boolean functions and its application in image encryption Neural Comput. Appl. 2015 27 677 685 10.1007/s00521-015-1887-y · DOI
  45. [49]Anees A. Ahmed Z. A Technique for Designing Substitution Box Based on Van der Pol Oscillator Wirel. Pers. Commun. 2015 82 1497 1503 10.1007/s11277-015-2295-4 · DOI
  46. [50]Lambić D. S-box design method based on improved one-dimensional discrete chaotic map J. Inf. Telecommun. 2018 2 181 191 10.1080/24751839.2018.1434723 · DOI
  47. [51]Silva-García V. Flores-Carapia R. Rentería-Márquez C. Luna-Benoso B. Aldape-Pérez M. Substitution box generation using Chaos: An image encryption application Appl. Math. Comput. 2018 332 123 135 10.1016/j.amc.2018.03.019 · DOI
  48. [52]Ahmed H.A. Zolkipli M.F. Ahmad M. A novel efficient substitution-box design based on firefly algorithm and discrete chaotic map Neural Comput. Appl. 2018 1 10 10.1007/s00521-018-3557-3 · DOI · Paper page
  49. [53]Millan W. How to Improve the Nonlinearity of Bijective S-Boxes Information Security and Privacy Lecture Notes in Computer Science Springer Heidelberg/Berlin, Germany 1998 181 192
  50. [54]Fuller J. Millan W. Dawson E. Multi-objective optimisation of bijective S-boxes New Gener. Comput. 2005 23 201 218 10.1007/BF03037655 · DOI
  51. [55]Laskari E. Meletiou G. Vrahatis M. Utilizing Evolutionary Computation Methods for the Design of S-Boxes Proceedings of the 2006 International Conference on Computational Intelligence and Security Guangzhou, China 3–6 November 2006

Cited by

Sorted by citations

Authors
Al Solami, E., Ahmad, M., Volos, C., Doja, M. N., & Beg, M. M. S.
Affiliations
  • Department of Information Technology, University of Jeddah, Jeddah 21589, Saudi Arabia
  • Jamia Millia Islamia (IN)
  • Aristotle University of Thessaloniki (GR)
  • Aligarh Muslim University (IN)
Corresponding author
Ahmad, Musheer
ORCID
Venue
Entropy
Publisher
MDPI AG
Published
2018-07-12
Volume
20
Issue
7
Pages
525
ISSN
1099-4300 (ISSN-L)
DOI
10.3390/e20070525
PMC
PMC7513051
OpenAlex
W2821393409
Cited by
113
References
3
Citations by year
2026: 2 · 2025: 8 · 2024: 2 · 2023: 19 · 2022: 17 · 2021: 22 · 2020: 23 · 2019: 16 · 2018: 4
FWCI
8.13
Type
journal article
Language
en
Open access
Licence
Locations
  • Entropy (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Topics
  • Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Chaos-based Image/Signal Encryption
  • Physical Sciences › Computer Science › Artificial Intelligence › Cryptographic Implementations and Security
  • Physical Sciences › Computer Science › Artificial Intelligence › Coding theory and cryptography
Keywords
hyperchaotic systems, substitution boxes, S-box design, bijective mapping, chaos-based cryptography, nonlinearity, differential uniformity, avalanche effect, block cipher design

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