Skip to main content
SpringerLink
Log in

Construction of new S-boxes based on triangle groups and its applications in copyright protection

  • Published:
Multimedia Tools and Applications Aims and scope Submit manuscript

Abstract

Substitution boxes with resilient cryptographic possessions are normally utilized in block ciphers to give the substantial property of nonlinearity. They are important to resist standard attacks such as linear and differential cryptanalysis. A cryptographically robust S-box must be sound with respect to cryptographic properties like nonlinearity, bit independent criteria, strict avalanche criteria, linear and differential approximation probability. In this paper, we have developed an innovative construction scheme of nonlinear component of block cipher based on the action of projective linear groups on the projective line, and the permutation triangle groups. This nonlinear component, namely S-box, is responsible for making the relation between plaintext and ciphertext intractable which is one of the most important requirements of any modern block ciphers. By widening the scope of the proposed S-boxes, we have applied these lightweight nonlinear components in watermarking scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bahrami S, Naderi M (2012) Image encryption using a lightweight stream encryption algorithm. Advances in Multimedia 2012:1–8

    Article  Google Scholar 

  2. Batool SI, Shah T, Khan M (2014) A color image watermarking scheme based on affine transformation and S4 permutation. Neural comput applic, Springer 25:2037–2045

    Article  Google Scholar 

  3. Baumslag G, Fine B, Xu X (2006) Cryptosystems using linear groups. AAECC 17(3–4):205–217

    Article  MathSciNet  MATH  Google Scholar 

  4. Belazi A, Khan M, Abd El-Latif AA, Belghith S (2017) Efficient cryptosystem approaches: S-boxes and permutation substitution-based encryption. Nonlinear Dyn 87:337–361

    Article  Google Scholar 

  5. Bogdanov A, Knudsen LR, Leander G, Paar C, Poschmann A, Robshaw MJB, Seurin Y, Vikkelsoe C (2007) PRESENT: an ultra-lightweight block cipher. In: Paillier P, Verbauwhede I (eds) The 9th international workshop on cryptographic hardware and embedded systems - CHES 2007, LNCS 4727. Springer, Berlin, pp 450–466

  6. Chawla G, Kamaldeep RS, Yadav R (2012) Classification of watermarking based upon various parameters. Int J Comput Appl Inf Technol 1:16–19

    Google Scholar 

  7. Cid C, Murphy S, Robshaw MJB (2005) Small scale variants of the AES, Proceedings of FSE 2005, LNCS, 145–162. Springer

  8. Conder M, Havas G, Newman MF. On one-relator quotients of the modular group, (preprint)

  9. Conder M (1987) Three-relator quotients of the modular group. Q J Math 38 (4):427–447

    Article  MathSciNet  MATH  Google Scholar 

  10. El-Sheikh HM, El-Mohsen OA, Elgarf T, Zekry A (2012) A new approach for designing key-dependent S-Box defined over G F(24) in AES. Int J Comput Theory Eng 4(2):158–164

    Article  Google Scholar 

  11. Everitt B (2000) Alternating quotients of Fuchsian groups. J Algebra 223:457–476

    Article  MathSciNet  MATH  Google Scholar 

  12. Gangadaril BR, Ahamed SR (2015) Analysis and algebraic construction of S-Box for AES algorithm using Irreducible Polynomials. In: 2015 8th international conference on contemporary computing (IC3). Noida0, pp 526–530

  13. Higman G, Mushtaq Q (1983) Coset diagrams and relations for P S L(2,z). Arab Gulf J Scient Res 1:159–164

    MathSciNet  MATH  Google Scholar 

  14. Isa H, Jamil N, Z’aba MR (2016) Construction of cryptographically strong S-Boxes inspired by bee waggle dance. New Generation Computing 34:221–238

    Article  Google Scholar 

  15. Khan M (2015) A novel image encryption scheme based on multi-parameters chaotic S-boxes. Nonlinear Dyn 82:527–533

    Article  Google Scholar 

  16. Khan M, Shah T (2015) An efficient construction of substitution box with fractional chaotic system. SIViP 9:1335–1338

    Article  Google Scholar 

  17. Khan M, Shah T (2015) A copyright protection using watermarking scheme based on nonlinear permutation and its quality metrics. Neural Comput Applic 26:845–855

    Article  Google Scholar 

  18. Khan M, Shah T, Batool SI (2016) A new implementations of chaotic S-boxes in CAPTCHA. SIViP 10:293–300

    Article  Google Scholar 

  19. Khan M, Shah T (2016) Construction and applications of chaotic S-boxes in image encryption. Neural Comput Applic 27:677–685

    Article  Google Scholar 

  20. Khan M., Asghar Z. (2018) A novel construction of substitution box for image encryption applications with Gingerbreadman chaotic map and S8 permutation. Neural Comput Applic 29:993–99

    Article  Google Scholar 

  21. Matsui M (1994) Linear cryptanalysis method for DES cipher, LNCS 765, Eurocrypt’93, pp 386–397

  22. Meier W, Staffelbach O (1990) Nonlinear criteria for cryptographic functions. In: Quisquater JJ, Vandewalle J (eds) Lecture Notes in Computer Science and Advances in Cryptology, EUROCRYPT 89, vol 434. Springer, Berlin, pp 549–562

  23. Mihajloska H, Gligoroski D (2012) Construction of optimal 4-bit S-boxes by Quasigroups of order 4. In: The 6th international conference on emerging security information, systems and technologies, SECURWARE 2012, Rome, Italy

  24. Mushtaq Q, Shaheen F (1987) Coset diagrams for a homomorphic image of △(2, 3, 6). Ars Combinatoria A 23:187–193

    MathSciNet  MATH  Google Scholar 

  25. Mushtaq Q (1988) Modular group acting on real quadratic fields. Bull Austral Math Soc 37:303–309

    Article  MathSciNet  MATH  Google Scholar 

  26. Mushtaq Q (1990) Coset diagrams for Hurwitz groups. Comm Algebra 18 (11):3857–3888

    Article  MathSciNet  MATH  Google Scholar 

  27. Mushtaq Q (1992) Parametrisation of all homomorphisms from \(PGL(2,\mathbb {Z})\) into P G L(2,q). Comm Algebra 20(4):1023–1040

    Article  MathSciNet  MATH  Google Scholar 

  28. Nakahara J Jr, de Freitas DS (2009) Mini-ciphers: a reliable testbed for cryptanalysis?, “symmetric cryptography”, seminar 09031. In: Dagstuhl S (ed) Dagstuhl Seminar Proceedings. Leibniz-Zentrum fuer Informatik, Germany, pp 1862–4405

  29. Phan RC-W (2002) Mini advanced encryption standard (Mini-AES): A testbed for cryptanalysis students. Cryptologia XXVI(4):283–306

    Article  Google Scholar 

  30. Pieprzyk J, Finkelstein G (1988) Towards effective nonlinear cryptosystem design. IEE proceedings 135, part E(6):325–335

    Google Scholar 

  31. Shannon C (1949) Communication theory of security systems. Bell Systems Technical Journal 28(4):656–715

    Article  MATH  Google Scholar 

  32. Siegenthaler T (1985) Decrypting a class of stream ciphers using Ciphertexts only. IEEE Trans Computer C34:81–85

    Article  Google Scholar 

  33. Singh AK, Kumar B, Singh G, Mohan A (2017) Medical image watermarking: Techniques and applications, Springer

  34. Steinwandt R (2001) Loopholes in two public key cryptosystems using the modular group, International workshop on public key cryptography, pp 180–189

  35. Tom MA (1976) Modular functions and Dirichlet series in number theory, 2nd edn. Springer, New York

    MATH  Google Scholar 

  36. Wang Z, Bovik AC (2002) A universal image quality index. IEEE Signal Process Lett 9(3):81–84

    Article  Google Scholar 

  37. Wilson JS (1999) Simple images of triangle groups. Quart Jour Math Ser 50:523–531

    Article  MathSciNet  MATH  Google Scholar 

  38. Yamamura A (1998) Public-key cryptosystems using the modular group, public key cryptography. In: 1st international workshop on practice and theory in public key cryptography, lecture notes in computer science, vol 1431. Springer, Berlin, pp 203–216

  39. Yamamura A (1999) A functional cryptosystem using a group action, information security and privacy. In: 4th australasian conference, lecture notes in computer science, vol 1587. Springer, Berlin, pp 314–325

Download references

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and suggestions, which improved significantly the presentation of the paper. The authors are thankful to the Higher Education Commission (HEC) of Pakistan for the financial support under-vide No. 21-1743/SRGP/R&D/HEC/2017.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad, Pakistan

    Ayesha Rafiq & Majid Khan

  2. Cyber and Information Security Lab (CISL), Institute of Space Technology, Islamabad, Pakistan

    Majid Khan

Authors
  1. Ayesha Rafiq
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Majid Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Majid Khan.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rafiq, A., Khan, M. Construction of new S-boxes based on triangle groups and its applications in copyright protection. Multimed Tools Appl 78, 15527–15544 (2019). https://doi.org/10.1007/s11042-018-6953-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11042-018-6953-x

Keywords

Search

Navigation