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

  • Ayesha Rafiq
  • Majid KhanEmail author


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.


S-boxes Projective linear groups Finite fields Modular group Triangle groups Algebraic analyses 



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.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsInstitute of Space TechnologyIslamabadPakistan
  2. 2.Cyber and Information Security Lab (CISL)Institute of Space TechnologyIslamabadPakistan

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