Designs, Codes and Cryptography

, Volume 80, Issue 1, pp 165–196 | Cite as

Constructions and analysis of some efficient \(t\)-\((k,n)^*\)-visual cryptographic schemes using linear algebraic techniques

  • Sabyasachi Dutta
  • Raghvendra Singh Rohit
  • Avishek Adhikari
Article
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Abstract

In this paper we put forward an efficient construction, based on linear algebraic technique, of a \(t\)-\((k,n)^*\)-visual cryptographic scheme (VCS) for monochrome images in which \(t\) participants are essential in a \((k,n)\)-VCS. The scheme is efficient in the sense that it only requires solving a system of linear equations to construct the required initial basis matrices. To make the scheme more efficient, we apply the technique of deletion of common columns from the initial basis matrices to obtain the reduced basis matrices. However finding exact number of common columns in the initial basis matrices is a challenging problem. In this paper we deal with this problem. We first provide a construction and analysis of \(t\)-\((k,n)^*\)-VCS. We completely characterize the case of \(t\)-\((n-1,n)^*\)-VCS, \(0 \le t \le n-1\), by finding a closed form of the exact number of common columns in the initial basis matrices and thereby deleting the common columns to get the exact value of the reduced pixel expansion and relative contrast of the efficient and simple scheme. Our proposed closed form for reduced pixel expansion of \((n-1,n)\)-VCS matches with the numerical values of the optimal pixel expansions for every possible values of \(n\) that exist in the literature. We further deal with the \((n-2,n)\)-VCS and resolve an open issue by providing an efficient algorithm for grouping the system of linear equations and thereby show that our proposed algorithm works better than the existing scheme based on the linear algebraic technique. Finally we provide a bound for reduced pixel expansion for \((n-2,n)\)-VCS and numerical evidence shows it achieves almost optimal pixel expansion.

Keywords

Linear algebra Pixel expansion Relative contrast  System of linear equations 

Mathematics Subject Classification

15A03 94A60 
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Notes

Acknowledgments

We would like to thank the anonymous reviewers for their important and helpful comments. Research of the first author is supported by CSIR PhD Fellowship, Government of India, Grant no.- 09/028(0808)/2010-EMR-I. Research of the third author is partially supported by National Board for Higher Mathematics, Department of Atomic Energy, Government of India (No 2/48(10)/2013/NBHM(R.P.)/R&D II/695). The third author is also thankful to the Centre of Excellence in Cryptology of Indian Statistical Institute.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sabyasachi Dutta
    • 1
  • Raghvendra Singh Rohit
    • 2
  • Avishek Adhikari
    • 1
  1. 1.Department of Pure MathematicsUniversity of CalcuttaKolkataIndia
  2. 2.Indian Institute of Science Education & Research KolkataPO Haringhata, NadiaIndia

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