Designs, Codes and Cryptography

, Volume 55, Issue 1, pp 19–35

Optimal (k, n) visual cryptographic schemes for general k


DOI: 10.1007/s10623-009-9327-6

Cite this article as:
Bose, M. & Mukerjee, R. Des. Codes Cryptogr. (2010) 55: 19. doi:10.1007/s10623-009-9327-6


In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L1-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.


Block design Kronecker algebra L1-norm Pixel expansion Progressive optimality Relative contrast Unbalanced VCS 

Mathematics Subject Classification (2000)

94A60 05B05 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Applied Statistics UnitIndian Statistical InstituteKolkataIndia
  2. 2.Indian Institute of Management CalcuttaKolkataIndia

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