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

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Abstract

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 L 1-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.

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Authors and Affiliations

  1. Applied Statistics Unit, Indian Statistical Institute, 203 B T Road, Kolkata, 700 108, India

    Mausumi Bose

  2. Indian Institute of Management Calcutta, Joka, Diamond Harbour Road, Kolkata, 700 104, India

    Rahul Mukerjee

Authors
  1. Mausumi Bose
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  2. Rahul Mukerjee
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Correspondence to Mausumi Bose.

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Communicated by P. Wild.

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Bose, M., Mukerjee, R. Optimal (k, n) visual cryptographic schemes for general k . Des. Codes Cryptogr. 55, 19–35 (2010). https://doi.org/10.1007/s10623-009-9327-6

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  • DOI: https://doi.org/10.1007/s10623-009-9327-6

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