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Optimal \((k, n)\) visual cryptographic schemes for general \(k\). (English) Zbl 1197.94179

In a \((k,n)\) visual cryptographic scheme (VCS), a secret image consisting of black and white pixels 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 identified (say by human eyes) if at least (but not less) any \(k\geq 2\) of these sheets are stacked on the top of each other. Employing a Kronecker algebra of Boolean matrices, necessary and sufficient conditions are proved for the existence of a \((k,n)\) VCS with a prior specification of relative contrast. Based on an \(L_1\)-norm and linear programming formulation, connections between visual cryptography and statistical model fitting are established with the help of which the conjectures made by C. Blundo et al. [SIAM J. Discrete Math. 16, No. 2, 224–261 (2003; Zbl 1039.94006)] for \(k=4\), and \(5\) are settled. Finally, it is shown how block designs can be used to construct optimal VCS with respect to the average and minimal relative contrasts in case \(k=3\).

MSC:

94A60 Cryptography
94A62 Authentication, digital signatures and secret sharing
05B05 Combinatorial aspects of block designs

Citations:

Zbl 1039.94006
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References:

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