Contrast-optimal k out of n secret sharing schemes in visual cryptography

Visual cryptography and (k,n)-visual secret sharing schemes were introduced by Naor and Shamir (Advances in Cryptology - Eurocrypt 94, Springer, Berlin, 1995, pp. 1-12). A sender wishing to transmit a secret message distributes n transparencies amongst n recipients, where the transparencies contain seemingly random pictures. A (k,n)-scheme achieves the following situation: If any k recipients stack their transparencies together, then a secret message is revealed visually. On the other hand, if only k - 1 recipients stack their transparencies, or analyze them by any other means, they are not able to obtain any information about the secret message. The important parameters of a scheme are its contrast, i.e., the clarity with which the message becomes visible, and the number of subpixels needed to encode one pixel of the original picture. Naor and Shamir constructed (k,k)-schemes with contrast 2(-(k-1)). By an intricate result from Linial (Combinatorica 10 (1990) 349-365), they were also able to prove the optimality of these schemes. They also proved that for all fixed k less than or equal to n, there are (k,n)-schemes with contrast (2e)(-k)/root 2 pi k. For k = 2,3,4 the contrast is approximately 1/105, 1/698 and 1/4380. In this paper, we show that by solving a simple linear program, one is able to compute exactly the best contrast achievable in any (k, n)-scheme. The solution of the linear program also provides a representation of a corresponding scheme. For small k as well as for k = n, we are able to analytically solve the linear program. For k = 2,3,4, we obtain that the optimal contrast is at least 1/4, 1/16 and 1/64. For k = n, we obtain a very simple proof of the optimality of Naor's and Shamir's (k,k)-schemes. In the case k = 2, we are able to use a different approach via coding theory which allows us to prove an optimal tradeoff between the contrast and the number of subpixels. (C) 2000 Published by Elsevier Science B.V. All rights reserved.