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

Abstract

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 ÿxed k6n, there are (k; n)-schemes with contrast (2e) −k = √ 2 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 di erent approach via coding theory which allows us to prove an optimal tradeo between the contrast and the number of subpixels.

Visual Cryptography (VC) with non-expanded shadow images: Hilbert-curve approach

“…[2] or [3] or other optimal-contrast VC methods. Let m be the traditional expansion rate of C0 and C1...…”

“…People can "view" the secret image by simply stacking any k of these n shadows (k ≤ n is a threshold value). In most of the reported VC schemes (for example, [1][2][3][4][5][6]), the size of each shadow is greater than that of S. By elegantly using the probabilistic approach, [7] and [8] are a few papers that generate only non-expanded shadows. In this paper, we use Hilbert curve [9] and two queues to present a VC scheme with the following properties: (1).…”

“…Each shadow image is as small as the input image S. (2). Many gracious (k, n) contrast-optimal algorithms (for example, [2][3]) can be directly and easily applied to our scheme. (3) Image quality is competitive.…”

Improving the perceived quality of half tone secret images in visual cryptography

“…Gizli görüntünün bu n adet pay görüntüden en az k tanesinin bir araya getirilmesiyle çözülmesi mümkün kılınabilir. Böylece farklı (n,k) şemaları oluşturulabilir [1], [3], [4]. Örneğin, gizlenmek istenen görüntünün 2 parçaya ayrılıp ancak bu 2 pay görüntünün biraya getirilmesiyle çözülmesini sağlayan şema (2,2)'lik bir şemadır.…”

A novel extended visual cryptography scheme using one shared image

“…The construction of their scheme is based on the theory of hypergraphy coloring. Afterward, some researchers made contribution to investigate the bounds and contrast of VCS [3][4][5]. Several constructions of VCS for colorful images were proposed in [6][7][8].…”