On the Extensions of (k, n)*-Visual Cryptographic Schemes
A deterministic (k, n)*-Visual cryptographic scheme (VCS) was proposed by Arumugam et.al [1] in 2012. Probabilistic schemes are used in visual cryptography to reduce the pixel expansion. In this paper, we have shown that the contrast of probabilistic (k,
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Abstract. A deterministic (k, n)*-Visual cryptographic scheme (VCS) was proposed by Arumugam et.al [1] in 2012. Probabilistic schemes are used in visual cryptography to reduce the pixel expansion. In this paper, we have shown that the contrast of probabilistic (k, n)*-VCS is same as that of deterministic (k, n)*- VCS. This paper also proposes a construction of (k, n)*-VCS with multiple essential participants. It is shown that in both deterministic and probabilistic cases the contrast of the (k, n)*-VCS with multiple essential participant is same as that of (k, n)*-VCS. Keywords: Visual Cryptography, Deterministic schemes, Probabilistic schemes.
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Introduction
Naor and Adi Shamir in 1995 developed a (k, n) OR based deterministic VCS [6] for sharing binary secret images. As an extension of Naor’s scheme a deterministic general access structure VCS was introduced by Ateniese et.al. [2] in 1996. Droste [4] in 1998 proposed a (k, n)-VCS with less pixel expansion than Noar’s scheme. In 2005 Tuyls et.al invented a XOR based deterministic VCS [7]. The basic parameters for determining the quality of VCS are pixel expansion and contrast. The pixel expansion is a measure of number of sub pixels used for encoding a pixel of secret image while contrast is the difference in grey level between black pixel and white pixel in the reconstructed image. The following are the definitions and notations. Let P = {P1, P2, P3,…, Pn} be the set of participants, and 2P denote the power set of P. Let us denote ΓQual as qualified set and ΓForb as forbidden set. Let ΓQual ∈ 2P and ΓForb ∈ 2P where ΓQual ∩ ΓForb = Ø. Any set A ∈ ΓQual can recover the secret image whereas any set A ∈ ΓForb cannot leak out any secret information. Let Γ0 = {A∈ ΓQual: A ′ ∉ ΓQual for all A ′ ⊆ A, A ′ ≠ A} be the set of minimal qualified subset of P. The pair Γ = (ΓQual, ΓForb) is called the access structure of the scheme. Let S be an n × m Boolean matrix and A ⊆ P, the vector obtained by applying the Boolean OR operation to the rows of S corresponding to the elements in A is denoted by SA. Let w(SA) denotes the Hamming weight of vector SA. Let W0 (resp. W1) is a set consist of G. Martínez Pérez et al. (Eds.): SNDS 2014, CCIS 420, pp. 231–238, 2014. © Springer-Verlag Berlin Heidelberg 2014
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K. Praveen, K. Rajeev, and M. Sethumadhavan
OR-ed value of any k tuple out of n tuple column vector V0 from S0 (resp. V1 from S1). Let X0 be a set of values consist of OR-ing 1 with all elements of W1 and 0 with all elements of W0. Let X1 be a set of values consist of OR-ing 1 with all elements of W0 and 0 with all elements of W1. Definition 1 [2]. Let Γ = (ΓQual, ΓForb) be an access structure on a set of n participants. Two collections of n × m Boolean matrices S0 and S1 constitute a (ΓQual, ΓForb, m) VCS if there exist a positive real number α and the set of thresholds {tA | A ∈ ΓQual} satisfying the two conditions: 1. Any qualified set A= {i1, i2,…,ip} ∈ ΓQual can recover the shared image by stacking their transparencies. Formally w (S0A) ≤ tA – α.m, whereas w(S1A) ≥
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