(t, k, n) XOR-based visual cryptography scheme with essential shadows

Abstract

Visual cryptography scheme (VCS) shares a binary secret image into multiple shadows, only qualified set of shadows can reveal the secret image by stacking operation. However, VCS suffers the problems of low visual quality of the revealed image and large shadow size. A (t, k, n) XOR-based visual cryptography scheme (XVCS) shares the secret image into n shadows including t essentials and n-t non-essentials. A qualified set of shadows contains any k shadows including t essentials. The revealing process is implemented by XOR operation on the involved shadows. In this paper, we propose a construction method for (t, k, n)-XVCS with essential shadows. The secret image can be revealed perfectly, and the shadow size is small compared with VCS. Theoretical analysis and experimental results show the security and effectiveness of the proposed scheme.

Introduction

Visual cryptography scheme (VCS) is a technique to protect a secret image among n participants proposed by Naor and Shamir [1]. In a (k, n) threshold VCS, a binary secret image is first shared into n noise-like shadow images, also called shadows. Each shadow is printed on a transparence and distributed to the corresponding participant. In the revealing process, any k out of n participants can cooperate together to reveal the secret image by stacking their shadows. One can perceive the content of the secret image by human visual system. However, with k−1 or less shadows, nothing about the secret image can be revealed even with computer resources.

The advantage of VCS is the stacking-to-see revealing property, while the disadvantages are degraded visual quality of the revealed secret image and large size of shadows. Many researchers tried to improve the visual quality of the revealed image based on XOR operation [2], [3], [4], [5], [6], [7] and reduce the shadow size expansion [8], [9], [10], [11], [12]. VCS based on XOR operation is also called XOR-based VCS (XVCS). Since XOR operation can be implemented by reversing and OR operation, it is reasonable to perform the revealing process using a copy machine with reversing function. We can also use some light weight computational devices to achieve perfect visual quality. Tuyls et al. [2] proposed the construction methods for (2, n)-XVCS and (k, n)-XVCS. Liu et al. [3] proposed XVCS for general access structure by using (2, 2)-XVCS as the building block. Wu and Sun [4] proposed XOR-based VCS without size expansion and no code book required. Fu el al. [5] proposed the perfect contrast XVCS via linear algebra. The minimal qualified sets are partitioned into multiple parts. For each part, a sub-shadow is generated for each participant. The final shadow is a concatenation of multiple sub-shadows. Yang and Wang [6] proved that the basis matrices of VCS can be used to implement XVCS. Recently, Li et al. [7] proposed a (2, 3)-XVCS for sharing two secret images. To reduce the shadow size, Shyu and Chen [8] acquired the optimal size expansion of (k, n)-VCS by solving an integer linear program. Probabilistic VCS [9], [10] and random grid based visual secret sharing methods [11], [12] were also proposed to achieve small size expansion. There were also many VCSs proposed with different features, like VCS sharing color image [13], [14], [15], VCS sharing more information [16], [17], [18], VCS with invariant aspect ratio [19], [20], and so on.

In (k, n)-VCS, each participant (shadow) has the same role in the revealing process. In another word, each shadow can be replaced by any other shadow. However, in real application, people usually act as different roles. For example, in a bank, there are 2 managers and 5 ordinary stuffs. The secret key can be revealed by at least 3 people with at least 1 manager. In this case, a manager can replace an ordinary stuff. However, an ordinary stuff cannot replace a manager in revealing process. The managers can be seen as essential participants, while the ordinary stuffs are non-essential participants. Similarly, in VCS a secret image is shared into n shadows. The first s shadows are much more important and represented as essentials, while the left n-s shadows are non-essentials. In the revealing process, we need at least k shadows including at least t essentials to reveal the secret image. This is so called (t, s, k, n)-VCS with essential shadows. Arumugam et al. [21] proposed the first (1, 1, k, n)-VCS. Guo et al. [22] extended Arumugam et al.’s scheme to (t, t, k, n)-VCS, also called (t, k, n)-VCS. However, these VCSs with essentials suffer from the large size expansion. Li et al. [23] proposed a secret image sharing method called (t, s, k, n) secret image sharing (SIS) scheme with essential shadows based on polynomial. Then Yang et al. [24] and Li et al. [25] proposed some improved (t, s, k, n)-SIS schemes. Recently, Li et al. proposed a (t, t, k, n)-SIS, also denoted as (t, k, n)-SIS [26]. The shadow size is further reduced. Although these polynomial-based SIS schemes can achieve small shadow size, they need complicate computation to reveal the secret image compared with VCS. Li et al. proposed a (t, s, k, n)-EVCS based on integer programming [27]. The shadows of EVCS is generated from a monotonic (K, N)-VCS. Since their scheme decodes the secret image by OR operation, the visual quality of the revealed image is poor.

In this paper, we propose an XOR-based (t, s, k, n)-VCS with t = s. It is also represented as (t, k, n)-XVCS. All essential shadows should be involved in the revealing process. The shadows are generated by using multiple pairs of basis matrices. The secret image can be perfectly revealed by XOR operation on qualified set of shadows. The rest of the paper is organized as follows. Some related works and concepts are introduced in Section 2. In Section 3, we propose an XOR-based (t, k, n)-XVCS, and theoretically analyze the security and contrast conditions of the proposed XVCS. Some experimental results and comparison are conducted in Section 4, and Section 5 makes a briefly conclusion.