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Perfect recovery of XOR-based visual cryptography scheme

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Abstract

Visual cryptography is an interesting secret sharing scheme, in which participants can observe the secret image by stacking their shares. However, the size expansion and distorted visual effect are two disadvantages of visual cryptography. In this paper, we focus on how to realize the perfect recovery by XOR-ing shares directly. First, we propose the definition of ideal access structure, which is the key point of perfect recovery of XOR-based visual cryptography scheme. The characteristics of ideal access structure are analyzed, and the construction algorithm of shares under ideal access structure is designed. Based on the ideal access structure, a new algorithm is proposed for dividing the general access structure into several ideal access structures, and the secret sharing and recovering algorithms for general access structure are presented. Furthermore, our method can also be utilized in color visual cryptography and multi-secret visual cryptography. The security and perfect recovery of our method have been proved theoretically. Compared with the previous schemes, the proposed scheme realizes the perfect recovery of secret image by XOR-ing shares directly, and the sizes of shares can be decreased efficiently compared with the previous schemes.

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Acknowledgements

The authors thank the anonymous reviewers for their valuable comments. This work was supported in part by the National Natural Science Foundation of China under Grant No.61602513 and the Outstanding Youth Foundation of Zhengzhou Information Science and Technology Institute under Grant No.2016611303.

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Authors and Affiliations

  1. Information Engineering University, Zhengzhou, China

    Zhengxin Fu, Yuqiao Cheng & Bin Yu

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  1. Zhengxin Fu
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  2. Yuqiao Cheng
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  3. Bin Yu
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Correspondence to Yuqiao Cheng.

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Appendix

Appendix

The division result on (k, n) threshold structure (1 < k < n < 7).

(2, 3) threshold structure:

Γ1 = {{1,2},{1,3}}; Γ2 = {{2,3}}.

(2, 4) threshold structure:

Γ1 = {{1,2},{2,3}, {1,4},{3,4}}; Γ2 = {{1,3},{2,4}}.

(2, 5) threshold structure:

Γ1 = {{1,2},{1,3},{1,4},{2,5},{3,5},{4,5}}; Γ2 = {{1,5},{2,3},{3,4}};

Γ3 = {{2,4}}.

(2, 6) threshold structure:

Γ1 = {{1,2},{1,3},{1,4},{1,5},{2,6},{3,6},{4,6},{5,6}}; Γ2 = {{2,5}, {3,4}};

Γ3 = {{1,6},{2,3},{2,4},{3,5},{4,5}}.

(3, 4) threshold structure:

Γ1 = {{1,2,3},{1,2,4}}; Γ2 = {{1,3,4},{2,3,4}}.

(3, 5) threshold structure:

Γ1 = {{1,2,3},{1,2,4},{1,2,5}}; Γ2 = {{1,3,4},{2,3,4},{1,3,5},{2,3,5}};

Γ3 = {{2,4,5},{3,4,5},{1,4,5}}.

(3, 6) threshold structure:

Γ1 = {{1,2,3},{1,2,4},{1,2,5},{1,3,6},{1,4,6},{1,5,6}};

Γ2 = {{1,3,4},{2,3,4},{3,4,5},{3,4,6}}; Γ3 = {{1,2,6},{2,4,6},{2,5,6},{2,3,6}};

Γ4 = {{1,3,5},{1,4,5},{2,3,5},{2,4,5},{4,5,6},{3,5,6}}.

(4, 6) threshold structure:

Γ1 = {{1,2,3,4},{1,2,3,5},{1,2,4,6},{1,2,5,6}}; Γ2 = {{1,2,3,6},{1,2,4,5}};

Γ3 = {{1,3,4,5},{1,3,4,6},{2,3,4,5},{2,3,4,6}}; Γ4 = {{3,4,5,6}};

Γ5 = {{1,3,5,6},{1,4,5,6},{2,3,5,6},{2,4,5,6}}.

(4, 5) threshold structure:

Γ1 = {{1,2,3,4},{1,2,3,5}}; Γ2 = {{1,2,4,5},{1,3,4,5}}; Γ3 = {{2,3,4,5}}.

(5, 6) threshold structure:

Γ1 = {{1,2,3,4,5},{1,2,3,4,6}}; Γ2 = {{1,2,3,5,6},{1,2,4,5,6}};

Γ3 = {{1,3,4,5,6},{2,3,4,5,6}}.

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Fu, Z., Cheng, Y. & Yu, B. Perfect recovery of XOR-based visual cryptography scheme. Multimed Tools Appl 78, 2367–2384 (2019). https://doi.org/10.1007/s11042-018-6364-z

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