# Fault-Tolerant Visual Secret Sharing Schemes without Pixel Expansion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}and G

_{2}, each called a share (also called sheet, shadow), of which one can be seen as a cipher text, and the other is a key to it. Stacking them is the only way to restore the hidden secret. Random grid-based VSS, invented by Kafri and Keren [2], received more attention in recent years, such as [3,4,5,6,7,8,9]. This method takes each pixel as a grid on the image and applies the concept of random variables to encrypt images.

## 2. Experimental Section

#### 2.1. Visual Cryptography Concepts

_{0}and C

_{1}. To share a white pixel, the dealer randomly chooses one of the matrices in C

_{0}, and to share a black pixel, the dealer randomly chooses one of the matrices in C

_{1}, The chosen matrix defines the color of the m subpixels in each one of the n transparencies. The solution is considered valid if the following three conditions are met:

- For any S in C
_{0}, the “or” V of any k out of n rows satisfies H(V) ≤ d − a × m. - For any S in C
_{1}, the “or” V of any k out of n rows satisfies H(V) ≥ d. - For any subset {i
_{1}, i_{2}, …, i_{q}} of {1, 2, …, n} with q < k, the two collections of q × m matrices D_{t}for t ∈ {0, 1} obtained by restricting each n × m matrix in C_{t}(where t = 0, 1) to row i_{1}, i_{2}, …, i_{q}are indistinguishable in the sense that they contain the same matrices with the same frequencies.

#### 2.2. Random Grid Encryption Algorithm

_{1}and G

_{2}. Let r

_{i}be a pixel in G

_{i}for i = 1, 2. The resulting value of the overlapped pixels r

_{1}and r

_{2}will be r

_{1}⊕ r

_{2}, where ⊕ stands for the Boolean “or” operation. All results when stacking any two pixels together are shown in Table 1.

r_{1} | r_{2} | r_{1} ⊕ r_{2} |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

_{1}and G

_{2}with the same size. We list these algorithms as follows, where Rand(0/1) is a function with output zero or one randomly and with equal probability.

Algorithm KK1: |

Generate a w × h random grid G_{1} |

For (i = 0; i < w; i++) |

For (j = 0; j < h; j++) |

If (S[i][j] == 0) |

G_{2}[i][j] = G_{1}[i][j]; |

else |

G_{2}[i][j] = 1 − G_{1}[i][j]$\overline{{\mathrm{G}}_{1}\left[i\right]\left[j\right]}$; |

Output (G_{1}, G_{2}) |

Algorithm KK2: |

Generate a w × h random grid G_{1} |

For (i = 0; i < w; i++) |

For (j = 0; j < h; j++) |

If (S[i][j] == 0) |

G_{2}[i][j] = G_{1}[i][j]; |

else |

G_{2}[i][j] = Rand(0/1); |

Output (G_{1}, G_{2}) |

Algorithm KK3: |

Generate a w × h random grid G_{1} |

For (i = 0; i < w; i++) |

For (j = 0; j < h; j++) |

If (S[i][j] == 0) |

G_{2}[i][j] = Rand(0/1); |

else |

G_{2}[i][j] = 1 – G_{1}[i][j]$\overline{{\mathrm{G}}_{1}\left[i\right]\left[j\right]}$; |

Output (G_{1}, G_{2}) |

S | Probability | G_{1} | G_{2} | G_{1}⊕ G_{2} | T(G_{1}⊕ G_{2}) |
---|---|---|---|---|---|

□ | 1/2 | □ | □ | □ | 1/2 |

1/2 | ■ | ■ | ■ | ||

■ | 1/2 | □ | ■ | ■ | 0 |

1/2 | ■ | □ | ■ |

#### 2.3. The Concept of Fault Tolerance

**Figure 2.**The patterns in Nakajima and Yamaguchi’s scheme [14].

## 3. Proposed Scheme

_{1}shifts one pixel right. All combinations for such a case when the pixel in the secret image is white or black are shown in Table 7 and Table 8, respectively. The way of stacking has been shown as Figure 3, where the red square represents the stacked position of the unit in G

_{1}.

Image | G_{1} | G_{2} | Stack | Image | G_{1} | G_{2} | Stack |
---|---|---|---|---|---|---|---|

□ | ■ | ||||||

Image | G_{1} | G_{2} | Stack | Image | G_{1} | G_{2} | Stack |
---|---|---|---|---|---|---|---|

□ | ■ | ||||||

Image | G_{1} | G_{2} | Stack | Image | G_{1} | G_{2} | Stack |
---|---|---|---|---|---|---|---|

□ | ■ | ||||||

Image | G_{1} | G_{2} | Stack | Image | G_{1} | G_{2} | Stack |
---|---|---|---|---|---|---|---|

□ | ■ | ||||||

**Table 7.**The analysis of stacked units with one pixel shift when the pixel in the secret image is white.

Image | G_{1} | G_{2} | Stack | Image | G_{1} | G_{2} | Stack |
---|---|---|---|---|---|---|---|

□ | □ | ||||||

□ | □ |

**Table 8.**The analysis of stacked units with one pixel shift when the pixel in the secret image is black.

Image | G_{1} | G_{2} | Stack | Image | G_{1} | G_{2} | Stack |
---|---|---|---|---|---|---|---|

■ | ■ | ||||||

■ | ■ |

_{1}that has been stacked with G

_{2}with one pixel shift. In this case, the ration of the number of white pixels to the total pixels of the stacked unit when the pixel in the secret image is white is (32 + 36 + 8 + 4)/256 = 80/256 = 5/16. Similarly, when the pixel in the secret image is black, this ratio is (16 + 12 + 8 + 12)/256 = 48/256 = 3/16. In summary, Table 9 gives the transmittance analysis for stacking two units for n = 3, 4, 5 or 6. In a perfect stacking, the resulting transmittance for a white secret pixel is 1/2, and because the design of the pattern for a black pixel is accordingly complementary, the resulting transmittance for the black secret pixel is zero. We also analyze the stacking results with different shifts: a one-pixel shift when n is three or four and up to a two-pixel shift when n is five or six. Besides, we also calculate the results for a one-pixel diagonal shift, that is a one-pixel right shift plus a one-pixel down shift. All of the results show that there are differences between the transmittance for the black and white pixels of the stacked image. When the transmittance for a white pixel of a secret image differs from that for a black pixel of a secret image, the original secret image can be recognized. Hence, the following theorem can be concluded.

## 4. Experimental Results

_{x}as the horizontal deviation (unit: pixel) and CI to measure the difference between the reconstructed images for a given deviation and no deviation. The correctness indices for black and white secret pixels, denoted as CI(B) and CI(W), respectively, are obtained by comparing the secret pixels between the reconstructed images of a deviation d

_{x}and no deviation; then, the CI is calculated as the average of all CI(B) and CI(W). For two shares G

_{1}and G

_{2}, using the parameter transmittance T

_{B}(G

_{1}⊕ G

_{2}) and T

_{W}(G

_{1}⊕ G

_{2}) for the black and white pixels of the secret image, we have:

_{B}(G

_{1}⊕ G

_{2}), CI(W) = 2T

_{W}(G

_{1}⊕ G

_{2}) and CI = (CI(B) + CI(W))/2

^{2}is used as the ratio of pixel expansion, that is α can be seen as the ratio of pixel expansion in one dimension, and PB is the percentages of the big subpixels in a share.

**Figure 8.**Comparison of our schemes with the misalignment-tolerant visual secret sharing (MTVSS) scheme for shifting the same pixels. CI, correctness index.

**Figure 9.**Comparison of our schemes with MTVSS scheme for shifting the same percentage of the shares.

_{x}/α). We use the result of the MTVSS scheme when PB = 1 (the best performance among different PBs) for different values of α. One can see that our results are better than the MTVSS scheme when n is bigger than three. In conclusion, both the theoretical analysis and simulation results demonstrate the effectiveness and practicality of our proposed schemes. In Table 10, we list some capabilities of our scheme compared to some previous works.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Naor, M.; Shamir, A. Visual cryptography. In Proceedings of the Advances in Cryptology, EUROCRYPT’94, Perugia, Italy, 9–12 May 1994; pp. 1–12.
- Kafri, O.; Keren, E. Encryption of pictures and shapes by random grids. Opt. Lett.
**1987**, 12, 377–379. [Google Scholar] [CrossRef] [PubMed] - Chang, J.J.-Y.; Juan, J.S.-T. Multi-VSS scheme by shifting random grids. In Proceedings of the World Academy of Science, Engineering and Technology, Tokyo, Japan, 29–30 May 2012; pp. 1277–1283.
- Chen, L.-C. Multi-Secret Images Sharing Schemes. Master’s Thesis, National Chi Nan University, Nantou, Taiwan, 10 July 2014. [Google Scholar]
- Chen, T.-H.; Tsao, K.-H.; Wei, K.-C. Multiple-image encryption by rotating random grids. In Proceedings of the Eighth International Conference on Intelligent Systems Design and Applications, ISDA’08, Kaohsiung, Taiwan, 26–28 November 2008; pp. 252–256.
- Nakajima, M.; Yamaguchi, Y. Extended visual cryptography for natural images. J. WSCG
**2002**, 10, 303–310. [Google Scholar] - Shyu, S.J. Image encryption by random grids. Pattern Recognit.
**2007**, 40, 1014–1031. [Google Scholar] [CrossRef] - Wang, D.S.; Dong, L.; Li, X. Towards shift tolerant visual secret sharing schemes. IEEE Trans. Inf. Forensics Secur.
**2011**, 6, 323–337. [Google Scholar] [CrossRef] - Yan, X.; Wang, S.; Niu, X.; Yang, C.N. Halftone visual cryptography with minimum auxiliary black pixels and uniform image quality. Digit. Signal Process.
**2015**, 38, 53–65. [Google Scholar] [CrossRef] - Blundo, C.; Santis, A. Visual cryptography schemes with perfect reconstruction of black pixels. Computer
**1998**, 22, 449–455. [Google Scholar] [CrossRef] - Blundo, C.; Bonis, A.; Santis, A. Improved schemes for visual cryptography. Des. Codes Cryptogr.
**2001**, 24, 255–278. [Google Scholar] [CrossRef] - Kobara, K.; Imai, H. Limiting the visible space visual secret sharing schemes and their application to human identification. In Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology, ASIACRYPT’96, Kyongju, Korea, 3–7 November 1996; pp. 185–195.
- Liu, F.; Wu, C.K.; Lin, X.J. The alignment problem of visual cryptography schemes. Des. Codes Cryptogr.
**2009**, 50, 215–227. [Google Scholar] [CrossRef] - Nakajima, M.; Yamaguchi, Y. Enhancing registration tolerance of extended visual cryptography for natural images. J. Electron. Imaging
**2004**, 13, 654–662. [Google Scholar] - Yang, C.N.; Peng, A.G.; Chen, T.S. MTVSS: Misalignment tolerant visual secret sharing on resolving alignment difficulty. Signal Process.
**2009**, 89, 1602–1624. [Google Scholar] [CrossRef]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Juan, J.S.-T.; Chen, Y.-C.; Guo, S. Fault-Tolerant Visual Secret Sharing Schemes without Pixel Expansion. *Appl. Sci.* **2016**, *6*, 18.
https://doi.org/10.3390/app6010018

**AMA Style**

Juan JS-T, Chen Y-C, Guo S. Fault-Tolerant Visual Secret Sharing Schemes without Pixel Expansion. *Applied Sciences*. 2016; 6(1):18.
https://doi.org/10.3390/app6010018

**Chicago/Turabian Style**

Juan, Justie Su-Tzu, Yung-Chang Chen, and Song Guo. 2016. "Fault-Tolerant Visual Secret Sharing Schemes without Pixel Expansion" *Applied Sciences* 6, no. 1: 18.
https://doi.org/10.3390/app6010018