# Verifiable (2, n) Image Secret Sharing Scheme Using Sudoku Matrix

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## Abstract

**:**

## 1. Introduction

## 2. Sudoku Matrix

## 3. Research Methods

#### 3.1. The Procedure of Hiding Secret Image and Authentication Codes to Cover Images

#### 3.2. The Procedure of Secret Image Reconstruction and Authentication

## 4. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic diagram of the proposed (2, n)-SIS scheme. (

**a**) Secret sharing process. (

**b**) Recovery process.

**Figure 3.**An example of our schematic diagram. (

**a**) Real share 1. (

**b**) Real share 2. (

**c**) Authentication result—fully recovered. (

**d**) Real share 1. (

**e**) Fake share 2. (

**f**) Authentication result—not recovered completely.

**Figure 4.**Schematic diagram of secret data embedded into a cover image. (

**a**) Schematic diagram. (

**b**) An embedding example.

**Figure 5.**Data embedding example with hidden number 6 changing ${\alpha}_{1}$ and ${\alpha}_{2}$ to 101 and 121.

**Figure 6.**Example of secret pixel reconstruction and authentication code extraction, where the same shape is used for the same parameters in the analytical equation.

**Figure 7.**Secret images and cover images. (

**a**) Secret image “Bird”. (

**b**) Secret image “Jet(F16)”. (

**c**) Cover image “Lena”. (

**d**) Cover image “Baboon”. (

**e**) Cover image “Pepper”. (

**f**) Cover image “Zelda”.

**Figure 8.**Reconstructed secret images. (

**a**) Recovered image “Bird” (PSNR = ∞). (

**b**) Recovered image “Jet(F16)” (PSNR = ∞).

**Figure 9.**An authentication example. (

**a**) Special image (fake logo). (

**b**) Standard image (fake logo). (

**c**) Uniform noise (fake logo). (

**d**) Image with tampering (fake cover). (

**e**) Image without tampering (real cover). (

**f**) Reconstructed secret image. (

**g**) Detected inauthentic region (painted in black).

Secret Images | Lena | Baboon | Pepper | Zelda | ||||
---|---|---|---|---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |

Bird | 49.88 | 0.993 | 49.88 | 0.998 | 49.88 | 0.993 | 49.89 | 0.990 |

Jet(F16) | 49.88 | 0.994 | 49.89 | 0.998 | 49.88 | 0.993 | 49.88 | 0.990 |

Features | Gao et al.’s Scheme [35] | Li et al.’s Scheme [38] | Liu et al.’s Scheme [33] | Liu et al.’s Scheme [36] | Gao et al.’s Scheme [34] | Proposed Scheme |
---|---|---|---|---|---|---|

Meaningful shares | Yes | Yes | Yes | Yes | Yes | Yes |

Reversibility | Yes | Yes | Yes | - | - | Yes |

Different cover images | - | - | - | Yes | Yes | Yes |

(k,n)-SIS | (2,2)-SIS | (3,3)-SIS | (2,2)-SIS | (2,2)-SIS | (2,3)-SIS | (2, n)-SIS |

Fault tolerance | Yes | - | - | - | Yes | Yes |

Average PSNR | 32.96 | 49.07 | 48.72 | 41.71 | 40.41 | 49.88 |

Embedding capacity (bits) | 786,432 | 624,215 | 524,288 | 785,525 | 1,310,720 | 786,432 |

Tampered Image | Gao et al.’s Scheme [34] | Proposed Scheme | |
---|---|---|---|

Special image | 0.684 | 0.942 | 0.997 |

Standard image | 0.084 | 0.834 | 0.995 |

Uniform noise | 0.097 | 0.836 | 0.995 |

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.-H.; Lee, J.-Y.; Chiang, M.-H.; Chen, S.-H.
Verifiable (2, *n*) Image Secret Sharing Scheme Using Sudoku Matrix. *Symmetry* **2022**, *14*, 1445.
https://doi.org/10.3390/sym14071445

**AMA Style**

Chen Y-H, Lee J-Y, Chiang M-H, Chen S-H.
Verifiable (2, *n*) Image Secret Sharing Scheme Using Sudoku Matrix. *Symmetry*. 2022; 14(7):1445.
https://doi.org/10.3390/sym14071445

**Chicago/Turabian Style**

Chen, Yi-Hui, Jia-Ye Lee, Min-Hsien Chiang, and Shih-Hsin Chen.
2022. "Verifiable (2, *n*) Image Secret Sharing Scheme Using Sudoku Matrix" *Symmetry* 14, no. 7: 1445.
https://doi.org/10.3390/sym14071445