# Progressive Secret Sharing with Adaptive Priority and Perfect Reconstruction

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## Abstract

**:**

## 1. Introduction

**Identity:**

**Performing XOR over “odd number” times:**

**Performing XOR over “even number” times:**

**Symmetric Inverse:**

**Commutative:**

**Associative:**

## 2. Former PVSS Scheme

#### 2.1. PVSS Scheme for Binary Image

Algorithm 1: Former Scheme [15]. |

Input: Secret image in binary format, $I$, of size $M\times N$ |

Number of shared images, $n$ |

Output: A set of generated shared images, $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n}\right\}$, each of size $M\times N$ |

Step 1: Based on priority weight ${w}_{j}$, determine the location set ${\mathcal{l}}_{j}$, for $j=1,2,\dots ,n$. |

Step 2: For Each Pixel $\left(x,y\right)$. Based on information ${\mathcal{l}}_{j}$, select two shared images ${r}_{1}$ and ${r}_{2}$. Do |

Step 3:${S}_{{r}_{1}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}{U}_{I}\left(0,1\right)$ |

Step 4:If$I\left(x,y\right)=0$, then ${S}_{{r}_{2}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}{S}_{{r}_{1}}\left(x,y\right)$ |

Step 5: Else${S}_{{r}_{2}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}\sim {S}_{{r}_{1}}\left(x,y\right)$ |

Step 6: For Each other shared images, ${S}_{i}$, with condition $1\le i\le n$ and $i\ne {r}_{1},{r}_{2}$ Do |

Step 7:${S}_{i}\left(x,y\right)\stackrel{\text{}}{\leftarrow}{S}_{{r}_{2}}\left(x,y\right)$ |

Step 8: Obtain $n$ generated shared images, $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n}\right\}$ |

#### 2.2. Limitation of PVSS Scheme

**Theorem**

**1.**

**Proof.**

## 3. Proposed PVSS Method

#### 3.1. Proposed Bitwise-Based PVSS Method

Algorithm 2: Proposed Bitwise-Based PVSS Method. |

Input: Secret image in binary format, $I$, of size $M\times N$ |

Number of shared images, $n$ |

Output: A set of generated shared images, $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n}\right\}$, each of size $M\times N$ |

Step 1: Based on priority weight ${w}_{j}$, determine the location set ${\mathcal{l}}_{j}$, for $j=1,2,\dots ,n$. |

Step 2: For Each Pixel $\left(x,y\right)$. Based on information of ${\mathcal{l}}_{j}$, select two shared images ${r}_{1}$ and ${r}_{2}$. Do |

Step 3:${S}_{{r}_{1}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}{U}_{I}\left(0,1\right)$ |

Step 4: If $I\left(x,y\right)=0$, Then ${S}_{{r}_{2}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}{S}_{{r}_{1}}\left(x,y\right)$ |

Step 5: Else ${S}_{{r}_{2}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}\sim {S}_{{r}_{1}}\left(x,y\right)$ |

Step 6: For Each Generated shared images, ${S}_{i}$, with the condition $1\le i\le n$ and $i\ne {r}_{1},{r}_{2}$ Do |

Step 7:${S}_{i}\left(x,y\right)\stackrel{\text{}}{\leftarrow}0$ |

Step 8: Obtain $n$ generated shared images, $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n}\right\}$ |

**Theorem**

**2.**

**Proof.**

#### 3.2. Proposed XOR-ed Based PVSS Method

Algorithm 3: Proposed XOR-ed Based PVSS Method. |

Input: A grayscale or color image as secret, $I$, of size $M\times N$ |

Number of shared images, $n$ |

Output: Full set of generated shared images, $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n}\right\}$, each of size $M\times N$ |

Step 1: Based on priority weight ${w}_{j}$, determine the location set ${\mathcal{l}}_{j}$, for $j=1,2,\dots ,n$. |

Step 2: For Each Pixel Position $\left(x,y\right)$. Based on the information in ${\mathcal{l}}_{j}$, decide the selected shared images ${r}_{1}$ and ${r}_{2}$. Do |

Step 3:$C\stackrel{\text{}}{\leftarrow}{U}_{I}\left(a,b\right)$ |

Step 4:${S}_{{r}_{1}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}I\left(x,y\right)\oplus C$ |

Step 5:${S}_{{r}_{2}}\left(x,y\right)\stackrel{\text{}}{\leftarrow}C$ |

Step 6: For Each other generated shared images, ${S}_{i}$, under the condition $1\le i\le n$ and $i\ne {r}_{1},{r}_{2}$ Do |

Step 7:${S}_{i}\left(x,y\right)\stackrel{\text{}}{\leftarrow}0$ |

Step 8: Obtain the $n$ generated shared images, $\left\{{S}_{1},{S}_{2},\dots ,{S}_{n}\right\}$ |

**Theorem**

**3.**

**Proof.**

## 4. Experimental Results

#### 4.1. Performance Evaluation

#### 4.2. Visual Evaluation on Binary Image

#### 4.3. Visual Investigation on Grayscale Image

#### 4.4. Visual Assessment of Color Image

#### 4.5. Performance Comparisons in Terms of Objective Image Quality Assessment

#### 4.6. Comparison of Algorithm Aspects for the Proposed Method and Other Schemes

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Examples of spatial pixel locations with various adaptive priority weights $\left\{{w}_{1}=0.5,{w}_{2}=0.3,{w}_{3}=0.1,{w}_{4}=0.1\right\}$: (

**a**) ${\mathcal{l}}_{1}$, (

**b**) ${\mathcal{l}}_{2}$, (

**c**) ${\mathcal{l}}_{3}$, and (

**d**) ${\mathcal{l}}_{4}$.

**Figure 5.**Four secret images in a binary format for experiment (

**a**) ${I}_{1}$, (

**b**) ${I}_{2}$, (

**c**) ${I}_{3}$, and (

**d**) ${I}_{4}$.

**Figure 6.**Generated binary shared images using the proposed bitwise-based PVSS method for $n=6$: (

**a**–

**f**) $\left\{{S}_{1},{S}_{2},\dots ,{S}_{6}\right\}$.

**Figure 7.**The results of stacking several shared images with $t=2,3,\dots ,6$, by setting $n=6$. The first colum is from the former scheme [15], while the second and third columns are from the proposed method.

**Figure 8.**Stacking several shared images $t=2,3,\dots ,5$, by setting $n=5$. The first column is from the former scheme [15], while the second and third columns are from the proposed method.

**Figure 9.**Four grayscale images as secret for experiment: (

**a**–

**d**) $\left\{{I}_{1},{I}_{2},{I}_{3},{I}_{4}\right\}$.

**Figure 10.**Generated shared images using the proposed eXclusive-OR (XOR)-based PVSS method: (

**a**–

**f**) $\left\{{S}_{1},{S}_{2},\dots ,{S}_{6}\right\}$, by setting $n=6$.

**Figure 11.**Stacking several shared images with $t=2,3,\dots ,6$ and $n=6$. The left and right columns are from the former scheme [15] and the proposed method, respectively.

**Figure 12.**Reconstructed secret images when several shared images are stacked, with $t=2,3,\dots ,5$ and $n=5$. The left and right columns are from the former scheme [15] and the proposed method, respectively.

**Figure 13.**A set of color images used as secret images in the experiment, denoted as: (

**a**) ${I}_{1}$, (

**b**) ${I}_{2}$, (

**c**) ${I}_{3}$, and (

**d**) ${I}_{4}$

**Figure 14.**Shared images obtained from the secret image in the color format using the proposed XOR-based PVSS method: (

**a**–

**f**) $\left\{{S}_{1},{S}_{2},\dots ,{S}_{6}\right\}$.

**Figure 15.**The results of stacking several shared images with $t=2,3,\dots ,6$ and $n=6$. The left and right columns are from the former scheme [15] and the proposed method, respectively.

**Figure 16.**The results of stacking several shared images $t=2,3,\dots ,5$ and $n=5$. The left and right columns are the recovered secret image from the former scheme [15] and the proposed method, respectively.

**Figure 17.**The average image contrast between the proposed method and the former scheme [15] of the binary secret image with (

**a**) $n=5$, and (

**b**) $n=6$.

**Figure 18.**The average bit error rate between the proposed method and the former scheme [15] for a binary secret image with: (

**a**) $n=5$, and (

**b**) $n=6$.

**Figure 19.**Comparisons between the proposed method and the former scheme [15] in terms of (

**a**,

**b**) PSNR, (

**c**,

**d**) SSIM, and (

**e**,

**f**) MAE values. The comparisons are conducted for a secret image in grayscale format.

**Figure 20.**Comparisons between the proposed method and former scheme [15] in terms of (

**a**,

**b**) PSNR, (

**c**,

**d**) SSIM, and (

**e**,

**f**) MAE values. Herein, the secret image is in a color format.

**Table 1.**Comparisons between the proposed method and the former scheme in terms of algorithm aspects.

Method | Share Style | Encoding Matrix | Pixel Expansion | Adaptive Priority | Quality |
---|---|---|---|---|---|

Fang’s Scheme [9] | Noise-Like Form | Require | Need | No | Lossless for $n$ is even |

Wang’s Scheme [10] | Noise-Like Form | Require | Need | No | - |

Hou’s Scheme [11] | Noise-Like Form | Require | No | No | - |

Hou’s Scheme [12] | Noise-Like Form | Require | No | Adaptive Priority | Lossy |

Lin’s Scheme [13] | Friendly Appearance | No | No | No | Lossy |

Yang’s Scheme [14] | Noise-Like Form | Require | No | Adaptive Priority | Lossy |

Former Scheme [15] | Noise-Like Form | No | No | Adaptive Priority | Lossy, if $n$ is oddLossless, if $n$ is even |

Prasetyo’s Scheme [16] | Noise-Like Form | No | No | No | Lossless for $n$ is odd or even |

Proposed Method | Noise-Like Form | No | No | Adaptive Priority | Lossless for $n$ is odd or even |

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## Share and Cite

**MDPI and ACS Style**

Prasetyo, H.; Hsia, C.-H.; Wicaksono Hari Prayuda, A.
Progressive Secret Sharing with Adaptive Priority and Perfect Reconstruction. *J. Imaging* **2021**, *7*, 70.
https://doi.org/10.3390/jimaging7040070

**AMA Style**

Prasetyo H, Hsia C-H, Wicaksono Hari Prayuda A.
Progressive Secret Sharing with Adaptive Priority and Perfect Reconstruction. *Journal of Imaging*. 2021; 7(4):70.
https://doi.org/10.3390/jimaging7040070

**Chicago/Turabian Style**

Prasetyo, Heri, Chih-Hsien Hsia, and Alim Wicaksono Hari Prayuda.
2021. "Progressive Secret Sharing with Adaptive Priority and Perfect Reconstruction" *Journal of Imaging* 7, no. 4: 70.
https://doi.org/10.3390/jimaging7040070