# A Novel Hierarchical Secret Image Sharing Scheme with Multi-Group Joint Management

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## Abstract

**:**

## 1. Introduction

- The secret image can be jointly managed by multiple groups.
- The proposed scheme has a hierarchical threshold access structure.
- Shares have a same size and same weight.

## 2. Preliminaries

#### 2.1. Review of Secret Sharing

#### 2.1.1. Shares Generation Procedure

**Step 1**.- Given a secret data s,the dealer chooses $t-1$ random number ${r}_{1},{r}_{2},\dots ,{r}_{t-1}$ and a prime number p, where $s\in {F}_{p}$ and ${r}_{1},{r}_{2},\dots ,{r}_{t-1}\in {F}_{p}$.
**Step 2**.- Construct a $t-1$ degree function $f\left(x\right)=s+{r}_{1}x+{r}_{2}{x}^{2}+\dots {r}_{t-1}{x}^{t-1}mod\phantom{\rule{3.33333pt}{0ex}}p$.
**Step 3**.- The outputs ${y}_{i}=f\left({x}_{i}\right)$ are regarded as shares ${S}_{i}$ and assigned to the shareholders ${U}_{i}$ in a secure channel.

#### 2.1.2. Secret Reconstruction Procedure

**Step 1**.- Given a subset of t disparate shares ${S}_{i}$, $i\in A$, $A=\{1,2,\dots ,t\}$, the $t-1$ degree polynomial can be reconstructed by Lagrange interpolation as formula (1):$$f\left(x\right)=\sum _{i=1}^{t}{y}_{i}\prod _{j=1,i\ne j}^{t}\frac{x-{x}_{j}}{{x}_{i}-{x}_{j}}mod\phantom{\rule{3.33333pt}{0ex}}p$$
**Step 2**.- The secret data s can be reconstructed by calculating $s=f\left(0\right)$.

#### 2.2. Birkhoff Interpolation

**Definition**

**1.**

**Birkhoff Interpolation Problem.**The Birkhoff interpolation problem that corresponds to the triplet $\langle X,E,C\rangle $ is a problem of finding a polynomial $P\left(x\right)\in {R}_{N-1}\left[x\right]$ that satisfies the N equalities

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

#### 2.3. Review of Guo’s Hierarchical Threshold Secret Image Sharing

#### 2.3.1. Shares Generation Procedure

**Step 1**.- Given a secret image I, a cover image O sized $M\times N$,and a set of n shareholders $U=\{{U}_{1},{U}_{2},\dots ,{U}_{n}\}$. The n shareholders are classified into $(m+1)$ levels ${L}_{0},{L}_{1},\dots ,{L}_{m}$ with the corresponding threshold requirement $\{{t}_{0},{t}_{1},\dots ,{t}_{m}\}$, where ${t}_{m}=t$, $0<{t}_{0}<{t}_{1}<\dots <{t}_{m}$.
**Step 2**.- Every ${t}_{m}$ non-lapped pixels $\{{s}_{0},{s}_{1},\dots ,{s}_{{t}_{m}-1}\}$ in the secret image I are grouped as a section. For each section, a $(t-1)$ degree function can be constructed as $f\left(x\right)={s}_{0}+{s}_{1}x+{s}_{2}{x}^{2}+\dots +{s}_{{t}_{m}-1}{x}^{{t}_{m}-1}mod\phantom{\rule{3.33333pt}{0ex}}p$, the outputs are shadow images of the first level ${L}_{0}$.
**Step 3**.- The shadow images in the hierarchy ${L}_{r}$ can be generated by computing ${f}^{({t}_{r}-1)}\left(x\right)$.
**Step 4**.- Denote the pixel values in the cover image O as ${o}_{j}$, where $j\in [1,M\times N]$. The dealer chooses a pair of parameters $(k,\sigma )$ satisfying the relation:$$\lfloor \frac{{o}_{j}}{k}\rfloor \times k+\sigma \le 255$$
**Step 5**.- Each shadow image $S{H}_{i}$ can be transformed a $\sigma $-bit stream as $S{H}_{i}={({y}_{i,1},{y}_{i,2},\dots ,{y}_{i,w})}_{\sigma}$, where $w=\lceil lon{g}_{\sigma}S{H}_{i}\rceil $. The w pixels ${o}_{i,j}$,$j\in [1,w]$ in stego images ${O}_{i}$ can be generated by replacing w pixels ${o}_{i,j}$ as:$$\begin{array}{c}{o}_{i,1}^{\prime}=\lfloor \frac{{o}_{i,1}}{k}\rfloor \times k+{y}_{i,1}\\ {o}_{i,2}^{\prime}=\lfloor \frac{{o}_{i,2}}{k}\rfloor \times k+{y}_{i,2}\\ \dots \\ {o}_{i,w}^{\prime}=\lfloor \frac{{o}_{i,w}}{k}\rfloor \times k+{y}_{i,w}\end{array}$$
**Step 6**.- The generated n stego images ${O}_{i}^{\prime}$ are regarded as the n shares ${S}_{i}$, $i\in 1,2,\dots ,t$ and sent to shareholders, respectively.

#### 2.3.2. Secret Image Reconstruction Procedure

**Step 1**.- Given a subset of t disparate shares, the t disparate shadow images $S{H}_{i}$, can be extracted from shares ${S}_{i}$, $i\in 1,2,\dots ,t$, according the formula (7)$$S{H}_{i}={y}_{i,1}\Vert {y}_{i,2}\Vert \dots \Vert {y}_{i,w}.$$
**Step 2**.- The secret image I can be reconstructed by employing Birkhoff interpolation with t disparate shadow images.

## 3. The Proposed Scheme

**Definition**

**2.**

#### 3.1. Shares Generation Procedure

**Step 1**.- Employ a reversible permutation operation on the secret image I to acquire a permuted secret image $\widehat{I}$. It is necessary to reduce the association between adjacent pixels.
**Step 2**.- Every t non-lapped pixels ${a}_{0},{a}_{1},\dots ,{a}_{t-1}$ are separated as a unit, and each unit can be used to construct a $t-1$ degree function $f\left(x\right)$ as follows:$$f\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\dots +{a}_{t-1}{x}^{t-1}mod\left({2}^{8}\right).$$
**Step 3**.- Repeat the step 2 until all pixels have been processed, and the intermediate shadows ${T}_{i}^{1}$ for ${G}_{1}$ can be generated by computing ${T}_{i}^{1}=f\left(i\right)$.
**Step 4**.- Calculate ${t}_{1}$-th derivative of $f\left(x\right)$ in the step 2, the derivation is shown as:$$g\left(x\right)={f}^{\left({t}_{1}\right)}\left(x\right)={b}_{0}+{b}_{1}x+\dots +{b}_{{t}_{2}-1}{x}^{{t}_{2}-1}mod\left({2}^{8}\right).$$The results of ${S}_{j}^{2}=g\left(j\right)$ are the pixel values in shares and the generated shares would be sent to the shareholders in ${G}_{2}$.
**Step 5**.- Obtain the mask shadow $R={b}_{0}\times {b}_{1}\times \dots \times {b}_{{t}_{2}-1}mod\left({2}^{8}\right)$.
**Step 6**.- The shares ${S}_{i}^{1}$ sent to ${G}_{1}$ can be generated by ${S}_{i}^{1}=({T}_{i}^{1}+R)mod\left({2}^{8}\right)$.

#### 3.2. Shares Generation Procedure

**Step 1**.- Given a subset of t disparate shares includes ${t}_{1}$ shares from ${G}_{1}$ and ${t}_{2}$ shares from ${G}_{2}$. Using ${t}_{2}$ shares from ${G}_{2}$, the coefficients of $g\left(x\right)$ can be reconstructed by Birkhoff interpolation and the mask shadow R can be obtained by $R={b}_{0}\times {b}_{1}\times \dots \times {b}_{{t}_{2}-1}mod\left({2}^{8}\right)$.
**Step 2**.- When ${G}_{2}$ has been calculated to generate the mask shadow R, the mask shadow R will be sent to ${G}_{1}$, and the ${t}_{1}$ intermediate shadows $T=({S}^{1}-R)mod\left({2}^{8}\right)$ in ${G}_{1}$ can be reconstructed.
**Step 3**.- By utilizing ${t}_{1}$ shares from ${G}_{1}$ and ${t}_{2}$ shares from ${G}_{2}$, the permuted secret image $\widehat{I}$ can be reconstructed by employing Birkhoff interpolation.
**Step 4**.- The secret image I can be reconstructed by employing the corresponding inverse-permutation on the permuted secret image $\widehat{I}$.

## 4. Security Analysis

**Case 1**. When $|{G}_{1}|<{t}_{1}$, $|{G}_{2}|\ge {t}_{2}$, $|{G}_{1}|+|{G}_{2}|\ge t$, the secret image cannot be reconstructed.

**Proof.**

**Case 2.**When $|{G}_{1}|\ge {t}_{1}$, $|{G}_{2}|<{t}_{2}$, $|{G}_{1}|+|{G}_{2}|\ge t$, the secret image cannot be reconstructed.

**Proof.**

**Case 3.**When $|{G}_{1}|\ge {t}_{1}$, $|{G}_{2}|\ge {t}_{2}$, $|{G}_{1}|+|{G}_{2}|\ge t$, the secret image can be reconstructed.

**Proof.**

## 5. Simulation and Comparison

#### 5.1. The Security

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 5.2. Histogram Analysis

#### 5.3. Correlation of Adjacent Pixels Analysis

#### 5.4. Resisting Noise Analysis

#### 5.5. Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) the test secret image “Lena”, (

**b**) the permuted secret image, (

**c**) the generated shares sent to ${G}_{1}$, (

**d**) the generated shares sent to ${G}_{2}$.

**Figure 2.**The simulation results in Examples 1, 2, 3, (

**a**) the reconstructed image in Example 1 (

**b**) the reconstructed image in Example 2 (

**c**) the reconstructed image in Example 3.

**Figure 3.**Histograms of Lena and the generated shares. (

**a**) The histogram of “Lena” (

**b**) the histogram of shares in ${G}_{1}$ (

**c**) the histogram of shares in ${G}_{2}$.

**Figure 4.**Histograms of Lena and the generated shares. (

**a**) The histogram of “Lena” (

**b**) the histogram of shares in ${G}_{1}$ (

**c**) the histogram of shares in ${G}_{2}$.

**Figure 5.**The results of resisting Gaussian noise. (

**a**) the share with 1% Gaussian noise and the decrypted result (

**b**) the share with 5% Gaussian noise and the decrypted result (

**c**) the share with 10% Gaussian noise and the decrypted result (

**d**) the share with 15% Gaussian noise and the decrypted result.

**Table 1.**The average correlation coefficients of shadows in $\left(\right(\mathrm{1,4}),4,4,8)$-PESIS scheme.

Lena | ${\mathit{S}}_{1}^{1}$ | ${\mathit{S}}_{1}^{2}$ | |
---|---|---|---|

Horizontal | 0.9746 | −0.0130 | −0.0131 |

Vertical | 0.9539 | −0.0089 | −0.0028 |

Diagonal | 0.9494 | −0.0101 | −0.0088 |

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**MDPI and ACS Style**

Wu, Z.; Liu, Y.; Jia, X.
A Novel Hierarchical Secret Image Sharing Scheme with Multi-Group Joint Management. *Mathematics* **2020**, *8*, 448.
https://doi.org/10.3390/math8030448

**AMA Style**

Wu Z, Liu Y, Jia X.
A Novel Hierarchical Secret Image Sharing Scheme with Multi-Group Joint Management. *Mathematics*. 2020; 8(3):448.
https://doi.org/10.3390/math8030448

**Chicago/Turabian Style**

Wu, Zhen, Yining Liu, and Xingxing Jia.
2020. "A Novel Hierarchical Secret Image Sharing Scheme with Multi-Group Joint Management" *Mathematics* 8, no. 3: 448.
https://doi.org/10.3390/math8030448