# An Efficient Essential Secret Image Sharing Scheme Using Derivative Polynomial

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## Abstract

**:**

## 1. Introduction

- (1)
- Same-sized shadows.
- (2)
- Smaller-sized shadows.
- (3)
- Effectiveness.

## 2. Preliminaries

#### 2.1. $\left(k,n\right)$ Secret Image Sharing Scheme

#### 2.1.1. Sharing Phase

**Step 1**: Set a prime number $p$; usually $p$ is set as 251.

**Step 2**: Permute each pixel’s position in $I$ by a permutation sequence. Then, all the pixels larger than 250 should be truncated to 250.

**Step 3**: The processed permuted image is divided into some units with $k$ pixels. These $k$ pixels then are to be used to construct a sharing function $f\left(x\right)={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\cdots +{a}_{k-1}{x}^{k-1}\mathrm{mod}p$, where ${a}_{0}$, ${a}_{1}$, …, ${a}_{k-1}$ are the pixels values in each unit.

**Step 4**: The output $f\left(1\right)$, $f\left(2\right)$, …, $f\left(n\right)$ are pixel values sequentially assigned to $n$ shadows ${O}_{1}$, ${O}_{2}$, …, ${O}_{n}$.

**Step 5**: Repeat step 3 and step 4 until each unit has been processed, the generated shadows ${O}_{1}$, ${O}_{2}$, …, ${O}_{n}$ are shared to $n$ participates ${P}_{1}$, ${P}_{2}$, …, ${P}_{n}$ respectively.

#### 2.1.2. Recovery Phase

**Step 1**: By collecting $k$ shadows from $k$ participates, the $\left(k-1\right)$ degree polynomial $f\left(x\right)$ can be reconstructed based on Lagrange’s interpolation. Therefore, the pixels in each unit can be recovery and the permuted secret image.

**Step 2**: Employ the corresponding inverse-permutation and the permuted secret image can obtain the secret image.

#### 2.2. $\left(t,s,k,n\right)$ Essential Secret Image Sharing Scheme

#### 2.2.1. Sharing Phase

**Step 1**: Encrypt $I$ by $(k,s+k-t)$-SIS scheme to generate $\left(s+k-t\right)$ shadows ${I}_{j}$ as intermediate shadows, where $1\le j\le s+k-t$. Set ${I}_{j}\left(1\le j\le s\right)$ as $s$ essential shadows ${O}_{i}\left(1\le i\le s\right)$ shared to corresponding essential participates ${P}_{i}$.

**Step 2**: Each remaining intermediate shadows ${I}_{j}\left(s+1\le j\le s+k-t\right)$ are further encrypted by $(j,n-s)$-SIS scheme and obtain $(n-s)$ sub-shadows ${S}_{j,1},$, ${S}_{j,2}$, …, ${S}_{j,n-s}$.

**Step 3**: The $(n-s)$ non-essential shadows ${O}_{i}\left(s+1\le i\le n\right)$ can be obtained by ${O}_{i}={S}_{j,i-s}\parallel {S}_{j+1,i-s}\parallel \cdots \parallel {S}_{s+k-t,i-s}$, and then shared to the corresponding non-essential participates ${P}_{i}\left(s+1\le i\le n\right)$.

#### 2.2.2. Recovery Phase

**Step 1**: Employ the Lagrange’s interpolation, the $(k-t)$ intermediate shadows ${I}_{j}\left(s+1\le j\le s+k-t\right)$ can be recovered by $(k-t)$ non-essential participates.

**Step 2**: Employ the Lagrange’s interpolation, the secret image can be recovered by $t$ essential shadows ${I}_{j}\left(1\le j\le t\right)$ from essential participates ${P}_{1}$, ${P}_{2}$, …, ${P}_{t}$.

## 3. Review and Analysis of Li et al.’s Scheme

#### 3.1. Review Li et al.’s Scheme

#### 3.1.1. Sharing Phase

**Step 1**: Permute each pixel’s position in $I$ by a permutation sequence to obtain the permuted secret image $\widehat{I}$.

**Step 2**: Employ the $(k,n)$-SIS scheme on $\widehat{I}$ to obtain $n$ intermediate shadows ${T}_{i}\left(1\le i\le n\right)$. And the mask shadow $R$ can be generated as $R={T}_{1}\oplus {T}_{2}\oplus \cdots \oplus {T}_{t}$, where $\oplus $ is denoted as the bit-wise XOR operation.

**Step 3**: The $t$ essential shadows ${O}_{i}={T}_{i}$ are shared to $t$ essential participants ${P}_{i}\left(1\le i\le t\right)$. And $\left(n-t\right)$ non-essential shadows ${O}_{i}=\left({T}_{i}+R\right)\mathrm{mod}\left(256\right)$ are shared to $\left(n-t\right)$ non-essential participates ${P}_{i}\left(t+1\le i\le n\right)$.

Algorithm 1 Sharing phase of Li et al.’s scheme |

Input: A secret image $\widehat{I}=P\left(I\right)$ and a pair of the parameter.Output: $n$ shadows: ${O}_{1}$, ${O}_{2}$, …, ${O}_{t}$ are essential shadows; ${O}_{t+1}$, ${O}_{t+2}$, …, ${O}_{n}$ are non-essential shadows. |

(A1-1): Permute $I$ to $\widehat{I}$ by $\widehat{I}=P\left(I\right)$;/*$P(\cdot )$: a reversible permutation operation */ (A1-2): Generate the intermediate shadows ${T}_{1}$, ${T}_{2}$, …, ${T}_{n}$, by applying $(k,n)$-SIS scheme on $\widehat{I}$;(A1-3): Compute the mask shadow $R={T}_{1}\oplus {T}_{2}\oplus \cdots \oplus {T}_{t}$, where $\oplus $ denotes the bit-wise XOR operation;(A1-4): Generate $t$ essential shadows ${O}_{1}={T}_{1}$, ${O}_{2}={T}_{2}$, …, ${O}_{t}={T}_{t}$ and $\left(n-t\right)$ non-essential shadows ${O}_{t+1}=\left({T}_{t+1}+R\right)\mathrm{mod}\left(256\right)$, ${O}_{t+2}=\left({T}_{t+2}+R\right)\mathrm{mod}\left(256\right)$, …, ${O}_{n}=\left({T}_{n}+R\right)\mathrm{mod}\left(256\right)$; |

#### 3.1.2. Recovery Phase

**Step 1**: By collecting $t$ essential shadows from $t$ essential participates, the mask shadow $R$ can be reconstructed by computing $R={O}_{1}\oplus {O}_{2}\oplus \cdots \oplus {O}_{t}$.

**Step 2**: Reconstruct $\left(n-t\right)$ non-essential shadows by $(k-t)$ non-essential participates ${T}_{i}=\left({O}_{i}-R+256\right)\mathrm{mod}\text{}256$ from $(k-t)$ non-essential participates, where $t+1\le i\le k$.

**Step 3**: By collecting $t$ essential shadows ${T}_{i}={O}_{i}\left(1\le i\le t\right)$ and $(k-t)$ non-essential shadows ${T}_{i}\text{}\left(t+1\le i\le k\right)$, the permuted secret image $\widehat{I}$ can be recovered by Lagrange’s interpolation.

**Step 4**: The secret image can be recovered by employing the corresponding inverse-permutation on the permuted secret image.

Algorithm 2 Recovery phase of the Li et al.’s scheme |

Input: $t$ essential shadows and any $\left(k-t\right)$ non-essential shadows./* say $t$ essential shadows are ${O}_{1}$, ${O}_{2}$, …, ${O}_{t}$ and $(k-t)$ non-essential shadows are ${O}_{t+1}$, ${O}_{t+2}$, …, ${O}_{k}$*/ Output: The secret image $I$. |

(A2-1): Collect $t$ essential shadows to compute the mask shadow $R={O}_{1}\oplus {O}_{2}\oplus \cdots \oplus {O}_{t}$;(A2-2): Compute $k$ intermediate shadows ${T}_{1}$, ${T}_{2}$, …, ${T}_{k}$, as: ${T}_{1}={O}_{1}$, ${T}_{2}={O}_{2}$, …, ${T}_{t}={O}_{t}$ and ${T}_{t+1}=\left({O}_{t+1}-R+256\right)\mathrm{mod}\left(256\right)$, …, ${T}_{k}=\left({O}_{k}-R+256\right)\mathrm{mod}\left(256\right)$;(A2-3): Since there are ${T}_{1}$, ${T}_{2}$, …, ${T}_{n}$, the permuted image $\widehat{I}$ can be obtained by employing Lagrange’s interpolation;(A2-4): Acquire the secret image by $I={P}^{-1}(\hat{I})$;$/*{P}^{(-1)}(\cdot ):$ the corresponding inverse-permutation of $P(\cdot )*/$ |

#### 3.2. Analysis Li et al.’s Scheme

## 4. The Proposed Scheme

#### 4.1. Sharing Phase

**Step 1**: Permute each pixel’s position in $I$ by a permutation sequence to obtain the permuted secret image $\widehat{I}$.

**Step 2**: Employ the $\left(k,k\right)$-SIS scheme on $\widehat{I}$ to obtain $k$ intermediate shadows ${T}_{i}\left(1\le i\le k\right)$.

**Step 3**: Construct the $(k-1)$-degree function $g\left(x\right)={w}_{0}+{w}_{1}x+\dots +{w}_{k-1}{x}^{k-1}\mathrm{mod}\left({2}^{8}\right)$; the coefficients in $g\left(x\right)$ are the pixel values at the same position in each intermediate shadow. And the outputs ${O}_{i}=g\left(i\right)$ are $s$ essential shadows shared to essential participates ${P}_{i}$, where $1\le i\le s$.

**Step 4**: Calculate $t$-th derivative of $g\left(x\right)$, a $(k-t-1)$-degree polynomial ${g}^{(t)}(x)$ can be constructed, and the outputs ${O}_{i}={g}^{\left(t\right)}\left(x\right)$ are non-essential shadows shared with the other non-essential participates ${P}_{i}$, where $s+1\le i\le n$.

Algorithm 3 Sharing phase of the proposed scheme |

Input: A secret image $I$ and a pair of the parameters $(t,s,k,n)$. Output: $n$ shadows: ${O}_{1}$, ${O}_{2}$, …, ${O}_{s}$ are essential shadows; ${O}_{s+1}$, ${O}_{s+2}$, …, ${O}_{n}$ are non-essential shadows. |

(A3-1): Obtain the permuted image $\widehat{I}$ by $\widehat{I}=P\left(I\right)$;(A3-2): Generate the intermediate shadows by applying $(k,k)$-SIS scheme on $\widehat{I}$;(A3-3): Construct the function $g\left(x\right)$, and the outputs ${O}_{i}=g\left(i\right)$ are $s$ essential shadows, where $1\le i\le s$;(A3-4): Calculate $t$-th derivative of $g\left(x\right)$ to obtain ${g}^{(t)}(x)$, and the outputs ${O}_{i}={g}^{\left(t\right)}\left(x\right)$ are non-essential shadows, where $s+1\le i\le n$; |

#### 4.2. Recovery Phase

**Step 1**: By collecting no less than $k$ shadows involved at least $t$ essential shadows form participants, the $k$ coefficients in the function $g\left(x\right)$ can be reconstructed by employing Lagrange interpolation, so that $k$ intermediate shadows ${T}_{i}$ also can be reconstructed.

**Step 2**: Employ the Lagrange interpolation to obtain the permuted secret image $\widehat{I}$ by at least $k$ intermediate shadows ${T}_{i}$.

**Step 3**: The secret image $I$ can be recovered by employing the corresponding inverse-permutation on the permuted secret image $\widehat{I}$.

Algorithm 4 Recovery phase of the proposed scheme |

Input: Any at least $k$ shadows and no less than $t$ essential shadows included.Output: The secret image $I$. |

(A4-1): The function $g\left(x\right)$ can be reconstructed by any $k$ involved shadows including at least $t$ essential shadows;(A4-2): The intermediate shadows ${T}_{1}$, ${T}_{2}$, …, ${T}_{k}$ can be reconstructed by the function $g\left(x\right)$;(A4-3): The permuted secret image $\widehat{I}$ can be reconstructed by ${T}_{1}$, ${T}_{2}$, …, ${T}_{k}$;(A4-4): Acquire the original secret image by $I={P}^{-1}(\widehat{I})$; |

## 5. Analysis

#### 5.1. The Security Analysis

- (1)
- Threshold condition: $\left|Q\right|\ge k$.
- (2)
- Essentiality condition: $\left|Q\backslash NEP\right|\ge t$.

**Theorem 1.**

**Proof.**

**Case 1.**

**Case 2.**

#### 5.2. The Analysis of Shadow Size Ratio

**Theorem 2.**

**Proof.**

## 6. Simulation Results and Comparison

#### 6.1. Simulation Results

#### 6.2. Comparison

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The simulation results of (1, 2, 3)-ESIS based on Li et al.’s scheme [27]. (

**a**) The test secret image; (

**b**) the revealed image which only utilizes two non-essential shadows.

**Figure 2.**The simulation results of the proposed (2, 3, 4, 6)-ESIS scheme. (

**a**) ‘Lena’ with 512 × 512 pixels; (

**b**) the permutated secret image with 512 × 512 pixels; (

**c**) four intermediate shadows with 512 × 128 pixels; (

**d**) three essential shadows with 512 × 128 pixels; (

**e**) three non-essential shadows with 512 × 128 pixels.

**Figure 3.**The simulation results of the recovered image in the proposed (2, 3, 4, 6)-ESIS scheme. (

**a**) The recovered image when the collected shadows violates the threshold condition; (

**b**) the recovered image when the collected shadows violates the essentiality condition; (

**c**) the secret image can be recovered when the collected shadows satisfy both the threshold condition and the essentiality condition.

**Figure 4.**(

**a**) The histogram of the secret image ‘Lena’; (

**b**) the histogram of essential shadows; (

**c**) the histogram of non-essential shadows.

**Figure 5.**The simulation results of the proposed (2, 2, 4, 6)-ESIS scheme. (

**a**) The secret image ‘Lena’; (

**b**) the generated two essential shadows ${O}_{1}$, ${O}_{2}$ and four non-essential shadows ${O}_{3}-{O}_{6}$; (

**c**) the recovered image with four non-essential shadows; (

**d**) the recovered image with two essential shadows and two non-essential shadows.

**Figure 6.**The different size ratios among the proposed scheme and Chen’s scheme [26] with different thresholds.

**Table 2.**Comparison on shadows’ size ratio among the proposed scheme and the other three related works.

[24] | [25] | [26] | Proposed Scheme | |
---|---|---|---|---|

Essential shadow size ratio | $\frac{1}{t}$ | $\frac{1}{t}$ | $\frac{ry}{\left(k-t\right)\times \left(x+y\right)}$ | $\frac{1}{k}$ |

Non-essential shadow size ratio | $\frac{1}{t}$ | $\frac{1}{t}$ | $\frac{ry}{\left(k-t\right)\times \left(x+y\right)}$ | $\frac{1}{k}$ |

$(\mathit{t},\mathit{s},\mathit{k},\mathit{n})$ | [24] | [25] | [26] ^{a} | Proposed Scheme | |
---|---|---|---|---|---|

$(2,3,4,6)$ | Essential shadow size ratio | $0.5$ | 0.5 | $0.3$ | $0.25$ |

Non-essential shadow size ratio | $0.5$ | 0.5 | $0.3$ | $0.25$ | |

$(3,6,8,10)$ | Essential shadow size ratio | $0.333$ | 0.333 | $0.163$ | $0.125$ |

Non-essential shadow size ratio | $0.333$ | 0.333 | $0.163$ | $0.125$ | |

$(5,7,9,11)$ | Essential shadow size ratio | $0.2$ | 0.2 | $0.538$ | $0.111$ |

Non-essential shadow size ratio | $0.2$ | 0.2 | $0.538$ | $0.111$ |

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## Share and Cite

**MDPI and ACS Style**

Wu, Z.; Liu, Y.-N.; Wang, D.; Yang, C.-N.
An Efficient Essential Secret Image Sharing Scheme Using Derivative Polynomial. *Symmetry* **2019**, *11*, 69.
https://doi.org/10.3390/sym11010069

**AMA Style**

Wu Z, Liu Y-N, Wang D, Yang C-N.
An Efficient Essential Secret Image Sharing Scheme Using Derivative Polynomial. *Symmetry*. 2019; 11(1):69.
https://doi.org/10.3390/sym11010069

**Chicago/Turabian Style**

Wu, Zhen, Yi-Ning Liu, Dong Wang, and Ching-Nung Yang.
2019. "An Efficient Essential Secret Image Sharing Scheme Using Derivative Polynomial" *Symmetry* 11, no. 1: 69.
https://doi.org/10.3390/sym11010069