Lossless and Efficient Secret Image Sharing Based on Matrix Theory Modulo 256
Abstract
:1. Introduction
2. Preliminaries
2.1. Shamir’s Polynomial-Based SS
2.2. Matrix Method for Polynomial-Based SS
2.3. The Method to Solve Inverse Matrix
3. The Proposed Scheme
3.1. The Basic Idea
- Condition 1:
- Any k row vectors of the matrix K are linearly independent.
- Condition 2:
- The determinant of any $k\times k$ submatrix is coprime with 256.$$\mathbf{K}=\left[\begin{array}{ccccc}{x}_{11}& {x}_{12}& {x}_{13}& \cdots & {x}_{1k}\\ {x}_{21}& {x}_{22}& {x}_{23}& \cdots & {x}_{2k}\\ {x}_{31}& {x}_{32}& {x}_{33}& \cdots & {x}_{3k}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {x}_{n1}& {x}_{n2}& {x}_{n3}& \cdots & {x}_{nk}\end{array}\right]$$
3.2. The Sharing Phase
Algorithm 1 The sharing phase of the proposed scheme. |
Input: The threshold parameters $(k,n)$, $n=k$ or $n=k+1$, and a grayscale secret image S with size of $M\times N$. Output: n shadows $S{C}_{1},S{C}_{2},\cdots ,S{C}_{n}$ and matrix K. Step 1: Generate an $n\times k$ matrix K randomly, and determine that the determinant of any $k\times k$ submatrix is not zero and is coprime with 256. If not, repeat Step 1. Step 2: For every secret pixel s in each position $S(i,j)$, $(i,j)\in \left\{(i,j)|1\u2a7di\u2a7dM,1\u2a7dj\u2a7dN\right\}$, repeat Step 3–4. Step 3: Generate a vector a$={({a}_{0},{a}_{1},\cdots ,{a}_{k-1})}^{T}$, set ${a}_{0}=s$, and generate ${a}_{1},\cdots ,{a}_{k-1}$ randomly in [0,255]. Step 4: Compute f = Ka (mod 256), where $S{C}_{1}(i,j)=f\left(1\right),\cdots ,S{C}_{n}(i,j)=f\left(n\right)$. Step 5: Output n shadows $S{C}_{1},S{C}_{2},\cdots ,S{C}_{n}$ and matrix K. |
3.3. The Recovery Phase
Algorithm 2 The recovery phase of the proposed scheme. |
Input: The k shadows which are randomly selected from n shadows $S{C}_{1},S{C}_{2},\cdots ,S{C}_{n}$ and corresponding k vectors ${k}_{i}$. Output:The original secret image S. Step 1: Construct a matrix K by k vectors ${k}_{i}$. Step 2: Calculate the adjoint matrix ${K}^{*}$ and determinant $\left|K\right|$ of matrix K. Compute the inverse matrix ${K}^{-1}$ according to Equation (7). Step 3: For each position $S(i,j)$, $(i,j)\in \left\{(i,j)|1\u2a7di\u2a7dM,1\u2a7dj\u2a7dN\right\}$, repeat Step 4–5. Step 4: Get a by $\mathbf{a}=\frac{{K}^{*}}{\left|K\right|}\mathbf{f}$. Step 5: Set the pixel $S(i,j)={a}_{0}$. Step 6: Output the secret image S. |
4. Theoretical Analysis
4.1. Threshold Analysis
- To make the determinant odd, we need to add an odd number of product terms with odd products. Corresponding to: to make the determinant to be 1, we need to add an odd number of product terms whose product is 1.
- To make the product term odd, all the factors need to be odd. Corresponding to: to make the product term to be 1, we need all 1 in the factor.
- There is one even number in the factor, then the product is even. Corresponding to: there is one 0 in the factor, then the product is 0.
4.2. Security Analysis
4.3. Complexity Evaluation
4.4. Lossless Recovery Analysis
5. Experiments and Comparisons
5.1. Image Illustration
5.2. Comparisons with Related Works
5.2.1. Illustration Comparison
5.2.2. Efficiency Comparison
5.3. Brief Summary
- The secret image can be reconstructed losslessly with k or more shadows and there is no leakage of secret information from the recovered image with less than k shadows.
- The shadows are noisy-like, thus every single shadow gives no clue about the secret. Pixel values of shadow are evenly distributed without security issues.
- The proposed scheme has obvious advantages in efficiency.
6. Conclusions
- To further exploit the secret image sharing scheme, we can consider various recommendation mechanisms that provide content to the end users [26].
- We can use the personalized content retrieval mechanisms [27], in order to exploit the content, i.e., images, that the users consume to further improve our secret image sharing scheme.
- Big data that are available in complex systems [28] can be exploited to improve our analysis and model.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Notations | Descriptions |
---|---|
$(k,n)$ | threshold, $k\u2a7dn$ |
$MS\left(256\right)$ | the integer space of modulo 256 |
p | generally a prime number, we take 256 in this paper |
K | a random matrix by a filter operation satisfying any k row vectors of the matrix K are linearly independent and the determinant of any $k\times k$ submatrix is coprime with 256 |
K | a $k\times k$ submatrix of K |
a | a vector in which ${a}_{0}$ is the secret pixel value and others are generated randomly in [0,255] |
f | a vector obtained by Ka = f, whose elements are $s{c}_{i}$ |
$s{c}_{i}$ | a pixel in shadow image |
$S{C}_{i}$ | a shadow image corresponding to the i-th participant |
$(\mathit{k},\mathit{k})$ | Time | $(\mathit{k},\mathit{k}+1)$ | Time |
---|---|---|---|
(2, 2) | 0.000499 | (2, 3) | 0.000998 |
(3, 3) | 0.000497 | (3, 4) | 0.001501 |
(4, 4) | 0.000501 | (4, 5) | 0.004497 |
(5, 5) | 0.000499 | (5, 6) | 0.009499 |
(6, 6) | 0.000499 | (6, 7) | 0.031500 |
$(\mathit{k},\mathit{n})$ | mod 257 | mod ${2}^{8}$ | mod 256 (Ours) |
---|---|---|---|
(2, 2) | 1.040 | 1.116 | 0.906 |
(2, 3) | 1.387 | 1.513 | 0.992 |
(3, 3) | 1.522 | 1.752 | 1.131 |
(3, 4) | 1.871 | 2.144 | 1.211 |
$(\mathit{k},\mathit{n})$ | mod 257 | mod ${2}^{8}$ | mod 256 (Ours) |
---|---|---|---|
(2, 2) | 1.146 | 0.934 | 0.785 |
(2, 3) | 1.139 | 0.923 | 0.789 |
(3, 3) | 1.743 | 1.562 | 0.891 |
(3, 4) | 1.746 | 1.561 | 0.878 |
$(\mathit{k},\mathit{n})$ | mod 257 | mod ${2}^{8}$ | mod 256 (Ours) |
---|---|---|---|
(2, 2) | 2.187 | 2.050 | 1.691 |
(2, 3) | 2.526 | 2.436 | 1.781 |
(3, 3) | 3.266 | 3.314 | 2.022 |
(3, 4) | 3.617 | 3.705 | 2.089 |
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Share and Cite
Yu, L.; Liu, L.; Xia, Z.; Yan, X.; Lu, Y. Lossless and Efficient Secret Image Sharing Based on Matrix Theory Modulo 256. Mathematics 2020, 8, 1018. https://doi.org/10.3390/math8061018
Yu L, Liu L, Xia Z, Yan X, Lu Y. Lossless and Efficient Secret Image Sharing Based on Matrix Theory Modulo 256. Mathematics. 2020; 8(6):1018. https://doi.org/10.3390/math8061018
Chicago/Turabian StyleYu, Long, Lintao Liu, Zhe Xia, Xuehu Yan, and Yuliang Lu. 2020. "Lossless and Efficient Secret Image Sharing Based on Matrix Theory Modulo 256" Mathematics 8, no. 6: 1018. https://doi.org/10.3390/math8061018