Lossless and Efficient PolynomialBased Secret Image Sharing with Reduced Shadow Size
Abstract
:1. Introduction
2. Preliminaries
2.1. PolynomialBased Secret Image Sharing
2.2. PSIS with Lossless Recovery
2.3. Generalized Arnold Permutation
3. The Proposed Scheme
3.1. Design Concept
3.2. The Permutation Process
 In Step 2, the formula to select the six numbers is fixed in advance.
 In Step 4, if in permutation phase before the sharing process, we evaluate M as Equation (7); else, in inverse permutation after the recovery process phase, we evaluate M as Equation (8).$$M={\left[\begin{array}{cc}1& \alpha \\ \beta & \alpha \beta +1\end{array}\right]}^{\theta}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}N$$$$M={\left[\begin{array}{cc}\alpha \beta +1& \alpha \\ \beta & 1\end{array}\right]}^{\theta}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}N$$$$\left[\begin{array}{c}{x}^{\prime}\\ {y}^{\prime}\end{array}\right]=M\left[\begin{array}{c}x\\ y\end{array}\right]\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}N$$
Algorithm 1 The permutation process 
Input: An image I with size of $N\times N$. 
Output: A permuted image $\widehat{I}$ with size of $N\times N$ 
Step 1. Count the numbers of each grayscale pixel value in the image I and sort them in ascending order. 
Step 2. Select three small numbers ${l}_{1}$, ${l}_{2}$, ${l}_{3}$ (${l}_{1}\le {l}_{2}\le {l}_{3}$) and three large numbers ${h}_{1}$, ${h}_{2}$, ${h}_{3}$ (${h}_{1}\ge {h}_{2}\ge {h}_{3}$) from the order according to a certain formula. 
Step 3. Generate the three parameters $\alpha $, $\beta $ and $\theta $ as Equation (6). 
Step 4. Evaluate the conversion matrix M as Equation (7) or Equation (8). 
Step 5. For each pixel P with the position $(x,y)$ in the image I, map P to a new position $({x}^{\prime},{y}^{\prime})$ according to Equation (9). 
Step 6. Output the permuted image $\widehat{I}$. 
3.3. The Sharing Process
 In Step 1, each section consists of $2(k1)$ pixels due to the first $k1$ coefficients in Equation (5) are utilized to embed secret values in Step 2 and each value consists of two adjacent pixel values in Step 3. Besides, to guarantee all pixels can be processed, the width of the image, N, should be an integer multiple of $2(k1)$.
 In Step 4, the last coefficient ${a}_{k1}$ is randomly assigned to improve the security.
 In Steps 5–7, we evaluate n shared values of each section. The screening operation occurs in Step 7 to guarantee none of the shared values is larger than 65,535.
 In Step 8, we obtain $2n$ shared pixels of each section.
 A sharing phase consists of Steps 3–8. In total, there are $N\times \frac{N}{2(k1)}$ sharing phases, and $N\times \frac{N}{k1}$ shared pixels for each shadow image are generated.
Algorithm 2 The proposed $(k,n)$ threshold PSIS scheme 
Input: A permuted secret image $\widehat{S}$ with size of $N\times N$; Threshold parameters $(k,n)$; n different serial numbers ${x}_{1},{x}_{2},\cdots ,{x}_{n}$. 
Output:n shadow images $S{C}_{1},S{C}_{2},\cdots ,S{C}_{n}$. 
Step 1. Divide the image S into $N\times \frac{N}{2(k1)}$ nonoverlapping sections, each of which consists of $2(k1)$ adjacent pixels. 
Step 2. For each $2(k1)$pixel section $Sec(i,j)=\{{P}_{m}(i,j)m\in \left[1,2(k1)\right]\}$, $i\in \left[1,N\right]$, $j\in \left[1,\frac{N}{2(k1)}\right]$, repeat Steps 3–8 until all sections have been processed. 
Step 3. Assign the coefficients ${a}_{0},{a}_{1},\cdots ,{a}_{k2}$ as follows.
$$\begin{array}{cc}{a}_{0}=& {P}_{1}(i,j)\times 256+{P}_{2}(i,j)\hfill \\ {a}_{1}=& {P}_{3}(i,j)\times 256+{P}_{4}(i,j)\hfill \\ & \cdots \hfill \\ {a}_{k2}=& {P}_{2(k2)+1}(i,j)\times 256+{P}_{2(k2)+2}(i,j)\hfill \end{array}$$

Step 4. Generate a random integer from $[0,\mathrm{65,536}]$ as the coefficient ${a}_{k1}$. 
Step 5. For each serial number ${x}_{t}$, $t\in \left[1,n\right]$, repeat Steps 6–7 until all n shared values have been evaluated. 
Step 6. Evaluate the shared value $f\left({x}_{t}\right)$ as follows.
$$f\left({x}_{t}\right)=({a}_{0}+{a}_{1}{x}_{t}+{a}_{2}{x}_{t}^{2}+\cdots +{a}_{k1}{x}_{t}^{k1})\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}\mathrm{65,537}$$

Step 7. If $f\left({x}_{t}\right)>\mathrm{65,535}$, return to Step 4 and redo Steps 4–7. Else continue. 
Step 8. For each shared value $f\left({x}_{t}\right)$, $t\in \left[1,n\right]$, generate two adjacent pixels in shadow image $S{C}_{t}$ as follows.
$$\begin{array}{c}\hfill S{C}_{t}(i,2j1)=f({x}_{t})/256\\ \hfill S{C}_{t}(i,2j)=f\left({x}_{t}\right)\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}256\end{array}$$

Step 9. Output n shadow images $S{C}_{1},S{C}_{2},\cdots ,S{C}_{n}$. 
3.4. The Recovery Process
 In Step 1, we take the first two nonused adjacent pixels from each of the k shadow images, to form a set with k pairs of shared pixels. The number of all sets is $N\times \frac{N}{2(k1)}$.
 Steps 2, 3 and 4 are the inverse operations of Steps 8, 6 and 3 in Algorithm 2, respectively.
 A recovery phase consists of Steps 2–4. In each recovery phase, we retrieve a $2(k1)$pixel section of the permuted secret image as mentioned in Algorithm 2. In total, there are $N\times \frac{N}{2(k1)}$ recovery phases.
Algorithm 3 Secret image recovery of the proposed scheme 
Input:k shadow images $S{C}_{1},S{C}_{2},\cdots ,S{C}_{k}$ with size of $N\times \frac{N}{k1}$; Threshold parameters $(k,n)$; k different serial numbers ${x}_{1},{x}_{2},\cdots ,{x}_{k}$. 
Output: A reconstructed permuted secret image $\widehat{{S}_{r}}$. 
Step 1. For each two nonoverlapping adjacent pixels $S{C}_{t}(i,2j1)$ and $S{C}_{t}(i,2j)$ in each shadow image $S{C}_{t}$, $i\in \left[1,N\right]$, $j\in \left[1,\frac{N}{2(k1)}\right]$, $t\in \left[1,k\right]$, repeat Steps 2–4 until all pairs pixels of the k shadow images have been processed. 
Step 2. Evaluate the k shared values $shar{e}_{t}(i,j)$, $t\in \left[1,k\right]$, as follows.
$$shar{e}_{t}(i,j)=S{C}_{t}(i,2j1)\times 256+S{C}_{t}(i,2j)$$

Step 3. Use the k serial numbers, k shared values and the Lagrange’s interpolation to obtain the $k1$ coefficients ${a}_{0},{a}_{1},\cdots ,{a}_{k2}$ in the the linear equations as Equation (10). 
Step 4. Obtain the $2(k1)$ pixels $\{{P}_{m}(i,j)m\in \left[1,2(k1)\right]\}$ corresponding to ${a}_{0},{a}_{1},\cdots ,{a}_{k2}$ as follows.
$$\begin{array}{cc}{P}_{1}(i,j)& ={a}_{0}/256\hfill \\ {P}_{2}(i,j)& ={a}_{0}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}256\hfill \\ & \cdots \hfill \\ {P}_{2(k1)1}(i,j)& ={a}_{k2}/256\hfill \\ {P}_{2(k1)}(i,j)& ={a}_{k2}\phantom{\rule{4pt}{0ex}}mod\phantom{\rule{4pt}{0ex}}256\hfill \end{array}$$

Step 5. Obtain all $N\times N$ pixels and reconstruct the permuted secret image $\widehat{{S}_{r}}$. 
Step 6. Output $\widehat{{S}_{r}}$. 
4. Performance Analyses
4.1. Lossless Recovery Analysis
4.2. Threshold Analysis
4.3. Security Analysis
5. Experiments and Comparisons
5.1. Image Illustration
 Lossless recovery: The secret image can be reconstructed losslessly with k or more shadow images.
 Security: The shadow images are noisylike, thus every single shadow is secure. Furthermore, there is no leakage of secret information from recovered images with less than k shadow images, which shows security of our scheme.
 Reduced shadow size: In the proposed $(k,n)$ threshold PSIS, the size of each shadow image is $\frac{1}{k1}$ of that of the secret image.
5.2. Comparisons with Related Works
 Lossless recovery: Classic PSISs can only achieve lossy recovery, while several other PSISs including our scheme with different solutions can achieve lossless recovery.
 Shadow size: Except ThienandLin’s and Our proposed PSISs, shadow size generated by other PSISs are the same or more than that of the secret image. The size of our PSIS is a little larger than that of ThienandLin’s, but the security and lossless recovery can be guaranteed. Furthermore, we can also utilize partial bits of the coefficient ${a}_{k1}$ to embed more secret values and assign remainder bits randomly, to further reduce the shadow size as well as to improve the efficiency.
 Random pixel expansion: Random pixel expansion may occur in ThienandLin’s lossless PSIS, so its generated shares can only be stored as data rather than images. In our scheme, n noiselike shares with size of $\frac{1}{k1}$ of that of the secret image are generated, which can be still stored as images.
 Preencryption and decryption: ThienandLin’s PSIS needs extra preencryption to avoid the leakage of secret information, so it results in more costs. Our scheme needs no extra permutation if there is no higher level of security requirement in general application scenarios.
 Computational complexity: In some PSISs, there is extra recombination or decryption after the recovery process, so only the complexity of secret recovery process is calculated here. Only the constant coefficient needs to be calculated by the Lagrange interpolation as the secret value in Shamir’s PSISs, while two or more coefficients as secret values in ThienandLin’s, Ding and coworkers’ and Our PSISs should be computed by solving equations. Therefore, the complexity of the latter PSISs is larger than that of the former PSISs. Yang and coworkers’ PSIS is based on Galois Field GF(${2}^{8}$), which lacks the theoretical calculation of computational complexity. However, the complexity of computations based on Galois Field GF(${2}^{8}$) is much larger than that of computations based on integers.
 The running time of our scheme is much shorter than that of Shamir’s and Ding and coworkers’ schemes, which indicates our scheme is more efficient than Shamir’s and Ding and coworkers’ schemes.
 The running time of our scheme is little longer than ThienandLin’s scheme. However, if the permutation process is removed in our scheme, the running time is approximately equal to or even slightly shorter than that of ThienandLin’s scheme. In fact, our scheme without permutation is sufficient to ensure security in general application scenarios.
 We can modify our scheme, specifying one pixel value as a secret value and 257 as the prime, with the same principle. As a result, the running time becomes longer than our original scheme’s. Therefore, to a certain degree, decreasing the number of secret values has improved the efficiency of sharing and recovery.
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Naor, M.; Shamir, A. Visual cryptography. In Workshop on the Theory and Application of of Cryptographic Techniques; Springer: Berlin, Germany, 1994; pp. 1–12. [Google Scholar]
 Weir, J.; Yan, W. A comprehensive study of visual cryptography. In Transactions on Data Hiding and Multimedia Security V; Springer: Berlin, Germany, 2010; pp. 70–105. [Google Scholar]
 Yan, X.; Liu, X.; Yang, C.N. An enhanced threshold visual secret sharing based on random grids. J. RealTime Image Process. 2015, 14, 61–73. [Google Scholar] [CrossRef]
 Shamir, A. How to share a secret. Commun. ACM 1979, 22, 612–613. [Google Scholar] [CrossRef]
 Xie, D.; Li, L.; Peng, H.; Yang, Y. A Secure and Efficient Scalable Secret Image Sharing Scheme with Flexible Shadow Sizes. PLoS ONE 2017, 12, e0168674. [Google Scholar] [CrossRef] [PubMed]
 Thien, C.C.; Lin, J.C. Secret image sharing. Comput. Graph. 2002, 26, 765–770. [Google Scholar] [CrossRef]
 Lin, P.Y.; Chan, C.S. Invertible secret image sharing with steganography. Pattern Recognit. Lett. 2010, 31, 1887–1893. [Google Scholar] [CrossRef]
 He, J.; Lan, W.; Tang, S. A secure image sharing scheme with high quality stegoimages based on steganography. Multimed. Tools Appl. 2017, 76, 7677–7698. [Google Scholar] [CrossRef]
 Mao, Q.; Bharanitharan, K.; Chang, C.C. Novel Lossless Morphing Algorithm for Secret Sharing via Meaningful Images. J. Inf. Hiding Multimed. Signal Process. 2016, 7, 1168–1184. [Google Scholar]
 Yang, C.N.; Ciou, C.B. Image secret sharing method with twodecodingoptions: Lossless recovery and previewing capability. Image Vis. Comput. 2010, 28, 1600–1610. [Google Scholar] [CrossRef]
 Li, P.; Yang, C.N.; Kong, Q.; Ma, Y.; Liu, Z. Sharing more information in gray visual cryptography scheme. J. Vis. Commun. Image Represent. 2013, 24, 1380–1393. [Google Scholar] [CrossRef]
 Li, P.; Yang, C.N.; Wu, C.C.; Kong, Q.; Ma, Y. Essential secret image sharing scheme with different importance of shadows. J. Vis. Commun. Image Represent. 2013, 24, 1106–1114. [Google Scholar] [CrossRef]
 Guo, C.; Chang, C.C.; Qin, C. A hierarchical threshold secret image sharing. Pattern Recognit. Lett. 2012, 33, 83–91. [Google Scholar] [CrossRef]
 Chen, C.C.; Tsai, Y.H. An Expandable Essential Secret Image Sharing Structure. J. Inf. Hiding Multimed. Signal Process. 2016, 7, 135–144. [Google Scholar]
 Chen, W.K.; Chen, H.P.; Tso, H.K. A Friendly and Verifiable Image Sharing Method. J. Netw. Intell. 2016, 1, 46–51. [Google Scholar]
 Zhou, Z.; Yang, C.N.; Cao, Y.; Sun, X. Secret Image Sharing Based on Encrypted Pixels. IEEE Access 2018, 6, 15021–15025. [Google Scholar] [CrossRef]
 Wu, X.; Yang, C.N.; Zhuang, Y.T.; Hsu, S. Improving recovered image quality in secret image sharing by simple modular arithmetic. Signal Process. Image Commun. 2018, 66, 42–49. [Google Scholar] [CrossRef]
 Bao, L.; Yi, S.; Zhou, Y. Combination of Sharing Matrix and Image Encryption for Lossless (k,n)Secret Image Sharing. IEEE Trans. Image Process. 2017, 26, 5618–5631. [Google Scholar] [CrossRef] [PubMed]
 Liu, L.; Lu, Y.; Yan, X.; Wang, H. Greyscaleimagesoriented progressive secret sharing based on the linear congruence equation. Multimed. Tools Appl. 2017, 1–28. [Google Scholar] [CrossRef]
 Yan, X.; Lu, Y.; Liu, L.; Wan, S.; Ding, W.; Liu, H. Chinese Remainder TheoremBased Secret Image Sharing for (k, n) Threshold. In Proceedings of the International Conference on Cloud Computing and Security, Nanjing, China, 16–18 June 2017; Springer: Berlin, Germany, 2017; pp. 433–440. [Google Scholar]
 Lin, S.J.; Lin, J.C. VCPSS: A twoinone twodecodingoptions image sharing method combining visual cryptography (VC) and polynomialstyle sharing (PSS) approaches. Pattern Recognit. 2007, 40, 3652–3666. [Google Scholar] [CrossRef]
 Ulutas, G.; Nabiyev, V.V.; Ulutas, M. Polynomial approach in a secret image sharing using quadratic residue. In Proceedings of the International Symposium on Computer and Information Sciences, Guzelyurt, Northern Cyprus, 14–16 September 2009; pp. 586–591. [Google Scholar]
 Yang, C.N.; Chen, T.S.; Yu, K.H.; Wang, C.C. Improvements of image sharing with steganography and authentication. J. Syst. Softw. 2007, 80, 1070–1076. [Google Scholar] [CrossRef]
 Ding, W.; Liu, K.; Yan, X.; Liu, L. PolynomialBased Secret Image Sharing Scheme with Fully Lossless Recovery. Int. J. Digit. Crime Forens. IJDCF 2018, 10, 120–136. [Google Scholar] [CrossRef]
 Jin, D.; Yan, W.Q.; Kankanhalli, M.S. Progressive color visual cryptography. J. Electr. Imaging 2005, 14, 033019. [Google Scholar] [CrossRef] [Green Version]
 Li, P.; Ma, P.J.; Su, X.H.; Yang, C.N. Improvements of a twoinone image secret sharing scheme based on gray mixing model. J. Vis. Commun. Image Represent. 2012, 23, 441–453. [Google Scholar] [CrossRef]
 Qi, D.; Wang, D.; Yang, D. Matrix transformation of digital image and its periodicity. Prog. Nat. Sci. Mater. Int. 2001, 11, 548–549. [Google Scholar]
Schemes  Lossless Recovery  Shadow Size  Random Pixel Expansion  PreEncryption and Decryption  Computational Complexity 

Shamir et al. [4]  No  1  No  No  $O\left(k{log}^{2}k\right)$ 
ThienandLin (lossy) [6]  No  $\frac{1}{k}$  No  Yes  $O\left({k}^{3}\right)$ 
ThienandLin (lossless) [6]  Yes  $\ge \frac{1}{k}$  Yes  Yes  $O\left({k}^{3}\right)$ 
Yang et al. [23]  Yes  1  No  No  High 
Ding et al. [24]  Yes  1  No  No  $O\left({k}^{3}\right)$ 
Our PSIS  Yes  $\frac{1}{k1}$  No  Yes  $O\left({k}^{3}\right)$ 
Our PSIS (without permutation)  Yes  $\frac{1}{k1}$  No  No  $O\left({k}^{3}\right)$ 
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Zhou, X.; Lu, Y.; Yan, X.; Wang, Y.; Liu, L. Lossless and Efficient PolynomialBased Secret Image Sharing with Reduced Shadow Size. Symmetry 2018, 10, 249. https://doi.org/10.3390/sym10070249
Zhou X, Lu Y, Yan X, Wang Y, Liu L. Lossless and Efficient PolynomialBased Secret Image Sharing with Reduced Shadow Size. Symmetry. 2018; 10(7):249. https://doi.org/10.3390/sym10070249
Chicago/Turabian StyleZhou, Xuan, Yuliang Lu, Xuehu Yan, Yongjie Wang, and Lintao Liu. 2018. "Lossless and Efficient PolynomialBased Secret Image Sharing with Reduced Shadow Size" Symmetry 10, no. 7: 249. https://doi.org/10.3390/sym10070249