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\begin{document}

\title{Threshold changeable secret image sharing using bivariate polynomial%\thanks{Grants or other notes
%about the article that should go on the front page should be
%placed here. General acknowledgments should be placed at the end of the article.}
}
%\subtitle{Do you have a subtitle?\\ If so, write it here}

%\titlerunning{Short form of title}        % if too long for running head

\author{Yanxiao Liu \and Ping Wu \and Qindong Sun \and Zhili Zhou
        %etc.
}

%\authorrunning{Short form of author list} % if too long for running head

\institute{Yanxiao Liu\at
              XI'AN University of Technology\\
              \email{liuyanxiao@xaut.edu.cn}           %  \\
%             \emph{Present address:} of F. Author  %  if needed
           \and
           Ping Wu\at
              XI'AN University of Technology\\
           \and
           Qindong Sun\at
              XI'AN University of Technology\\
              \and
           Zhili Zhou\at
              Nanjing University of Information Science and Technology\\
}

\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor


\maketitle

\begin{abstract}
In secret image sharing (SIS) scheme, a confidential image is encrypted into multiple shadows, any group of shadows that reaches the threshold, otherwise nothing can be reconstructed at all. Most existing SIS schemes have a fixed threshold, however in this work, we consider more complicated cases that the threshold may be adjusted due to the changeable security environment. In this paper, we construct a $(k\leftrightarrow h,n)$ threshold changeable SIS (TCSIS) scheme using bivariate polynomial, which has $h-k+1$ possible thresholds $k,k+1,...,h$. During image reconstruction, each participant can update the his shadow according to the current threshold $T$ only based on his initial shadow. Comparing with previous TCSIS schemes, the proposed scheme achieves unconditional security, and can overcome the information disclosure problem caused by homomorphism.
\keywords{secret image sharing \and threshold changeable \and bivariate polynomial}
% \PACS{PACS code1 \and PACS code2 \and more}
% \subclass{MSC code1 \and MSC code2 \and more}
\end{abstract}

\section{Introduction}
\label{intro}
Researching on image related secure issues are important in the field of information security, such as image based data hiding [1-3], water marking technologies [4] or other aspects [5-7]. SIS scheme is an important issue in information security that can protect confidential images among multiple participants. Most SIS schemes satisfy a $(k,n)$ threshold, such that an image is encrypted into $n$ shadows, any $k$ or more shadows can reconstruct the image, less than $k$ shadows get nothing. There are two mainly approaches for SIS, visual cryptography based SIS schemes [8-10] and polynomial based SIS schemes [11-13]. Visual cryptography based SIS uses human visual system to recover image, but the shadow size is greatly expanded from original image and the reconstructed image is quality loss; Polynomial based SIS is capable to recover lossless image and the shadow size is reduced from original image, but the computation for image reconstruction is more complicated than visual cryptography based SIS. Many research issues are developed on SIS, such as progressive SIS [14-16], SIS with essential shadows [17][18], SIS with authentication [19].

Most existing SIS schemes consider a single security policy, and the threshold is fixed. However, the security environment for image reconstruction is probably changeable in fact, and therefore it is more reasonable to design a SIS scheme with the capability of threshold changeability. The necessity for threshold changeability includes: (1) the confidential level of secret image changes. (2) the number of total participants varies. (3) the power of adversary increases. (4) information disclosure caused by some malicious participants. The discussion of changing threshold in  traditional secret sharing [20-22] have already been proposed. However, SIS and traditional secret sharing are different, the researches on threshold changeable secret sharing could not be directly copied into TCSIS. In fact, the discussion on TCSIS is not enough, two TCSIS schemes [23][24] have been constructed. In TCSIS scheme [23], there are $N$ possible thresholds $T_{1},T_{2},...,T_{N}$, but extra two-variable one way functions are included for image reconstruction, the computational complexity is greatly increased, and the security is based on the assumption of one-way functions, it is unconditional secure; the other TCSIS scheme [24] can reduce the computational complexity, but it has only three possible thresholds $(k^{'},k,k^{''})$. Besides, it requires the dealer involved in shadow updating, and also suffers the problem of information leakage.

In this paper, we construct a $(k\leftrightarrow h,n)$ TCSIS scheme which has $h-k+1$ possible thresholds $(k,k+1,..,h-1,h)$. Three different secure levels are considered in our scheme. The shadows are encrypted using bivariate polynomial, and the participant only keeps one initial shadow from the dealer. During image reconstruction, each participant can update his shadow according to the current threshold $T$, $(k<T<h)$. In addition, the process of shadow updating is only based on the initial shadow, the dealer does not need to participate in this process. All the computations in proposed scheme are polynomial based computation which has high efficiency in computational complexity.

The rest of this paper is organized as follows. In next section, some preliminaries are prepared which includes Thien-Lin's polynomial based $(k,n)$ SIS and some results of previous TCSIS. Our proposed scheme is described in section 3 together with the corresponding theoretical proofs. Comparisons between proposed scheme and previous TCSIS schemes and experimental results will be shown in section 4. The conclusion of our work is proposed in section 5 at last.

\section{Methods}
\subsection{Related works}
\subsubsection{Polynomial based $(k,n)$ SIS}

 In 2002, Thien and Lin proposed a polynomial based $(k,n)$ threshold SIS scheme, which is foundation of later polynomial based SIS schemes. Thien-Lin's $(k,n)$ SIS consists of two phases: \emph{\textbf{Shadow Encryption Phase}} and \emph{\textbf{Image Reconstruction Phase}}. In first phase, a confidential image $O$ is encrypted into $n$ shadows $S_{1},S_{2},...,S_{n}$; during second phase, $k$ or more shadows are able to recover the image $O$.\\

\noindent\textbf{Scheme 1}: \emph{Thien-Lin's $(k,n)$ SIS}\\
\emph{\textbf{Shadow Encryption Phase}}:\\ Input: image $O$, Output: $n$ shadows $S_{1},S_{2},...,S_{n}$
\begin{enumerate}
\item[1] The dealer divides $O$ into $l$-non-overlapping $k$-pixel groups, $G_{1},G_{2},...,G_{l}$.
\item[2] For $k$ pixels $p_{j,0},p_{j,1},...,p_{j,k-1}$ in each group $G_{j},j\in[1,l]$, the dealer builds a $k-1$ degree polynomial $f_{j}(x)=p_{j,0}+p_{j,1}x+p_{j,2}x^{2}+...,+p_{j,k-1}x^{k-1}$.
\item[3] Computing $n$ sub-shadows, $s_{j,1}=f_{j}(1),s_{j,2}=f_{j}(2),...,s_{j,n}=f_{j}(n),j\in[1,l]$.
\item[4] Outputs $n$ shadows $S_{i}=s_{1,i}\parallel s_{2,i}\parallel,...,\parallel s_{l,i}, i=1,2,...,n$.
\end{enumerate}

\noindent \textbf{\emph{Image Reconstruction Phase}}: \\Input: $m$ shadows $S_{1},S_{2},...,S_{m}$.$(m\geq k)$. Outputs: secret image $O$.
\begin{enumerate}
\item[1] Reconstructing $f_{j}(x)$ from $s_{1,j},s_{2,j},...,s_{m,j},j\in[1,l]$ using Lagrange interpolation:
\begin{equation}
f_{j}(x)=\sum_{i=1}^{m}[s_{i,j}\ast \prod_{w=1,w\neq i}^{m}\frac{x-w}{i-w}]
\end{equation}
then the block $G_{j}$ is recovered from all $k$ coefficients in $f_{j}(x)$.
\item[2] Outputs $O=G_{1}\parallel G_{2}\parallel ,...,\parallel G_{l}$.
\end{enumerate}

\subsubsection{Results on TCSIS}
\label{sec:2}
In this part, we give a definition on TCSIS and then describe some results on previous TCSIS schemes.

The model of TCSIS scheme consists of two phases: \textbf{\emph{Shadow Encryption Phase}} and \emph{\textbf{Image Reconstruction Phase}}, which have the following steps in detail.\\

\textbf{\emph{Shadow Encryption Phase}}
\begin{enumerate}
\item[1] A dealer encrypts a confidential image $O$ into initial shadows $S_{1},S_{2},...,S_{n}$.
\item[2] Each initial shadow $S_{i}$ is sent to participant $\mathcal{P}_{i}$ through secure channel.
\end{enumerate}

\emph{\textbf{Image Reconstruction Phase}}
\begin{enumerate}
\item[1] Selecting a threshold $T$ from the set of all possible thresholds ${T_{1},T_{2},...,T_{d}}$.
\item[2] Each participant $\mathcal{P}_{i}$ updates the shadow according to current threshold $T$.
\item[3] Any group of participants that satisfy the access structure can reconstruct the image $O$ using updated shadows.
\end{enumerate}

The difference between SIS model and TCSIS model is that during \emph{\textbf{Image Reconstruction Phase}}, each participant needs to update the shadow according to the current threshold $T$.

The scheme [23] is a polynomial based scheme that satisfies the model of TCSIS, however some one-way functions are adopted in [23] to changing threshold. Therefore the security of [23] is based on the security assumption of those one-way functions, and the computational complexity is much higher than the computation in polynomial interpolation. The TCSIS scheme [24] can reduce computational complexity from [23], but there are only three possible thresholds and the dealer has to involve in shadow updating, which is not reasonable. Besides, the problem of information leakage would be caused in [24] due to the property of homomorphism.

\subsection{Aim and design motivation}
In real applications, security conditions are probably changeable after dealer sends shadows in a SIS scheme to all participants. In this work, we assume there more multiple security levels for image reconstruction. For instance, a confidential image should be reconstructed immediately due to emergency cases such as medical or traffic. However, the number of available participants does not satisfy the access structure for image reconstruction, it would cause loss or even loss of life. Therefore a SIS with capability of changing threshold is more reasonable since it can reconstruct image under multiple security levels. The design concept of our work can be shown in Fig.1.

\begin{figure}[!t]
\centering
\includegraphics[width=4.0in]{fig1.eps}
%where an .eps filename suffix will be assumed under latex,
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{Thresholds for different schemes: (a) $(k,n)$ SIS scheme (b) $(k\leftrightarrow h, n)$ TCSIS scheme}
\label{fig_sim}
\end{figure}

\subsection{Proposed TCSIS using bivariate polynomial}

In this part, we describe our $(k\leftrightarrow h,n)$ TCSIS using bivariate polynomial, which has $h-k+1$ possible thresholds $\{k,k+1,...,h\}$.

\noindent\textbf{Scheme 2}: $(k\leftrightarrow h,n)$ TCSIS\\
\emph{\textbf{Shadow Encryption Phase}}:
\begin{enumerate}
\item[1] The dealer $\mathcal{D}$ divides $O$ into $l$ non-overlapping $kh$-pixel groups, $G_{1},G_{2},...,G_{l}$.
\item[2] For $kh$ pixels
$\left\{\begin{array}{ll}
 p_{0,0},p_{0,1},...,p_{0,h-1},\\
 p_{1,0},p_{1,1},...,p_{1,h-1},\\
...\\
p_{k-1,0},p_{k-1,1},...,p_{k-1,h-1}
\end{array}\right.$
in each block $G_{r},r\in[1,l]$, $\mathcal{D}$ builds a bivariate polynomial:
    \begin{equation}
    F_{r}(x,y)=\left\{\begin{array}{ll}
 p_{0,0}+p_{0,1}y+,...,+p_{0,h-1}y^{h-1},\\
 p_{1,0}x+p_{1,1}xy+,...,+p_{1,h-1}xy^{h-1},\\
...\\
p_{k-1,0}x^{k-1}+p_{k-1,1}x^{k-1}y+,...,+p_{k-1,h-1}x^{k-1}y^{h-1}
\end{array}\right.
    \end{equation}
\item[3] For each each block $G_{r}$ in $\{G_{1},G_{2},...,G_{l}\}$, $\mathcal{D}$ computes $f_{r,i}(y)=F_{r}(i,y), g_{r,i}(x)=F_{r}(x,i)$. The sub-shadow $s_{i,r}$ for $\mathcal{P}_{i}$ is $s_{i,r}=(f_{r,i}(y),g_{r,i}(x))$.
\item[3] The initial shadow $S_{i}$ for each participant $\mathcal{P}_{i}$ is $S_{i}=s_{i,1}||s_{i,2}||...||s_{i,l}$.

\end{enumerate}

\noindent \textbf{\emph{Image Reconstruction Phase}}:


 \begin{enumerate}
  \item[1] Selecting a threshold $T$ from the set $\{k,k+1,...,h\}$.

    \begin{enumerate}
    \item If current threshold is $T=k$, each participant $\mathcal{P}_{i}$ updates his sub-shadows by $s^{k}_{i,r}=f_{r,i}(y),r=1,2,...,l$, then the updated shadow is $S^{k}_{i}=s^{k}_{i,1}||s^{k}_{i,2}||...||s^{k}_{i,l}$.
    \item If current threshold is $T=h$, each participant $\mathcal{P}_{i}$ updates his sub-shadows by $s^{h}_{i,r}=g_{r,i}(x),r=1,2,...,l$, then the updated shadow is $S^{h}_{i}=s^{h}_{i,1}||s^{h}_{i,2}||...||s^{h}_{i,l}$.
    \item If current threshold $T$ satisfies $k<T<h$, the participants select $h-T+1$ integers $e_{1},e_{2},...,e_{h-T+1}$ other than $1,2,...,n$. Each participant $\mathcal{P}_{i}$ computes $f_{r,i}(e_{1}),f_{r,i}(e_{2}),...,f_{r,i}(e_{h-T+1}), r=1,2,...,l$, and the updated shadow $S^{T}_{i}$ is:

        $\begin{array}{ll}
        S^{T}_{i}=&S^{h}_{i}||(f_{1,i}(e_{1}),f_{1,i}(e_{2}),...,f_{1,i}(e_{h-T+1}))\\
        &||(f_{2,i}(e_{1}),f_{2,i}(e_{2}),...,f_{2,i}(e_{h-T+1}))\\
        &||...\\
        &||(f_{l,i}(e_{1}),f_{l,i}(e_{2}),...,f_{l,i}(e_{h-T+1}))
        \end{array}$

    \end{enumerate}
 \item[2] The any group of $T$ participants can reconstruct original image $O$ using Lagrange Interpolation.

\end{enumerate}

The thresholds for image reconstructions in our proposed scheme are analyzed in following theorems. Without loss of generality, we only describe the thresholds on the first $kh$-block $G_{1}$ from different shadows. Since each shadow consists of similar information on each block $G_{i},i=1,2,...,l$, the threshold for $G_{1}$ is identical to the entire image $O=G_{1}||G_{2}||...||G_{l}$.

\begin{theorem}
The threshold $T$ for updated shadows $\mathcal{S}^{k}_{1},\mathcal{S}^{k}_{2},...\mathcal{S}^{k}_{n}$ on $G_{1}$ is $T=k$.
\end{theorem}
\begin{proof}
The sub-shadow in $\mathcal{S}^{k}_{i}$ for $G_{1}$ is $s^{k}_{i,1}=f_{1,i}(y)=F_{1}(i,y)$. $F_{1}(x,y)$ in Eq.(2) can be rewrote as:
\begin{equation}
F_{1}(x,y)=u_{0}(x)+u_{1}(x)y+u_{2}(x)y^{2}+,...,+u_{h-1}(x)y^{h-1}
\end{equation}
where $u_{0}(x),u_{1}(x),...,u_{h-1}(x)$ are all $k-1$ degree univariate polynomials. Suppose that
\begin{equation}
s^{k}_{i,1}=F_{1}(i,y)=b_{i,0}+b_{i,1}y+,...,b_{i,h-1}y^{h-1},i=1,2,...,n
\end{equation}
 Comparing E.q.(3) E.q.(4), we can observe that $(b_{1,0},b_{2,0},...,b_{n,0})$ are interpolations on $u_{0}(x)$ that $b_{i,0}=u_{0}(i),i=1,2,...,n$. Since $u_{0}(x)$ is $k-1$ degree polynomial, the threshold on $(b_{1,0},b_{2,0},...,b_{n,0})$ to reconstruct $u_{0}(x)$ is $k$, and the reconstruction can be executed using Lagrange Interpolation E.q.(1). Because each $b_{i,0}$ comes from sub-shadow $s^{k}_{i,1}$, the threshold on $(s^{k}_{1,1},s^{k}_{2,1},...,s^{k}_{n,1})$ for $u_{0}(x)$ is $k$. By the same way, the threshold for the other polynomials $u_{1}(x),u_{2}(x),...,u_{h-1}(x)$ on $(s^{k}_{1,1},s^{k}_{2,1},...,s^{k}_{n,1})$ are also $k$. In sum, the threshold for the $kh$-pixel block $G_{1}$ from $(\mathcal{S}^{k}_{1},\mathcal{S}^{k}_{2},...,\mathcal{S}^{k}_{n})$ is $T=k$.
\end{proof}

\begin{theorem}
The threshold $T$ for updated shadows $\mathcal{S}^{h}_{1},\mathcal{S}^{h}_{2},...\mathcal{S}^{h}_{n}$ is $T=h$.
\end{theorem}
\begin{proof}
The sub-shadow in $\mathcal{S}^{h}_{i}$ for $G_{1}$ is $s^{h}_{i,1}=g_{1,i}(x)=F_{1}(x,i)$. $F_{1}(x,y)$ in Eq.(2) can be rewrote as:
\begin{equation}
F_{1}(x,y)=v_{0}(y)+v_{1}(y)x+v_{2}(y)x^{2}+,...,+v_{k-1}(y)x^{k-1}
\end{equation}
where $v_{0}(y),v_{1}(y),...,v_{k-1}(y)$ are all $h-1$ degree univariate polynomials. Suppose that
\begin{equation}
s^{h}_{i,1}=F_{1}(x,i)=c_{i,0}+c_{i,1}x+,...,c_{i,k-1}x^{k-1},i=1,2,...,n
\end{equation}
 Comparing E.q.(5) E.q.(6), we can observe that $(c_{1,0},c_{2,0},...,c_{n,0})$ are interpolations on $v_{0}(y)$ that $c_{i,0}=v_{0}(i),i=1,2,...,n$. Since $v_{0}(y)$ is $h-1$ degree polynomial, the threshold on $(c_{1,0},c_{2,0},...,c_{n,0})$ to reconstruct $v_{0}(y)$ is $h$, and the reconstruction can be executed using Lagrange Interpolation E.q.(1). Because each $c_{i,0}$ comes from sub-shadow $s^{h}_{i,1}$, the threshold on $(s^{h}_{1,1},s^{h}_{2,1},...,s^{h}_{n,1})$ for $v_{0}(y)$ is $h$. By the same way, the threshold for the other polynomials $v_{1}(y),v_{2}(y),...,v_{k-1}(y)$ on $(s^{h}_{1,1},s^{h}_{2,1},...,s^{h}_{n,1})$ are also $h$. In sum, the threshold for the $kh$-pixel block $G_{1}$ from $(\mathcal{S}^{h}_{1},\mathcal{S}^{h}_{2},...,\mathcal{S}^{k}_{n})$ is $T=h$.
\end{proof}

\begin{theorem}
The threshold on updated shadows $(\mathcal{S}^{T}_{1},\mathcal{S}^{T}_{2},...,\mathcal{S}^{T}_{n})$ when $k<T<h$ is $T$.
\end{theorem}
\begin{proof}
The sub-shadow in $\mathcal{S}^{T}_{i}$ for $G_{1}$ is $s^{T}_{i,1}=s^{h}_{i,1}||(f_{1,i}(e_{1}),f_{1,i}(e_{2}),...,f_{1,i}(e_{h-T}))$. Without loss of generality, there are $T$ sub-shadows $s^{T}_{1,1},s^{T}_{2,1},...,s^{T}_{T,1}$ on $G_{1}$. Since $f_{1,i}(e_{j})=F_{1}(i,e_{j})$, $f_{1,i}(e_{j})$ is an interpolation on $g_{e_{j}}(x)=F_{1}(x,e_{j})$. On the other hand, $T>k$ and $g_{e_{j}}(x)$ is of degree $k-1$, it can be reconstructed from $f_{1,i}(e_{j}),i=1,2,...,T$. As a result, $F_{1}(x,e_{j}),F_{1}(x,e_{2}),...,F_{1}(x,e_{h-T})$ can be reconstructed from $T$ sub-shadows $s^{T}_{1,1},s^{T}_{2,1},...,s^{T}_{T,1}$. According to E.q.(5), $F_{1}(x,e_{j}),j=1,2,...,h-T$ can be presented as:
\begin{equation}
F_{1}(x,e_{j})=v_{0}(e_{j})+v_{1}(e_{j})x+v_{2}(e_{j})x^{2}+,...,+v_{k-1}(e_{j})x^{k-1}
\end{equation}
Therefore, $h-T$ extra Interpolations can be obtained on each polynomial in $v_{0}(y),v_{1}(y),...,v_{k-1}(y)$. On the other hand, other $T$ Interpolations can be obtained from $s^{h}_{1,1},s^{h}_{2,1},...,s^{h}_{T,1}$, there are totally $h-T+T=h$ Interpolations on each polynomial $v_{0}(y),v_{1}(y),...,v_{k-1}(y)$. Then $F_{1}(x,y)$ can be reconstructed. When there are $T-1$ or less shadows, at most $h-1$ interpolations can be gathered on $v_{0}(y),v_{1}(y),...,v_{k-1}(y)$, $F_{1}(x,y)$ cannot be reconstructed. In sum, the threshold for the $kh$-pixel block $G_{1}$ from $(\mathcal{S}^{T}_{1},\mathcal{S}^{T}_{2},...,\mathcal{T}^{k}_{n})$ when $k<T<h$ is $T$.
\end{proof}

\section{Results}

\subsection{Experiments}
In this part, we use examples and experimental results to show the performance of proposed scheme, and then gives comparison between proposed scheme and previous TCSIS schemes.

Suppose the secret image is \\$O=(97,46,253,12,165,19,247,251,214,142,191,180,210,172,152)$. \\
We construct a proposed $(3\leftrightarrow 5,7)$ TCSIS on this image. Our proposed scheme is based on the computation over a $GF(P)$, here $P=251$ and $P=2^{8}$ are adopted in our examples. When using $P=251$, all pixels that larger than $250$ are transformed to $250$ instead, and the computation is over $mod(251)$, therefore the reconstructed image is quality-loss from the original image; when using $P=2^{8}$, no distortion would be caused from reconstructed image, but each pixel needs to be transferred into a polynomial, and the computation is over $mod(x^{8}+x^{4}+x^{3}+x+1)$, which is much more complicated than the computation in $mod(251)$.\\\\
\textbf{Example 1:} Proposed $(3\leftrightarrow 5,7)$ TCSIS on $O$ over $GF(251)$.\\

First the original image $O$ is transformed into image $O^{'}$ where the pixels larger than $250$ is transformed to $250$. Then we get \\
$O^{'}=(97,46,250, 12,165,19,247,250,214,142,191,180,210,172,152)$. \\
Next, a bivariate $F(x,y)$ with degree $2$ on $x$ and degree $4$ on $y$ is constructed based on $O^{'}$.
    \begin{equation}
    F(x,y)=\left\{\begin{array}{ll}
 97+46y+250y^{2}+12y^{3}+165y^{4},\\
 19x+247xy+250xy^{2}+214xy^{3}+142xy^{4},\\
191x^{2}+180x^{2}y+210x^{2}y^{2}+172x^{2}y^{3}+152x^{2}y^{4}
\end{array}\right.
    \end{equation}
Then the dealer computes $f_{i}(y)=F(i,y),g_{i}(x)=F(x,i),i=1,2,...,7$ over $GF(251)$, the initial shadow $\mathcal{S}_{i}=(f_{i}(y),g_{i}(x))$ is sent to each participant $P_{i},i=1,2,...,7$ confidentially. The initial shadows $\mathcal{S}_{1},\mathcal{S}_{2},...,\mathcal{S}_{7}$ are listed in Eq(9).
 \begin{equation}
    \left\{\begin{array}{ll}
\mathcal{S}_{1}: f_{1}(y)=56+222y+208y^{2}+147y^{3}+208y^{4},g_{1}(x)=68+119x+152x^{2}\\
\mathcal{S}_{2}: f_{2}(y)=146+5y+84y^{2}+124y^{3}+53y^{4},g_{2}(x)=160+226x+179x^{2}\\
\mathcal{S}_{3}: f_{3}(y)=116+148y+129y^{2}+194y^{3}+202y^{4},g_{3}(x)=110+210x+250x^{2}\\
\mathcal{S}_{4}: f_{4}(y)=217+149y+92y^{2}+106y^{3}+153y^{4},g_{4}(x)=101+86x+226x^{2}\\
\mathcal{S}_{5}: f_{5}(y)=198+8y+224y^{2}+111y^{3}+157y^{4},g_{5}(x)=9+14x+102x^{2}\\
\mathcal{S}_{6}: f_{6}(y)=59+227y+23y^{2}+209y^{3}+214y^{4},g_{6}(x)=156+48x+7x^{2}\\
\mathcal{S}_{7}: f_{7}(y)=51+53y+242y^{2}+149y^{3}+73y^{4},g_{7}(x)=55+136x+204x^{2}
\end{array}\right.
    \end{equation}
During image reconstruction, suppose the threshold is $T$, and $P_{1},P_{2},..,P_{T}$ are involved.
 \begin{enumerate}
    \item If $T=3$, $P_{1},P_{2},P_{3}$ publishes $S^{3}_{i}=f_{i}(y),i=1,2,3$, all coefficients in $F(x,y)$ can be computed using Lagrange interpolation according to Theorem 1, then the image $O^{'}$ can be reconstructed.
    \item If $T=4$, $P_{1},P_{2},P_{3},P_{4}$ publish $S^{4}_{i}=g_{i}(x)||f_{i}(e_{1}),i=1,2,3,4$. Here $e_{1}=8$. The interpolation polynomial on $f_{i}(e_{1}),i=1,2,3,4$ is $g_{e_{1}}(x)=F(x,e_{1})=167+120x+86x^{2}$ for Example 1. Then, all coefficients in $F(x,y)$ can be computed using Lagrange interpolation according to Theorem 3, then the image $O^{'}$ can be reconstructed.
    \item If $T=5$, $P_{1}-P_{5}$ publish $S^{5}_{i}=g_{i}(x),i=1,2,...,5$, all coefficients in $F(x,y)$ can be computed using Lagrange interpolation according to Theorem 1, then the image $O^{'}$ can be reconstructed.

    \end{enumerate}

\textbf{Example 2:} Proposed $(3\leftrightarrow 5,7)$ TCSIS on $O$ over $GF(2^{8})$.\\

 A bivariate $F(x,y)$ with degree $2$ on $x$ and degree $4$ on $y$ is constructed based on $O$ as follows.
    \begin{equation}
    F(x,y)=\left\{\begin{array}{ll}
 97+46y+253y^{2}+12y^{3}+165y^{4},\\
 19x+247xy+251xy^{2}+214xy^{3}+142xy^{4},\\
191x^{2}+180x^{2}y+210x^{2}y^{2}+172x^{2}y^{3}+152x^{2}y^{4}
\end{array}\right.
    \end{equation}
Then the dealer computes $f_{i}(y)=F(i,y),g_{i}(x)=F(x,i),i=1,2,...,7$ over $GF(2^{8})$, the initial shadow $\mathcal{S}_{i}=(f_{i}(y),g_{i}(x))$ is sent to each participant $P_{i},i=1,2,...,7$ confidentially. The initial shadows $\mathcal{S}_{1},\mathcal{S}_{2},...,\mathcal{S}_{7}$ are listed in Eq(11).
 \begin{equation}
    \left\{\begin{array}{ll}
\mathcal{S}_{1}: f_{1}(y)=205+109y+212y^{2}+118y^{3}+179y^{4},g_{1}(x)=27+71x+237x^{2}\\
\mathcal{S}_{2}: f_{2}(y)=141+61y+117y^{2}+61y^{3}+244y^{4},g_{2}(x)=58+245x+253x^{2}\\
\mathcal{S}_{3}: f_{3}(y)=33+126y+92y^{2}+71y^{3}+226y^{4},g_{3}(x)=104+99x+106x^{2}\\
\mathcal{S}_{4}: f_{4}(y)=40+106y+179y^{2}+87y^{3}+232y^{4},g_{4}(x)=168+119x+50x^{2}\\
\mathcal{S}_{5}: f_{5}(y)=132+41y+154y^{2}+45y^{3}+254y^{4},g_{5}(x)=34+153x+200x^{2}\\
\mathcal{S}_{6}: f_{6}(y)=196+121y+59y^{2}+102y^{3}+185y^{4},g_{6}(x)=168+219x+2x^{2}\\
\mathcal{S}_{7}: f_{7}(y)=104+58y+18y^{2}+28y^{3}+175y^{4},g_{7}(x)=10+247x+61x^{2}
\end{array}\right.
    \end{equation}
The reconstruction with different thresholds are similar as Example 1. Here we only emphsize that when the threshold is $T=4$, the participants can decide $e_{1}=8$, and compute $g_{e_{1}}(x)=13+248x+98x^{2}$ in $GF(2^{8})$, which is different from the $g_{e_{1}}(x)$ in Example 1.


The following Fig.2 shows the experimental results of proposed $(5\leftrightarrow 7,9)$ over $GF(251)$, where original image and initial shadows are included.

\begin{figure}[!t]
\centering
\includegraphics[width=4.0in]{fig2.eps}
%where an .eps filename suffix will be assumed under latex,
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{Original images and shadows using proposed scheme}
\label{fig_sim}
\end{figure}

\subsection{Comparison and discussion}
In this part, we give the discussion of comparisons between proposed TCSIS with previous two TCSIS schemes [23] and [24] in detail. First we illustrate the information disclosure problem of scheme [24]. The initial shadows $S_{i}$ for each participant $P_{i}$ in [24] is generated from a $k-1$ degree polynomial $F_{i}^{*}(x)=F(x)+f_{i}(x)$, where $F(x)$ is of degree $k-1$, $f_{i}(x)$ is of degree $k^{'}-1$, and $f_{i}(x)$ contains the pixels of secret image $O$. When the threshold is $k^{'}$, each participant $P_{i}$ modifies his initial shadow by $S^{'}_{i}=S_{i}-F(i)$, thus the threshold of updated shadows is reduced to $k^{'}$ from $k$. However, any $k^{'}$ participants can recover a distortion image without updating their shadows, based on the homm, $S_{i}-S_{j}$ is generated from $F^{*}_{i}(x)-F^{*}_{j}(x)=f_{i}(x)-f_{j}(x)$. Since $f_{i}(x)-f_{j}(x)$ is of degree $k^{'}-1$, any $k^{'}$ participants can recover $f_{i}(x)-f_{j}(x)$. As a result, a distortion image can be recovered from the pixel information in $f_{i}(x)-f_{j}(x)$. The experimental results of information disclosure problem in [23] are shown in Fig.3.

\begin{figure}[!t]
\centering
\includegraphics[width=3.4in]{fig3.eps}
%where an .eps filename suffix will be assumed under latex,
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{(a-1,b-1,c-1): original images, (a-2,b-2,c-2): quality lossy images with $(k^{'}=2,k=3)$, (a-3,b-3,c-3): quality lossy images with $(k^{'}=3,k=4)$, (a-4,b-4,c-4): quality lossy images with $(k^{'}=4,k=5)$}
\label{fig_sim}
\end{figure}

The proposed TCSIS scheme generates initial shadows using bivariate polynomial $F(x,y)$, where image pixels are hidden in all coefficients in $F(x,y)$. Thus the proposed scheme avoids such information disclosure problem in [24]. On the other hand, the scheme [24] only provides three potential thresholds $k^{'},k,k^{''}$ for low, media and high security levels, and our scheme can provide $h-k+1$ thresholds $k,k+1,k+2,...,h$ to satisfy more complicated security requirements. The previous scheme [23] can also provides more thresholds than scheme [24], but there are two weaknesses of scheme [23]. One is that the dealer needs to publish necessary information when changing threshold, the participation of dealer in this process would not only reduces the efficiency of image reconstruction, but also brings risk of information leakage from the communication between dealer and participants. Such problem also exists in the scheme [24]. The other problem in [23] is that the security is based on the security assumption on two-verifiable one way functions, it is not unconditional secure. In addition, the computation in one way functions is much more complicated than the computation of polynomial interpolation. The comparisons between proposed scheme and previous TCSIS schemes [23][24] is listed in Tab.1.
\begin{table}
% table caption is above the table
\caption{Comparisons of three TCSIS schemes}
\label{tab:1}       % Give a unique label
% For LaTeX tables use
\begin{tabular}{c|c|c|c|c|c}
\hline
Schemes & Thresholds & Changing threshold & Security level & Main computation & Shadow size\\
\hline
Scheme [24]& $k^{'},k,k^{''}$ & Dealer involves & Unconditional & Polynomial interpolation & $\frac{1}{k^{'}}$\\
\hline
Scheme [23] & $T_{1},T_{2},...,T_{N}$ & Dealer involves & Conditional & One way function & $\frac{N}{T_{N}}$\\
\hline
Proposed scheme & $k,k+1,...,h$ & Without dealer & Unconditional & Polynomial interpolation & $\frac{k+h}{kh}$\\
\hline
\end{tabular}
\end{table}

\section{Conclusion}
In this paper, we construct a $(k\leftrightarrow h,n)$ TCSIS scheme based on bivariate polynomial, that provides $h-k+1$ potential thresholds. During image reconstruction, participants can modify their initial shadows to adjust the threshold from $\{k,k+1,...,h\}$. Comparing with previous TCSIS schemes, our scheme has following advantages:
 \begin{enumerate}
    \item our scheme provides more thresholds than scheme [24].
    \item our scheme does not require the dealer involve in changing threshold, that has lower cost than scheme [23][24] reduces the risk of information leakage.
    \item our scheme does not adopt one-way functions, it achieves unconditional secure.
    \item the computation is only based on polynomial interpolation, which has higher efficiency than previous scheme [23]
 \end{enumerate}

\subsection*{Abbreviations}
SIS: secret image sharing; TCSIS: threshold changeable secret image sharing.
\subsection*{Acknowledgements}
We want to thank Professor Chingnung Yang from National DongHwa University for his help of English improvement
\subsection*{Authors�� contributions}
Yanxiao Liu provides the main concept, Ping Wu and Qindong Sun design the algorithms and Zhili Zhou gives the experiments and comparisons.
\subsection*{Funding}
The research presented in this paper is supported in part by the National Natural Science Foundation (No.: 61571360), the Youth Innovation Team of Shaanxi Universities, the Innovation Project of Shaanxi Provincial Department of Education (No.17JF023), and the Shaanxi Provincial Natural Science Basic Project (No.2019JQ-736), and Xi'an Science and technology plan projects GXYD14.12 and GXYD14.13.

\subsection*{Availability of data and materials}
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
\subsection*{Competing interests}
The authors declare that they have no competing interests.
\subsection*{Author details}
Yanxiao Liu, Ping Wu and Qindong Sun are at Department of computer science and engineering, XI'AN University of Technology; Zhili Zhou is at School of computer and software, Nanjing University of Information Science and Technology.
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