(k,n) secret image sharing scheme capable of cheating detection
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Abstract
In a (k,n) threshold secret image sharing scheme, a secret image is encrypted into n imageshadows that satisfy the following: (1) any less than k imageshadows get no information on the image and (2) any k or more imageshadows can reconstruct the entire image. Cheating problem is an important issue in traditional secret sharing scheme. However, this issue has not been discussed sufficiently in the field of secret image sharing. In this paper, we consider the scenario of cheating behavior in secret image sharing scheme and construct a (k,n) threshold secret image sharing scheme which is capable of cheating detection. Our proposed scheme is able to detect the cheating behavior from up to k − 1 cheaters, and the size of imageshadow is almost same as the imageshadow in the original secret image sharing scheme.
Keywords
Secret sharing Secret image sharing Cheating detection1 Introduction
(k,n) threshold secret sharing scheme was first proposed by Shamir [1] in 1979 for safeguarding secret information among a group of participants. In [1], a secret s is encrypted into n shares v_{1},v_{2},...,v_{ n } using a k − 1th degree polynomial in such a way that less than k shares get no information on the secret s; any k or more shares can recover the secret s efficiently. Secret sharing scheme is a fundamental tool for other cryptographic protocols [2]. In 2002, Thien and Lin extended Shamir’s secret sharing and proposed a (k,n) threshold secret image sharing scheme [3] by regarding an image as secret information. The (k,n) secret image sharing schemes can be mainly divided into two categories: polynomialbased schemes [4, 5, 6] and visual cryptography schemes [7, 8, 9]. Polynomialbased secret image sharing schemes can reconstruct a lossless image with reduced shadow size; the image reconstruction in visual cryptography schemes can be simply accomplished by human visual system without any computation. However, a reconstructed image is lossy and the size of shadow is greatly expanded from the original image.
The cheating scenario in secret sharing scheme was first proposed in 1989 by Tompa and Woll [7]. They considered the scenario that some dishonest participants (cheaters) pool fake shares when reconstructing the secret. In this way, the cheaters could recover a secret exclusively, and the honest participants can only recover a forged secret. Many works have focused on solving the cheating problem in secret sharing schemes. Some [10, 11, 12] were interested in detecting the cheating behavior, and some works [13, 14, 15] focused on not only detecting the cheating behavior, but also identifying the cheaters. The cheating identifiable schemes have stronger capability to resist cheating; it results that the shares are larger and the schemes are more complicated than those cheating detectable schemes. Polynomialbased secret image sharing scheme was extended from Shamir’s scheme [1]. As a result, the problem of cheating is also an important topic in polynomialbased secret image sharing. However, this issue has not been discussed sufficiently so far. In [16, 17, 18, 19], some secret image sharing schemes with authentication and steganography were capable of detecting the cheating. However, these secret image sharing schemes were not extensions of Shamir’s scheme and the capabilities of cheating detection were not strong enough to prevent cheating.
In this work, we consider the problem of cheating in the fundamental polynomialbased secret image sharing scheme [3]. Since cheating identifiable scheme requires large size expansion on shadows and much more complicated identification algorithm, we consider the cheating detection to prevent cheating behavior in this work. A (k,n) threshold secret image sharing scheme capable of detecting k − 1 cheaters is constructed. In addition, the size of imageshadow in our scheme is almost same as the shadow size in the scheme [3]. The computational complexity of cheating detection is efficient. The rest of this paper is organized as follows: Some preliminaries which include Shamir’s (k,n) secret sharing scheme, polynomialbased secret image sharing scheme, and the model of cheating detection in secret sharing scheme are introduced in Section 2. In the next section, a (k,n) threshold secret image sharing scheme with cheating detection is proposed. The properties of proposed scheme and experimental results will be shown in Section 4, and we make a conclusion in Section 5.
2 Preliminaries
2.1 Shamir’s (k,n) secret sharing scheme
 1
Sharing phase: The dealer \(\mathcal {D}\) encrypts secret s into n shares v_{1},v_{2},…,v_{ n } and sends each share v_{ i } to a participant P_{ i }.
 2
Reconstruction phase: k or more participants submit their shares to recover secret.
 1
Correctness: Any group of at least k shares can recover the valid secret s.
 2
Secrecy: Fewer than k shares get no information on the secret s.
Shamir’s scheme was based on interpolation polynomial which is shown in Scheme 1.

Sharing phase:
1 The dealer \(\mathcal {D}\) chooses a k − 1th degree polynomial f(x)∈GF(q)[ X] which satisfies s = f(0)∈GF(q).
2 The dealer \(\mathcal {D}\) generates n shares v_{ i } = f(i),i = 1,2,…,n and sends each v_{ i } to a participant P_{ i }.

Reconstruction phase:
1 m(≥ k) participants (say P_{1},P_{2},…,P_{ m }) submit their shares v_{1},v_{2},...,v_{ m } together.
2 Computing the interpolated polynomial f(x) on v_{1},v_{2},…,v_{ m } by the equation \(f(x)~=~\sum ^{m}_{i~=~1}\left (v_{i}\prod _{u\neq i}\frac {x~~u}{i~~u}\right)\). Then the secret s = f(0).
2.2 Cheating detection in secret sharing scheme
 1
Sharing phase: The dealer \(\mathcal {D}\) divides the secret s into n shares v_{1},v_{2},….,v_{ n } and sends each share v_{ i } to a participant P_{ i }.
 2Reconstruction phase: A group of m participants (m ≥ k) submit their shares.
 (1)
A public cheating detection algorithm is applied on these shares to detect cheating.
 (2)
 −
If cheating is detected, stop the Reconstruction phase and output ⊥.
 −
Else, reconstruct the secret s from these shares and output s.
 −
 (1)
2.3 Polynomialbased secret image sharing scheme
In [3], a remarkable (k,n) secret image sharing scheme was proposed by Thien and Lin which was based on Shamir’s scheme. An image I is made up of multiple pixels (p_{1},p_{2},…,p_{ w }), where each pixel p_{ i } can be presented as its gray value in GF(251). If all the pixels in an image are treated as secrets, a secret image sharing scheme can be extended from original secret sharing scheme. The scheme [3] consists of two phases: shadow generation phase and image reconstruction phase. In the first phase, a dealer encrypts I into n imageshadows S_{1},S_{2},…,S_{ n }; in the second phase, any set of m imageshadows k ≤ m ≤ n reconstructs the secret image I.
Scheme 2: ThienLin’s secret image sharing
Shadow generation phase:
 1
The dealer divides I into lnonoverlapping kpixels blocks, B_{1},B_{2},…,B_{ l }.
 2
For k pixels a_{j,0},a_{j,1},…,a_{j,k−1}∈GF(251) in each block B_{ j },j∈ [ 1,l], the dealer generates a k − 1th degree polynomial f_{ j }(x)∈GF(251)[X], such that f_{ j }(x) = a_{j,0} + a_{j,1}x + a_{j,2}x^{2}+…,+a_{j,k − 1}x^{k − 1}, and computes n shares v_{j,1} = f_{ j }(1),v_{j,2} = f_{ j }(2),…,v_{j,n} = f_{ j }(n),j∈ [ 1,l] as Shamir’s scheme.
 3
Outputs n shadows S_{ i } = v_{1,i}∥v_{2,i}∥,…,∥v_{l,i},i = 1,2,…,n.
Image reconstruction phase:
 1
Extract v_{1,j},v_{2,j},…,v_{m,j},j∈ [ 1,l] from S_{1},S_{2},…,S_{ m }.
 2
Using the approach of Shamir’s scheme, reconstruct the polynomial f_{ j }(x) = a_{j,0} + a_{j,1}x + a_{j,2}x^{2}+….,+a_{j,k −}x^{k − 1} from v_{1,j},v_{2,j},…,v_{m,j},j∈ [ 1,l]. The block B_{ j } = a_{j,0}∥a_{j,1}∥,…,∥a_{j,k−1}.
 3
Outputs I = B_{1}∥B_{2}∥,…,∥B_{ l }.
It is obvious that Scheme 2 satisfies kthreshold property that k or more imageshadows are capable of recovering the entire image; fewer than k imageshadows get noting on the image. The size of imageshadow in Scheme 2 is \(\frac {1}{k}\) times of the image I.
3 (k,n) secret image sharing with cheating detection
The problem of cheating in secret image sharing is considered in this part, such that some cheaters submit forged imageshadows during image reconstruction phase. It results that these cheaters are able to recover secret image exclusively, and the honest participants recover only a fake image. In order to solve this problem, we construct a (k,n) threshold secret image sharing scheme capable of detect cheating under the model in Section 2.2. Our scheme is extended from ThienLin’s fundamental scheme which can be adopted in other polynomialbased secret image sharing schemes. Our scheme is shown in Scheme 3.
 Shadow Generation Phase: Input a secret image I, output n imageshadows S_{1},S_{2},…,S_{ n }

The dealer divides I into tnonoverlapping 2k−2pixel blocks, B_{1},B_{2},...,B_{ t }.

For each block B_{ i },i∈ [ 1,t], there are 2k − 2 secret pixels a_{i,0},a_{i,1},…,a_{i,k−1} and b_{i,2},b_{i,3},…,b_{i,k−1}∈GF(251). The dealer generates a k−1th degree polynomial f_{ i }(x) = a_{i,0} + a_{i,1}x+,…,+a_{i,k − 1}x^{k − 1}∈GF(251)[ X].

The dealer chooses a random integer r_{ i }, and computes two pixels b_{i,0},b_{i,1} which satisfy that: r_{ i }a_{i,0} + b_{i,0} = 0,r_{ i }a_{i,1} + b_{i,1} = 0 over GF(251). Then the dealer generates another k − 1th degree polynomial g_{ i }(x) = b_{i,0} + b_{i,1}x+…,+b_{i,k − 1}x^{k−1}.

For each block B_{ i },i∈ [ 1,t], the dealer computes subshadow v_{i,j} = {m_{i,j},d_{i,j}},m_{i,j} = f_{ i }(j),d_{i,j} = g_{ i }(j),j=1,2,…,n for each participant P_{ j }. The shadow S_{ j } for P_{ j } is S_{ j } = v_{1,j}∥v_{2,j}∥,…,∥v_{t,j}.

 Image Reconstruction Phase: Input k shadows, without loss of generality (S_{1},S_{2},…,S_{ k })

Extract v_{i,j} = (m_{i,j},d_{i,j}),i=1,2,…,t,j = 1,2,…,k from S_{1},S_{2},…,S_{ k }.

For each group of v_{i,1},v_{i,2},…,v_{i,k},i∈ [ 1,t], reconstruct f_{ i }(x) and g_{ i }(x) from m_{i,1},m_{i,2},…,m_{i,k} and d_{i,1},d_{i,2},…,d_{i,k} using Lagrange interpolation respectively.
 Let a_{i,0},a_{i,1},b_{i,0} and b_{i,1} be the coefficients of x^{0} and x in f_{ i }(x) and g_{ i }(x) respectively.

If there exist a common integer r_{ i } which satisfies that r_{ i }a_{i,0} + b_{i,0} = 0 and r_{ i }a_{i,1}+b_{i,1} = 0, recover the 2k − 2–pixel block \( B_{i}\! =\!\left \{a_{i,0},a_{i,1},\ldots,\right. \left. a_{i,k1},b_{i,2},_{i,3} \!,\ldots,\!b_{i,k\,\,1}\right \}\), the secret image I is I = B_{1}∥B_{2}∥,…,∥B_{ t }.

Else, there are fake shadows participating in image reconstruction; the cheating is detected, output ⊥.


Notice that in ThienLin’s scheme, the size on imageshadow is \(\frac {1}{k}\) times of the image. In our scheme, the share v_{i,j} = (m_{i,j},d_{i,j}) are generated from each 2k−2pixel block; therefore, the size on imageshadow in our scheme is \(\frac {2}{2k2}=\frac {1}{k1}\) times of image I.
The features of our scheme will be described in following theorems. Theorem 1 proves that our scheme satisfies the property of (k,n) threshold, such that less than k image shadows get no information on secret image; k or more imageshadows are able to recover secret image. In Scheme 3, the secret image I is cut into 2k−2pixels blocks B_{1},B_{2},…,B_{ t }. Each block is encrypted by the same approach; we only need to prove that the n shares v_{1,j},j=1,2,…,n on block B_{1} satisfy the (k,n) threshold property. The property of detect cheating will be analyzed in Theorem 2.
It seems that the relationship between a_{0},a_{1},b_{0},b_{1} would leak some information on the secret. However, the following Theorem 1 will prove that a_{0},a_{1},b_{0},b1 leak no information about the secret at all, and the proposed scheme is a perfect (k,n) threshold secret image sharing scheme that satisfies the threshold property. The capability of cheating detection of proposed scheme is discussed in Theorem 2.
Theorem 1
The proposed scheme satisfies the property of (k,n) threshold.
Proof
In our scheme, any 2k−2 pixels in a block B_{ i } are encrypted into n shares v_{i,j},j=1,2,…,n using Shamir’s scheme; it is easy to prove that k or more shares can recover all the 2k−2 pixels in B_{ i }. □

all k−1 participants use all possible shares on the kth participant and generates p=251 corresponding interpolation polynomials \(f^{u}_{i}(x),u=1,2,\ldots,251\) and \(g^{u}_{j}(x),u=1,2,\ldots,251\).

If \(f^{u^{*}}_{i}(x)\) and \(g^{v^{*}}_{i}(x)\) satisfy \(r_{i}a^{'}_{0}+b^{'}_{0}=0\) and \(r_{i}a^{'}_{1}+b^{'}_{1}=0\) where \(a^{'}_{0},b^{'}_{0},a^{'}_{1},b^{'}_{1}\) are coefficients of x^{0},x in \(f^{u^{*}}_{i}(x)\) and \(g^{v^{*}}_{j}(x)\). Then \(f^{u^{*}}_{i}(x)\) and \(g^{v^{*}}_{i}(x)\) would be the original polynomials selected by dealer, and all the 2k−2 pixels can be gotten from \(f^{u^{*}}_{i}(x)\) and \(g^{v^{*}}_{i}(x)\).
Assume that m^{∗}(k) is the share of kth participant that is randomly selected, then the k−1 participants generates a k−1th degree polynomial f_{ i }(x) from \((1,m_{1}),(2,m_{2}),\ldots,(k1,m_{k1}),\left (k,m^{*}_{k}\right)\). Let \(a^{'}_{0},a^{'}_{1}\) be the corresponding coefficients in f_{ i }(x). According to the method of exhaustion described previously, if there exists a k−1th degree polynomial \(g_{j}(x)=b^{'}_{0}+b^{'}_{1}x+,\ldots,+b^{'}_{k1}x^{k1}\) which is interpolated by \((1,d_{1}),(2,d_{2}),\ldots,(k1,d_{k1}),\left (k,d^{*}_{k}\right)\), satisfies that \(r^{'}a^{'}_{0}+b^{'}_{0}=0, r^{'}a^{'}_{1}+b^{'}_{1}=0\) (\(\phantom {\dot {i}\!}r^{'}\) could be any value in GF(251)), then f_{ i }(x) and g_{ i }(x) are the two polynomials selected by the dealer. \(b^{'}_{0},b^{'}_{1},\ldots,b^{'}_{k1}\) and \(\phantom {\dot {i}\!}r^{'}\) can be regarded as k+1 unknowns, then k+1 linear equations on these unknowns can be established: \(g^{'}(i)=d_{i},i=1,2,\ldots,k1, r^{'}a^{'}_{0}+b^{'}_{0}=0, r^{'}a^{'}_{1}+b^{'}_{1}=0\). (Here \(a^{'}_{0},a^{'}_{1}\) are known to the k−1 participants.) Therefore, all the unknowns \(b^{'}_{0},b^{'}_{1},\ldots,b^{'}_{k1}\) can be obtained from these equations; we can also get the polynomial g_{ i }(x). It means that the k−1 participants will find that each possible share could be the valid share of the kth participant. This proves that the approach of exhaustion cannot work in the proposed scheme.
An example is used to show the approach of exhaustion in proposed scheme. Assume k=4 and two polynomials f(x)=1+3x+4x^{2}+5x^{3} and g(x)=4+5x+x^{2}+3x^{3} over GF(7) are chosen by the dealer. It is obvious that 3a_{0}+b_{0}=0,3a_{1}+b_{1}=0. Let P_{1}.P_{2},P_{3},P_{4} be the four participants; the shares of them are P_{1}: (m_{1}=6,d_{1}=6), P_{2}: (m_{2}=0,d_{2}=0), P_{3}: (m_{3}=6,d_{3}=4), and P_{4}: (m_{4}=5,d_{4}=1). Suppose P_{1},P_{2},P_{3} try to recover secret using approach of exhaustion. As described previously, they can randomly assume the subshare of P_{4}: \(m^{*}_{4}=0\) and generate an interpolation polynomial \(\phantom {\dot {i}\!}f^{'}(x)=6+2x+2x^{2}+3x^{3}\) correspondingly. Then they try each possible subshare \(d^{*}_{4}\) of P_{4} and verify whether it is fit. For instance, when they use \(d^{*}_{4}=2\), the interpolating polynomial is \(\phantom {\dot {i}\!}g^{'}(x)=3+x+2x^{3}\). The coefficients \(a^{'}_{0},a^{'}_{1},b_{0}^{'},b_{1}^{'}\) satisfy that \(3a^{'}_{0}+b^{'}_{0}=0,3a^{'}_{1}+b^{'}_{1}=0\). Then they get the conclusion that \(\phantom {\dot {i}\!}f^{'}(x),g^{'}(x)\) would be the valid polynomials and \(\phantom {\dot {i}\!}s^{'}=f^{'}(0)=6\) is the secret. In fact, they recover a wrong secret. End of proof.
The capability of cheating detection in our scheme is analyzed in Theorem 2.
Theorem 2
Our scheme is able to detect cheating from k−1 cheaters.
Proof
Suppose P_{1},P_{2},...,P_{ k } participate in secret reconstruction phase and P_{1},P_{2},...,P_{k−1} are cheaters. Since cheating detection algorithm is used in each block B_{ i },i∈[ 1,t], we only analyze the cheating detection in decoding B_{1} in this theorem. Suppose the fake shares from cheaters are \(v^{*}_{i}=\left (m_{i}+m^{*}_{i},d_{i}+d^{*}_{i}\right),i=1,2,\ldots,k1\). They can get two polynomials f^{∗∗}(x)=f(x)+f^{∗}(x),g^{∗∗}(x)=g(x)+g^{∗}(x) in image reconstruction phase, where \(f^{*}(x)=a^{*}_{0}+a^{*}_{1}x+\cdots +a^{*}_{k1}x^{k1}\) and \(g^{*}(x)=b^{*}_{0}+b^{*}_{1}x+\cdots +b^{*}_{k1}x^{k1}\) are interpolated polynomials on the k points \((1,m^{*}_{1}),\left (2,m^{*}_{2}\right),\ldots, \left (k\,\,1,m^{*}_{k1}\right),(k,0)\) and \(\left (1,d^{*}_{1}\right),\left (2,d^{*}_{2}\right),\ldots,\left (k1,d^{*}_{k1}\right), (k,0)\) respectively. Since f^{∗}(x) and g^{∗}(x) can be decided by cheaters exclusively, they can select a random number r^{∗}, and satisfy that \(r^{*}a^{*}_{0}+b^{*}_{0}=0, r^{*}a^{*}_{1}+b^{*}_{1}=0\). According to our algorithm, if there exists a common number \(\phantom {\dot {i}\!}r^{'}\), satisfying \(r^{'}\left (a_{0}+a^{*}_{0}\right)+b_{0}+b^{*}_{0}=0, r^{'}\left (a_{1}+a^{*}_{1}\right)+b_{1}+b^{*}_{1}=0\), the cheating avoids detection. We can easily observe that the cheating succeeds only when r^{∗}=r. As analyzed in Theorem 1, these k−1 cheaters have no information on r; the possibility of r^{∗}=r is \(\frac {1}{251}\). As a result, the successful cheating probability is \(\epsilon =\frac {1}{251}\). Since all the pixels are in GF(251), the successful cheating possibility \(\epsilon =\frac {1}{251}\) means that our scheme is effective to detect the cheating. End of proof. □
4 Results and discussion
In this part, we use an example to describe the cheating detection using our scheme. Let (k,n)=(4,7) and the secret image I is divided into t blocks where each block includes 2k−2=6 secret pixels. Assume the first block B_{1} consists of the following 6 pixels: (57,68,90,231,42,89), the dealer selects an integer r_{1}=9, and generates two k−1=3th degree polynomials: f_{1}(x)=57+68x+90x^{2}+231x^{3} and g_{1}(x)=161+104x+42x^{2}+89x^{3}, where 57+9∗161=0(mod251),68+9∗104=0(mod251). The n=7 shares from B_{1} are v_{1,1}=(195,145),v_{1,2}=(142,245),v_{1,3}=(29,242),v_{1,4}=(238,168),v_{1,5}=147,55,v_{1,6}=(138,186),v_{1,7}=(91,91).
Suppose P_{1},P_{2},P_{3}, and P_{4} participate in image reconstruction and P_{1},P_{2},P_{3} are k−1=3 cheaters. They submit forged shares \(v^{*}_{1,1}=(105,87),v^{*}_{1,2}=(162,31),v^{*}_{1,3}=(23,98)\) in image reconstruction. As a result are two polynomials \(f^{*}_{1}(x)=55+188x+105x^{2}+8x^{3}\) and g^{∗}(x)=135+167x+56x^{2}+231x^{3}. Since there is no common integer r_{1} satisfying that 55r_{1}+135=0(mod251) and 188r_{1}+167=0(mod251), the cheating is successfully detected.
Comparisons between cheating detectable secret sharing schemes
5 Conclusions
In this paper, we consider the wellknown cheating problem in polynomialbased (k,n) secret image sharing scheme, such that a group of malicious participants submit fake shadows during image reconstruction. In order to prevent such cheating behavior, we construct a (k,n) secret image sharing scheme with cheating detection under the model of cheating detectable secret sharing scheme. Our scheme is capable of detecting the cheating from up to k−1 cheaters with only Lagrange interpolations. In addition, the proposed scheme is based on the landmark ThienLin’s polynomialbased secret image sharing which can be easily extended into other polynomialbased secret image sharing schemes. The size of shadow in our scheme is only \(\frac {1}{k1}\) times of the secret image, which is almost same as the shadow size in original (k,n) secret image sharing scheme.
6 Method
In this work, we aim to solve the cheating problem in polynomialbased secret image sharing scheme. Since polynomialbased secret image sharing is extended from traditional secret sharing scheme, we also used a cheating detection algorithm in traditional secret sharing in the field of secret image sharing. The experimental results are generated using Matlab software.
Notes
Funding
The research presented in this paper is supported in part by the China National Natural Science Foundation (No. 61502384, 61571360), Shaanxi Science and Technology Coordination and Innovation Project (No. 2016KTZDGY0509), and the Innovation Project of Shaanxi Provincial Department of Education (No. 17JF023).
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Authors’ contributions
YXL contributed in algorithm designing; QDS was responsible for experiment and analysis; and CNY took part in correctness checking. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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