# Threshold Meaningful Secret Image Sharing Scheme Based on QR Code

Authors: Fei Hu; Yuanzhi Yao; Weihai Li; Nenghai Yu
Journal: Security and Communication Networks
 (2022)

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## Abstract

In the traditional secret image sharing scheme (SIS), the noise-like cover image is vulnerable and thus suspicion is easily aroused by malicious users. Therefore, the meaningful secret image sharing scheme deserves investigation. In this paper, we introduce a newk,n threshold meaningful secret image sharing scheme based on QR code (MSIS-QR). In the proposed MSIS-QR, the secret image is shared with n QR codes, where each share of the scheme is a valid QR code and can be decoded by a QR code scanner. When k or more shares are collected, the secret image can be losslessly recovered by the Chinese remainder theorem. Experimental results indicate the advantages of our scheme in terms of recovered image quality and decoding time.


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## Body

## 1. Introduction

In [1], Shamir created a k,n threshold secret share scheme which aims to divide data into n different shares, and users can easily reconstruct the original data from k shares. Originated from the above idea, Naor and Shamir proposed the visual cryptography scheme (VCS) in 1994 [2], and they first used the binary image as the secret data. Furthermore, to apply Shamir’s scheme to the grayscale image, Thien and Lin [3] introduced a new method of secret image sharing (SIS) to share an image as secret information into n shares, which is also called shadow images. And, any k of the n shares can reconstruct the original secret image. Different from the recovery progress of VCS that depends on the human visual system, the SIS scheme reconstructs the secret by Lagrange interpolation and the reconstructed secret has high image quality. Besides, the Chinese remainder theorem (CRT) based secret image sharing scheme can achieve the same result with low calculation and no distortion [4–10]. In [4], Yan et al. proposed a k,n threshold CRT-based SIS scheme in which the secret images can be losslessly reconstructed. Hu et al. [11] proposed another lossless sharing method that combines the chaotic map and CRT. Wang et al. [12] proposed a weighted SIS scheme with lossless recovery.Secret image sharing is increasingly applied in many scenarios [13–19], such as watermarking embedding [13, 15], authentication [16], and data hiding [17], and so on.Nowadays, quick response code (QR code) is widely used in our daily life. The QR code was invented by Denso Wave Incorporated [20] in 1994 which consists of many black and white blocks. The popularity of the QR code has drawn much attention from many researchers [21–24]. Considering the popularity of QR codes, Cheng et al. [21] proposed a two-level QR code-based VSC, which makes the shares of the scheme look like a regular QR code. Fu et al. [25] proposed a k,n-VCS with all shares being QR codes with specific meanings. Different from [21], they embedded the generated shares into QR codes by using the redundancy of the QR code encoding phase.In this paper, we propose ak,n threshold meaningful secret image sharing scheme based on QR codes. In our scheme, the grayscale secret image is shared by the Chinese remainder theorem-based secret share method, and the output QR codes which contain information can be decoded by a QR code scanner like ordinary QR codes. The use of the CRT method can help us realize a losslessly recovering scheme, while the QR code share reduces the risk of attacks by suspicious users.The major contributions of this paper can be summarized as follows:(1)
A novelk,n threshold meaningful secret image sharing scheme based on QR code (MSIS-QR) is proposed(2)
The secret image can be losslessly recovered by the Chinese remainder theorem(3)
Extensive experiments demonstrate the merits of the proposed schemeThe remainder of the paper is organized as follows. Section2 introduces the Chinese remainder theorem-based secret information sharing scheme and QR code. And, Section 3 illustrates our proposed k,n threshold meaningful secret image sharing scheme based on the QR code. Section 4 gives some security analysis of our scheme. Section 5 shows the experimental result of our scheme and some comparison with some related works, and Section 6 concludes this paper.


## 2. Related Work

### 2.1. Chinese Remainder Theorem

The Chinese remainder theorem aims to solve the general linear congruent equation:(1)x≡a1modn1,x≡a2modn2,…,x≡ak−1modnk−1,x≡akmodnk,As shown in (1), nii=1,2,…,k are set as integers are pairwise coprime, which meets the condition of gcdni,nj=1 while i≠j, and the modules of aii=1,2,…,k are integers that ai<ni for every i. Then, for a given N, where N=∏i=1kni, there is one and only one solution x that meets (1) while 0≤x<N.To solve the above equation, we first letNi=N/ni. And, because ni is coprime, Ni and ni are also coprime. According to Bézout’s identity, we can find 2 integers Mi and mi such that(2)MiNi+mini=1.And, based on (2), we can get a solution of (1).(3)x=∑i=1kaiMiNi.Furthermore, we define an integerNi−1 which subjects to the condition of NiNi−1≡1modni, and we can get a solution to (1):(4)x≡a1N1N1−1+a2N2N2−1+⋯+akNkNk−1modN.


### 2.2. Threshold Secret Sharing Scheme Based on Chinese Remainder Theorem

According to Asmuth and Bloom’s scheme [5], a threshold secret sharing method is introduced. The basic method of secret sharing is illustrated as follows.For a (k, n) threshold secret sharing, where k denotes the threshold and n denotes the number of shares, a set of integers p,m1<m2<⋯<mn satisfying the following conditions:(1)
gcdmi,mj=1fori≠j(2)
gcdp,mi=1for alli≠j(3)
∏i=1kmi>p∏i=1k−1mn−i+1For convenience, we denoteM=∏i=1kmi and N=∏i=1k−1mn−i+1; thus, condition (3) is equal to M>pN. If a secret x is to be shared, here we assume that 0≤x<p, and let y=x+Ap, where A is a random integer that makes 0≤y<M. Then, compute ai≡ymodmi and let sci=aifori=1,2,…,n be the shares of the secret sharing scheme. After all these, the output n shares sc1,sc2,…,scn are ready.


### 2.3. Quick Response Code

In 1994, Denso Wave Incorporate invented a new kind of two-dimensional code which consists of many black and white squares. This type of new code is named as quick response code (QR code). As shown in Figure1, a standard QR code consists of several obligatory elements: position patterns, version information, timing patterns, formation patterns, alignment patterns, data, and error correction keys. And, Figure 1 shows these patterns which are required in a complete QR code.Figure 1
The structure of QR codes. There is no formally established standard of QR code. However, a QR code must consist of some obligatory shown in the picture.To locate the square and verify the structure of the QR code, the position patterns, alignment patterns, and timing patterns are required, while the version of a QR code is encoded and stored in the version information patterns. The versions of it are so important which are directly connected with the size of the QR code and the number of bits that can be stored in the QR code.If a standard QR code is damaged in a proper range, the error correction capability can help a code recover the lost information. To realize the error correction function, Reed–Solomon codes are added to the original data. There are four error correction levels available for choosing which areL, M, Q, and H, and the corresponding error correction capacity is 7%, 15%, 25%, and 30%, respectively.


## 2.1. Chinese Remainder Theorem

The Chinese remainder theorem aims to solve the general linear congruent equation:(1)x≡a1modn1,x≡a2modn2,…,x≡ak−1modnk−1,x≡akmodnk,As shown in (1), nii=1,2,…,k are set as integers are pairwise coprime, which meets the condition of gcdni,nj=1 while i≠j, and the modules of aii=1,2,…,k are integers that ai<ni for every i. Then, for a given N, where N=∏i=1kni, there is one and only one solution x that meets (1) while 0≤x<N.To solve the above equation, we first letNi=N/ni. And, because ni is coprime, Ni and ni are also coprime. According to Bézout’s identity, we can find 2 integers Mi and mi such that(2)MiNi+mini=1.And, based on (2), we can get a solution of (1).(3)x=∑i=1kaiMiNi.Furthermore, we define an integerNi−1 which subjects to the condition of NiNi−1≡1modni, and we can get a solution to (1):(4)x≡a1N1N1−1+a2N2N2−1+⋯+akNkNk−1modN.


## 2.2. Threshold Secret Sharing Scheme Based on Chinese Remainder Theorem

According to Asmuth and Bloom’s scheme [5], a threshold secret sharing method is introduced. The basic method of secret sharing is illustrated as follows.For a (k, n) threshold secret sharing, where k denotes the threshold and n denotes the number of shares, a set of integers p,m1<m2<⋯<mn satisfying the following conditions:(1)
gcdmi,mj=1fori≠j(2)
gcdp,mi=1for alli≠j(3)
∏i=1kmi>p∏i=1k−1mn−i+1For convenience, we denoteM=∏i=1kmi and N=∏i=1k−1mn−i+1; thus, condition (3) is equal to M>pN. If a secret x is to be shared, here we assume that 0≤x<p, and let y=x+Ap, where A is a random integer that makes 0≤y<M. Then, compute ai≡ymodmi and let sci=aifori=1,2,…,n be the shares of the secret sharing scheme. After all these, the output n shares sc1,sc2,…,scn are ready.


## 2.3. Quick Response Code

In 1994, Denso Wave Incorporate invented a new kind of two-dimensional code which consists of many black and white squares. This type of new code is named as quick response code (QR code). As shown in Figure1, a standard QR code consists of several obligatory elements: position patterns, version information, timing patterns, formation patterns, alignment patterns, data, and error correction keys. And, Figure 1 shows these patterns which are required in a complete QR code.Figure 1
The structure of QR codes. There is no formally established standard of QR code. However, a QR code must consist of some obligatory shown in the picture.To locate the square and verify the structure of the QR code, the position patterns, alignment patterns, and timing patterns are required, while the version of a QR code is encoded and stored in the version information patterns. The versions of it are so important which are directly connected with the size of the QR code and the number of bits that can be stored in the QR code.If a standard QR code is damaged in a proper range, the error correction capability can help a code recover the lost information. To realize the error correction function, Reed–Solomon codes are added to the original data. There are four error correction levels available for choosing which areL, M, Q, and H, and the corresponding error correction capacity is 7%, 15%, 25%, and 30%, respectively.


## 3. The Proposed Scheme

In this section, we propose a new meaningful secret sharing scheme based on QR codes. We use the Chinese remainder theorem to realize the secret image share and recover scheme.In our scheme, we share a secret image S with the size ofH×W into n shares, denoted as SI1,SI2,…,SIn, and any k or more of the n shares can recover the original secret image. When generating n shares, n cover QR codes QR1,QR2,…,QRn with the same size of secret images are also required. For a meaningful secret image sharing scheme, it should be subjected to the following conditions:(1)
Security condition: no information about the secret image should be got with less thank shares(2)
Contrast condition: the secret image can be reconstructed successfully with no less thank shares(3)
Meaningful condition: then shares are generated according to n QR codes, so these shares are readable by the QR code scannerIn our paper, we evaluate our work by calculating the following parameters: peak signal-to-noise ratio (PSNR) and weighted PSNR (WPSNR).The PSNR is calculated to evaluate the similarity of the original imageI and the recovered image I′, the equation is illustrated as follows:(5)PSNR=10log102552MSE,where MSE equals to the mean square error of I and I’(6)MSE=1W×H∑i=1W∑j=1HI′i,j−Ii,j2,where Ii,j denotes the grayscale value of the image and the WPSNR is used to weight the spatial frequency of the error image using contrast sensitivity function (CSF), and the error image is the difference between two images.(7)WPSNR=10log101∑i=1W∑j=1HF2i,j/WH,where F denotes the contrast sensitivity function of the error image, which is detailly elaborated in [26].


### 3.1. (k, n) Threshold Secret Image Sharing Scheme Based on Chinese Remainder Theorem

According to Section2.2 and Yan et al. [4], the CRT-based SIS can recover secret image losslessly with low computational complexity. The main idea of the scheme is described as follows.We first choose a set of integers that conforms to128≤p<m1<m2<⋯<mn≤256, and those integers are subject to the conditions of:(1)
gcdmi,mj=1fori≠j(2)
gcdp,mi=1for alli≠j(3)
∏i=1kmi>p∏i=1k−1mn−i+1which have already been introduced in Section 2.2. And, we release M=∏i=1kmi, N=∏i=1k−1mn−i+1, and p to all participants. Furthermore, we compute T=⌊M/p−1⌋−⌈N/p⌉/2 and release it to all participants, too.For each pixel of the secret image with the value ofx, if 0≤x<p, we randomly choose an arbitrary integer A in the range of T+1,⌊M/p⌋−1 and compute y=x+Ap; otherwise, we select a random integer A from 0,T and then compute y=x−p+Ap. Then, ai≡ymodmi is computed and assigned to n participants. Based on these operations, we can share all the range of pixel values x into n shares regardless of those 0≤x<p.To recover the secret image,k or more shares are required, for each pixel, we can get the solution of the general linear congruent equation. Traversing all pixels, the secret image can be recovered losslessly.


### 3.2. Our Threshold Meaningful Secret Image Sharing Scheme Based on QR Codes

The main idea of the proposed MSIS-QR algorithm is illustrated in Figure2. In the sharing phase, a grayscale image S with the size of H×W is shared into n grayscale image shares SI1,SI2,…,SIn with the same size of S, the threshold k,n and n cover image of QR codes QR1,QR2,…,QRn are prepared to revise the pixel value of SI. The detailed secret sharing steps are listed in Algorithm 1 and the recovery scheme is presented in Algorithm 2.Figure 2
The flow chart of meaningful secret image sharing scheme based on QR code.Algorithm 1: Threshold meaningful secret image sharing scheme based on QR codes.
(i)
Input: a grayscale secret image S with the size ofH×W; participant number n; secret recovery threshold k; cover image of QR codes QR1,QR2,…,QRn and tolerant threshold TH.(ii)
Output:n grayscale images SI1,SI2,…,SIn.(1)
Choose a set of integers that conform to128≤p<m1<m2<⋯<mn≤256 and those numbers are subject to the following:(1)
gcdmi,mj=1,i≠j(2)
gcdmi,p=1,fori=1,2,…,n(3)
∏i=1kmi>p∏i=1k−1mn−i+1 then make M=∏i=1kmi, N=∏i=1k−1mn−i+1 and p public to all participants(2)
ComputeT=⌊M/p−1⌋−⌈N/p⌉/2 and make it public to all participants, too(3)
For each pixel in the secret image of coordinatesh,w where 1≤h≤H,1≤w≤W, let x=Sh,w and repeat steps 4-5(4)
If0≤x<p, then random choose an arbitrary integer A in the range of T+1,⌊M/p⌋−1 and compute y=x+Ap, else select a random integer A from 0,T and then compute y=x−p+Ap(5)
fori = 1 to n, compute ai≡ymodmi(6)
IfCOMPai,TH=QRih,w,i=1,2,…,n, assign ai to SIiw,h and go to next position, else go to step 4, where COMPai,TH=1,ifai≥128+TH andQRih,w=1,0,ifai≤128−TH andQRih,w=0.(7)
Outputn decodable QR codes with secret sharing information SI1,SI2,…,SInAlgorithm 2: The recovery phase of threshold meaningful secret image sharing based on QR codes.
(i)
Input: anyt grayscale shares SIr1,SIr2,…,SIrt, the corresponding t private modular integers mr1,mr2,…,mrt, p, and T.(ii)
Output: the recovered secret grayscale imageS′.(1)
For every input imageSIrii=1,2,…,t, we scan them by QR code scanner and verify if the information is the same as the given information. If not, discard the shared image and let t=t−1.(2)
Ift≥k, for each pixel in the secret image of coordinates h,w where 1≤h≤H,1≤w≤W, let ari=SIrih,w and repeat steps 3–5.(3)
Fori=1,2,…,k, solve the following equation: y≡ar1modmr1y≡ar2modmr2⋯y≡arkmodmrk.(4)
CalculateT∗=⌊y/p⌋. If T∗≥T, let x≡ymodp; otherwise x≡ymodp+p.(5)
SetS′h,w=x.(6)
Output the reconstructed secret imageS′ with the size of H×W.As for Algorithm 1, for each pixel of the secret imageSh,w, we first use the CRT-based SIS to generate n shares that are subject to the (k, n) threshold condition. And then, the generated n shares are compared with the given QR codes, respectively. For each pixel in the shared image, we compare it with the corresponding QR code pixel. If the difference is in the tolerant range of the given threshold, we keep the sharing result; otherwise, we go back and compute a set of new sharing values.In our scheme, we add a condition that requires the pixel value of the shared image is near the cover QR code in the same coordinate. Thus, we can regenerate a new QR code that contains public parametersM,N,p, and T before step 6 for more convenient access by QR code scanner; thus, we can omit the step of making those parameters public.And the recovery scheme is illustrated in Algorithm 2. Whentk≤t≤n shares are collected, we first extract the information in QR code to verify if the share is a true share that generates from Algorithm 1. For every pixel in the same location of shares, a general linear congruent equation can be obtained, and the solution of the equation is the corresponding pixel value of the secret image.


## 3.1. (k, n) Threshold Secret Image Sharing Scheme Based on Chinese Remainder Theorem

According to Section2.2 and Yan et al. [4], the CRT-based SIS can recover secret image losslessly with low computational complexity. The main idea of the scheme is described as follows.We first choose a set of integers that conforms to128≤p<m1<m2<⋯<mn≤256, and those integers are subject to the conditions of:(1)
gcdmi,mj=1fori≠j(2)
gcdp,mi=1for alli≠j(3)
∏i=1kmi>p∏i=1k−1mn−i+1which have already been introduced in Section 2.2. And, we release M=∏i=1kmi, N=∏i=1k−1mn−i+1, and p to all participants. Furthermore, we compute T=⌊M/p−1⌋−⌈N/p⌉/2 and release it to all participants, too.For each pixel of the secret image with the value ofx, if 0≤x<p, we randomly choose an arbitrary integer A in the range of T+1,⌊M/p⌋−1 and compute y=x+Ap; otherwise, we select a random integer A from 0,T and then compute y=x−p+Ap. Then, ai≡ymodmi is computed and assigned to n participants. Based on these operations, we can share all the range of pixel values x into n shares regardless of those 0≤x<p.To recover the secret image,k or more shares are required, for each pixel, we can get the solution of the general linear congruent equation. Traversing all pixels, the secret image can be recovered losslessly.


## 3.2. Our Threshold Meaningful Secret Image Sharing Scheme Based on QR Codes

The main idea of the proposed MSIS-QR algorithm is illustrated in Figure2. In the sharing phase, a grayscale image S with the size of H×W is shared into n grayscale image shares SI1,SI2,…,SIn with the same size of S, the threshold k,n and n cover image of QR codes QR1,QR2,…,QRn are prepared to revise the pixel value of SI. The detailed secret sharing steps are listed in Algorithm 1 and the recovery scheme is presented in Algorithm 2.Figure 2
The flow chart of meaningful secret image sharing scheme based on QR code.Algorithm 1: Threshold meaningful secret image sharing scheme based on QR codes.
(i)
Input: a grayscale secret image S with the size ofH×W; participant number n; secret recovery threshold k; cover image of QR codes QR1,QR2,…,QRn and tolerant threshold TH.(ii)
Output:n grayscale images SI1,SI2,…,SIn.(1)
Choose a set of integers that conform to128≤p<m1<m2<⋯<mn≤256 and those numbers are subject to the following:(1)
gcdmi,mj=1,i≠j(2)
gcdmi,p=1,fori=1,2,…,n(3)
∏i=1kmi>p∏i=1k−1mn−i+1 then make M=∏i=1kmi, N=∏i=1k−1mn−i+1 and p public to all participants(2)
ComputeT=⌊M/p−1⌋−⌈N/p⌉/2 and make it public to all participants, too(3)
For each pixel in the secret image of coordinatesh,w where 1≤h≤H,1≤w≤W, let x=Sh,w and repeat steps 4-5(4)
If0≤x<p, then random choose an arbitrary integer A in the range of T+1,⌊M/p⌋−1 and compute y=x+Ap, else select a random integer A from 0,T and then compute y=x−p+Ap(5)
fori = 1 to n, compute ai≡ymodmi(6)
IfCOMPai,TH=QRih,w,i=1,2,…,n, assign ai to SIiw,h and go to next position, else go to step 4, where COMPai,TH=1,ifai≥128+TH andQRih,w=1,0,ifai≤128−TH andQRih,w=0.(7)
Outputn decodable QR codes with secret sharing information SI1,SI2,…,SInAlgorithm 2: The recovery phase of threshold meaningful secret image sharing based on QR codes.
(i)
Input: anyt grayscale shares SIr1,SIr2,…,SIrt, the corresponding t private modular integers mr1,mr2,…,mrt, p, and T.(ii)
Output: the recovered secret grayscale imageS′.(1)
For every input imageSIrii=1,2,…,t, we scan them by QR code scanner and verify if the information is the same as the given information. If not, discard the shared image and let t=t−1.(2)
Ift≥k, for each pixel in the secret image of coordinates h,w where 1≤h≤H,1≤w≤W, let ari=SIrih,w and repeat steps 3–5.(3)
Fori=1,2,…,k, solve the following equation: y≡ar1modmr1y≡ar2modmr2⋯y≡arkmodmrk.(4)
CalculateT∗=⌊y/p⌋. If T∗≥T, let x≡ymodp; otherwise x≡ymodp+p.(5)
SetS′h,w=x.(6)
Output the reconstructed secret imageS′ with the size of H×W.As for Algorithm 1, for each pixel of the secret imageSh,w, we first use the CRT-based SIS to generate n shares that are subject to the (k, n) threshold condition. And then, the generated n shares are compared with the given QR codes, respectively. For each pixel in the shared image, we compare it with the corresponding QR code pixel. If the difference is in the tolerant range of the given threshold, we keep the sharing result; otherwise, we go back and compute a set of new sharing values.In our scheme, we add a condition that requires the pixel value of the shared image is near the cover QR code in the same coordinate. Thus, we can regenerate a new QR code that contains public parametersM,N,p, and T before step 6 for more convenient access by QR code scanner; thus, we can omit the step of making those parameters public.And the recovery scheme is illustrated in Algorithm 2. Whentk≤t≤n shares are collected, we first extract the information in QR code to verify if the share is a true share that generates from Algorithm 1. For every pixel in the same location of shares, a general linear congruent equation can be obtained, and the solution of the equation is the corresponding pixel value of the secret image.


## 4. Security Analysis

In this section, we give the security analysis of the MSIS-Q scheme.Lemma 1.
No information of the pixel value for the secret image can be obtained from any share generated by Algorithm 1.Proof.
For each pixel value ofx=Sh,w, we will compute y=x+Ap or y=x+Ap−p and then compute ai≡ymodmi; after that, a modification phase is applied to make sure that the share value is near to the cover image’s value in an appropriate range (noted as R). Because the parameter A is randomly chosen, the final result will be a random number in R,mi for every share. Therefore, the share images are random for the secret image.Lemma 2.
The secret image S can be losslessly recovered withk or more shares.Proof.
We assume thatSI1,SI2,…,SIk are obtained generally. To recover the secret pixel value x=Sh,w, we should find y firstly. Based on CRT, for the collected equations of given k images, we can get solution y modulo M=∏i=1kmi, and the solution is unique.
As is described in Section2.1, the integers of mii=1,2,…,n are coprime with each other, and we define Mi=M/mi which is coprime with mi. Thus, Mi has a unique multiplicative inverse of modulo mi, denoted by Mi−1, and we have MiMi−1=1modmi.
Wheni≠j, we can get mi|Mj, videlicet, Mj≡0modmi. Therefore,(8)Mj×Mj−1modmjmodmi≡Mjmodmi×Mj−1modmjmodmimodmi≡0,whileMi×Mi−1modmimodmi≡Mi×Mi−1modmi≡1,so y≡a1M1M1−1+a2M2M2−1+⋯+akMkMk−1modM is the solution to the linear congruent equation ai≡ymodmi for i∈1,2,…,k.
And then, we will prove thaty is the only solution to the equation. Assuming that y0 is another solution to the equation(9)ai≡y0modmii=1,2,…,k,and because of ai≡ymodmi, we can get y0−y≡0modmi, or mi|y0−y. Furthermore, mi is pairwise coprime, which leads to M|y0−y, that is,(10)y0−y≡0modM,so y0modM=ymodM, which means y is the unique solution modulo M.
Once we get the uniquey by the Chinese remainder theorem, we can easily get x according to the equation to y=x+Ap or y=x+Ap−p. So, we can get unique x for k shares.Lemma 3.
No information of the pixel value for the secret image can be obtained from less thank shares generated by Algorithm 1.Proof.
Whenk−1 shares are collected, generally, we assume that SI1,SI2,…,SIk−1 are given, the pixel value of the same coordinate noted as y1,y2,…,yk−1, and the product of corresponding parameters is M′=∏i=1k−1mi.
According to CRT, all we can get is the solution(11)yi≡y0modM′M′=∏i=1k−1mi,wherey0∈0,M′−1,while the true y generated by Algorithm 1 is in the range of 0,M−1, we can easily find that y and y0 are located in different ranges.
Furthermore, according to the given requirements, we haveM/M′>p and gcdM′,p=1; thus, we have a set(12)y′|y′modM′=y0,which can cover all the solutions to the k−1 linear congruent equations. In the other words, y′=y0+b∏i=1k−1mi are also solutions to the equation for b=1,2,…,mk−1. So, we can get more than one solution.Lemma 4.
SIi stores the same information of the cover QR code QRi for i=1,2,…,n and can be read by the scanner.Proof.
The pixel value ofSIi is in the appropriate distance from QRi. Benefiting from the stability of the QR code, the generated shares can be losslessly decoded if the difference of QRi and SIi is in a tolerable range.


## 5. Experimental Results and Discussion

In this section, we will display the experimental results of our scheme, and some comparisons with other schemes will be done.


### 5.1. Experimental Results

In our experiments, we apply 3 different sizes of secret images and cover QR code images. We set the size of secret images with41×41, 256×256 and 512×512, respectively, and set the corresponding QR code with the same size.


#### 5.1.1. The Visual Quality of the Share Images and the Recovered Images

Figures3 and 4 show the result of the proposed (k, n) threshold meaningful secret share scheme based on QR codes. In these schemes, we set k=3, n=4, m1=251, m2=253, m3=254, m4=255, and p=131. In Figure 3, we set TH=40 and the size of the secret image with 512×512. In Figure 4, TH=6, and the size of the secret image with 41×41.Figure 3
The experimental result of proposed (3, 4) MSIS-QR. The parameter in this experiment is illustrated above. And, for a better view, the size of the images is scaled to a proper size. (a) Secret imageS, (b) cover QR code 1, (c) cover QR code 2, (d) cover QR code 3, (e) cover QR code 4, (f) SI1, (g) SI2, (h) SI3, (i) SI4, (j) SI12′, (k) SI13′, (l) SI14′, (m) SI23′, (n) SI24′, (o) SI34′, (p) SI123′PSNR=∞, (q) SI124′PSNR=∞, (r) SI134′PSNR=∞ (s) SI234′PSNR=∞, and (t) SI1234′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)(r)(s)(t)Figure 4
The result of (3, 4) MSIS-QR: the public parameter is the same as those in Figure3 while TH = 6. (b)–(e) are the shares, and (f) is the reconstruction result by (b) and (d) which is deservedly a random picture while (g) and (h) can reconstruct the secret image with no less than 3 shares. (a) secret image S, (b) SI1, (c) SI2, (d) SI3, (e) SI4, (f) SI13′, (g) SI234′PSNR=∞, and (h) SI1234′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)Figure3(a) shows the grayscale secret image, 3(b)–3(e) show the cover QR code images, 3(f)–3(i) show the output of Algorithm 1, which can be decoded by QR code scanner. 3(j)–3(o) are the recover results by Algorithm 2 for any two shares and the rest are the recover results with more than two shares.In Figure3, we can find that, (1) for those sets that the number of participants is less than k (i.e., 1 and 2 in this experiment), we cannot get any information about the secret image except the image size. And, for those sets that no less than k participants are involved, the secret image is losslessly reconstructed, the recovery image is the same as the secret image with PSNR=∞.Figure4 shows the secret image, QR code similarity shares generated by Algorithm 1 and some reconstruction results of Algorithm 2. And, we can find that the shares can be more random with low TH with the performance of secret sharing staying the same.Figure5 shows the result of (2, 3) MSIS-QR with m1=251, m2=253, m3=255, and p=128. (a) displays a Lena that samples to 256×256, and (b)–(d) are three cover QR codes that with the same size of (a). (e)–(g) are 3 share images generated by our scheme, and the information in these pictures are exactly the same with the corresponding codes in (b)–(d). The reconstructed results of different sets of (e)–(g) are displayed in (h)–(k), as we can see that the reconstructed image is lossless with any 2 or more shares.Figure 5
Experimental result of (2, 3) MSIS-QR scheme. The shares (e)–(g) reveal no information of the secret image (a), and they can be decoded by a QR code scanner. And, any two or more shares can losslessly reconstruct the secret image. (a) Secret image S, (b) cover QR code 1, (c) cover QR code 2, (d) cover QR code 3, (e)SI1, (f) SI2, (g) SI3, (h) SI12′PSNR=∞, (i) SI13′PSNR=∞, (j) SI23′PSNR=∞, and (k) SI123′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)


#### 5.1.2. Error Correction Ability of Shadows

As the shadow images of our scheme are restricted by the QR code, and we do not change the error correction capacity of the original QR code. For a shadow image, the damage and the noise in the appropriate range can be decoded correctly.Figure6 illustrates the ability of error correction if the shadows suffered from damage. (a), (d), and (g) are the generated shadows of our scheme, (a) and (g) are (2, 3) threshold results with TH=50.5 and TH=−26, respectively, while (d) is (3, 4) scheme with TH=25. And, (b) is image (a) with salt and pepper noise, (e) and (h) are (d) and (g) with Gaussian noise, respectively, and (c) and (f) are damaged shares of (a) and (d).Figure 6
Error correction ability. (a), (d), and (g) are the generated shadows. (b), (e), and (h) are corresponding shadows added with noise, and (c) and (f) are damaged shadows. (a) Shadow image, decodable, (b) shadow distorted with noise, decodable, (c) damaged shadow, decodable, (d) shadow image, decodable, (e) shadow distorted with noise, decodable, (f) damaged shadow, decodable, (g) shadow image, decodable, and (h) shadow distorted with noise, nondecodable.
(a)(b)(c)(d)(e)(f)(g)(h)As is shown in Figure6, the damaged shadows and noise added shadows of our paper can be correctly decoded when the value of threshold is well selected. For image (h), the shadow image can be decoded correctly, while (g) cannot be recognized with noise added. The bigger of threshold, the better performance of error correction ability will be kept.


#### 5.1.3. Evaluation for Visual Quality

In our scheme, the tolerant threshold TH is the parameter that plays an important role in the sharing phase. As is illustrated in step 6 of Algorithm 1 (seen in (13)), we will evaluate that if the result of the pixel in a secret image shared by CRT is in tolerance. If the result satisfies the condition, we keep the shares’ value, else recalculate it.(13)COMPai,TH=1,ifai≥128+THandQRih,w=1,0,ifai≤128−THandQRih,w=0.From the threshold-sharing time curve in Figure7, we can find that the sharing time is almost increasing with the rise of the threshold, and the increasing speed is faster than the growth of the threshold. For practicality, if we want to get a share that is similar to QR code, we can apply a big threshold, otherwise, if we want to let the shares look more random, we can use a relatively small threshold to make the value of the pixel has a wider range.Figure 7
The sharing time influenced by different thresholds. In this figure,t1 is the curve of sharing time of the secret image with 41×41, t2 with 256×256, and t3 with 512×512.Based on the previous introduction, the secret image can be losslessly recovered by our scheme. And here, we calculate the value of PSNR and WPSNR of the secret image and the reconstructed image, and the result is shown in Table1. And here, we present 3 different sharing schemes. PSNR1 and WPSNR1 are the curve of (3, 4) MSIS-QR with the secret image of 41×41, PSNR2 and WPSNR2 are the (2, 3) MSIS-QR with the image size of 256×256, and PSNR3 and WPSNR3 are the (3, 4) MSIS-QR with image size of 512×512. And, because the sharing time is too long for the secret image of 512×512 with a big threshold, the experiment ends at the threshold of 80. As we can see in the result of PSNR and WPSNR, the secret image can be losslessly recovered when TH<80 with all sizes of the secret image.Table 1
The value of PSNR/WPSNR about the threshold.
Threshold1020305080100PSNR1∞∞∞∞∞∞WPSNR1∞∞∞∞∞∞PSNR2∞∞∞∞∞∞WPSNR2∞∞∞∞∞∞PSNR3∞∞∞∞∞-WPSNR3∞∞∞∞∞-


### 5.2. Comparison with Related Schemes

In this section, we will compare our work with Yan et al. [27], Fu et al. [25], and Jiang et al. [28]. And, the experimental results of these schemes are depicted in Figure 8 [27], Figure 9 [25], and Figure 10 [28] separately.Figure 8
Experimental results of the introduced (k, n) threshold SIS scheme of [27], where k = 2, n = 3, m1 = 251, m2 = 253, m3 = 255, p = 128, and TH = 24. (a) The grayscale secret image. (b–d) Three grayscale comprehensible shares SC1, SC2, and SC3. (e) Grayscale secret image reconstructed with the first two shares. (f–h) Recovered binary cover images. (a) Secret image S, (b) SC1, PSNR=12.3, (c) SC2, PSNR=13.2, (d) SC3, PSNR=12.7, (e) SI12′PSNR=∞, (f) C1′, PSNR=∞, (g) C2′, PSNR=∞, (h) C3′, and PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Experimental results of [25]: (a)–(d) are shares; (e) is the decoding of (a); (f) is the secret image, and (g) is the XOR-ed result of (a) and (b), while (h) is the XOR-ed result of (a) and (d).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Results of (2, 3)-threshold CRTQR-SIS with meaningful share proposed by [28], where (a) is the input grayscale secret image Lena; (b–d) are three shares SC1, SC2, and SC3; (e–h) are retrieved results by different shares. (a) Lena, (b) SC1, (c) SC2, (d) SC3, (e) S1,2′, (f) S1,3′, (g) S2.3′, and (h) S1,2,3′.
(a)(b)(c)(d)(e)(f)(g)(h)Yan et al. [27] proposed a RISS algorithm with a (k, n)-threshold based on the CRT, and the secret image is losslessly recovered, and Figure 8 shows the result of (2, 3) threshold of [27]. The secret image can be reconstructed losslessly with 2 or more shares. But in Figures 8(b) and 8(d), we can easily find some extra outlines which do not belong to the Cameraman and the Elaine.Compared with [27], the cover image of QR code can conceal more information than a grayscale image in our scheme. Benefiting from QR code, we can write some parameters that should be released to all participants to it and omit the operation of publication. And, in (2, 3) threshold schemes, the shares of [27] have inconspicuous outline of the secret image while ours do not with a proper threshold.Fu et al. [25] proposed a (k, n)-VCS in which all shares are QR codes with specific meaning. Figure 9 shows the experimental result of (2, 4)-VCS. In their scheme, the generated shares can be decoded correctly, and they can reconstruct secret image with 2 shares by XOR operation.However, limited by the features of VCS, the secret image is a binary image and the recovered image may not be losslessly compared with the secret image, only a proper set of shares can recover secret image losslessly by XOR operation. And, their scheme embeds the shares into the cover image by using the encoding redundancy of QR codes, so the process of the encoding phase is more complicated than ours.Jiang et al. [28] proposed a Chinese Remainder theorem and QR code-based extended secret image sharing with lossless recovery for k,n threshold. Figure 10 gives the experimental results for QR code version 5 with error correction level H by (2, 3) threshold scheme of [28]. By using their algorithm, these shares can be decoded correctly, and the secret image can retrieve with efficient shares together.But as we can see in Figures10(b)–10(d), these shares have clearly outline of 10(a), which may leak information about secret image. This problem will be relieved with appropriate parameter setting (i.e., QR coded with version 26, error correction level H, and (3, 3)-SIS scheme), but still cannot be avoided, which is undesirable in secret image sharing scheme. Compared to their scheme, our shares do not have the problem of information leakage.The feature comparisons between our scheme with related schemes are shown in Table2.Table 2
The comparison with related schemes.
SchemeSecret imageLossless recoveryShare imagesShare authenticationYan et al. [27]GrayscaleYesGrayscale images but have secret image shadows in some (2, 3)-SISNoFu et al. [25]BinaryNoDecodable QR codesYesJiang et al. [28]GrayscaleYesDecodable QR codes but have secret image shadowsYesOur schemeGrayscaleYesDecodable QR codesYes


### 5.3. Discussion

There are two main methods of secret image sharing: the CRT-based SIS and polynomial-based SIS. The CRT-based SIS is modular operation in the reconstruction phase, while polynomial-based SIS requires Lagrange interpolation. So, the CRT-based SIS has lower computational cost. According to [5], the reconstruction complexity of our scheme is Ok, while the complexity of methods using the method of Lagrange interpolation is Oklogk.Restricted by Asmuth and Bloom’s algorithm, the number of participants is limited. With the growth ofn, the suitable number of mi decreases. According to [27], in CRT-based secret image sharing scheme, the number of participants is generally small, such as n≤6. To solve the problem, Ning et al. [28] proposed an ideal secret sharing scheme based on CRT for polynomial ring over finite field, and this will be our future work.


## 5.1. Experimental Results

In our experiments, we apply 3 different sizes of secret images and cover QR code images. We set the size of secret images with41×41, 256×256 and 512×512, respectively, and set the corresponding QR code with the same size.


### 5.1.1. The Visual Quality of the Share Images and the Recovered Images

Figures3 and 4 show the result of the proposed (k, n) threshold meaningful secret share scheme based on QR codes. In these schemes, we set k=3, n=4, m1=251, m2=253, m3=254, m4=255, and p=131. In Figure 3, we set TH=40 and the size of the secret image with 512×512. In Figure 4, TH=6, and the size of the secret image with 41×41.Figure 3
The experimental result of proposed (3, 4) MSIS-QR. The parameter in this experiment is illustrated above. And, for a better view, the size of the images is scaled to a proper size. (a) Secret imageS, (b) cover QR code 1, (c) cover QR code 2, (d) cover QR code 3, (e) cover QR code 4, (f) SI1, (g) SI2, (h) SI3, (i) SI4, (j) SI12′, (k) SI13′, (l) SI14′, (m) SI23′, (n) SI24′, (o) SI34′, (p) SI123′PSNR=∞, (q) SI124′PSNR=∞, (r) SI134′PSNR=∞ (s) SI234′PSNR=∞, and (t) SI1234′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)(r)(s)(t)Figure 4
The result of (3, 4) MSIS-QR: the public parameter is the same as those in Figure3 while TH = 6. (b)–(e) are the shares, and (f) is the reconstruction result by (b) and (d) which is deservedly a random picture while (g) and (h) can reconstruct the secret image with no less than 3 shares. (a) secret image S, (b) SI1, (c) SI2, (d) SI3, (e) SI4, (f) SI13′, (g) SI234′PSNR=∞, and (h) SI1234′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)Figure3(a) shows the grayscale secret image, 3(b)–3(e) show the cover QR code images, 3(f)–3(i) show the output of Algorithm 1, which can be decoded by QR code scanner. 3(j)–3(o) are the recover results by Algorithm 2 for any two shares and the rest are the recover results with more than two shares.In Figure3, we can find that, (1) for those sets that the number of participants is less than k (i.e., 1 and 2 in this experiment), we cannot get any information about the secret image except the image size. And, for those sets that no less than k participants are involved, the secret image is losslessly reconstructed, the recovery image is the same as the secret image with PSNR=∞.Figure4 shows the secret image, QR code similarity shares generated by Algorithm 1 and some reconstruction results of Algorithm 2. And, we can find that the shares can be more random with low TH with the performance of secret sharing staying the same.Figure5 shows the result of (2, 3) MSIS-QR with m1=251, m2=253, m3=255, and p=128. (a) displays a Lena that samples to 256×256, and (b)–(d) are three cover QR codes that with the same size of (a). (e)–(g) are 3 share images generated by our scheme, and the information in these pictures are exactly the same with the corresponding codes in (b)–(d). The reconstructed results of different sets of (e)–(g) are displayed in (h)–(k), as we can see that the reconstructed image is lossless with any 2 or more shares.Figure 5
Experimental result of (2, 3) MSIS-QR scheme. The shares (e)–(g) reveal no information of the secret image (a), and they can be decoded by a QR code scanner. And, any two or more shares can losslessly reconstruct the secret image. (a) Secret image S, (b) cover QR code 1, (c) cover QR code 2, (d) cover QR code 3, (e)SI1, (f) SI2, (g) SI3, (h) SI12′PSNR=∞, (i) SI13′PSNR=∞, (j) SI23′PSNR=∞, and (k) SI123′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)


### 5.1.2. Error Correction Ability of Shadows

As the shadow images of our scheme are restricted by the QR code, and we do not change the error correction capacity of the original QR code. For a shadow image, the damage and the noise in the appropriate range can be decoded correctly.Figure6 illustrates the ability of error correction if the shadows suffered from damage. (a), (d), and (g) are the generated shadows of our scheme, (a) and (g) are (2, 3) threshold results with TH=50.5 and TH=−26, respectively, while (d) is (3, 4) scheme with TH=25. And, (b) is image (a) with salt and pepper noise, (e) and (h) are (d) and (g) with Gaussian noise, respectively, and (c) and (f) are damaged shares of (a) and (d).Figure 6
Error correction ability. (a), (d), and (g) are the generated shadows. (b), (e), and (h) are corresponding shadows added with noise, and (c) and (f) are damaged shadows. (a) Shadow image, decodable, (b) shadow distorted with noise, decodable, (c) damaged shadow, decodable, (d) shadow image, decodable, (e) shadow distorted with noise, decodable, (f) damaged shadow, decodable, (g) shadow image, decodable, and (h) shadow distorted with noise, nondecodable.
(a)(b)(c)(d)(e)(f)(g)(h)As is shown in Figure6, the damaged shadows and noise added shadows of our paper can be correctly decoded when the value of threshold is well selected. For image (h), the shadow image can be decoded correctly, while (g) cannot be recognized with noise added. The bigger of threshold, the better performance of error correction ability will be kept.


### 5.1.3. Evaluation for Visual Quality

In our scheme, the tolerant threshold TH is the parameter that plays an important role in the sharing phase. As is illustrated in step 6 of Algorithm 1 (seen in (13)), we will evaluate that if the result of the pixel in a secret image shared by CRT is in tolerance. If the result satisfies the condition, we keep the shares’ value, else recalculate it.(13)COMPai,TH=1,ifai≥128+THandQRih,w=1,0,ifai≤128−THandQRih,w=0.From the threshold-sharing time curve in Figure7, we can find that the sharing time is almost increasing with the rise of the threshold, and the increasing speed is faster than the growth of the threshold. For practicality, if we want to get a share that is similar to QR code, we can apply a big threshold, otherwise, if we want to let the shares look more random, we can use a relatively small threshold to make the value of the pixel has a wider range.Figure 7
The sharing time influenced by different thresholds. In this figure,t1 is the curve of sharing time of the secret image with 41×41, t2 with 256×256, and t3 with 512×512.Based on the previous introduction, the secret image can be losslessly recovered by our scheme. And here, we calculate the value of PSNR and WPSNR of the secret image and the reconstructed image, and the result is shown in Table1. And here, we present 3 different sharing schemes. PSNR1 and WPSNR1 are the curve of (3, 4) MSIS-QR with the secret image of 41×41, PSNR2 and WPSNR2 are the (2, 3) MSIS-QR with the image size of 256×256, and PSNR3 and WPSNR3 are the (3, 4) MSIS-QR with image size of 512×512. And, because the sharing time is too long for the secret image of 512×512 with a big threshold, the experiment ends at the threshold of 80. As we can see in the result of PSNR and WPSNR, the secret image can be losslessly recovered when TH<80 with all sizes of the secret image.Table 1
The value of PSNR/WPSNR about the threshold.
Threshold1020305080100PSNR1∞∞∞∞∞∞WPSNR1∞∞∞∞∞∞PSNR2∞∞∞∞∞∞WPSNR2∞∞∞∞∞∞PSNR3∞∞∞∞∞-WPSNR3∞∞∞∞∞-


## 5.1.1. The Visual Quality of the Share Images and the Recovered Images

Figures3 and 4 show the result of the proposed (k, n) threshold meaningful secret share scheme based on QR codes. In these schemes, we set k=3, n=4, m1=251, m2=253, m3=254, m4=255, and p=131. In Figure 3, we set TH=40 and the size of the secret image with 512×512. In Figure 4, TH=6, and the size of the secret image with 41×41.Figure 3
The experimental result of proposed (3, 4) MSIS-QR. The parameter in this experiment is illustrated above. And, for a better view, the size of the images is scaled to a proper size. (a) Secret imageS, (b) cover QR code 1, (c) cover QR code 2, (d) cover QR code 3, (e) cover QR code 4, (f) SI1, (g) SI2, (h) SI3, (i) SI4, (j) SI12′, (k) SI13′, (l) SI14′, (m) SI23′, (n) SI24′, (o) SI34′, (p) SI123′PSNR=∞, (q) SI124′PSNR=∞, (r) SI134′PSNR=∞ (s) SI234′PSNR=∞, and (t) SI1234′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)(r)(s)(t)Figure 4
The result of (3, 4) MSIS-QR: the public parameter is the same as those in Figure3 while TH = 6. (b)–(e) are the shares, and (f) is the reconstruction result by (b) and (d) which is deservedly a random picture while (g) and (h) can reconstruct the secret image with no less than 3 shares. (a) secret image S, (b) SI1, (c) SI2, (d) SI3, (e) SI4, (f) SI13′, (g) SI234′PSNR=∞, and (h) SI1234′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)Figure3(a) shows the grayscale secret image, 3(b)–3(e) show the cover QR code images, 3(f)–3(i) show the output of Algorithm 1, which can be decoded by QR code scanner. 3(j)–3(o) are the recover results by Algorithm 2 for any two shares and the rest are the recover results with more than two shares.In Figure3, we can find that, (1) for those sets that the number of participants is less than k (i.e., 1 and 2 in this experiment), we cannot get any information about the secret image except the image size. And, for those sets that no less than k participants are involved, the secret image is losslessly reconstructed, the recovery image is the same as the secret image with PSNR=∞.Figure4 shows the secret image, QR code similarity shares generated by Algorithm 1 and some reconstruction results of Algorithm 2. And, we can find that the shares can be more random with low TH with the performance of secret sharing staying the same.Figure5 shows the result of (2, 3) MSIS-QR with m1=251, m2=253, m3=255, and p=128. (a) displays a Lena that samples to 256×256, and (b)–(d) are three cover QR codes that with the same size of (a). (e)–(g) are 3 share images generated by our scheme, and the information in these pictures are exactly the same with the corresponding codes in (b)–(d). The reconstructed results of different sets of (e)–(g) are displayed in (h)–(k), as we can see that the reconstructed image is lossless with any 2 or more shares.Figure 5
Experimental result of (2, 3) MSIS-QR scheme. The shares (e)–(g) reveal no information of the secret image (a), and they can be decoded by a QR code scanner. And, any two or more shares can losslessly reconstruct the secret image. (a) Secret image S, (b) cover QR code 1, (c) cover QR code 2, (d) cover QR code 3, (e)SI1, (f) SI2, (g) SI3, (h) SI12′PSNR=∞, (i) SI13′PSNR=∞, (j) SI23′PSNR=∞, and (k) SI123′PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)


## 5.1.2. Error Correction Ability of Shadows

As the shadow images of our scheme are restricted by the QR code, and we do not change the error correction capacity of the original QR code. For a shadow image, the damage and the noise in the appropriate range can be decoded correctly.Figure6 illustrates the ability of error correction if the shadows suffered from damage. (a), (d), and (g) are the generated shadows of our scheme, (a) and (g) are (2, 3) threshold results with TH=50.5 and TH=−26, respectively, while (d) is (3, 4) scheme with TH=25. And, (b) is image (a) with salt and pepper noise, (e) and (h) are (d) and (g) with Gaussian noise, respectively, and (c) and (f) are damaged shares of (a) and (d).Figure 6
Error correction ability. (a), (d), and (g) are the generated shadows. (b), (e), and (h) are corresponding shadows added with noise, and (c) and (f) are damaged shadows. (a) Shadow image, decodable, (b) shadow distorted with noise, decodable, (c) damaged shadow, decodable, (d) shadow image, decodable, (e) shadow distorted with noise, decodable, (f) damaged shadow, decodable, (g) shadow image, decodable, and (h) shadow distorted with noise, nondecodable.
(a)(b)(c)(d)(e)(f)(g)(h)As is shown in Figure6, the damaged shadows and noise added shadows of our paper can be correctly decoded when the value of threshold is well selected. For image (h), the shadow image can be decoded correctly, while (g) cannot be recognized with noise added. The bigger of threshold, the better performance of error correction ability will be kept.


## 5.1.3. Evaluation for Visual Quality

In our scheme, the tolerant threshold TH is the parameter that plays an important role in the sharing phase. As is illustrated in step 6 of Algorithm 1 (seen in (13)), we will evaluate that if the result of the pixel in a secret image shared by CRT is in tolerance. If the result satisfies the condition, we keep the shares’ value, else recalculate it.(13)COMPai,TH=1,ifai≥128+THandQRih,w=1,0,ifai≤128−THandQRih,w=0.From the threshold-sharing time curve in Figure7, we can find that the sharing time is almost increasing with the rise of the threshold, and the increasing speed is faster than the growth of the threshold. For practicality, if we want to get a share that is similar to QR code, we can apply a big threshold, otherwise, if we want to let the shares look more random, we can use a relatively small threshold to make the value of the pixel has a wider range.Figure 7
The sharing time influenced by different thresholds. In this figure,t1 is the curve of sharing time of the secret image with 41×41, t2 with 256×256, and t3 with 512×512.Based on the previous introduction, the secret image can be losslessly recovered by our scheme. And here, we calculate the value of PSNR and WPSNR of the secret image and the reconstructed image, and the result is shown in Table1. And here, we present 3 different sharing schemes. PSNR1 and WPSNR1 are the curve of (3, 4) MSIS-QR with the secret image of 41×41, PSNR2 and WPSNR2 are the (2, 3) MSIS-QR with the image size of 256×256, and PSNR3 and WPSNR3 are the (3, 4) MSIS-QR with image size of 512×512. And, because the sharing time is too long for the secret image of 512×512 with a big threshold, the experiment ends at the threshold of 80. As we can see in the result of PSNR and WPSNR, the secret image can be losslessly recovered when TH<80 with all sizes of the secret image.Table 1
The value of PSNR/WPSNR about the threshold.
Threshold1020305080100PSNR1∞∞∞∞∞∞WPSNR1∞∞∞∞∞∞PSNR2∞∞∞∞∞∞WPSNR2∞∞∞∞∞∞PSNR3∞∞∞∞∞-WPSNR3∞∞∞∞∞-


## 5.2. Comparison with Related Schemes

In this section, we will compare our work with Yan et al. [27], Fu et al. [25], and Jiang et al. [28]. And, the experimental results of these schemes are depicted in Figure 8 [27], Figure 9 [25], and Figure 10 [28] separately.Figure 8
Experimental results of the introduced (k, n) threshold SIS scheme of [27], where k = 2, n = 3, m1 = 251, m2 = 253, m3 = 255, p = 128, and TH = 24. (a) The grayscale secret image. (b–d) Three grayscale comprehensible shares SC1, SC2, and SC3. (e) Grayscale secret image reconstructed with the first two shares. (f–h) Recovered binary cover images. (a) Secret image S, (b) SC1, PSNR=12.3, (c) SC2, PSNR=13.2, (d) SC3, PSNR=12.7, (e) SI12′PSNR=∞, (f) C1′, PSNR=∞, (g) C2′, PSNR=∞, (h) C3′, and PSNR=∞.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Experimental results of [25]: (a)–(d) are shares; (e) is the decoding of (a); (f) is the secret image, and (g) is the XOR-ed result of (a) and (b), while (h) is the XOR-ed result of (a) and (d).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Results of (2, 3)-threshold CRTQR-SIS with meaningful share proposed by [28], where (a) is the input grayscale secret image Lena; (b–d) are three shares SC1, SC2, and SC3; (e–h) are retrieved results by different shares. (a) Lena, (b) SC1, (c) SC2, (d) SC3, (e) S1,2′, (f) S1,3′, (g) S2.3′, and (h) S1,2,3′.
(a)(b)(c)(d)(e)(f)(g)(h)Yan et al. [27] proposed a RISS algorithm with a (k, n)-threshold based on the CRT, and the secret image is losslessly recovered, and Figure 8 shows the result of (2, 3) threshold of [27]. The secret image can be reconstructed losslessly with 2 or more shares. But in Figures 8(b) and 8(d), we can easily find some extra outlines which do not belong to the Cameraman and the Elaine.Compared with [27], the cover image of QR code can conceal more information than a grayscale image in our scheme. Benefiting from QR code, we can write some parameters that should be released to all participants to it and omit the operation of publication. And, in (2, 3) threshold schemes, the shares of [27] have inconspicuous outline of the secret image while ours do not with a proper threshold.Fu et al. [25] proposed a (k, n)-VCS in which all shares are QR codes with specific meaning. Figure 9 shows the experimental result of (2, 4)-VCS. In their scheme, the generated shares can be decoded correctly, and they can reconstruct secret image with 2 shares by XOR operation.However, limited by the features of VCS, the secret image is a binary image and the recovered image may not be losslessly compared with the secret image, only a proper set of shares can recover secret image losslessly by XOR operation. And, their scheme embeds the shares into the cover image by using the encoding redundancy of QR codes, so the process of the encoding phase is more complicated than ours.Jiang et al. [28] proposed a Chinese Remainder theorem and QR code-based extended secret image sharing with lossless recovery for k,n threshold. Figure 10 gives the experimental results for QR code version 5 with error correction level H by (2, 3) threshold scheme of [28]. By using their algorithm, these shares can be decoded correctly, and the secret image can retrieve with efficient shares together.But as we can see in Figures10(b)–10(d), these shares have clearly outline of 10(a), which may leak information about secret image. This problem will be relieved with appropriate parameter setting (i.e., QR coded with version 26, error correction level H, and (3, 3)-SIS scheme), but still cannot be avoided, which is undesirable in secret image sharing scheme. Compared to their scheme, our shares do not have the problem of information leakage.The feature comparisons between our scheme with related schemes are shown in Table2.Table 2
The comparison with related schemes.
SchemeSecret imageLossless recoveryShare imagesShare authenticationYan et al. [27]GrayscaleYesGrayscale images but have secret image shadows in some (2, 3)-SISNoFu et al. [25]BinaryNoDecodable QR codesYesJiang et al. [28]GrayscaleYesDecodable QR codes but have secret image shadowsYesOur schemeGrayscaleYesDecodable QR codesYes


## 5.3. Discussion

There are two main methods of secret image sharing: the CRT-based SIS and polynomial-based SIS. The CRT-based SIS is modular operation in the reconstruction phase, while polynomial-based SIS requires Lagrange interpolation. So, the CRT-based SIS has lower computational cost. According to [5], the reconstruction complexity of our scheme is Ok, while the complexity of methods using the method of Lagrange interpolation is Oklogk.Restricted by Asmuth and Bloom’s algorithm, the number of participants is limited. With the growth ofn, the suitable number of mi decreases. According to [27], in CRT-based secret image sharing scheme, the number of participants is generally small, such as n≤6. To solve the problem, Ning et al. [28] proposed an ideal secret sharing scheme based on CRT for polynomial ring over finite field, and this will be our future work.


## 6. Conclusion

In this paper, a threshold meaningful secret image sharing scheme based on QR code is proposed by using the Chinese remainder theorem. In our scheme, then shares are QR codes with specific information embedded by users, while the recovery phase is with low computation and lossless. And, the QR code shares can reduce the risk of malicious attacking. Experimental results show that our scheme performs well in the above features and is superior to related schemes. However, how to break the limit of participants’ number and improve the hidden capacity is our future work that should be investigated.



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*Source: 7864235-2022-04-23.xml*