This paper investigates the problem of sharing the secret document containing a secret data between leader and participant(s) depending on specific conditions and rules. The participant(s) can retrieve part of the secret document but will not be able to retrieve any secret data without the leader. At the same time, the leader may have a little information about the secret document but cannot retrieve the secret data and the secret document without cooperating with participant(s). To evaluate the proposed model and the system efficiency, four tests are suggested, which are concatenation and sharing data test, leader visual test, information entropy analysis, and correlation analysis. Results show that the proposed model is efficient in sharing the data between the leader and participant(s) and the model can achieve our concept of the data sharing between leader and participant(s). However, by analyzing the proposed model using numerical tests and visual tests, the results show that the visual tests will not give attackers useful information about the original data, while the numerical tests show that the entropy attacks are not possible and the correlation between the adjacent pixels will not give useful information. Finally, the results show that the proposed model is strong against different types of attacks.
The secret sharing mechanism is a mechanism used in the large network to share the secret key between participants in the network, in which each participant has his own shadow. The purpose of secret sharing is to secure the key between different participants, to allow the authorized participants to retrieve the secret information, and to recover the secret key if some shadows are lost or distorted. Therefore, the key could be retrieved if and only if a specific number of participants collaborated together by using their shadows. In 1979, Shamir [
In this paper, we try to reformulate the problem to a new problem called leader and participant sharing puzzle as follows.
Eleven scientists are working on a secret project containing a secret data. One of them is a team leader and the rest are the team members. They wish to lockup the document in a cabinet so that the cabinet can be opened if and only if at least one leader and five participants are present. At the same time, the team leader or any five members can retrieve part of the secret project but will never retrieve the secret data.
The dealer of this system has two parts of information: the secret document and the secret data in the document. The solution could be described as twolevel cabinet, one for participants as a first level and another one for leader as a second level (one inside another one). The participants can access the first cabinet that contains partial document which describes part of the information not the whole information, where the leader cannot access either document or the secret data but it may have a little information about the document. In addition, neither leader nor participant(s) will ever retrieve the secret data in the document alone. The participants need to collaborate with leader to retrieve the whole secret document and the secret data in the document and vice versa.
In this scheme, a dealer can encode and divide secret document into two parts: the participant part and the leader part. The dealer then distributes the
The proposed system could be found in many of our social life. Assume that we have a store with two doors: iron door and glass door, one behind the other one. If we assume that the glass door is a leader and the iron door is a participant(s), then the leader and the participant(s) should be existing at the same time to open the two doors and to access the store resources. With only the participant(s) information, the participant(s) can see what is inside the store through the glass door and will not have a permission to access the store without the leader part, where the leader cannot access the whole information about the store and it may access a little information only.
The rest of this paper is organized as follows. Related work is discussed in Section
Many researchers in the data sharing field focused on sharing the secret images between the participants in the network [
Lin and Chan [
This paper aims to design a new concept in the data sharing by assuming two types of users: leader and participant. The dealer can share the secret document with the secret data between the participant(s) and the leader, in which the participant(s) can retrieve part of the secret document but will not be able to retrieve any secret data without the leader. At the same time, the leader may have a little information about the secret document but cannot retrieve the secret data and the secret document without cooperating with participant(s).
The following subsections are dedicated to explain Shamir’s (
To share a secret
The coefficients
The irreducible polynomial could be found for most degrees. For example, there are eight different irreducible polynomials of degree 8. In Table
The possible primitive polynomials for different degrees.

Primitive polynomial 

2  7 
3  11 
4  19 
5  37, 61, 55 
6  67, 103, 109 
7  137, 143, 157, 247, 191, 213, 131, 203, 229 
8  285, 361, 487, 299, 357, 351, 451, 355 
The secret data would be hidden within the sharing data by using the secret data as a coefficients of the (
In this section, the proposed sharing model and retrieving model will be discussed, in which the sharing model is divided into three core phases, which are startup phase; to create a leader and the participant, hiding phase to hide the secret data within the participant data, and sharing phase to share the participant data between the participants, where, to retrieve the original data and the secret information, the reverse calculations will be applied.
The proposed sharing model is divided into three phases which are startup, hiding, and sharing phases as shown in the detailed subsections.
The original data will be divided into two parts: leader part and participant part, in which the generated parts have smaller size than the original data and could describe part of the original data. The proposed model will divide the original data using the following:
The participant and the leader parts have the same size of the original image, where each block contains 4 bits only. Therefore, each adjacent block will be concatenated to create one block with 8 bits each as shown in (
Before sharing the secret data, a linear independence relationship between the leader and the secret data will be created. The simple method to create a linear independence relationship is to use XOR operation between two sides as shown in the following:
Creating the linear independence between the participant and leader and between secret data and leader will increase the randomness of the transmitted data, which will add a new protection level on the transmitted data. Moreover, the modified secret data (Sec_{ L}) will use a Shamir model to hide the data as shown in the next phase.
The dealer will share the two parts: leader and participant, to the corresponding users on the network. The leader data will be transmitted to the leader using different media: CD, video tape, USB flash, and so forth, where the participant data will be shared using the Shamir model in (
The dealer can share the original data between the users by using (
The dealer can divide the data into two parts using (
The original data will be retrieved by using the reverse order of the sharing process. So, the leader and the participant need to be collaborated with each other to retrieve the data and the secret information. First, the participants need to be collaborated with each other to retrieve the participant part by using the langrage interpolation in a finite field. Afterwards, the leader and the participant will be collaborated with each other to retrieve the secret data and the original data using (
All the calculations in (
The participants can retrieve part of the original data using the langrage interpolation in the finite field [
The data is divided into two parts: participant and leader; since the participant will use (
The participant can retrieve the original data and the secret data if and only if leader and participant are used.
The linear independence relationship between the leader and the participant and between the leader and the secret data can be solved only if the two parts are used. So, the participant will not be able to retrieve the original data and the secret data without other parts from another side.
The participant can retrieve the original data and the secret data using the langrage interpolation in the finite field if and only if leader data is used.
Proofed in the above discussion.
To evaluate the performance of the proposed model different types of images are used. Figure
The four
Due to the page limit, the peppers image is used only in this test. Using the proposed model, the data will be divided into two parts: leader and participant. The leader part will be sent to the desired person using any media such as CD, video tape, and internet, where the participant part will be shared between the participants on the network. Figures
Peppers image using the proposed model.
Two shares for the participant part using (
The visual test is used to validate if the leader part will give any useful information about the original image. In this test, four images are used: peppers, Lena, montage, and cameraman with row concatenation type. To increase a randomization in the image, we can use a random concatenation or a distance concatenation (using the faraway rows) instead of the three mentioned methods. The Figures (Figures
Leader visual test.
The entropy could be defined depending on the field of science. In the data transmission and information theory, the entropy is defined as a measure of the loss of information in a transmitted signal, whereas, in the statistical mechanics, it is defined as a measure of the randomness of the microscopic constituents of a thermodynamic system. In this part, we are interested in the randomness of leader and participant images, where the true random variable should generate 2^{8} symbols with equal probability and the entropy value equals 8. To check the randomness of the image the following is used:
The information entropy for the leader part.
Image  Leader_{ R}  Leader_{ C}  Leader_{ D}  Average 

Lena  7.9592  7.9202  7.9646 

Peppers  7.9349  7.9045  7.9483 

Montage  6.4283  6.5454  6.7207 

Cameraman  7.2078  7.2691  7.4107 

In Table
The information entropy for the participant part.
Image  Part_{ R}  Part_{ C}  Part_{ D}  Average 

Lena  5.3331  5.0588  5.4728 

Peppers  5.2263  5.1187  5.4652 

Montage  4.5669  4.4087  4.8000 

Cameraman  4.4983  4.4841  4.7128 

It is known that some algorithms were broken by using correlations analysis between the adjacent pixels, so the correlation coefficient will be calculated for all possible cases. To find a correlation between the adjacent pixels the correlation coefficient is calculated by using the following formula:
Table
Correlation coefficients of adjacent pixels.
Name  Leader  Vertical  Horizontal  Diagonal 

Lena 
Row  0.1855  0.1482  0.0986 
Column  0.1215  0.1183  0.0717  
Diagonal  0.0702  0.0908  0.0959  


Montage 
Row  0.4253  0.2808  0.2561 
Column  0.3141  0.4792  0.2476  
Diagonal  0.2443  0.4286  0.1823  


Cameraman 
Row  0.4947  0.4531  0.4145 
Column  0.4536  0.5314  0.4856  
Diagonal  0.4726  0.5007  0.4522  


Peppers 
Row  0.1612  0.1404  0.0793 
Column  0.1757  0.1702  0.0399  
Diagonal  0.0604  0.2041  0.1352 
This paper addressed the problem of leader and participant sharing puzzle. The puzzle defined how the leader and the participant(s) can share the secret information with the secret data on the network. The puzzle assumes the following conditions to share the data: the participant(s) can retrieve part of the secret document but will not be able to retrieve any secret data without the leader. At the same time, the leader may have a little information about the secret document but cannot retrieve the secret data and the secret document without cooperating with participant(s).
After evaluating the proposed system by using four tests which are concatenation and sharing data test, leader visual test, information entropy analysis, and correlation analysis, the results indicate that proposed model is efficient in sharing the data between the leader and the participant(s). However, after analyzing the proposed model using numerical tests and visual tests, the tests indicate that the proposed model is strong against different types of attacks and useful to be used on the internet. Finally, in our future work we will address the problem of multilevel leaders that applied in the huge networks.
The authors declare that there is no conflict of interests regarding the publication of this paper.