E–Voting System based on Secret Sharing Scheme

- The electoral process is considered as one of the important and sensitive operations that take place from time to time in all countries of the world and need to be protected. The importance of the electronic voting came because, it provides a maintaining for the secrecy of the vote, as well as, the speed, accuracy and credibility of the vote counts. This is due to the growing of the technology that always needs for electronic process and for new approaches to achieve high security. In this paper, a new E-voting system is designed based on secret sharing scheme. The new protocol is implemented to show its efficiency in terms of computational time and cost.


Introduction
Protecting the confidential data is a necessary aspect emerges nowadays with the growing needs for electronic process. Electronic-voting is one of the important processes that need to be protected. The distribution of shares and the computation of votes for each candidate are sensitive issue that needs to be secure in such a way that each casted vote can be modified and tracked by the particular candidate only, where the vote representation in the E-voting process is as bitwise pattern. In this work, secret sharing scheme is used to design a secure E-voting system, such that, the voter's identity is protected, where each casted vote is divided into shares to be distributed to multiple parties. By using a secret sharing scheme that based on edge dominating set in a given graph, the random share given to each participant according to one of the elements in the minimum edge dominating set, and do not give any information about the voter, which resulted in a perfectly secure system. The proposed secret sharing scheme that is based on minimum edge dominating set is applied for the E-voting system, such that, this system is represented as a regular graph where each candidate represents an edge in the graph. To ensure secure data storage from eavesdropping and substituting, and to overcome the threats of destruction attack and troubles of storage devices is achieved by making as many copies of the secret as possible. Having many copies of the secret leads to leak out, which necessitate that, the number of the copies to be as small as possible. To achieve this goal, a new protocol is proposed independently in 1979 by Shamir [1] and Blakley [2] called secret sharing scheme. It is a method to split secret information into pieces called shares , each of them has no information about the secret , but collecting several numbers of them may reconstruct the secret . It allows the secret to be shared among a set of participants [3]. A secret sharing scheme is called threshold scheme [1], if all groups of participants of at least some fixed size are qualified to reconstruct the secret. A special participant is called the dealer is responsible of choosing the secret , he is also responsible for distributing the shares. The collection of subsets of participants that used to reconstruct the secret is called access structure . It is called monotone if for and then . A minimal qualified subset is a subset of participants such that for all . The basis of denoted by , is the family of all minimal qualified subset. Let denotes the collection of all subsets of . For any , the closure of is defined as ( ) * +. Therefore, an access structure is the same as the closure of its basis ( ( )). The important relation between an access structure and secret sharing scheme motivates many researchers to propose different approaches to use suitable access structure in order to obtain a perfect and an efficient secret sharing scheme (see, [4, 5, and 6]). The approach that investigates an access structure based on graph theory considered as an attractive; it is called graph access structure. In this approach, the vertices and the edges of the graph represent the participant, and the access structure (see, [5, 6, 7, and 8]), where the access structure of the graph G is decomposed into a number of subgraphs, in such a way that at least λ subgraphs of them should contain each edge of . This idea is called λ-decomposition, it was introduced by Blundo et al. [5] and Brickell and Stinson [9]. It also leads to an increase in the number of subgraphs that increased with the increment of the number of vertices and edges (participants), which resulted in an exponential increment of the construction time. In 2016, a new secret sharing scheme is proposed by Alsaidi et al. [10], it is based on dominating set of edges as an access structure. The classical approaches that based on graph theory used to consider the set of vertices in the graph as a set of participants. In [10], the set of participants * + corresponds to the set of edges of the graph , * +, and the minimum edge dominating set in represent the minimum access structure . The current work is organized as follows: In Section 2, related work of E -voting is abstracted. In Section 3, two types of secret sharing scheme based on graph access structure is presented. The proposed E -voting system is presented in section 4. System implementation is given in section 5. The paper finally concluded in section 6.

2-Related work
The first E-voting system based on a secret sharing scheme was introduced by Benaloh [11] in 1987. A voting system based on multiple key ciphers was proposed by Boyd et al., in 1990 [12]. In 1992, Iverson and Kenneth [13] proposed a secure election system based on secret sharing technique and zero knowledge protocol. In 1993, Fujioka et al. [14] presented a practical secret voting scheme for large scale elections. In 1999, Schoenmakers [15] presented a secure E-voting system based on a publicly verifiable secret sharing scheme. In 2007, Iftene et al. [16] proposed a general secret sharing based on Chinese remainder theorem with application to e-voting. In 2010, Damagaard et al. [17] proposed a generalization of the Pailliar's cryptosystem and its application to the voting schemes. In 2014, Chen et al. [18] suggested a scheme based on discrete logarithm problem and secret sharing. In 2014, Pan et al. [19] introduced an improved scheme with high privacy and confidentiality.

3-Secret Sharing Scheme based on graph access structure
In this section, two types of secret sharing scheme based on graph access structure is presented, they are: I.Sun et al. Scheme [20] A new construction of a perfect secret sharing scheme of rank m is proposed, such that P= {p 1 ,p 2 ,…,p n } is a set of participants and  is a uniform access structure of rank m on those participants, where  0 is the basis of . The decomposition of  0 is  i 's, for 1 i n where * +. Thus, ( ) ( ) ( ) then * * + + id defined, where each cl( i *) is a uniform access structure of rank m-1. The secret K=(k 1 ,k 2 ,…,k m }, where each k i , 1 i m is taken randomly over GF (q h(m-1) ), which is considered as the space of the secret. A polynomial f(x) of degree m.h(m-1)-1with coefficients K is selected by a dealer to compute y i =f(i-1)mod q, for i=1, …n.h(m-1). Thus, the secret can be recovered if one can get m.h(m-1) or more y i 's. If one has no knowledge of any y i , no information about the secret can be obtained. Random numbers r 1 ,r 2 ,…,r n are also selected by the dealer over GF(q h(m-1) ). They presumed that there exists a secret sharing scheme realizing cl( i * ), such that, the secret is r i +y i and the share of participant p j is S j ( i * ), which is given by: The reconstruction of the secret is done when the authorized participants collect their share together. [21] Another secret sharing scheme is proposed by represented the set of participants by the set of vertices in graph G, but the minimum access structure is represented by the minimum dominating set of vertices (MID). The scheme is summarized as follows: The set of vertices V={v 1 ,v 2 ,…,v n } corresponds to the set of participants P={p 1 ,p 2 ,…,p n }, and the set of all MID in G is corresponds to the minimum access structure  0 . They decomposed the graph G into nsubgraphs ( ), i=1,2,…,n, in such a way that * , -+. The set  0 is also decomposed into where * + They defined * * + +. The coefficients of the polynomial f(x) are chosen randomly over GF(q (m-1)! ) and used to represent the secret K=(k 1 ,k 2 ,…,k m }. Hence, ( ) ( ). The secret K can be reconstructed by getting m or more y i 's, where y i 's are computed using y i =f(i) mod q, i=1,2,…,n, and the share for each participant p i is calculated after selecting r random numbers r 1 ,r 2 ,…,r n by the dealer such that:

II.Alsaidi et al. Scheme
When the authorized participants pool their share together, the secret can be reconstructed.

4-The Proposed E -Voting system
In traditional elections, there are things that must be present, such as democracy, privacy and so on, which ensures free elections and a safe. For this, we need to use the electronic voting, because it ensures all of the above. The electoral process is considered as one of the important and sensitive operations that attract many researchers to work on. The importance of the electronic voting came because, it provides what we have mentioned earlier in terms of maintaining the secrecy of the vote, as well as, the speed, accuracy and simplicity of the vote counts. This is due to the growing needs for technology that need for new approaches to achieve high security. In the proposed algorithm for E-voting, a secret sharing scheme that based on minimum edge dominating set of a graph is used for this purpose. It is explained in the following algorithm: This algorithm can be illustrated through the following three phases:

I.Key generation phase
In this phase, we choose the number of candidate and the number of voters and then use the edge dominating algorithm given in [10] to find .

II.The decomposition phase (encoding)
To encoding the votes, we first choose a value of over ( ) where , is the number of edges, construct the polynomial ( ) , where is the value of the vote, such that; the value of depends on the number of voters and the number of candidates.

III.The reconstruction phase (decoding)
For each voter, the share for all candidates is computed by ( ), then the share is sent to the collection center ( ), after that the shares of each candidate are summed in the collection center.
In the construction part, we use the sum in the collection center and applied the Lagrange interpolation formula to obtain the polynomial f(x), and finally, from its constant term, the number of votes for each candidate is found.

System Implementation
In the following we explain how the process of voting is implemented and we compute the votes for each candidate. Let the number of voters equal 10 and the number of candidates 10, to find the votes of each candidate, we apply algorithm 1. Since we have 10 candidates, then a graph of regular two with 10 edges is constructed as shown in Figure 1, the rank of the graph is 4 and the minimum edge dominating set is:  , where is the value of vote for candidate The 40 bit vote pattern is initially set to 0. When a voter votes for candidate 1, bit is set to 1, similarly for candidate 2, bit is set to 1 and so on for candidate 10, bit is set to 1. ( ) . The first voter votes for candidate 1, then

6-Conclusion
The voting scheme in this paper is a new method that serves to compute the individual vote for each candidate. It provides good security and efficiency when the number of voters is small. The property of the graph theory called edge domination is used to compute the access structure which represents the minimum edge dominating set in the graph; it is used to compute the number of vote for each candidate.