Multi-authority E-voting System Based on Group Blind Signature

—Targeting at some problems such as the control of a single authority and guarantee of privacy of multi-authority, this paper proposes a double-blind multi-authority e-voting system based Ghadafi’s group blind signature and SP-signature. By introducing multi-authority, this paper solves the control problem of a single authority. By using the SP-signature, voters cannot vote for others. The new system voter can protect confidentiality of voting information and the privacy of authority. Finally, the security of the new e-voting system is analyzed which includes eligibility, privacy, universal verifiability, uncoercibility, unreusability and fairness.


INTRODUCTION
Election and voting are the reflection of democracy of a country. And the electronic voting system is an important tool for fair voting. It includes three voting modes: traditional voting, electronic voting and network voting. Traditional voting costs time and labor, and has high error probability. Network voting is seldom used because of the insecurity of network. Electronic voting is widely used because it is secure and saves time and energy.
Due to the importance of electronic voting, many scholars have conducted researches on it. Chaum [1]first introduced the concept of electronic voting. In 2011, Philip et al. [2]constructed an multi-authority electronic voting system in which the identity of authority was not confidential. In 2011, Okediran et al. [3]constructed a electronic voting system framework open to several authorities whose identities were public. The areas voters belonging to could be known, which weakens the privacy of voters. In 2012, Olusola et al. [4] summarized the basic concepts of electronic voting system, depict the development of electronic voting and concludes the characteristics ofelectronic voting system. In 2013, Shubhangi et al. [8] built a safe electronic voting system (simply called "e-voting system" in the following) by homomorphic technology. But there is only one authority in this system. Thus the voting result could be easily controlled by the authority.
From the above analysis, it can be known that e-voting system is either single-authority or multi-authority. The voting result is easily controlled in the former, but the identity of authority is public in the latter, which leaks part of voters' information and damages a certain privacy of voters. Based on Ghadafi's group blind signature agreement and Abe's structural signature, this paper constructs a multi-authority and double-private e-voting system to guarantee both the reliability of voting result and the identity of authority so as to protect voters. This paper proposes the basic definitions in chapter 2, constructs e-voting system in chapter 3, analyzes the security of new system in chapter 4 and gives the summary in chapter 5.  Definition 5 GS Certification [9] The certification of non-interactive zero-knowledge in relative bilinear group equations is given in standard model. It is suitable for various bilinear group element in group equation, including: bilinear product equation, scalar multiplication equation and quadratic equation. This paper just introduces the bilinear product equation under the SXDH hypothesis, as follows, 3 , , i j n G Z ! " are given, and the bilinear equation is listed in the following. 1 1 , , , , ,

II. BASIC DEFINITIONS
are variates. In order to verify that the variates satisfy the above bilinear equation, GS commitment has to be made first, that is, ( ) n y , simply as . Then bilinear product equation is used for certification. Finally, the certifier send variate commitment and the relative certification to verifier, and verifier can prove whether the given variates can satisfy bilinear equation.
Definition 6 Group Signature [11] Ghadafi has constructed a group signature in 2013 to protect not only the correctness of message but also the privacy of signers. The specific group signature process is as follows: Join the agreement: by using SP signature in Definition 4, signers can acquire signature certificate, the group manager saves the corresponding certificates and public key.
Group Signature: a user selects Here s is the secret key for the signer. If

III. ELECTRONIC VOTING SYSTEM
Electronic voting system consists of four participants: the trusted third party TTP, group manager GM, voter i U and authority AU . The trusted third party is responsible for extracting the voter who repeats voting; the group manager is in charge of the participation of authority and the counting of voting; voters are the one who conduct voting; the authority is in charge of collecting ballots from voters in one area. There is one authority in one area. He is in charge of collecting the ballots from voters in this area and then hand them over to the group manager. The group manager is responsible for the whole voting statistics. If there is voter who repeats voting, the group manager will send him to the trusted third party who will recover the identity of the repeat voter.

A. Basic Parameters
In the following, there are the basic parameters for multi-authority voting system. ! is the safety parameter,

B. Join the Agreement
The agreement allows the legal authority join in the group and he can acquire the certificate from the group manager and uses it to send voting qualification to voters, as follows: (1) Authority ! Group Manager: authority selects (2) Group Manager ! Authority: group manager verifies the public key of the authority to judge whether the authority is the legal regional manager and whether he will be granted the certificate. If one of the condition can not be satisfied, the agreement will terminate; or the group manager divide i pk into 1 2 ( , ) S S to judge whether SP blind signature is correct. If it is not, the agreement terminates; or he will use his own private key and use SP blind signature to generate the authority's public key i pk into signature ! i , and finally the group manager send certificate i ! to the authority.
(3) Authority: the authority judges whether the certificate is correct, and if it is not, the agreement will terminate; or saves his own certificate i ! .

C. Register the Agreement
This agreement allows voters to register to their regional authority to obtain the right to vote. Only the registered users have the right to vote. The agreement is as follows: (1) Voter ! Authority: a voter chooses a random number * p S Z ! , and uses the commitment key given by the trusted third party to generate a commitment of the random number S C and the corresponding certificate S ! , and then sends , , i S U S C pk ! to the authority.
(2) Authority ! Voter: the authority verifies the correctness of S ! and the qualification of the voter (such as age). If one of them is not correct, the agreement will terminate; or the authority generates the voter's commitment into a group blind signature ( ) S GGBS C .
(3) Voter: the voter verifies the correctness of the group blind signature. If it is not correct, the agreement will terminate; or the voter obtains the ( ) S GGBS C of the random number's commitment.

D. Voting Agreement
This agreement allows voters to vote for authority. In order to avoid other voters pretending this voter to vote, the voter generates SP signature for his ballot. It is not mandatory because SP signature cannot be forged. The randomization proved by Groth-Sahai commitment and zero-knowledge can be used to randomize to the authority.
(3) Authority: he can grant the right to verify the correctness of the group blind signature '( ) S GGBS C . If it is correct, it means the voter is a legal user who can be certificated by the electronic voting system. Then the authority verifies whether the random number R is fresh. If it is not fresh, it means that i U repeats the voting; or it means that U i 's vote is exclusive. Thus, it prevents i U from repeating the vote.

E. Vote Counting Agreement
None of voters can vote when the voting time is over. The agreement allows the regional authority to count the ballots voted by users of this region. If submitted the group blind signature, the certification of corresponding commitment and the SP signature are correct, the voter is legal and has voted right ballot, as follows: (1) Authority ! Group Manager: the authority sends the collected legal votes Piao to the group manager.
(2) Group Manager ! Trusted Third Party: the group manager verifies whether the votes submitted by the authority is legal. If it is legal, he will submit the votes to the trusted third party.

IV. ANALYSIS OF SECURITY
If the Ghadafi group blind signature cannot be forged, SP signature cannot be fake, discrete logarithm problem is difficult and Groth-Sahai certification is zero-knowledge, the new multi-authorized electronic voting system can meet the requirement of authentication, confidentiality, overall verifiability, non-mandatory, non-repeatablity, and fairness, as follows: PAPER MULTI-AUTHORITY E-VOTING SYSTEM BASED ON GROUP BLIND SIGNATURE (1) Authentication Authentication means that only the legal voters can vote.
In the registered agreement, the authority should authenticate the legality of voter's identity. If it is legal, the authority will distribute a group blind signature to the voter. The generated group blind signature can be turned into an anonymous code for the voter. Because Ghadafi group blind signature cannot be forged, no attacker can forge the legal identity of the user, which can protect the authentication of the voter.
(2) Confidentiality Confidentiality means no attacker can link the vote with its voter.
In the registered agreement, the voter submits his own public key and random number commitment and in the voting agreement, what the user submitted is the update random number commitment and the certification of corresponding correctness. Because Groth-Sahai certification is zero-knowledge, no attacker can link the update commitment and certification with the former commitment and certification. Thus, the new multiauthorized electronic voting system is confidential.
(3) Overall Verifiability The overall verifiability means that any voters can verify the legality of others' votes.
In the voting agreement, what the users submitted are Ghahafi group blind signature and SP blind signature. Because group blind signature is pubic for verification, everyone can verify the legality of users' votes.
(4) Non-mandatory Non-mandatory means any attacker cannot force users to generate unwilling votes.
In the voting agreement, the user will generate SP signature for his own vote. Because SP signature cannot be forged, no attacker can force the user to vote against his will.
(5) Non-repeatability Non-repeatability means any voter cannot vote twice for the same message.
In the voting agreement, the voter needs to submit a security tag which is implanted in user's public key and a random number of the authority. If the user votes twice for the same vote, there will be two different random numbers corresponding to the authority. Thus, the user's public key will be counted by the authority, which guarantee the nonrepeatability of the multi-authorized electronic voting system, as follows: If the user votes twice for the message S, the security tags contained in the votes are  In the voting system, the votes are commitment and the corresponding blind signature. The blind signatures are certificates for Groth-Sahai zero-knowledge and both Groth-Sahai commitment and zero-knowledge certificate are zero-knowledge, so no one can obtain any message from the commitment and corresponding blind signature before the end of voting. Thus, the new multi-authority electronic voting system is fair.

V. CONCLUSION
This paper first constructs a double-blind multiauthority electronic voting system by Ghadafi group blind signature and SP blind signature. In the new system, the confidentiality of the votes and the authority can be protected. Several authorities are introduced to solve the problem that one single authority can easily control the voting result. Meanwhile, the new system can meet the requirement of authentication, confidentiality, overall verifiability, non-mandatory, non-repeatablity, and fairness, which improves the security of the electronic voting system.