Electronic voting systems for defending free will and resisting bribery and coercion based on ring anonymous signcryption scheme

Abstract

Vote by ballot is the feature in a democratic society and the process of decision-making, tending to achieve the philosophy of democratic politics by having the public who are eligible to vote for competent candidates or leaders. With the rapid development of technologies and network applications, electronization has been actively promoted globally during the social transformation period that the concept of electronic voting is further derived. The major advantages of electronic voting, comparing with traditional voting, lie in the mobility strength of electronic voting, reducing a large amount of election costs and enhancing the convenience for the public. Electronic voting allows voters completing voting on the Internet that not only are climate and location restrictions overcome, but the voter turnout is also increased and the voting time is reduced for the public. With the development in the past three decades, electronic voting presents outstanding performance theoretically and practically. Nevertheless, it is regrettable that electronic voting schemes still cannot be completely open because of lures by money and threats. People to lure by money and threats would confirm the voters following their instructions through various methods that more factors would appear on election results, affecting the quality and fairness of the election. In this study, this project aims to design an electronic voting scheme which could actually defend voters’ free will so that lure of money and threats would fail. Furthermore, an electronic voting system based on Elliptic Curve Cryptography is proposed to ensure the efficiency and security, and Ring Signature and Signcryption are applied to reducing the computing costs. Moreover, this project also focuses on applying voting system to mobile devices. As the system efficiency and security are emphasized, voters do not need to participate in the election, but simply complete voting with smart phones, iPads, and computers. The votes would be automatically calculated and verified the results that the ballots are not necessarily printed, the printing of election mails is reduced, and manual handling is canceled. Such a method would effectively reduce voting costs and enhance the economic efficiency.

Keywords

Electronic voting free will elliptic curve cryptography ring signature signcryption

Introduction

Vote by ballot is a feature in democratic societies. Traditional voting has taken place in public places such as schools, temples, or activity centers and the activities are the decision-making process, in which citizens eligible to vote elect competent candidates or leaders to achieve the idea of democratic politics. Traditional election is completed through three steps, namely, authentication of identity, secret ballot, and ballot count, and presents three characteristics of security, including one vote per person, fair and just, and secret ballot. Nevertheless, the voting does not conform to real economic benefits, and the voting processes of identity authentication, ballot receipt, candidate ticking, casting ballot into the box, and leaving the voting booth consume a lot of human resources and time and result in the waste of paper. Besides, the ballot verification result requires manpower to call out the name selected in a ballot, which takes a lot of time and energy, and the personnel negligence could cause insufficient justice and accuracy.

The rapid development of technologies and the advance of various network application technologies have the Internet present the characteristics of high efficiency, low costs, and convenience. In the social transformation process, electronization has been actively promoted globally; traditional paper-based operation is gradually eliminated, and the application of computer-based operation becomes apparent, such as e-commerce, social networking, e-bank, e-medicine, and e-government. Vote by ballot is one of the development items that the idea of electronic voting, which was first proposed by Chaum,¹ is derived. Comparing it with traditional voting, the functions and contents do not appear large differences. However, the major difference lies in the mobility strength of electronic voting, which could largely reduce election costs and enhance convenience for the public. Electronic voting allows voters completing the votes through network. Such a method not only overcomes the restrictions to climate and location but also largely enhances the voter turnout and reduces the voting time and costs for the public.

The electronic voting scheme is derived from the three characteristics of traditional voting, and the security requirements for electronic voting contain the following seven items:

Anonymity refers to no one being able to recognize the correlations between ballots and voters.

Fairness indicates that no one could acquire or count the temporary votes of each candidate before the election results are announced.

Uniqueness allows each legitimate voter voting only once in each election.

Legitimacy indicates that merely the citizens eligible to vote could participate in the election.

Verifiability refers to a voter being able to check the ballot being correctly counted in the election results.

Mobility shows the advantage of ubiquity that voters could vote online anywhere.

Non-bribery and Coercion refer to voters with free will being able to elect the ideal candidate, without being coerced and bribed.

Observing the currently proposed electronic voting schemes, the former six security requirements have been conformed, while the requirement for Non-bribery and Coercion has not been completely solved. It enables bribers or coercers to check the voters’ ballot contents through various tactics or methods to result in voters not being able to freely elect the ideal candidates. For instance, they might confirm the e-voting receipt, which could also be the advantage and the blind spot of most electronic voting schemes; the election system would issue a receipt, which is the certificate of the ballot being counted in the results, after the voter completes voting. However, when a voter is bribed or coerced, the election receipt would become the most beneficial tool to confirm the voter following the will of bribers or coercers that it could hardly be guarded.

Furthermore, some schemes might apply other irrelevant election receipts to verifying the ballots so that coercers or bribers, in order to confirm the voters following their instructions, catch the transmitted parameter value of ballots by blocking the network after the voter transmits the ballot or acquire the election-related keys or transmission parameters from the voters in advance to compare and judge whether the voters do what they expect.

Finally, in order to destroy some electronic voting schemes allowing a voter transmitting the voting intention to the election center through anonymity or non-wiretap channel before the authentication of identity in the election or at the voting stage and to prevent from bribery and coercion, coercers or bribers might compulsively request to monitor the entire vote by ballot process and even acquire the vote-related passwords or keys with various tricks to directly substitute for participating in the election and completing all election procedures. Such two behaviors are hard to prevent and would significantly affect the election results.

The above tricks show that coercers or bribers try to compulsively request voters to follow their instructions. Either passive authentication or active interference would impact the expansion and healthy development of electronic election.

In this case, this project aims to design an electronic voting scheme which could surely guard a voter’s free will. Once the voter’s free will is guaranteed, bribery and coercion would not work. Moreover, to ensure the existence of both efficiency of the scheme and the seven security characteristics, the key length for the RSA public key encryption system is requested for being above 1024 bits. It therefore results in great computation burden for the voting system. An electronic voting system based on Elliptic Curve Cryptography and assisted with Ring Signature and Signcryption is therefore proposed to reduce the computation cost.

Chaum¹ proposed an anonymous digital mail delivery scheme in 1981 and applied digital pseudonym and anonymous e-mails to anonymously deliver mails. Furthermore, Chaum applied it to the electronic voting scheme, in which digital pseudonym could be regarded as the public key, and a voter could register the election body with the pseudonym through the anonymous mail scheme and further sign the legitimate ballot with personal private key. The registered digital pseudonym could be treated as the public identity that it presented public verifiability. Moreover, a ballot was encrypted in the transmission process, and the ballot content was protected that no one could know the voter’s choice, and the anonymity and the ballot privacy were guaranteed. The fundamental requirements for an election were therefore satisfied. It was considered as the beginning of electronic election.

In early 1990s, Fujioka et al.,² Wagner³ and Nurmi et al.⁴ separately proposed seven points of view for the standardized requirements for an electronic voting scheme. Nurmi et al. proposed the following items: (1) merely legitimate citizens could vote, (2) voters could vote through computers and the Internet, (3) merely voters knew the personal ballot contents, (4) a voter could merely vote once in an election, (5) a voter could change the ballot content in the election period, (6) a voter could verify the personal ballot being correctly counted by the election center into the election results, and (7) a voter could question the election center when the personal ballot was not correctly counted.

Nevertheless, the standardized items proposed by Nurmi et al.⁴ did not consider the security. For this reason, Fujioka proposed an improved electronic election scheme in the following year and set seven electronic election requirements for the security. (1) Vote stability, a voter violating the rules could not interfere in election; (2) ballot non-repeatability, each voter could merely cast one ballot; (3) voter legitimacy, merely legitimate voters could vote; (4) justice, nothing could affect the proceeding of election; (5) ballot privacy, all ballots were kept private, without divulgence; (6) count of vote integrity, all valid ballots were correctly counted; and (7) public verifiability, a voter could examine the ballot being correctly counted after the announcement of the election result.

Similar electronic voting schemes were proposed;^2,4–7 however, most of them were simple models because of the brief equipment that coercion and bribery could not be effectively defended. Benaloh and Tuinstra⁸ first proposed an electronic voting scheme without election receipt in 1994, in which no election receipt was used for the certificate of vote so as to block coercers and bribers from knowing whether the voters followed their instructions. Nonetheless, the scheme required the application to election booths that the specific mobility strength of electronic voting schemes was lost. Besides, it required a large amount of costs for establishing the scheme that the theory was not suitable for practical applications.

Cranor and Crtron⁹ indicated in 1997 that an electronic election scheme tending to resist lures and bribery would mutually conflict with the requirements for security verifiability of electronic voting schemes. For instance, the schemes proposed by Fujioka et al.,² Cramer et al.,¹⁰ and Cetinkaya and Doganaksoy¹¹ allowed bribers and coercers easily verifying the election receipt so that the requirement for resistance was not achieved. In this case, a favorable electronic election scheme required special methods to assist in the authentication so that coercers and bribers could not know the behaviors of voters. For example, blank doors or hidden communication channels aimed to prevent anonymity from being destroyed. Accordingly, researchers provided distinct election systems to resist bribery and coercion. Okamoto¹² proposed to have a voter precede the secret ballot through non-wiretap channel; however, the idea tended to the theory and was hard to be implemented. Fan and Sun¹³ used multiple election receipts; but, a ballot box (as used in traditional election) was utilized in the scheme that the cost of physical equipment was high, and the mobility strength was neglected.

Furthermore, Juels et al.¹⁴ proposed an electronic voting scheme being able to resist bribery and coercion and allowing a voter forging a fake voting certificate with which to confuse bribers and coercers with the voter following their instructions, while the voter indeed preceded the secret ballot with the real certificate through the anonymous communication channel. Although such a method looked perfect, coercers and bribers could constantly monitor the behavior of the voter during the voting to disable the secret ballot and result in the scheme not completely reaching the security requirements without bribery and coercion. Moreover, Chen et al.¹⁵ proposed an electronic voting scheme constructed with a secret sharing scheme; as the ballot content in the scheme could be computed in advance and compared with the voted ballot, coercers and bribers could easily verify whether the voter followed their instructions. In this case, the scheme could not completely meet the requirements without bribery and coercion.

In past years, Chung and Wu¹⁶ proposed an electronic voting system with subliminal channels, with which a voter could skillfully avoid the real-time verification of coercers and bribers and vote with personal free will; however, the substitutive coercive voting still revealed success probability.

Chung and Wu¹⁷ proposed an e-voting system with passwords in 2012, with which to distinguish whether a voter was bribed and coerced and to solve the substitutive coercive voting. Nevertheless, the scheme was based on the RSA public key algorithm, which required longer key to guarantee the security that the efficiency would be unfavorable in large-scale elections.

In sum, the electronic voting scheme is still worth doing to prevent coercion and bribery, with the theoretical research and the practical development, and the prospect is anticipated.^3,11,16–19 This project aims to design an electronic voting scheme with efficiency and being able to maintain a voter’s free will.

The rest of this article is arranged as follows. Section “Related technologies” introduces the techniques used in our proposal. Section “Proposed electronic voting scheme” illustrates the proposed electronic voting scheme that can defend free will and resist bribery and coercion. Security analysis is exhibited in section “Security analysis.” Finally, the conclusion is drawn in section “Conclusion.”

Related technologies

Elliptic curve cryptosystem

To ensure high efficiency and security, the proposed scheme is established based on elliptic curve cryptosystem (ECC), which can achieve the equal security to DSA²⁰ or RSA in the prime factorization problem²⁰ and presents lower computation overhead and smaller key size. The mathematic background of ECC^21–23 is defined as follows.

Ring signature scheme

The initial ring signature²⁴ allowed signers signing documents anonymously so as to protect the identity of signers. The concept of ring signature is similar to that of Fuzzy Theory. To generate a signature as ring for a particular message, a signer can dynamically choose members and the number of members according to the condition and then uses the public key of other members and the individual secret key. A manager is not necessary to handle affairs in such a system that a verifier can only determine the group in which the signer joins, but not the identity related to the signer.

Proposed electronic voting scheme

An electronic voting scheme which could resist bribery and coercion is proposed in this study. In the scheme, Signcryption and Ring Signature²⁵ are utilized for the characteristics of anonymity and fair vote, and passwords are applied to resisting coercion and bribery with blank doors. Moreover, ECC is used for enhancing the electronic voting efficiency and reducing the costs. With other applied technologies, such as blind signature, asymmetric encryption, and hash function, the scheme achieves the integrity.

For the system architecture, it is composed of two organizations which are authentication center (AC) and ballot tabulation center (BTC). An AC is a reliable certification authority, responsible for authenticating the legitimate identity of a voter and issuing legitimate ballots. A BTC, on the other hand, is responsible for collecting ballots, verifying and counting votes after the end of voting, and finally announcing the election result.

The election procedure consists of three phases of authentication phase, voting phase, and result announcing phase. In the authentication phase, the AC confirms the voter legitimacy through the voter’s private certificate and issues legitimate ballot to the voter after the confirmation for participating in the voting. In the voting phase, a voter votes the ballot to the considered ideal candidate and casts the ballot to the tabulation center. Finally, for the announcement of election results, the BTC would verify all ballots for counting and announce the correct number of ballots and the election results. The entire procedure is shown in Figure 1.

Figure 1.

Election procedure of the electronic voting scheme.

System construction and voter registration phase

In the beginning, registration, the total number of eligible voters in the election, and enabling the election of key rights to the voters must be confirmed before a formal election. AC will confirm all eligible voters in the database, including renewing the data of new migration and eligible citizens and removing the voters after death, and then all eligible voters will be informed to register, verify, and own the right for using the election key pairs to vote as follows:

Step 1. Let q denote a large prime number, E denote an elliptic curve, P denote a base point on the elliptic curve E with order q, and H denote a one-way hash function for resisting collision, where $q, E, P, and H$ are public parameters and $Z_{q}$ is a finite field with q elements.

Step 2. Let the eligible voters set be $V = {U_{1}, U_{2}, \dots, U_{n}}$ under the ECC, and the private keys of $U_{1}, U_{2}, \dots, U_{n}$ are $d_{1}, d_{2}, \dots, d_{n}$ , respectively. The corresponding public keys $Q_{1}, Q_{2}, \dots, Q_{n}$ satisfy $Q_{i} = d_{i} P$ where i = 1, 2, …, n. The private and public keys of AC $U_{AC}$ are $d_{AC}$ and $Q_{AC} = d_{AC} P$ , respectively.

Step 3. A temporary public key $Q_{BTCS} = d_{BTCS} P$ is issued for one round election. Note that the temporary private key $d_{BTCS}$ is obtained by the BTC only after the voting period ends.

Authentication phase

When the election begins, voters carry out the authentication and voting phases where they will have to interact with AC and BTC. During the authentication phase, AC authenticates the voter legitimacy by checking their certificates Cert_v and then blind signs [] the unique number of their votes, that is, signing validly without knowledge to the number, for voting property of uniqueness:

Step 1. The voter calculates a blind message Bs = Blind(number) for the voted unique number with AC’s public key Q_AC.

Step 2. The voter uses his/her private key d_v to sign the value of Bs and obtains the signature Sig_v(Bs) for the message.

Step 3. The voter sends the certificate Cert_v, Bs, and Sig_v(Bs) to AC.

Step 4. AC verifies the correctness of Cert_v. If it is true, AC blind signs Bs using the private key d_AC, and the blind signature Bs′ is returned to the voter.

Step 5. The voter eliminates the blind factor of Bs′ to obtain the signature of the voted number Ns which is signed by AC.

Voting phase

Each voter will generate his personal legitimate ballot Vs for voting. He or she is asked to send the Signcryption text of chosen candidate and signed the unique number Ns for BTC. The process of generating Signcryption text is as follows:

Step 1. AC publishes a list of the candidates’ information, M = {m₁, m₂, …, m_n}.

Step 2. The voter U_i randomly selects $k \in_{R} [1, q - 1] and r \in_{R} [1, q - 1]$ .

Step 3. The voter U_i calculates $(x_{i}, y_{i}) = T_{i} = kP, (x_{r}, y_{r}) = R = rP, and (x_{e}, y_{e}) = T_{e} = r Q_{BTCS}$ .

Step 4. When $t = 1, and then t - 1 = n .$ Let $t = i + 1, i + 2, \dots, n, 1, \dots, i - 1,$ the voter U_i selects $s_{t} \in_{R} [1, q - 1],$ and calculates $c_{t} = H (m_{i} | | x_{t - 1})$ and $(x_{t}, y_{t}) = T_{t} = s_{t} P + c_{t} O_{t}$ .

Step 5. The voter $U_{i}$ calculates $c_{i} = H (m_{i} | | x_{i - 1}) and s_{i} = k - d_{i} c_{i} (mod q)$ .

Step 6. The voter $U_{i}$ encrypts the candidate number m_i and generates the formal ballot $Vs = E_{x_{e}} (m_{i} | | announcing number)$ using the symmetric secret key $x_{e}$ .

Step 7. The voter $U_{i}$ sends all messages of vote $σ = (Ns, Vs, c_{1}, s_{1}, s_{2}, \dots, s_{n}, R)$ to the BTC.

Announcing phase

On receiving the messages $σ = (Ns, Vs, c_{1}, s_{1}, s_{2}, \dots, s_{n}, R)$ , BTC performs the following steps to verify the vote, count it, and announce the election result:

Step 1. To ensure that BTC will not know the immediate election result by using the decrypted key d_BTCS to decrypt Vs during the voting period, BTC gets d_BTCS from Election Certification authority only after the voting phase.

Step 2. Let $(x_{r}, y_{r}) = R,$ BTC calculates $(x_{d}, y_{d}) = d_{BTCS} R$ and $(m_{i} | | announcing number)$ is decrypted by $E_{x_{d}} (Vs)$ .

Step 3. Let $t = 1, 2, \dots, n - 1$ , BTC calculates $(x_{t}, y_{t}) = T_{t} = s_{t} P + c_{t} Q_{t}$ and $c_{t + 1} = H (m_{i} | | x_{t})$ .

Step 4. BTC calculates $(x_{n}, y_{n}) = T_{n} = s_{n} P + c_{n} Q_{n} and c'_{1} = H (m_{i} | | x_{n})$ .

Step 5. Once BTC confirms $c'_{1} = c_{1}$ , $(Vs, c_{1}, s_{1}, s_{2}, \dots, s_{n}, R)$ is a valid anonymous Signcryption ballot from the group $V = {U_{1}, U_{2}, \dots, U_{n}}$ ; otherwise, reject the vote. In addition, BTC checks for repeated Ns to avoid double voting.

Step 6. BTC counts the total amount of m_i and then announces the election result as given in Table 1, including the outcome and each candidate’s number of votes by announcing number, for example, N₃₈ and N₁₉.

Table 1.

Election result.

Candidates	m ₁	m ₂	m ₃	…	m_n
Gained votes	N ₁₉ N ₅₃₂ …	N ₈₄ N ₁₀₁ …	N ₆₅ …	…	N ₃₈ N ₆₆₇ …

Total votes: $x = (\sum_{1}^{n} votes of m_{i}), 1 \leq i \leq n$ .

Percentage vote: $x / all voters$ .

Off-line investigation phase

In the election, coercers and bribers confirm the electronic election receipts, acquire the vote-related passwords or keys, request for monitoring the entire voting process of a voter, and directly substitute for all election procedures with various tactic; either passive verification or active interference, such people aim to make voters follow their instructions that it would impact the expansion of electronic election and the healthy development.

To avoid the above compulsive bribery behaviors, this project tends to add the idea of Emergency Number in the e-voting system for the convenient operation and exposure of voters and effectively stopping such bad behaviors. First, the voter is requested to set the prefix passwords with different length so that the coercers and bribers cannot know the correct passwords. Second, two options are provided in the log-in frame. One is an ordinary situation, where the system would precede general voting after the voter keys in personal account and password. The other is a special situation, where an emergency number is prepared in the system for the voter keys in the number in addition to the password when being bribed or coerced. The system would identify the password as “password plus emergency number” and present the identical frame as the general voting frame, where the ballot would not be counted in the correct database but written to a specific ballot database for recording the bribery and coercion. Such data would not be included in the formal ballot box opening so that the actual count of vote and the results would not be affected.

After the election, investigators could calculate and trace the ballot database from bribery and coercion to realize the bribery situation in the election and tabletop exercise the possible backstage manipulators and suspects.

Moreover, in order to reach the voting verifiability and avoid coercers and bribers threatening the victims to inquire the votes being successful, the ballots, either the normally voted ones or the ones being bribed and threatened, could be searched with the random number acquired after the voting. Besides, to prevent coercers from finding out the number of successful votes being different from the total ballots opened and questioning the vote, the final enquiry system is modified to show the enquiry of single datum so that the ballot content could merely inquired by the voter keying in the personal random number (Figure 2).

Figure 2.

The e-voting procedure with resistance to coercion and bribery.

Security analysis

This electronic voting scheme can satisfy all needed security requirements, including eligibility, anonymity, uniqueness, fairness, mobility, verifiability, and uncoercibility. How each requirement is met is analyzed in detail as follows.

Eligibility

Eligibility means the prerequisite of a citizen to participate in an election. During the authentication phase in this scheme, each voter must transmit individual certificate Cert_v to AC for the verification. The information on Cert_v is used for checking the correctness of signature Sig_v(Bs), the validity of the voter’s public key, and other related messages such as personal data, identifier, and issuer of the certificate. Hence, any citizens passing this phase are considered as legitimate voters.

Anonymity

Anonymity denotes a status when the relation between a common voter and his ballot cannot be traced by anyone, including AC, BTC, and other voters. Considering the anonymity between a voter and BTC, due to the property of ring signature, on receiving the Signcryption messages during announcing phase, BTC enables to authenticate the validity of the messages, but disables to identify the sender (voter). As to the anonymity between a voter and the other party, a third party can only check which group the voter belongs to and whether the signature is issued by a particular member in the group after the election, but cannot determine the identity of the voter. In other words, neither BTC nor a third party can identify which voter uses the Signcryption messages.

Uniqueness

Uniqueness means that each eligible voter can vote only once. In this proposal, a legitimate voter only has one signature Ns of a unique number for voting, and the same Ns is sent again in order to cast the ballot twice (since this signed value is made by AC’s secret key that no voter can make a different one easily). Hence, BTC can detect if a voter has violated the “one vote per person” rule by checking for the repetitions of Ns during the verification of votes.

Fairness

Fairness means that no one can know the candidates’ vote count trends before voting is closed so that no voter’s choice is affected or made liable to upsetting the election results before the results are officially announced. Such a requirement is essential for a fair election as opportunists may try to influence the will of other voters through coercion or bribery turning the tide of the election to their favor and eventually winning it when such a situation is permitted before the end of voting. The absence of fairness will increase the probability of coercion or bribery behaviors and eventually render the election result to be unfair.

In this scheme, voted message m_i and announcing number is sent in ciphertext form Vs so that it can only be decrypted by those with a secret session key. Moreover, BTC cannot decrypt the Vs until the announcing phase when the temporary private key d_BTCS is obtained from Election Certification authority. Without the decryption key, it is very difficult to crack a ballot within a short period, so fairness is guaranteed.

Mobility

This scheme is designed to be run on the current Internet. Voters would require only the basic equipments such as a low-end personal computer, simple calculating machine, personal digital assistant (PDA), smart phone, Internet equipment such as Ethernet or Optical device, and legal certificates of election. With this setup, voters can easily cast their ballots irrespective of where they are. Hence, the proposal fulfills mobility.

Verifiability and uncoercibility

Verifiability means the ability of a voter to verify that the ballot has been counted. In the proposed scheme, BTC announces the election result with the voted announcing number, thus voters can easily verify their ballots.

Uncoercibility is the ability to prevent possible verification behaviors practiced by bribers and coercers. Examination of recent published studies on electronic election shows that only few studies bring up the investigation of these verification behaviors. However, this is important in designing and implementing a bribery and coercion preventive e-voting scheme.

In the scheme, the off-line investigation phase, as described in section “Off-line investigation,” utilizes the emergency number to inspect individual vote to reach the verifiability and avoid coercers and bribers threatening the voter to inquire the vote being successful that the uncoercibility requirements are met.

Comparison

Table 2 shows a comparison between this scheme and nine proposed schemes, including the Fujioka–Okamoto–Ohta scheme,² the Benaloh–Tuinstra scheme,⁸ the Sako–Kilian scheme,²⁶ the Cranor–Crtron scheme,⁹ the Chen–Jan–Chen scheme,¹⁵ the Fan–Sun scheme,¹³ the Mohanty–Majhi scheme,²⁷ and the Chung–Wu scheme.¹⁷

Table 2.

The comparison between seven schemes.

	1	2	3	4	5	6	7	8
FOO scheme (1992)	ˇ	ˇ	ˇ	ˇ	ˇ	ˇ	Partial prevention	RSA
BT scheme (1994)	ˇ	ˇ	ˇ		ˇ		Full prevention	RSA
SK scheme (1995)	ˇ	ˇ	ˇ		ˇ	ˇ	Partial prevention	DLP
CC scheme (1997)	ˇ	ˇ	ˇ	ˇ	ˇ	ˇ	Partial prevention	RSA
CJC scheme (2004)	ˇ	ˇ	ˇ	ˇ	ˇ		Partial prevention	RSA
FS scheme (2006)	ˇ	ˇ	ˇ		ˇ	ˇ	Full prevention	RSA
MM scheme (2010)	ˇ	ˇ	ˇ	ˇ	ˇ	ˇ	Partial prevention	RSA
CW scheme (2012)	ˇ	ˇ	ˇ	ˇ	ˇ	ˇ	Full prevention	RSA
Proposed scheme (2016)	ˇ	ˇ	ˇ	ˇ	ˇ	ˇ	Full prevention	ECDLP

DLP: discrete logarithm problem; ECDLP: elliptic curve discrete logarithm problem; FOO: Fujioka–Okamoto–Ohta; BT: Benaloh–Tuinstra; SK: Sako–Kilian; CC: Cranor–Crtron; CJC: Chen–Jan–Chen; FS: Fan–Sun; MM: Mohanty–Majhi; CW: Chung–Wu.

Anonymity; 2. Eligibility; 3. Fairness; 4. Mobility; 5. Uniqueness; 6. Verifiability; 7. Uncoercibility; 8. Based Security Hard Problem.

From the table, it can be observed that this scheme can satisfy the demands in all respects, especially to reach complete uncoercibility. The scheme protects against bribery and coercion behaviors, including getting all ballot information, checking the encrypted ballot, verifying the election receipt, and obtaining the authorized signature that allows the coercer/briber to vote in place of the voter. Although the BT-scheme and the FS-scheme also meet full uncoercibility requirements, both of them require a physical voting booth, and hence, they cannot conform to mobility, feasibility, or verifiability, and they are not easy to put into practice. Furthermore, this scheme is based on ECC, which, compared to RSA or DLP, merely requires fewer bits to achieve certain security. The complexity of time, the use of frequency width, and the space for memory could be largely reduced.

Conclusion

An electronic voting scheme which could stop bribery and coercion is designed in this study. In the scheme, Signcryption and Ring Signature are utilized for the characteristics of anonymity and fair vote, and passwords are applied to forming a blank door to resist coercion and bribery behaviors. Moreover, ECC is based and relevant applied technologies, such as blind signature, asymmetric encryption, and hash function, are used for reinforcing the voting efficiency and reducing the costs so that the scheme is worth practicing.

This voting system is extremely suitable for mobile devices or small-scale election, like the ones in research foundations, department associations and committees in schools, and securities’ portfolio management institutes. Since the system presents both efficiency and security, voters do not need to go to the voting booths when using the system, but merely complete the voting, count of vote, and vote verification through mobile phones, computers, or PDA. Electoral workers do not need to print ballots and election mails and manually move such printings that the voting costs could be effectively reduced to enhance the economic benefits.

Footnotes

Academic Editor: Stephen D Prior

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Education Department of Fujian Province (Project No. JA15031, Z1525120 and Z15P0066). The Science & Technology Planning Fund of Quanzhou (Project No. 2016T009 andZ1624063). Taiwan with grant no. MOST 105-2410-H-129-002.

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