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Selected Papers: Research Activities in Laboratories of New NTT Fellows
Part II

Quantum Voting Scheme Based on Conjugate Coding

Tatsuaki Okamoto, Koutarou Suzuki, and Yuuki Tokunaga

Abstract

We describe a quantum voting scheme in which a ballot is an unknown quantum state for a voter. Such a ballot is hard to forge because it is difficult in principle to make a copy of an unknown quantum state. Moreover, our quantum voting scheme guarantees anonymity because the ballot can be randomized by voters. We also describe a distributed variant of our scheme that is robust against a dishonest administrator. This scheme should remain usable in the coming era of quantum computers, which are expected to break most existing cryptographic schemes.

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NTT Information Sharing Platform Laboratories
Musashino-shi, 180-8585 Japan

1. Introduction

There are several types of voting systems. Open ballot voting is a system in which voters write their own names on their ballots for self-identification and to prevent double voting. In such a voting system, the privacy of the voters is not guaranteed. If voter privacy is desired, an anonymous voting system is better than open ballot voting. To achieve fair anonymous voting, the voting scheme should satisfy the following requirements.

  1. Correctness: A valid vote by an authorized voter should be counted correctly and an invalid vote should not be counted. This is known as theone-voter one-vote principle.
  2. Anonymity: Voters should be able to vote anonymously.
  3. Receipt-freeness: Voters should not be able to prove how they voted to prevent a buyer or coercer from corrupting voters.

Unlike in open ballot voting, in anonymous voting we need to achieve correctness while keeping anonymity. Since this is a challenging issue for cryptography, there has been a lot of research on cryptographic anonymous voting schemes. However, almost all existing practical electronic voting schemes that satisfy these requirements are based on the computational complexity assumption of discrete logarithm or integer factoring, which will be broken by quantum algorithms. Therefore, when a quantum computer is made in the future, we will lose almost all our practical electronic voting schemes.

On the other hand, quantum information technology (QIT), which includes quantum communications and quantum computation, should be suitable for constructing a voting scheme because it is difficult to make a copy of an unknown quantum state in quantum physics. That is, if an unknown quantum state is used for a ballot, it is hard to forge, which is very suitable for a voting scheme.

In this paper, we describe the new concept ofquantum voting. The anonymity of the protocol is guaranteed unconditionally because the ballots can be randomized by voters. Correctness, i.e., the one-voter one-vote principle, is guaranteed by a quantum complexity assumption called one-more unforgeability, which means that it is impossible to forge an additional valid ballot.

We also present a cut-and-choose protocol and a distributed protocol for preventing the administrator from dishonestly executing procedures. In a distributed version of our scheme, no central entity (center) knows enough of the whole secret to issue blank ballots, which is better in terms of keeping secrecy.

1.1 Related work

Quantum voting: There have been a few proposals of quantum voting schemes [1], [2] for specialized situations. The key technique of these voting schemes is distributing quantum entanglement for obtaining anonymity. Unlike those schemes, ours does not use quantum entanglement, so it should be easier to achieve and be more flexible for various circumstances.

Electronic voting: There have been many proposals of classical electronic voting schemes. They can be classified into three approaches: (1) blind-signature-based schemes [3]–[5], (2) mix-net-based schemes [6]–[10], and (3) homomorphic-encryption-based schemes [11]–[14]. Some of these achieve receipt-freeness [12] under the assumption of a one-way untappable channel [5], [8], [15] or two-way untappable channel [12], [16], [17].

2. Our quantum voting scheme

LetH be a 2-dimensional Hilbert space, i.e., the space of a 1-qubit state and letHk be the k-times tensor product of H, i.e., the space of the k-qubit state. We define 1-qubit state Ψa, bH as

The value of b determines the basis: if b is 0, then a is encoded in basis Z; if b is 1, then a is encoded in basis X. This encoding is the same as BB84 quantum cryptography [18] or conjugate coding [19]. Similarly, the qubit measurement is performed in basis Z or X if the measurement basis b is specified to be 0 or 1, respectively.

For basis K = (b1, ..., bn+1) ∈{0, 1}n+1 and r = (a1, ..., an+1) ∈{0, 1}n+1, we denote

We call state φr, K a blank piece with respect to basis K if an+1 = a1 ... an. A blank piece is a fraction of a blank ballot, as described later.

For the measurement of an (n+1)-qubit state ρ ∈ H(n+1) in basis K = (b1, ..., bn+1) ∈{0, 1}n+1, we say that the measurement result is valid if the result ã1, ..., ãn+1 satisfies ãn+1 = ã1 ... ãn. Note that if we measure blank piece φr, K in basis K, the measurement result is always valid.

2.1 One-more unforgeability

Now, we define the one-more-unforgeability assumption, on which our quantum voting scheme is constructed. We consider the following game involving adversary F that models an adversarial voter who tries to forge blank pieces, i.e., tries to create a total of (w + 1) blank pieces when given w blank pieces.

Let adversary F be a polynomial-time quantum Turing machine and n be a security parameter. At the beginning of the game, basis K = (b1, ..., bn+1) ∈ {0, 1}n+1 is selected uniformly. Adversary F is given w (which is polynomial in n) blank pieces Φrj, K H(n+1) (j = 1, ..., w) with respect to basis K. Here, rj = (aj, 1, ..., aj, n+1)∈{0, 1}n+1, aj, 1, ..., aj, n are selected uniformly, and aj, n+1 = aj, 1 ... aj, n (j = 1, ..., w). Note that F is not given K and rj. Finally, adversary F outputs a (w+1)(n+1)-qubit state ρ ∈ H (w+1)(n+1).

We define the advantage Adv(F) of adversary F as

where (ãj, 1, ..., ãj, n+1) are the measurement results of the j-th (n+1) qubits of ρ in basis K and the probability is taken over the choice of K, the choice of rj (j = 1, ..., w), and the postulate of quantum measurement.

Assumption. (One-more unforgeability) We say that the one-more unforgeability assumption holds if, for every polynomial-time quantum adversary F, the advantage Adv(F) is negligible with respect to security parameter n.

2.2 Quantum voting protocol

In this subsection, we describe our quantum voting protocol. The anonymity of this protocol is guaranteed unconditionally, and correctness is guaranteed if the one-more-unforgeability assumption holds.

Let n be a security parameter and m = O(n) be the bit length of the voting message. There are an administrator A, counter C, and υ voters Vi (i = 1, ...,υ). We assume (1) an authenticated channel, where sender and receiver are authenticated, from administrator A to voter Vi and (2) a sender-anonymous channel, where the sender is anonymous and the receiver is authenticated, from voter Vi to counter C. For simplicity, we assume that these channels are secure, i.e., error-free. Counter C is assumed to carry out protocols correctly; otherwise, the result of voting could be manipulated as C desired. C just acts as a fair procedure and other aspects of security do not especially depend on his existence, i.e., C does not have any secrets of his own and all results of measurement are published.

Issuing. Administrator A uniformly selects a secret K = (b1, ..., bn+1) ∈{0, 1}n+1. Then, for i = 1, ..., υ, administrator A selects ri = (ri1, ..., rim) = (ai1, 1, ..., ai1, n+1, ..., aim, 1, ..., aim, n+1) ∈{0, 1}m(n+1), where aij, 1 ... aij, n (j = 1, ..., m) are selected uniformly and aij, n+1 = aij, 1 ... aij, n (j = 1, ..., m), and constructs a blank piece Φrij, K and a blank ballot

A blank ballot is depicted in Fig. 1.

Administrator A sends the blank ballot ηi to voter Vi through the authenticated channel.


Fig. 1. Quantum blank ballot ηi.

Randomization. Voter Vi receives the blank ballot ηi from administrator A through the authenticated channel. He selects ti = (ti1, ..., tim) = (di1, 1, ..., di1, n+1, ..., dim, 1, ..., dim, n+1) ∈{0, 1}m(n+1), where dij, 1 ... dij, n (j = 1, ..., m) are selected uniformly and dij, n+1 = dij, 1 ... dij, n (j = 1, ..., m), and obtains a randomized blank ballot

where U(ti) = U1(ti1) ... Um(tim) and Uj(tij) = Ydij, 1 ...

Ydij, n+1, where Y = XZ = and Y0 = I = .

A randomized blank ballot is depicted in Fig. 2.


Fig. 2. Randomized quantum blank ballot ηi.

Remark: The unitary transformation Y flips a for both bases Z and X. Sometimes, it also changes the global phase, but no one can distinguish the difference in global phase. Consequently, the global phase change caused by Y does not affect the user's anonymity.

Remark: Since each blank ballot has a randomly chosen pattern of information ri, if a voter utilizes a blank ballot without any modification, the administrator may be able to trace the ballot by recording ri, so the voter's privacy cannot be guaranteed.

Voting. Let M ⊂ {0, 1}m be the set of all the valid voting messages, where the number of valid voting messages |M| is constant for security parameter n and m = O(n), so the probability that a random bit string becomes a valid voting message is negligible for n. On the randomized blank ballot ηi, voter Vi writes his/her voting message ci = (ci1, ..., cim) ∈M ⊂ {0, 1}m, where ci is the name of the candidate for whom voter Vi wants to vote. Voter Vi makes his/her voting ballot

where U(ci) = U1(ci1) ... Um(cim), Uj(cij) = I ... I Ycij. A voting ballot is depicted in Fig. 3. Voter Vi sends the ballot η(ci) to counter C through the sender-anonymous channel.


Fig. 3. Quantum voting ballot η(ci).

Counting. Counter C receives secret K from administrator A through a classical secure channel after all voters have sent their votes. He receives all ballots η(ci) (i = 1, ..., υ) from all voters Vi (i = 1, ...,дн) through the sender-anonymous channel. He measures all ballots η(ci) (i = 1, ...,дн) and obtains the measurement results (˜aij, 1, ..., ˜aij,n+1) (i = 1, ...,дн, j = 1, ..., m), where (˜aij,1, ..., ˜aij,n+1) are the measurement results of j-th (n+1) qubits of η(ci) in basis K = (b1, ..., bn+1). He computes ˜cij = ˜aij,1 ⊕ ... ⊕ ˜aij, n+1 (i = 1, ...,дн, j = 1, ..., m), and checks whether or not ˜ci = (˜ci1, ..., ˜cim) ∈M ⊂ {0, 1}m. If ˜ciM, then ˜ci is counted as the result of the voting; otherwise, ˜ci is discarded.

2.3 Cut-and-choose protocol for issuing

If a malicious administrator issues invalid blank pieces, correctness and anonymity can be violated because the administrator can nullify and/or trace a ballot by mixing invalid pieces into it. To avoid the risk of this, we can use a cut-and-choose technique to verify the validity of blank pieces issued by the administrator. Our cut-and-choose protocol is explained below.

In the issuing stage, administrator A creates 2mυ blank pieces φr, K and puts them in register R. Each voter Vi randomly picks m blank pieces as his/her blank ballot ηi.

Before the counting stage, after all voters have cast their votes, counter C receives secret K from administrator A through a classical secure channel and performs the following check. Counter C measures mv blank pieces left in register R using secret basis K and checks whether or not all the results of these measurements are valid. If some are found to be invalid, we judge that administrator A issued invalid blank pieces and abort the voting process.

Note that a malicious administrator can mingle a small number of invalid blank pieces into a voter's blank ballot without detection even if we use the cut-and-choose protocol.

2.4 The t-out-of-l threshold protocol for issuing

Here, we present a threshold distributed protocol based on threshold quantum cryptography. If an administrator who knows secret K illegally casts a valid voting ballot in the voting stage, or dishonestly gives secret K to adversaries, then the adversaries can freely forge ballots as they like. To avoid the risk of this, a distributed scheme is very effective. In such a scheme, several centers each hold a share of secret K, and these centers collaborate to create blank pieces.

In this t-out-of-l threshold distributed protocol, each center chooses its secret by itself and distributes shares of its own secret. Once the secrets have been shared among l centers, an arbitrary t-out-of-l centers can issue valid blank pieces. After the preliminary secret distribution phase, even if lt centers happen to be down or out of service, at least t centers will still work properly, so the whole scheme works correctly. Moreover, if the number of dishonest centers is smaller than t, they cannot make valid quantum ballots. We apply a threshold technique such as Shamir's secret sharing scheme [21] (i.e., t-out-of-l scheme) for this threshold distributed protocol. We also apply Pederson's idea [22] that several centers share their own secrets.

Distribution. For l centers to distribute shares of their secrets, all centers Pj out of the l centers perform the following procedure.

Pj chooses its own secret σj = (bj,1, bj, 2, ..., bj,n+1), where bj, k are uniformly chosen from {0, 1}.

The center Pj then makes l shares, Sj,1, ..., Sj, l, of σj by using Shamir's secret sharing scheme over GF(2N), where N = n+1. Let fj (x) be a secret (t−1)-th-degree polynomial, Sj,i = fj (xj, i) for i = 1, ..., l, and σj = fj(0) over GF(2N), where xj, i for i = 1, ..., l are l distinct points in GF(2N) and are published. The center Pj sends Sj, i to a center Pi secretly for each i = 1, ..., l.

Precomputation. For simplicity, we assume that the t centers P1, ..., Pt collaborate to issue blank pieces.

For each i = 1, ..., t, Pi calculates and secretly stores the following value using the Lagrange interpolation formula

Let

be the binary representation of Ki in GF(2N), where bk[i] are in {0, 1}. The whole secret is

Note that even in the collaboration procedure, Ki and σj are kept secret in Pi, and K is not recovered.

Issuing. The t centers P1, ..., Pt collaborate to construct a blank piece φ. Below, we describe the sequential protocol from P1 to Pt, but the order is not essential: any order is possible.

P1 generates a quantum state

where r[1] = (a1[1], ..., an[1]), ∈{0, 1}n is a random string uniformly picked by P1 for each blank piece, and a[1]n+1 = a1[1] ⊕ ... ⊕ an[1]. Here, ψai[1]bi[1] is defined in the same manner as in Eq. (1).

Next, P1 sends φ[1] to P2. When Pi receives φ[i−1] from Pi−1, Pi follows the following procedure to generate φ[i] and sends it to Pi+1, where i = 2, ..., l and Pl+1 is a voter.

Then, Pi obtains φ[i] by applying the following unitary transformation W[i] to φ[i−1].


where

Here, a[i]n+1 = a1[i] ⊕ ... ⊕ an[i], r[i] = (a1[i], ..., an[i]), ∈{0, 1}n is a random string uniformly picked by Pi for each blank piece, unitary transformation Y follows the previous definition in Subsection. 2.2, and

After repeating this procedure, finally Pl sends φ[l] to a voter as a blank piece.

Correctness. φ[1] encodes string a1[1], ..., a[1]n+1 using secret bases K[1]. By unitary transformation W[2], φ[1] is transformed into φ[2] (here, we ignore the global phase), which encodes a1[1]a1[2] , ..., a[1]n+1a[2]n+1 using basis K[1]K[2]. Finally, φ[t] encodes a1[1] ⊕ ... ⊕ a1[l] , ..., a[1]n+1 ⊕ ... ⊕ a[l]n+1 using basis K = K[1]K[2] ⊕ ... ⊕ K[l]. Thus, the quantum ballot φ[l] is valid.

Security. If we assume that the communication among l parties during their collaboration is protected from adversaries and that the l parties are honest, then the security of the threshold scheme is at least on the same level as the original voting scheme.

3. Security of our quantum voting scheme

3.1 Security considerations for voting requirements

Correctness. First, we consider an active attack by a voter who tries to forge ballots. In this case, security will depend on the quantum computational complexity problem, i.e., one-more unforgeability (the Assumption in Subsection 2.1), whose hardness is not clearly understood yet, so conclusive security is beyond the scope of this paper. Here, we show that several naïve attacks cannot work well. It is impossible to simply make a copy of an unknown quantum state in principle according to the no-cloning theorem. If a forger (including a voter) tries to make a quantum ballot without knowing secret K, i.e., randomly makes a forged quantum ballot, the forged ballot ψ is accepted with probability |M|/2m. Since the number of candidates |M| is constant, the probability of successful forging is negligible. If K is derived from valid blank ballots, an adversary can forge valid ballots freely. However, deriving secret key K is intractable, as described in Subsection 3.2. So, it is difficult for a dishonest voter to forge quantum ballots in our scheme.

Next, we consider an active attack by the administrator. If the administrator mixes k invalid blank pieces into 2mv blank pieces in the issuing stage, we can detect invalid blank pieces with probability 1–1/2k using our cut-and choose protocol for issuing. If an invalid blank piece is detected, we abort the voting. The invalid blank pieces pass through the checking stage with probability 1/2k, so in this case, the administrator can nullify at most k unspecified votes.

Anonymity. A voter cannot break the privacy of another voter because he is isolated from the other voter's procedures and the communication channels are protected to make them secure. Thus, we consider only an active attack by the administrator.

Since all quantum voting ballots are sent through an anonymous channel, if there is no identifiable information in a ballot, then the privacy of voting is kept. The administrator can record randomness r encoded in blank ballot η in order to try and break the privacy of voters. To prevent this, the voter randomizes η by applying random unitary transformation U with randomness t and obtains randomized blank ballot η with randomness r′ = rt. Since t is uniformly selected by the voter, the original randomness r is unconditionally concealed and privacy is protected.

Although the administrator can mix k invalid blank pieces into 2mυ blank pieces in the issuing stage to trace votes and break privacy, we can detect the invalid blank pieces in the register with probability 1–1/2k using our cut-and choose protocol. If an invalid blank piece is detected, we abort the voting and discard the quantum voting ballots held by counter C. Therefore, in this case, votes are never revealed and privacy is protected. The invalid blank pieces pass through the checking stage with probability 1/2k. In this case, the administrator can invalidate at most k unspecified votes and break the privacy of at most k unspecified voters.

Receipt-freeness. A voter might try to embed some information into his/her voting ballot η(c) in order to prove his/her vote to a buyer/coercer. However, since the voter does not know randomness r encoded in blank ballot η, he/she cannot control randomness r′ encoded in blank ballot η′. If the voter mixes errors into the name of candidate c encoded in his/her voting ballot η(c), then voting ballot η(c) can be traced, but it will be discarded and not counted.

A voter might try to copy his/her voting ballot η(c) to make a receipt for the vote buyer/coercer. However, it is difficult to copy quantum voting ballot η(c), as already described in Correctness.

3.2 Intractability of finding secret key K

Let us consider a straightforward individual attack on our voting scheme. An adversary F is given a blank ballot φ = φr1, K ... φrm, K and examines the basis K by measuring quantum bits individually as follows. First, F guesses K from {0, 1}n+1 and measures φrj, K with the basis and gets the result (˜a,j, 1, ..., ˜aj, n+1). If the guess of K is correct, then all the results are valid i.e., ˜aj, n+1 = ˜aj, 1 ⊕ ... ⊕ ˜aj, n for all j. Then, F is assured that his guess is correct and he has obtained the correct K. The guess of bk of K = (b1, ..., bn+1) is correct with probability 1/2, so the probability that guesses of all bk for k = 1, ..., n+1 are correct is only 1/2n+1. Therefore, the secrecy of K against the simple individual attack is proven.

It has been pointed out that a Grover-type attack is applicable to this construction [23] if we allow F to use a general (coherent) attack. However, it still needs exponential time quantum computation. Grover's algorithm [24] has been proven to be optimal within the query-type algorithm to find a unique solution from a database [25]–[27]. Thus, F needs another type of algorithm to get secret bases efficiently, but no efficient attack has been found yet. In our scheme, the parity of random bits aj, 1 ⊕ ... ⊕ aj, n = aj, n+1 is encoded in ψaj, n+1, bn+1 of φrj, K and the rest of φrj, K is a totally random state for voters who do not know K. The construction of our scheme is very simple and it seems to be difficult to find another type of algorithm that is not a query-type algorithm. Although the difficulty has not been proven, it might be reduced to the difficulty of a well-known computational complexity problem such as [28], which is believed to be difficult even for a quantum computer. We leave this complexity as an open problem.

4. Concluding remarks

In this paper, we described a new cryptographic protocol concept, quantum voting. The anonymity of the protocol is unconditionally guaranteed, and correctness is guaranteed if the one-more-unforgeability assumption holds. We also presented a cut-and-choose protocol and a distributed protocol for preventing the administrator from dishonestly executing procedures.

In this paper, we just assumed that the quantum channel is secure (error-free) for simplicity. We could eliminate this assumption by constructing a secure quantum channel using a quantum one-time pad [29], [30] or using some other means.

We leave the analysis of quantum computational complexity as an open problem. Although the security of our protocol is based on quantum computational complexity and does not have unconditional security, it should have many applications in quantum information technology.

To implement a quantum voting scheme, we need a quantum register/memory. Since our scheme does not make use of quantum entanglement, a sufficient register for our scheme is one of the most basic elements of quantum computers, that is, a 1-qubit register that need not keep quantum correlation between two quantum bits. (The situation will be different when we implement quantum error correction.) Even such a quantum register is still beyond our current technologies. However, such technologies are being extensively studied and developed. In addition, the operation needed for our scheme is very simple, only 1-qubit unitary transformation, which is already feasible.

Finally, we emphasize that the use of a quantum state to achieve a voting scheme should be an attractive and fruitful research approach. The work described in this paper is the first step in this direction.

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Tatsuaki Okamoto
Research Fellow, Okamoto Research Laboratory, NTT Information Sharing Platform Laboratories.
He received the B.E., M.E., and Dr.Eng. degrees from the University of Tokyo, Tokyo, in 1976, 1978, and 1988, respectively. He is a member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan and the International Association for Cryptologic Research.
Koutarou Suzuki
Research Scientist, Information Security Project, NTT Information Sharing Platform Laboratories.
He received the B.S., M.S., and Ph.D. degrees from the University of Tokyo, Tokyo, in 1994, 1996, and 1999, respectively. He joined NTT Information Sharing Platform Laboratories in 1999. He has been engaged in research on public key cryptography, especially on cryptographic protocols and digital signatures. He is a member of IEICE and the Information Processing Society of Japan. He received the SCIS Paper Award from IEICE in 2002.
Yuuki Tokunaga
Researcher, Okamoto Research Laboratory, NTT Information Sharing Platform Laboratories.
He received the Bachelor's degree for integrated human studies in mathematical information science from Kyoto University in 1999, the M.S. degree in information science from the University of Tokyo in 2001, and the D.S. degree in material physics from Osaka University in 2007. He joined NTT Information Sharing Platform Laboratories in 2001. He has been engaged in research on quantum information science, especially on algorithms, cryptography, and optics. Recently, he has been engaged in research on optical quantum information processing. He is a member of the Physical Society of Japan.

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