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The enhancement of three-party simultaneous quantum secure direct communication scheme with EPR pairs
Optics Communications 284 (2011) 515–518
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
The enhancement of three-party simultaneous quantum secure direct communication scheme with EPR pairs Song-Kong Chong, Tzonelih Hwang ⁎ National Cheng Kung University, Department of Computer Science and Information Engineering, No. 1, Ta-Hsueh Road, Tainan City 701, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 2 April 2010 Received in revised form 4 August 2010 Accepted 16 August 2010 Keywords: Quantum secure direct communication Dense coding Einstein–Podolsky–Rosen pairs
a b s t r a c t Recently, Wang et al. proposed a three-party simultaneous quantum secure direct communication (3P-SQSDC) scheme with EPR pairs, which enables three involved parties to exchange their secret messages simultaneously by using an EPR pair. This work proposed an enhancement on Wang et al.'s scheme. With the enhancement, the communications in the improved 3P-SQSDC can be paralleled and thus improves the protocol efficiency. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The quantum secure direct communication (QSDC) is one whereby a party, say Alice, can transmit her messages securely via photons to a receiver, say Bob, without first establishing secret keys. It is one of the most attractive research topics in the quantum cryptography. Many QSDC schemes have been proposed recently [1–8]. In 2002, Beige et al. [1] proposed a QSDC scheme based on single photon. Boströem and Felbinger [2] proposed a ping-pong QSDC protocol based on Einstein–Podolsky–Rosen (EPR) pairs. Later [9] indicated a security problem on [2] when in a noisy channel. In 2003, Deng et al. [3] proposed a two-step QSDC protocol by using a block of EPR pairs. Later, Deng and Long [7] used single-qubit states to propose a novel QSDC protocol. In 2005, Gao et al. [10] proposed a three-party QSDC scheme. In their scheme, a party can obtain the other two parties' messages simultaneously with Greenberger–Horne–Zeilinger (GHZ) states and entanglement swapping. Based on the idea of ping-pong protocol [2,9,11], Chamoli [5] also presented a three-party QSDC based on GHZ states. Later, Naseri [12] showed that any dishonest party in [5] can obtain the other one's secret message without being detected and then proposed a simple improvement to resist this attack. In order to enable three parties to exchange messages simultaneously, Jin et al. [13] proposed a three-party simultaneous QSDC (3P-SQSDC) based on GHZ states. However, their scheme has been proved insecure by Man et al. [14] in that 1 bit of a secret message of a party could be leaked without eavesdropping. Man et al. also presented a scheme to
⁎ Corresponding author. E-mail address: [email protected] (T. Hwang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.037
improve the security of Jin et al.'s. In 2007, Wang et al. [15] proposed a 3P-SQSDC. In their scheme, each party can obtain the other two users' messages simultaneously through EPR pairs. Compared with [13,14], Wang et al.'s 3P-SQSDC is very easy to carry out because three parties can exchange their respective secret messages by using only EPR pairs. The qubit transmissions in Wang et al.'s 3P-SQSDC [15] unfortunately are sequential because Bob and Charlie have to encode their messages respectively on the same qubit of an EPR pair. That is, Alice sends a qubit of an EPR pair to Bob, and then Bob will transmit that qubit to Charlie after he encodes his message on the qubit. After that, Charlie also encodes her message into the received qubit and then returns it to Alice (i.e., Alice → Bob → Charlie → Alice). Since every party in the scheme needs to wait for the other's response, the transmissions have to be in sequential. Fig. 1 shows the qubit transmissions of Wang et al.'s 3P-SQSDC, which is finished in seven processing steps, and Alice has to store the other particle of an EPR pair for the same amount of time duration (i.e., seven processing steps). This study shows that by encoding on distinct particles on an EPR pair by users, the qubit transmissions in Wang et al.'s 3P-SQSDC [15] can be paralleled. In the newly enhanced 3P-SQSDC scheme, Alice will send the first qubit of the EPR pair to Bob, and the second qubit to Charlie simultaneously. After encoding, both Bob and Charlie will return the encoded qubits to Alice simultaneously (i.e., Alice→ (Bob and Charlie), and (Bob and Charlie)→Alice). Since Bob and Charlie can encode their messages on distinct qubits of the EPR pair, the communication as well as the encoding for each user can be done in parallel. Fig. 2 shows the qubit transmissions of the newly proposed 3P-SQSDC, which can be finished in five processing steps and, as will be described later, Alice has to store the returned photons for the time duration of public discussion only.
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4. The quantum memory of Alice in Wang et al.'s scheme requires to store photons for much longer time duration than that of the proposed scheme.
Fig. 1. Wang et al.'s qubit transmissions.
Since every party has to wait for the responses of the others in Wang et al.'s scheme, their scheme needs 3T to perform the photons transmissions (see Fig. 1, Steps 2, 4, and 6), where T denotes the time duration required for one transmission. However, in the proposed 3P-SQSDC, it needs only 2T to do the job because the qubits are transmitted in parallel (see Fig. 2, Steps 2 and 4). Similarly, Wang et al.'s scheme needs 4W to perform the operations (see Fig. 1, Steps 1, 3, 5, and 7), where W denotes the time duration required for each user to carry out the coding or quantum generating operations. However, the proposed 3P-SQSDC needs only 3W to complete the work (see Fig. 2, Steps 1, 3, and 5). Therefore, the parallel transmissions are more efficient than the sequential transmissions in protocol execution. Compare with Wang et al.'s 3P-SQSDC, this newly proposed scheme has the following advantages: 1. In Wang et al.'s scheme, since Bob and Charlie have to encode messages on the same photon, the processes in their scheme can only be performed sequentially. However, in the proposed 3PSQSDC, because Bob and Charlie encode on distinct photons, the processes can be done in parallel. 2. The quantum channel between Bob and Charlie is required in Wang et al.'s 3P-SQSDC. However, the quantum channel between Bob and Charlie can be omitted in the enhanced protocol. This also makes the enhanced protocol rather practical (e.g., in quantum telephone) because arbitrary two users can execute the QSDC via a server (Alice) without establishing any communication channel between them at all. 3. In Wang et al.'s scheme, Alice and Charlie are required to equip with a quantum generator respectively. However, in the proposed 3P-SQSDC, Alice is the only one who is required to equip with a quantum generator. This advantage also makes the proposed scheme feasible for practical applications.
To show the novelty of the newly proposed protocol, it is compared with which is also a block-type 3P-SQSDC. The new protocol does provide better performance, e.g., the time in communications and photon processing is more efficient; the qubit efficiency and the quantum memory utilization are also better than their scheme. The rest of this paper is structured as follows. Section 2 reviews the encoding of Wang's 3P-SQSDC. Then, the encoding of this work is presented. Section 3 proposes the improved 3P-SQSDC scheme. Moreover, the security analysis and a comparison to another blocktype SQSDC (i.e., Man et al.'s improvement) are also presented in this section. Finally, Section 4 concludes the result. 2. The related works In Wang et al.'s 3P-SQSDC, each user has a secret message to exchange respectively (e.g., ma, mb, and mc). Alice will generate an EPR 1
pair in jΨþ 〉ht = pffiffiffið j01〉+ j10〉Þht , where h denotes the first qubit of 2
+
|Ψ 〉 and t denotes the second qubit. After that, Alice will transmit the qubit t to Bob. According to mb, Bob encodes his message into the received qubit t by performing one of the two unitary operations, Ui, on t as follows (e.g., Umb(t) = t ′). t ′ is then sent to Charlie. Umb =
I; if mb =0; σx ; if mb =1:
ð1Þ
On the other hand, Charlie can also operate one of the two unitary operations on the received qubit t ′ (e.g., Umc(t ′) = t ″). t ″ is then sent to Alice. Umc =
I; σz ;
if mc =0; if mc =1;
ð2Þ
where I = |0〉〈0|+ |1〉〈1|, σ z = |0〉〈0| − |1〉〈1|, and σ x = |0〉〈1| − |1〉〈0|. Table 1 summarizes the encoding of Wang et al.'s 3P-SQSDC. According to her measuring result jΨþ 〉; jΨ− 〉 = 1 pffiffiffi 2
ð j00〉 + j11〉Þ or jΦ− 〉 =
1 pffiffiffi 2
1 pffiffiffi 2
ð j01〉− j10〉Þ; jΦþ 〉 =
ð j00〉− j11〉Þ, Alice can decode Bob's
and Charlie's mb and mc respectively. For instance, if Alice's measuring result is |Φ+〉 on (h,t″), then according to Table 1, she knows U mb =σ x and U mc =I, and thus mb =1, mc =0 can be derived. In Wang et al.'s 3P-SQSDC, Bob and Charlie perform unitary operations on the second qubit t sequentially. Therefore one has to wait for the work of the other. And thus, the protocol can only be performed sequentially. In the following, a distinct encoding is devised to improve the transmission efficiency of Wang et al.'s 3P-SQSDC scheme. The detail is described as follows. Similar to the original scheme, Alice will generate an EPR pair in |Ψ+〉ht. Then she sends the qubit h to Bob, and qubit t to Charlie simultaneously. Bob (Charlie) will encode mb (mc) into h (t) according to Eq. (1) (Eq. (2)). After that, Bob and Charlie will return the encoded qubits h ′ and t ′ respectively to Alice simultaneously. By performing a Bell measurement on h ′ and t ′, Alice can decode Bob's mb and Charlie's mc. Table 2 summarizes the new encoding. It is obvious to Table 1 The encoding of Wang et al.'s 3P-SQSDC.
Fig. 2. The qubit transmissions of this study.
I•I I•σz σx•I σx•σz
|Ψ+〉 |Ψ+〉 |Ψ−〉 |Ψ+〉 |Φ−〉
S.-K. Chong, T. Hwang / Optics Communications 284 (2011) 515–518 Table 2 The new encoding. |Ψ+〉 |Ψ+〉 |Ψ−〉 |Ψ+〉 |Φ−〉
I⊗I I ⊗ σz σx ⊗ I σx ⊗ σz
note that although Bob and Charlie perform their unitary operations on h and t respectively, the Bell measurement results of Tables 1 and 2 are the same. Consequently, Alice can decode mb and mc through Table 2. After that, Alice sends both x = mb ⊕ ma and y = mc ⊕ ma to Bob and Charlie. According to x, y, Bob can first obtain the secret message ma of Alice from x (because he knows mb), and then use ma to obtain Charlie's mc from y. Similarly, Charlie can also obtain ma and mb from x, y respectively. 3. The proposed 3P-SQSDC scheme This section proposes an improvement of Wang et al.'s 3P-SQSDC scheme. While Wang et al.'s scheme is processed in stream type, the proposed 3P-SQSDC is performed in block by block way. Moreover, the related security analysis and comparisons are also given in this section. 3.1. The improved scheme Again let Alice, Bob, and Charlie be three parties in the 3P-SQSDC scheme. Alice is with a Bell state generator, a single-qubit generator, and a quantum memory. Bob is equipped with two local unitary operators, I and σx. Charlie is equipped with I and σz. They want to exchange their secret messages simultaneously based on Alice's EPR pairs. Without loss of generality, the length of their messages is assumed to be the same. Similar to Wang's 3P-SQSDC [15], the classical channels are assumed to be authenticated. The steps involved in the proposed 3P-SQSDC scheme are described as follows (see also Fig. 3): 1
Step 1. Alice generates n EPR pairs in jΨþ 〉hi ti = pffiffiffið j01〉+ j10〉Þhi ti . 2 She divides these EPR pairs into two sequences, Qh and Qt, where Qh = [h1, h2,..., hn] is the set of the first qubits of all EPR pairs, and Qt = [t1, t2,..., tn] is the set of the second qubits of all EPR pairs. Moreover, Alice prepares four sets of decoy photons, CB1, CB2, CC1 and CC2, randomly chosen from |0〉, |1〉, | + 〉, and | − 〉. Step 2. Alice randomly inserts decoy photons in CB1 and CB2 into Qh to form a new sequence Qh⁎. Similarly, she randomly inserts all
517
decoy photons in CC1 and CC2 into Qt to form a new sequence Q⁎t . Alice sends Qh⁎ to Bob, and Q⁎t to Charlie simultaneously through quantum channels. Step 3. Upon receiving Qh⁎ (Q⁎t ), Bob (Charlie) replies “OK” to Alice via authenticated classical channel. Step 4. Alice announces the positions of CB1 and CB2 and the states of CB1 to Bob; the positions of CC1 and CC2 and the states of CC1 to Charlie. This can be done in parallel via the authenticated classical channels. Step 5. Bob (Charlie) verifies CB1 (CC1) to check the existence of eavesdropping in the quantum channels. If there is no eavesdropper, Bob (Charlie) encodes his message MB (MC) into Qh (Qt), where MB, MC ∈ {0, 1}n, Qh = Qh⁎ − CB1 − CB2, which denotes the remained bits of Q h⁎ after removing the corresponding bits in CB1 and CB2; and Qt = Q⁎t − CC1 − CC2. Let Q′h, Q′t be the encoded qubits by Bob and Charlie respectively. Bob (Charlie) permutes the qubit sequence Q′h (Q′t) with CB2 (CC2) to form a new qubit sequence Q″h (Q″t). After that, Bob (Charlie) returns Q″h (Q″t) to Alice through the quantum channel. Again, these can be done in parallel by Bob and Charlie respectively. Step 6. Upon receiving Q″h and Q″t from Bob and Charlie respectively, Alice replies “OK” to them in parallel via the authenticated classical channels. Step 7. Bob (Charlie) announces the positions and permutation of CB2 (CC2) to Alice via the authenticated classical channel. Step 8. Alice checks the existence of eavesdropping by verifying CB2 and CC2. If there is no eavesdropper, Alice sends “OK” to Bob and Charlie in parallel via the authenticated classical channels. Step 9. Bob (Charlie) sends Alice the permutation of Q′h (Q′t) via the authenticated classical channel. Step 10. According to the received information, Alice can recover the original encoded qubit sequences Q′h and Q′t . By performing Bell measurements on Q′h and Q′t , Alice obtains MB and MC of Bob and Charlie simultaneously. Step 11. Alice encodes her message MA into MB and MC, e.g., X = MB ⊕ MA, Y = MC ⊕ MA, where MA ∈ {0, 1}n. Then, she sends both X, Y to Bob and Charlie in parallel via the authenticated classical channel. Step 12. According to X, Y, Bob and Charlie can obtain the other two users' messages simultaneously. In the proposed 3P-SQSDC scheme, Bob and Charlie communicate with Alice directly and there is no communication between them. Therefore the communication channels (e.g., both quantum channel and authenticated classical channel) are not required between them. This advantage makes the scheme suitable for the client–server environment because it allows any two users to communicate without pre-establishing any communication channel between them at all. Although Bob and Charlie require two quantum communications with Alice respectively (i.e., in total, four quantum communications are required), the parallel transmissions save time in photons communications as well as photon processing.
3.2. Security analysis
“ “
” denotes the quantum channel ” denotes the authenticated classical channel
Fig. 3. The proposed 3P-SQSDC scheme.
The security which the improved 3P-SQSDC scheme is based on is similar to that of Wang et al.'s 3P-SQSDC scheme, i.e., they are based on the security of the public discussion. In the proposed 3P-SQSDC, four decoy photons sets (i.e., CB1, CB2, CC1 and CC2) are used for detecting the presence of eavesdroppers. In Step 5, if the eavesdropping checks fail, Bob and Charlie will terminate the communications directly. Furthermore, if the eavesdropping checks in Step 8 fail, Bob
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Table 3 Comparison. Man et al.'s 1. Quantum resources 2. 3. 4. 5.
GHZ states Parallel transmission Partial Classical channel between Bob and Charlie Yes Qubit efficiency 25% Time duration for the usage of quantum Longer memory
The proposed scheme EPR pairs + single photons Yes No 37.5% Shorter
and Charlie will not announce their secret permutations on Q hV and Q tV to Alice. Therefore, the eavesdropper cannot obtain MB and MC without knowing the permutations of Q hV and Q tV. On the other hand, since there is a round trip quantum transmission (Step 2 and Step 5) in the proposed scheme, the eavesdropper could perform the invisible photon attack [6] or delay-photon Trojan horse attack [16] to obtain the messages of Bob and Charlie without being detected. However, by using a wavelength filter [17] and a photon number splitter [17], these attacks can be prevented. Accordingly, the improved 3P-SQDC scheme is secure against eavesdropping. 3.3. Comparisons Since the security weakness of Jin et al.'s 3P-SQSDC [13] has been improved by Man et al. [14], this section mainly compares the proposed 3P-SQSDC with the three-party case of Man et al.'s scheme, which is also performed in block by block way. Table 3 presents the result of comparison of these two schemes. In Man et al.'s scheme, the processes cannot be done fully in parallel, and the classical channel between Bob and Charlie is required. That is, in the eavesdropping check process, though Bob (or Charlie) can inform Charlie (or Bob) and the central party, Alice, the positions, the measuring bases and the measuring results of these checking qubits simultaneously, Bob has to wait for the eavesdropping check between Charlie and Alice, and this process cannot be done in parallel. However, in the proposed 3P-SQSDC, the decoy photons used for eavesdropping check are pre-determined by the central party, Alice. Both Bob and Charlie communicate with Alice directly. Therefore, the processes can be done fully in parallel, and the classical channel is not required between Bob and Charlie. In Man et al.'s 3P-SQSDC, two three-particle GHZ states are required to share three secret bits, i.e., one GHZ state is used for messages exchange, and the other GHZ state is used for checking the security of the first quantum transmissions (i.e., Alice→(Bob, and Charlie)). However, when Bob and Charlie return the encoded qubits to Alice via the quantum channels, as suggested by [13], Bob and Charlie can collaborate to announce publicly part of their secret messages (e.g., half of their secret messages) to Alice for eavesdropping check. Therefore, the qubit efficiency of Man et al.'s scheme is 25%. In the proposed 3P-SQSDC, one the other hand, eight photons are required to communicate three secret bits, i.e., four qubits (one qubit of an EPR pair plus three decoy qubits) for Bob, and another four qubits for Charlie. After that, Bob (Charlie) and Alice will use two decoy photons to check the security of the first quantum transmission, and then use the remained decoy photons to check the security of the second quantum transmission. Therefore, the qubit efficiency of the proposed scheme is 37.5%. Furthermore, in Man et al.'s 3P-SQSDC, Alice has to store the first qubits of the GHZ states in the memory from the very beginning, therefore the time duration for her to store the qubits is much longer
than that of the proposed scheme. Furthermore, since the EPR pairs and single photons are easily prepared in the experiment than GHZ states [15,18], the proposed 3P-SQSDC is very easier to carry out. 4. Discussions and conclusions A practical and efficient 3P-SQSDC is proposed based on Wang et al.'s 3P-SQSDC. The quantum channel as well as the classical channel between Bob and Charlie can be omitted here. The parallel transmission of the proposed scheme provides efficiency in communications and photons processing. Moreover, the qubit efficiency and the usage of quantum memory of the proposed scheme are also better than that of Man et al.'s scheme. The proposed 3P-SQSDC is very suitable for the client–server environment. It can then be considered as a new version of secure quantum telephone [19–21] in that Bob and Charlie can communicate with each other with the help of Alice who represents a trusted telephone company. Compare with [19–21], the proposed 3P-SQSDC is more flexible in practice because the clients (e.g., Bob and Charlie) only need to establish the quantum channels and the authenticated classical channels with the telephone company (e.g., Alice). It enables any two clients to communicate freely without pre-establishing any communication channels. On the other hand, in [19–21], the quantum channels and the authenticated classical channels are required between any two clients. Moreover, the clients can process their messages simultaneously in the proposed 3P-SQSDC. Thus it is more efficient in contrast to a sequential flow in [19–21]. Since the security of the proposed 3P-SQSDC is based on the security of the public discussion (see Section 3.2), the eavesdropping attack mentioned in [20] does not work is the proposed scheme. Acknowledgments The authors would like to thank the anonymous reviewers' valuable comments and suggestions to improve the clarity and readability of this article. The extension of the proposed scheme to quantum telephone is due to one of the anonymous reviewers. This article is financially supported by the National Science Council of the Republic of China, under the Contract No. NSC 98-2221-E-006-097-MY3. References [1] A. Beige, B.-G. Englert, C. Kurtsiefer, H. Weinfurter, Acta Physica Polonica A 101 (3) (2002) 357. [2] K. Boströem, T. Felbinger, Physical Review Letter 89 (2002) 187902. [3] F.G. Deng, G.L. Long, X.S. Liu, Physical Review A 68 (2003) 042317. [4] Z.X. Man, Z.J. Zhang, Y. Li, Chinese Physics Letters 22 (1) (2005) 22. [5] A. Chamoli, C.M. Bhandari, Quantum Information Processing 8 (4) (2009) 347. [6] Q.Y. Cai, Physics Letters A 351 (2006) 23. [7] F.G. Deng, G.L. Long, Physical Review A 69 (5) (2004) 052319. [8] Q.Y. Cai, Physical Review Letters 91 (10) (2003) 109801. [9] A. Wójcik, Physical Review Letters 90 (15) (2003) 157901. [10] T. Gao, F.L. Yan, Z.X. Wang, Journal of Physics A 38 (2005) 5761. [11] Z.J. Zhang, Z.X. Man, Y. Li, Physics Letters A 333 (2004) 46. [12] M. Naseri, Quantum Information Processing (2009), doi:10.1007/s11128-009-0157-2. [13] X.R. Jin, X. Ji, S. Zhang, S.K. Hong, K.H. Yeon, C.I. Um, Physics Letters A 354 (1–2) (2006) 67. [14] Z.X. Man, Y.J. Xia, Chinese Physics Letters 24 (1) (2007) 15. [15] M.Y. Wang, F.L. Yan, Chinese Physics Letters 24 (9) (2007) 2486. [16] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Reviews of Modern Physics 74 (2002) 145. [17] L. Dong, X.M. Xiu, Y.J. Gao, F. Chi, Optics Communications 281 (2008) 6135. [18] M. Lucamarini, S. Mancini, Physical Review Letter 94 (2005) 140501. [19] M. Naseri, Optics Communications 282 (16) (Aug. 2009) 3375. [20] Y. Sun, Q.Y. Wen, F. Gao, F.C. Zhu, Optics Communications 282 (11) (Jun. 2009) 2278. [21] X.J. Wen, Y. Liu, N.R. Zhou, Optics Communications 275 (1) (Jul. 2007) 278.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
The enhancement of three-party simultaneous quantum secure direct communication scheme with EPR pairs Song-Kong Chong, Tzonelih Hwang ⁎ National Cheng Kung University, Department of Computer Science and Information Engineering, No. 1, Ta-Hsueh Road, Tainan City 701, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 2 April 2010 Received in revised form 4 August 2010 Accepted 16 August 2010 Keywords: Quantum secure direct communication Dense coding Einstein–Podolsky–Rosen pairs
a b s t r a c t Recently, Wang et al. proposed a three-party simultaneous quantum secure direct communication (3P-SQSDC) scheme with EPR pairs, which enables three involved parties to exchange their secret messages simultaneously by using an EPR pair. This work proposed an enhancement on Wang et al.'s scheme. With the enhancement, the communications in the improved 3P-SQSDC can be paralleled and thus improves the protocol efficiency. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The quantum secure direct communication (QSDC) is one whereby a party, say Alice, can transmit her messages securely via photons to a receiver, say Bob, without first establishing secret keys. It is one of the most attractive research topics in the quantum cryptography. Many QSDC schemes have been proposed recently [1–8]. In 2002, Beige et al. [1] proposed a QSDC scheme based on single photon. Boströem and Felbinger [2] proposed a ping-pong QSDC protocol based on Einstein–Podolsky–Rosen (EPR) pairs. Later [9] indicated a security problem on [2] when in a noisy channel. In 2003, Deng et al. [3] proposed a two-step QSDC protocol by using a block of EPR pairs. Later, Deng and Long [7] used single-qubit states to propose a novel QSDC protocol. In 2005, Gao et al. [10] proposed a three-party QSDC scheme. In their scheme, a party can obtain the other two parties' messages simultaneously with Greenberger–Horne–Zeilinger (GHZ) states and entanglement swapping. Based on the idea of ping-pong protocol [2,9,11], Chamoli [5] also presented a three-party QSDC based on GHZ states. Later, Naseri [12] showed that any dishonest party in [5] can obtain the other one's secret message without being detected and then proposed a simple improvement to resist this attack. In order to enable three parties to exchange messages simultaneously, Jin et al. [13] proposed a three-party simultaneous QSDC (3P-SQSDC) based on GHZ states. However, their scheme has been proved insecure by Man et al. [14] in that 1 bit of a secret message of a party could be leaked without eavesdropping. Man et al. also presented a scheme to
⁎ Corresponding author. E-mail address: [email protected] (T. Hwang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.037
improve the security of Jin et al.'s. In 2007, Wang et al. [15] proposed a 3P-SQSDC. In their scheme, each party can obtain the other two users' messages simultaneously through EPR pairs. Compared with [13,14], Wang et al.'s 3P-SQSDC is very easy to carry out because three parties can exchange their respective secret messages by using only EPR pairs. The qubit transmissions in Wang et al.'s 3P-SQSDC [15] unfortunately are sequential because Bob and Charlie have to encode their messages respectively on the same qubit of an EPR pair. That is, Alice sends a qubit of an EPR pair to Bob, and then Bob will transmit that qubit to Charlie after he encodes his message on the qubit. After that, Charlie also encodes her message into the received qubit and then returns it to Alice (i.e., Alice → Bob → Charlie → Alice). Since every party in the scheme needs to wait for the other's response, the transmissions have to be in sequential. Fig. 1 shows the qubit transmissions of Wang et al.'s 3P-SQSDC, which is finished in seven processing steps, and Alice has to store the other particle of an EPR pair for the same amount of time duration (i.e., seven processing steps). This study shows that by encoding on distinct particles on an EPR pair by users, the qubit transmissions in Wang et al.'s 3P-SQSDC [15] can be paralleled. In the newly enhanced 3P-SQSDC scheme, Alice will send the first qubit of the EPR pair to Bob, and the second qubit to Charlie simultaneously. After encoding, both Bob and Charlie will return the encoded qubits to Alice simultaneously (i.e., Alice→ (Bob and Charlie), and (Bob and Charlie)→Alice). Since Bob and Charlie can encode their messages on distinct qubits of the EPR pair, the communication as well as the encoding for each user can be done in parallel. Fig. 2 shows the qubit transmissions of the newly proposed 3P-SQSDC, which can be finished in five processing steps and, as will be described later, Alice has to store the returned photons for the time duration of public discussion only.
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4. The quantum memory of Alice in Wang et al.'s scheme requires to store photons for much longer time duration than that of the proposed scheme.
Fig. 1. Wang et al.'s qubit transmissions.
Since every party has to wait for the responses of the others in Wang et al.'s scheme, their scheme needs 3T to perform the photons transmissions (see Fig. 1, Steps 2, 4, and 6), where T denotes the time duration required for one transmission. However, in the proposed 3P-SQSDC, it needs only 2T to do the job because the qubits are transmitted in parallel (see Fig. 2, Steps 2 and 4). Similarly, Wang et al.'s scheme needs 4W to perform the operations (see Fig. 1, Steps 1, 3, 5, and 7), where W denotes the time duration required for each user to carry out the coding or quantum generating operations. However, the proposed 3P-SQSDC needs only 3W to complete the work (see Fig. 2, Steps 1, 3, and 5). Therefore, the parallel transmissions are more efficient than the sequential transmissions in protocol execution. Compare with Wang et al.'s 3P-SQSDC, this newly proposed scheme has the following advantages: 1. In Wang et al.'s scheme, since Bob and Charlie have to encode messages on the same photon, the processes in their scheme can only be performed sequentially. However, in the proposed 3PSQSDC, because Bob and Charlie encode on distinct photons, the processes can be done in parallel. 2. The quantum channel between Bob and Charlie is required in Wang et al.'s 3P-SQSDC. However, the quantum channel between Bob and Charlie can be omitted in the enhanced protocol. This also makes the enhanced protocol rather practical (e.g., in quantum telephone) because arbitrary two users can execute the QSDC via a server (Alice) without establishing any communication channel between them at all. 3. In Wang et al.'s scheme, Alice and Charlie are required to equip with a quantum generator respectively. However, in the proposed 3P-SQSDC, Alice is the only one who is required to equip with a quantum generator. This advantage also makes the proposed scheme feasible for practical applications.
To show the novelty of the newly proposed protocol, it is compared with which is also a block-type 3P-SQSDC. The new protocol does provide better performance, e.g., the time in communications and photon processing is more efficient; the qubit efficiency and the quantum memory utilization are also better than their scheme. The rest of this paper is structured as follows. Section 2 reviews the encoding of Wang's 3P-SQSDC. Then, the encoding of this work is presented. Section 3 proposes the improved 3P-SQSDC scheme. Moreover, the security analysis and a comparison to another blocktype SQSDC (i.e., Man et al.'s improvement) are also presented in this section. Finally, Section 4 concludes the result. 2. The related works In Wang et al.'s 3P-SQSDC, each user has a secret message to exchange respectively (e.g., ma, mb, and mc). Alice will generate an EPR 1
pair in jΨþ 〉ht = pffiffiffið j01〉+ j10〉Þht , where h denotes the first qubit of 2
+
|Ψ 〉 and t denotes the second qubit. After that, Alice will transmit the qubit t to Bob. According to mb, Bob encodes his message into the received qubit t by performing one of the two unitary operations, Ui, on t as follows (e.g., Umb(t) = t ′). t ′ is then sent to Charlie. Umb =
I; if mb =0; σx ; if mb =1:
ð1Þ
On the other hand, Charlie can also operate one of the two unitary operations on the received qubit t ′ (e.g., Umc(t ′) = t ″). t ″ is then sent to Alice. Umc =
I; σz ;
if mc =0; if mc =1;
ð2Þ
where I = |0〉〈0|+ |1〉〈1|, σ z = |0〉〈0| − |1〉〈1|, and σ x = |0〉〈1| − |1〉〈0|. Table 1 summarizes the encoding of Wang et al.'s 3P-SQSDC. According to her measuring result jΨþ 〉; jΨ− 〉 = 1 pffiffiffi 2
ð j00〉 + j11〉Þ or jΦ− 〉 =
1 pffiffiffi 2
1 pffiffiffi 2
ð j01〉− j10〉Þ; jΦþ 〉 =
ð j00〉− j11〉Þ, Alice can decode Bob's
and Charlie's mb and mc respectively. For instance, if Alice's measuring result is |Φ+〉 on (h,t″), then according to Table 1, she knows U mb =σ x and U mc =I, and thus mb =1, mc =0 can be derived. In Wang et al.'s 3P-SQSDC, Bob and Charlie perform unitary operations on the second qubit t sequentially. Therefore one has to wait for the work of the other. And thus, the protocol can only be performed sequentially. In the following, a distinct encoding is devised to improve the transmission efficiency of Wang et al.'s 3P-SQSDC scheme. The detail is described as follows. Similar to the original scheme, Alice will generate an EPR pair in |Ψ+〉ht. Then she sends the qubit h to Bob, and qubit t to Charlie simultaneously. Bob (Charlie) will encode mb (mc) into h (t) according to Eq. (1) (Eq. (2)). After that, Bob and Charlie will return the encoded qubits h ′ and t ′ respectively to Alice simultaneously. By performing a Bell measurement on h ′ and t ′, Alice can decode Bob's mb and Charlie's mc. Table 2 summarizes the new encoding. It is obvious to Table 1 The encoding of Wang et al.'s 3P-SQSDC.
Fig. 2. The qubit transmissions of this study.
I•I I•σz σx•I σx•σz
|Ψ+〉 |Ψ+〉 |Ψ−〉 |Ψ+〉 |Φ−〉
S.-K. Chong, T. Hwang / Optics Communications 284 (2011) 515–518 Table 2 The new encoding. |Ψ+〉 |Ψ+〉 |Ψ−〉 |Ψ+〉 |Φ−〉
I⊗I I ⊗ σz σx ⊗ I σx ⊗ σz
note that although Bob and Charlie perform their unitary operations on h and t respectively, the Bell measurement results of Tables 1 and 2 are the same. Consequently, Alice can decode mb and mc through Table 2. After that, Alice sends both x = mb ⊕ ma and y = mc ⊕ ma to Bob and Charlie. According to x, y, Bob can first obtain the secret message ma of Alice from x (because he knows mb), and then use ma to obtain Charlie's mc from y. Similarly, Charlie can also obtain ma and mb from x, y respectively. 3. The proposed 3P-SQSDC scheme This section proposes an improvement of Wang et al.'s 3P-SQSDC scheme. While Wang et al.'s scheme is processed in stream type, the proposed 3P-SQSDC is performed in block by block way. Moreover, the related security analysis and comparisons are also given in this section. 3.1. The improved scheme Again let Alice, Bob, and Charlie be three parties in the 3P-SQSDC scheme. Alice is with a Bell state generator, a single-qubit generator, and a quantum memory. Bob is equipped with two local unitary operators, I and σx. Charlie is equipped with I and σz. They want to exchange their secret messages simultaneously based on Alice's EPR pairs. Without loss of generality, the length of their messages is assumed to be the same. Similar to Wang's 3P-SQSDC [15], the classical channels are assumed to be authenticated. The steps involved in the proposed 3P-SQSDC scheme are described as follows (see also Fig. 3): 1
Step 1. Alice generates n EPR pairs in jΨþ 〉hi ti = pffiffiffið j01〉+ j10〉Þhi ti . 2 She divides these EPR pairs into two sequences, Qh and Qt, where Qh = [h1, h2,..., hn] is the set of the first qubits of all EPR pairs, and Qt = [t1, t2,..., tn] is the set of the second qubits of all EPR pairs. Moreover, Alice prepares four sets of decoy photons, CB1, CB2, CC1 and CC2, randomly chosen from |0〉, |1〉, | + 〉, and | − 〉. Step 2. Alice randomly inserts decoy photons in CB1 and CB2 into Qh to form a new sequence Qh⁎. Similarly, she randomly inserts all
517
decoy photons in CC1 and CC2 into Qt to form a new sequence Q⁎t . Alice sends Qh⁎ to Bob, and Q⁎t to Charlie simultaneously through quantum channels. Step 3. Upon receiving Qh⁎ (Q⁎t ), Bob (Charlie) replies “OK” to Alice via authenticated classical channel. Step 4. Alice announces the positions of CB1 and CB2 and the states of CB1 to Bob; the positions of CC1 and CC2 and the states of CC1 to Charlie. This can be done in parallel via the authenticated classical channels. Step 5. Bob (Charlie) verifies CB1 (CC1) to check the existence of eavesdropping in the quantum channels. If there is no eavesdropper, Bob (Charlie) encodes his message MB (MC) into Qh (Qt), where MB, MC ∈ {0, 1}n, Qh = Qh⁎ − CB1 − CB2, which denotes the remained bits of Q h⁎ after removing the corresponding bits in CB1 and CB2; and Qt = Q⁎t − CC1 − CC2. Let Q′h, Q′t be the encoded qubits by Bob and Charlie respectively. Bob (Charlie) permutes the qubit sequence Q′h (Q′t) with CB2 (CC2) to form a new qubit sequence Q″h (Q″t). After that, Bob (Charlie) returns Q″h (Q″t) to Alice through the quantum channel. Again, these can be done in parallel by Bob and Charlie respectively. Step 6. Upon receiving Q″h and Q″t from Bob and Charlie respectively, Alice replies “OK” to them in parallel via the authenticated classical channels. Step 7. Bob (Charlie) announces the positions and permutation of CB2 (CC2) to Alice via the authenticated classical channel. Step 8. Alice checks the existence of eavesdropping by verifying CB2 and CC2. If there is no eavesdropper, Alice sends “OK” to Bob and Charlie in parallel via the authenticated classical channels. Step 9. Bob (Charlie) sends Alice the permutation of Q′h (Q′t) via the authenticated classical channel. Step 10. According to the received information, Alice can recover the original encoded qubit sequences Q′h and Q′t . By performing Bell measurements on Q′h and Q′t , Alice obtains MB and MC of Bob and Charlie simultaneously. Step 11. Alice encodes her message MA into MB and MC, e.g., X = MB ⊕ MA, Y = MC ⊕ MA, where MA ∈ {0, 1}n. Then, she sends both X, Y to Bob and Charlie in parallel via the authenticated classical channel. Step 12. According to X, Y, Bob and Charlie can obtain the other two users' messages simultaneously. In the proposed 3P-SQSDC scheme, Bob and Charlie communicate with Alice directly and there is no communication between them. Therefore the communication channels (e.g., both quantum channel and authenticated classical channel) are not required between them. This advantage makes the scheme suitable for the client–server environment because it allows any two users to communicate without pre-establishing any communication channel between them at all. Although Bob and Charlie require two quantum communications with Alice respectively (i.e., in total, four quantum communications are required), the parallel transmissions save time in photons communications as well as photon processing.
3.2. Security analysis
“ “
” denotes the quantum channel ” denotes the authenticated classical channel
Fig. 3. The proposed 3P-SQSDC scheme.
The security which the improved 3P-SQSDC scheme is based on is similar to that of Wang et al.'s 3P-SQSDC scheme, i.e., they are based on the security of the public discussion. In the proposed 3P-SQSDC, four decoy photons sets (i.e., CB1, CB2, CC1 and CC2) are used for detecting the presence of eavesdroppers. In Step 5, if the eavesdropping checks fail, Bob and Charlie will terminate the communications directly. Furthermore, if the eavesdropping checks in Step 8 fail, Bob
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S.-K. Chong, T. Hwang / Optics Communications 284 (2011) 515–518
Table 3 Comparison. Man et al.'s 1. Quantum resources 2. 3. 4. 5.
GHZ states Parallel transmission Partial Classical channel between Bob and Charlie Yes Qubit efficiency 25% Time duration for the usage of quantum Longer memory
The proposed scheme EPR pairs + single photons Yes No 37.5% Shorter
and Charlie will not announce their secret permutations on Q hV and Q tV to Alice. Therefore, the eavesdropper cannot obtain MB and MC without knowing the permutations of Q hV and Q tV. On the other hand, since there is a round trip quantum transmission (Step 2 and Step 5) in the proposed scheme, the eavesdropper could perform the invisible photon attack [6] or delay-photon Trojan horse attack [16] to obtain the messages of Bob and Charlie without being detected. However, by using a wavelength filter [17] and a photon number splitter [17], these attacks can be prevented. Accordingly, the improved 3P-SQDC scheme is secure against eavesdropping. 3.3. Comparisons Since the security weakness of Jin et al.'s 3P-SQSDC [13] has been improved by Man et al. [14], this section mainly compares the proposed 3P-SQSDC with the three-party case of Man et al.'s scheme, which is also performed in block by block way. Table 3 presents the result of comparison of these two schemes. In Man et al.'s scheme, the processes cannot be done fully in parallel, and the classical channel between Bob and Charlie is required. That is, in the eavesdropping check process, though Bob (or Charlie) can inform Charlie (or Bob) and the central party, Alice, the positions, the measuring bases and the measuring results of these checking qubits simultaneously, Bob has to wait for the eavesdropping check between Charlie and Alice, and this process cannot be done in parallel. However, in the proposed 3P-SQSDC, the decoy photons used for eavesdropping check are pre-determined by the central party, Alice. Both Bob and Charlie communicate with Alice directly. Therefore, the processes can be done fully in parallel, and the classical channel is not required between Bob and Charlie. In Man et al.'s 3P-SQSDC, two three-particle GHZ states are required to share three secret bits, i.e., one GHZ state is used for messages exchange, and the other GHZ state is used for checking the security of the first quantum transmissions (i.e., Alice→(Bob, and Charlie)). However, when Bob and Charlie return the encoded qubits to Alice via the quantum channels, as suggested by [13], Bob and Charlie can collaborate to announce publicly part of their secret messages (e.g., half of their secret messages) to Alice for eavesdropping check. Therefore, the qubit efficiency of Man et al.'s scheme is 25%. In the proposed 3P-SQSDC, one the other hand, eight photons are required to communicate three secret bits, i.e., four qubits (one qubit of an EPR pair plus three decoy qubits) for Bob, and another four qubits for Charlie. After that, Bob (Charlie) and Alice will use two decoy photons to check the security of the first quantum transmission, and then use the remained decoy photons to check the security of the second quantum transmission. Therefore, the qubit efficiency of the proposed scheme is 37.5%. Furthermore, in Man et al.'s 3P-SQSDC, Alice has to store the first qubits of the GHZ states in the memory from the very beginning, therefore the time duration for her to store the qubits is much longer
than that of the proposed scheme. Furthermore, since the EPR pairs and single photons are easily prepared in the experiment than GHZ states [15,18], the proposed 3P-SQSDC is very easier to carry out. 4. Discussions and conclusions A practical and efficient 3P-SQSDC is proposed based on Wang et al.'s 3P-SQSDC. The quantum channel as well as the classical channel between Bob and Charlie can be omitted here. The parallel transmission of the proposed scheme provides efficiency in communications and photons processing. Moreover, the qubit efficiency and the usage of quantum memory of the proposed scheme are also better than that of Man et al.'s scheme. The proposed 3P-SQSDC is very suitable for the client–server environment. It can then be considered as a new version of secure quantum telephone [19–21] in that Bob and Charlie can communicate with each other with the help of Alice who represents a trusted telephone company. Compare with [19–21], the proposed 3P-SQSDC is more flexible in practice because the clients (e.g., Bob and Charlie) only need to establish the quantum channels and the authenticated classical channels with the telephone company (e.g., Alice). It enables any two clients to communicate freely without pre-establishing any communication channels. On the other hand, in [19–21], the quantum channels and the authenticated classical channels are required between any two clients. Moreover, the clients can process their messages simultaneously in the proposed 3P-SQSDC. Thus it is more efficient in contrast to a sequential flow in [19–21]. Since the security of the proposed 3P-SQSDC is based on the security of the public discussion (see Section 3.2), the eavesdropping attack mentioned in [20] does not work is the proposed scheme. Acknowledgments The authors would like to thank the anonymous reviewers' valuable comments and suggestions to improve the clarity and readability of this article. The extension of the proposed scheme to quantum telephone is due to one of the anonymous reviewers. This article is financially supported by the National Science Council of the Republic of China, under the Contract No. NSC 98-2221-E-006-097-MY3. References [1] A. Beige, B.-G. Englert, C. Kurtsiefer, H. Weinfurter, Acta Physica Polonica A 101 (3) (2002) 357. [2] K. Boströem, T. Felbinger, Physical Review Letter 89 (2002) 187902. [3] F.G. Deng, G.L. Long, X.S. Liu, Physical Review A 68 (2003) 042317. [4] Z.X. Man, Z.J. Zhang, Y. Li, Chinese Physics Letters 22 (1) (2005) 22. [5] A. Chamoli, C.M. Bhandari, Quantum Information Processing 8 (4) (2009) 347. [6] Q.Y. Cai, Physics Letters A 351 (2006) 23. [7] F.G. Deng, G.L. Long, Physical Review A 69 (5) (2004) 052319. [8] Q.Y. Cai, Physical Review Letters 91 (10) (2003) 109801. [9] A. Wójcik, Physical Review Letters 90 (15) (2003) 157901. [10] T. Gao, F.L. Yan, Z.X. Wang, Journal of Physics A 38 (2005) 5761. [11] Z.J. Zhang, Z.X. Man, Y. Li, Physics Letters A 333 (2004) 46. [12] M. Naseri, Quantum Information Processing (2009), doi:10.1007/s11128-009-0157-2. [13] X.R. Jin, X. Ji, S. Zhang, S.K. Hong, K.H. Yeon, C.I. Um, Physics Letters A 354 (1–2) (2006) 67. [14] Z.X. Man, Y.J. Xia, Chinese Physics Letters 24 (1) (2007) 15. [15] M.Y. Wang, F.L. Yan, Chinese Physics Letters 24 (9) (2007) 2486. [16] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Reviews of Modern Physics 74 (2002) 145. [17] L. Dong, X.M. Xiu, Y.J. Gao, F. Chi, Optics Communications 281 (2008) 6135. [18] M. Lucamarini, S. Mancini, Physical Review Letter 94 (2005) 140501. [19] M. Naseri, Optics Communications 282 (16) (Aug. 2009) 3375. [20] Y. Sun, Q.Y. Wen, F. Gao, F.C. Zhu, Optics Communications 282 (11) (Jun. 2009) 2278. [21] X.J. Wen, Y. Liu, N.R. Zhou, Optics Communications 275 (1) (Jul. 2007) 278.