Multiparty Quantum Secret Sharing via Introducing Auxiliary Particles Using a Pure Entangled State

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1 Commun. Theor. Phys. (Beijing, China) 49 (2008) pp c Chinese Physical Society Vol. 49, No. 6, June 15, 2008 Multiparty Quantum Secret Sharing via Introducing Auxiliary Particles Using a Pure Entangled State XIA Yan, SONG Jie, SONG He-Shan, and HUANG Xiao-Li School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian , China (Received June 8, 2007; Revised July 13, 2007) Abstract We propose a new multiparty quantum secret sharing protocol via introducing auxiliary particles using a non-maximally entangled (pure) two-particle state without a Bell measurement. The communication parties utilize decoy particles to check eavesdropping. After ensuring the security of the quantum channel, the sender encodes the secret message and transmits it to the receiver by using controlled-not operation and von Neumann measurement. If and only if all the agents agree to collaborate, they can read out the secret message. PACS numbers: Hk, Ud Key words: multiparty quantum secret sharing, auxiliary particle, controlled-not operation, non-maximally entangled two-particle state The principles of quantum mechanics supplied many interesting application in the field of information in the last decade. Quantum secret sharing (QSS), an important branch of quantum communication, is the generalization of classical secret sharing [1] into a quantum scenario and has attracted much attention. [2 10] The basic idea of secret sharing in a simple case is that Alice wants to send a secret message to two distant parties, Bob and Charlie. One of them, Bob or Charlie, is not entirely trusted by Alice, but she knows that if the two of them exist, the honest one can keep the dishonest one from doing any damage. Instead of giving the entire secret messages to either of them, Alice splits the secret messages into two encrypted parts and sends each one a part so that neither individual is able to obtain all of the original information unless they collaborate. So far, many QSS protocols [2 29] have been proposed in both theoretical and experimental aspects. Hillery et al. proposed a QSS protocol [2] using a three-particle or a four-particle Greenberger Horne Zeilinger (GHZ) state for distributing a private key among some agents and sharing classical information. Li et al. proposed a multiparty quantum secret sharing (MQSS) protocol [20] using a multi-particle GHZ-basis measurement, which is difficult to realize. Zhang et al. proposed a MQSS protocol [19] based on entanglement swapping by using three Einstein Podolsky Rosen (EPR) pairs and a Bell measurement. Zhang et al. proposed a multiparty secret sharing of quantum information protocol [19] via cavity QED. The secret messages are deterministically sent through the quantum channel in their protocol. In protocols of Refs. [2] [29] if they want to realize a faithful MQSS, the quantum channel must be a maximally entangled. But during the transmission, storage and processing, the entanglement of quantum state will unavoidably decrease because of noises. In practical case, entangled states distributed among distant locations are usually non-maximally entangled resulting from noises, so it is hard to realize a faithful MQSS. In this paper, we propose a faithful MQSS protocol via introducing auxiliary particles using a non-maximally entangled two-particle state without a Bell measurement. The secret messages are imposed on the auxiliary particles. The transmitted particles do not carry any useful secret messages, the sender encodes the secret message and transmits it to the receiver by using controlled-not (CNOT) operation and von Neumann measurement, which ensures the security of our protocol. Instead of performing unitary operations to recover Alice s secret message, the last agent measures the particles in a fixed measuring basis and recovers Alice s secret message according to the correlation between the measurement results. Only single-particle measurements need to be performed on the particles in our protocol. On the other hand, our protocol provides a higher-efficiency eavesdropping check. Now let us turn to depict our MQSS protocol. For convenience, let us first describe a three-party QSS protocol and then generalize it to the case of M agents. Suppose Alice wants to send a secret message to two distant parties Bob and Charlie. One of them, Bob or Charlie, is not entirely trusted by Alice, and she knows that if the two guys coexist, the honest one will keep the dishonest on from doing any damage. The two receivers, Bob and Charlie, can infer the secret message only by their mutual assistance. The three-party QSS protocol can be implemented in detail as follows: The project supported by National Natural Science Foundation of China under Grant No xia-208@163.com hssong@dlut.edu.cn

2 No. 6 Multiparty Quantum Secret Sharing via Introducing Auxiliary Particles Using a Pure Entangled State 1469 (S1) Alice prepares N ordered non-maximally entangled two-particle states in the state φ + HT = (a 00 + b 11 ) HT, (1) where a 2 + b 2 = 1. H stands for home, T stands for travel. We denote the ordered N EPR pairs with {[P 1 (H), P 1 (T)], [P 2 (H), P 2 (T)],..., [P N (H), P N (T)]}, where the subscript indicates the pair s order in the sequence, and H and T represent the two particles of each non-maximally entangled two-particle state. Alice takes one particle from each non-maximally entangled two-particle state to form an ordered non-maximally entangled two-particle state partner particle sequence [P 1 (H), P 2 (H),..., P N (H)], called the S H sequence. The remaining non-maximally entangled two-particle state partner particles compose the S T sequence, [P 1 (T), P 2 (T)],..., P N (T)]. (S2) Alice prepares a large number (x) of decoy photons (D x ) for eavesdropping check. Each of the decoy particles is randomly in one of the four states { 0, 1 } or { +, }, where { + = (1/ 2)( ), = (1/ 2)( 0 1 )}. Alice inserts the prepared decoy particles D x in S T sequence randomly and sends the new S T sequence to Bob and keeps the S H sequence for herself. (S3) Bob inserts a filter in front of their devices to filter out the photo signal with an illegitimate wavelength, and then chooses a sufficiently large subset of photons randomly. They split each sampling signal with a PNS and measure the two signals after the PNS with the two measure bases σ z and σ x chosen randomly. If the multiphoton rate is unreasonable high, Bob terminates the transmission and repeats the communication from the beginning. Otherwise, they continue to the next step. (S4) After confirming that Bob has received S T sequence via a classical communication, the eavesdropping check can be completed with the following procedures. (a1) Alice takes some decoy particles D y (y x) and publishes the position and basis of the decoy particles D y in the new S T sequence. Bob picks the correlated particles D y in the new S T sequence. (a2) Bob preforms von Neumann measurement on the decoy particles D y according to the corresponding measuring basis and announces publicly his measurement result. (a3) According to Bob s result, Alice can then analyze the error rate during the transmission of the new S T sequence. If the error rate is below the threshold they preset, Alice and Bob can conclude that there are no eavesdroppers on the line. Alice and Bob continue to next step. Otherwise, they have to discard their transmission and abort the communication. (S5) For the particles in the S T sequence, Bob randomly chooses one of the two local unitary operations U i (i = 0, 1) (U 0 I = , U 1 σ x = ) to encrypt each of the particles (say U B ), where I is the 2 2 identity operator and σ x is the Pauli operators. The two local unitary operations U i can transform the non-maximally entangled state φ + HT into another: U 0 φ + HT = (a 00 + b 11 ) HT = φ + HT, (2) U 1 φ + HT = (a 01 + b 10 ) HT = ψ + HT. (3) Then, Bob sends the new S T (P N (T) and D w, (w x, w y)) sequence to Charlie in order. (S6) Charlie confirms to Alice and Bob that he has received the new sequence S T, Alice announces the positions of D w. Alice lets Bob announce his operation on decoy particles D w first, and then she announces the basis of the decoy particles D w to Charlie. Charlie preforms von Neumann measurement on the decoy particles D w according to the corresponding measuring basis. According to his result, Charlie can then analyze the error rate during the transmission of the new S T sequence from Bob to him. If no eavesdropping exists, their results should be completely the same. After that, if the error rate is below the threshold they preset, Alice and Charlie can conclude that there are no eavesdroppers on the line. Alice and Charlie continue to next step. Otherwise, they have to discard their transmission and abort the communication. (S7) Alice introduces some auxiliary particles P N (a) for each particle of the S H sequence. If Alice wants to let Bob and Charlie know her secret bit 0 (1), she prepares the P N (a) particle in the state 0 ( 1 ). Suppose that Alice s secret bit is 0, she then prepares the particle P N (a) in the state 0. Since Bob has randomly performed one of the two unitary operations U i on each particle of the S T sequence, the initial state φ + HT may be transformed into φ + HT, φ HT, and the state of the particle P N (a), P N (H), and P N (T) may have two patterns: Φ 0 aht = 0 a (a 00 + b 11 ) HT, (4) Φ 1 aht = 0 a (a 01 + b 10 ) HT, (5) where the subscript a denotes the particle P N (a). If Alice s secret bit is 1, she prepares the particle P N (a) in the state 1 ; then the state of the particle P N (a), P N (H), and P N (T) may also have two patterns: Ψ 0 aht = 1 a (a 00 + b 11 ) HT, (6) Ψ 1 aht = 1 a (a 01 + b 10 ) HT. (7) (S8) Alice performs a CNOT gate operation on the particles P N (a) and P N (H) (P N (H) is the controller; P N (a) is the target). After the CNOT gate operation, equations (4) (7) become: and Φ 0 aht = (a 000 aht + b 111 aht, (8) Φ 1 aht = (a 011 aht + b 100 aht, (9) Ψ 0 aht = (a 001 aht + b 110 aht, (10) Ψ 1 aht = (a 010 aht + b 101 aht. (11) (S9) After having done CNOT operation, Alice and Charlie measure particles P N (a) and P N (T) in Z-basis,

3 1470 XIA Yan, SONG Jie, SONG He-Shan, and HUANG Xiao-Li Vol. 49 { 0, 1 }, respectively. At this time, although Charlie obtains the measurement results, he cannot recover Alice s secret message without Bob s cooperation (because Charlie does not know which operation Bob has performed on each particle of sequence S T in (S4) and Alice s measurement results). (S10) If Alice is sure that it is secure, she publicly announces the measurement results and the position of particle P N (a) to Bob and Charlie. (S11) If Bob and Charlie agree to collaborate, they can read out Alice s secret messages. For example, if Alice s measurement result is 0, Charlie s measurement result is 0 and Bob s unitary operation is U 0. Then Bob and Charlie can conclude that Alice s secret message is 0 (see Table 1 for the detail). On the other hand, if Bob and Charlie do not collaborate, neither of them can get access to Alice s secret message with 100% certainty (see Fig. 1). Fig. 1 Quantum circuit implementing MQSS. PNS is a 50/50 filter; U B {I, σ x }; PM { 0, 1 }; CM { 0, 1, +, }; QM {I, σ x }. The dash line denotes the case that Bob and Charlie collaborate with each other. Table 1 Recovery of Alice s secret message. Bob s operation (U) Alice s result [P N (a)] Charlie s result Alice s secret message U (I) U (σ x ) So far we have proposed the three-party QSS protocol via introducing auxiliary particles using a non-maximally entangled two-particle state. Now, let us generalize the three-party QSS protocol to the case of M agents. For simplicity, we let the boss be Alice and the M agents (Bob, Charlie, Dick,..., Zach). This MQSS protocol can be implemented in a simple way by modifying the process in the case of two agents. We describe it after (S5) discussed above. (S6*) Charlie inserts a filter in front of their devices to filter out the photo signal with an illegitimate wavelength, and then chooses a sufficiently large subset of photons randomly. They split each sampling signal with a PNS and measure g the two signals after the PNS with the two measuring bases σ z and σ x chosen randomly. If the multiphoton rate is unreasonable high, Charlie terminates the transmission and repeats the communication from the beginning. Otherwise, they continue to the next step. (S7*) Charile confirms to Alice and Bob that he has received the new sequence S T, Alice announces the positions of decoy particles D q (q w). Alice lets Bob announce his operations on the decoy particles D q first, and then Alice announces the basis of the decoy particles D q to Charlie. Charlie preforms von Neumann measurement on the decoy particles D q according to the corresponding measuring basis. According to his result, Charlie can then analyze the error rate during the transmission of the new S T sequence from Bob to him. If no eavesdropping exists, their results should be completely the same. After that, if the error rate is below the threshold they preset, Alice and Charlie can conclude that there are no eavesdroppers on the line. Charlie continues to the next step. Otherwise, they have to discard their transmission and abort the communication. (S8*) Charlie randomly performs one of the two unitary operations U i on each remaining particle of the S T (including D s (s w, s q y)) sequence. Then Charlie sends S T to next agent. (S9*) After Zach receives the S T sequence, (S6*) and (S8*) have been repeated M 1 times. (S10*) Alice introduces some auxiliary particles for each particle of the S H sequence according to the remaining particles of the sequence S T that were not measured

4 No. 6 Multiparty Quantum Secret Sharing via Introducing Auxiliary Particles Using a Pure Entangled State 1471 during the process of the eavesdropping check. Alice prepares these particles in the state 0 ( 1 ) according to her secret bit 0 (1). Then, Alice performs a CNOT gate operation and measures the particle P N (a) under the measuring basis { 0, 1 }. Zach measures his particle under the measuring basis { 0, 1 }. Alice publicly announces the measurement results and the position of P N (a) to all the Bobs. (S11*) If Zach and the other M 1 agents (Bob, Charlie, Dick,..., the (n 1)-th receiver) collaborate, they can obtain Alice s secret message by using correct measuring basis. On the other hand, if all the receivers do not collaborate, then none of them can get access to Alice s secret message with 100% certainty. So far we have established an M-party QSS protocol via introducing auxiliary particles using a non-maximally entangled two-particles state. The security of the present M-party QSS protocol is the same as the security of threeparty QSS protocol, which is also unconditionally secure. According to Stinespring dilation theorem, [30] Eve s attack can be realized by a unitary operation Ê on a large Hilbert space, H AB H E. The state of decoy particles and Eve s probe state is Ê 0, ε = α 0, ε 00 + β 1, ε 01, (12) Ê 1, ε = β 0, ε 10 + α 0, ε 11, (13) Ê +, ε = 1 2 [α 0, ε 00 + β 1, ε 01 + β 0, ε 10 + α 0ε 11 ] = 1 2 [ + (α ε 00 + β ε 01 + β ε 10 + α ε 11 ) + (α ε 00 β ε 01 + β ε 10 α ε 11 )], (14) Ê, ε = 1 2 [α 0, ε 00 + β 1, ε 01 β 0, ε 10 α 0, ε 11 ] = 1 2 [ + (α ε 00 + β ε 01 β ε 10 α ε 11 ) + (α ε 00 β ε 01 β ε 10 + α ε 11 )], (15) where ε 00, ε 01, ε 10, and ε 11 denote Eve s probe state. The probe operator can be written as ( ) α β Ê = β α. (16) As Ê is a unitary operation, the complex numbers α, β, α and β must satisfy ÊÊ = I and we can obtain the relations α 2 = α 2, β 2 = β 2. (17) The error rate introduced by Eve is e = β 2 = 1 α 2. Our protocol is based on non-maximally entangled two-particle state, the proof of the security of the eavesdropping check of the present three-party MQSS protocol is the same as the BBM92 QKD protocol, [31] whose security has been proven in both an ideal condition [32] and a practical condition, [33] so the process for the transmission of the sequence S T is secure for any eavesdropper, including the agents. The operation done by agents on each particle in the sequence S T is equivalent to encryption of the particle with a random key, which allows no one else to have the ability to read out the secret messages on the particles. Compared with protocol, [20] there are three advantages in our protocol. (i) In our protocol, during the whole process of transmission, the transmitted particles do not carry any useful secret messages; the secret messages are imposed on the auxiliary particles. Once it is secure after the eavesdropping check, there is no chance for Eves (eavesdroppers) to attack the secret messages. However, in protocol, [20] it is necessary to send the qubits carrying the secret messages in the public channel. Therefore, Eve can attack the qubits in transmission. Thus, our protocol is more secure than protocol. [20] (ii) In our protocol, the participants only need to perform a single-particle measurement under the measuring basis { 0, 1 }, which is very easy to realize. However, in protocol, [20] they must perform a multi-particle GHZ-basis measurement on the particles, which is difficult to realize. (iii) Our protocol is based on non-maximally entangled two-entangled state and single-particle state, but the protocol of Ref. [20] is based on a GHZ state. It is known that according to the present-day technologies a GHZ state might be synthesized from two Bell states, while the synthesization efficiency is lower (not greater that 50%). Hence, it is experimentally easier to prepare a non-maximally entangled two-entangled state and single-particle state in our protocol as compared to a GHZ state. (iv) Compared with the M (M 4)-party protocol in protocol of Ref. [20], our protocol has distinct advantages. When the secret sharing protocol is applied to secret message splitting, the advantage is also clear. For instance, as far as a 10-party protocol is concerned, in their protocol, [20] they need nine Bell states. However, in the present protocol, we only need one non-maximally entangled two-particle state. This means that in the present protocol the experimental difficulties in preparing initial states and in discriminating some entangled states are greatly reduced. According to the discussions mentioned above, our protocol is superior to the protocol of Refs. [21] too. It should be pointed out that the present MQSS protocol seems to be designed only for ideal quantum channels. In this protocol, the scheme of two parties reliably sharing an entangled qubit pair is very important and necessary. It is well known that when a qubit of an entangled pair travels in a noisy quantum channel, part of the initial entanglement between the two particles might be lost. Hence, a security problem for this protocol in a noisy channel seems to arise. Fortunately, it has been proven that over any long distance, two parties can reliably share a non-maximally entangled pair by using the quantumrepeater technique, containing entanglement purification

5 1472 XIA Yan, SONG Jie, SONG He-Shan, and HUANG Xiao-Li Vol. 49 and teleportation. [34] Once two parties have shared an entangled qubit pair, then in the process of an eavesdropping check, any eavesdropping can be detected by using the method of two MBs. Hence, even in a noisy channel, the present protocol works securely. In summary, we have proposed an MQSS protocol via introducing auxiliary particles using a non-maximally entangled two-particle state. This protocol has the advantage of a practical entangled source and of high security with decoy photons, compared with those protocols using EPR pairs. During the whole process of transmission, the transmitted particles do not carry any useful secret messages, and the secret messages are imposed on the auxiliary particles. After ensuring the security of the quantum channel, the sender encodes the secret message and transmits it to the receiver by using controlled-not operation and von Neumann measurement. Instead of performing unitary operations to recover Alice s secret message, the last agent measures the particles in a fixed measuring basis and recovers Alice s secret message according to the correlation between the measurement results. Only singleparticle measurements need to be performed on the particles in our protocol. Once it is secure after the process of the eavesdropping check, there is no chance for Eves to attack the secret messages. Even if Alice finds that Bob and Charlie are both dishonest after steps (S1 S11), neither of them can get accurate messages with a probability of 100% as long as Alice does not publish her measurement results. Hence, our protocol is unconditionally secure. It is experimentally easier to prepare a non-maximally entangled two-entangled state quantum channel as compared to a GHZ state and Bell state. So this protocol is simple and realizable with present technology. References [1] G.R. Blakley in Proceeding of the American Federation of Information Processing National Computer Conference, American Federation of Information Processing, Arlington, VA (1979) p. 313; A. Shamir, Commun. ACM 22 (1979) 612. [2] M. Hillery, V. Buzek, and A. Berthiaume, Phys. Rev. A 59 (1999) [3] A. Karlsson, M. Koashi, and N. Imoto, Phys. Rev. A 59 (1999) 162. [4] A.C.A. Nascimento, J. Mueller-Quade, and M. Imai, Phys. Rev. A 64 (2001) [5] G.P. Guo and G.C. Guo, Phys. Lett. A 310 (2003) 247. [6] A.M. Lance, T. Symul, W.P. Bowen, B.C. Sanders, and P.K. Lam, Phys. Rev. Lett. 92 (2004) [7] F.G. Deng, H.Y. Zhou, and G.L. Long, Phys. Lett. A 337 (2005) 329. [8] F.G. Deng, G.L. Long, and H.Y. Zhou, Phys. Lett. A 340 (2005) 43. [9] L.Y. Hsu and C.M. Li, Phys. Rev. A 71 (2005) [10] A.M. Lance, T. Symul, W.P. Bowen, et al., Phys. Rev. A 71 (2005) [11] W. Tittel, H. Zbinden, and N. Gisin, Phys. Rev. A 63 (2001) [12] F.G. Deng, G.L. Long, Y. Wang, and L. Xiao, Chin. Phys. Lett. 23 (2004) [13] S. Bandyopadhyay, Phys. Rev. A 62 (2000) [14] H.F. Chau, Phys. Rev. A 66 (2002) (R). [15] L. Xiao, G.L. Long, F.G. Deng, and J.W. Pan, Phys. Rev. A 69 (2004) [16] Z.J. Zhang, Phys. Lett. A 60 (2005) 342. [17] F.L. Yan and T. Gao, Phys. Rev. A 72 (2005) [18] Z.X. Man, Z.J. Zhang, and Y. Li, Chin. Phys. Lett. 22 (2005) [19] Y.Q. Zhang, X.R. Jin, and S. Zhang, Phys. Lett. A 341 (2005) 380. [20] N.Y. Li, K.S. Zhang, and K.C. Peng, Phys. Lett. A 324 (2004) 420. [21] Z.J. Zhang and Z.X. Man, Phys. Rev. A 72 (2005) [22] F.G. Deng, P. Zhou, X.H. Li, C.Y. Li, and H.Y. Zhou, Chin. Phys. Lett. 23 (2006) [23] F.G. Deng, X.Y. Li, Y.S. Li, H.Y. Zhou, and Y. Wang, Phys. Rev. A 72 (2005) [24] F.G. Deng, X.H. Li, C.Y. Li, P. Zhou, and H.Y. Zhou, Phys. Rev. A 72 (2005) [25] Z.J. Zhang, Y. Li, and Z.X. Man, Phys. Rev. A 71 (2005) ; F.G. Deng, X.H. Li, H.Y. Zhou, and Z.J. Zhang, Phys. Rev. A 72 (2005) [26] Z.J. Zhang, G. Gao, X. Wang, L.F. Han, and S.H. Shi, Opt. Commun. 269 (2007) 418. [27] Z.Y. Wang, H. Yuan, S.H. Shi, and Z.J. Zhang, Eur. Phys. J. D 41 (2007) 371. [28] L.F. Han, Y.M. Liu, S.H. Shi, and Z.J. Zhang, Phys. Lett. A 361 (2007) 24. [29] D. Wang, Y.M. Liu, G. Gao, and Z.J. Zhang, Commun. Theor. Phys. (Beijing, China) 47 (2007) 437. [30] K. Bostrom and T. Felbinger, Phys. Rev. Lett. 89 (2002) [31] C.H. Bennett, G. Brassard, and N. D. Mermin, Phys. Rev. Lett. 68 (1992) 557. [32] H. Inamori, L. Ralan, and V. Vedral, J. Phys. A 34 (2001) [33] E. Waks, A. Zeevi, and Y. Yamamoto, Phys. Rev. A 65 (2002) [34] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54 (1996) 3824.