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1、SCIENCE CHINA Information Sciences . RESEARCH PAPER . February 2018, Vol. 61 022501:1022501:12 doi: 10.1007/s11432-016-9151-x A secure rational quantum state sharing protocol Zhao DOU , Gang XU 1 1,2 , Xiu-Bo CHEN 1,5* , Xin LIU 3,4 & Yi-Xian YANG 4 2 Information Security Center, State Key Laborator
2、y of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China; School of Software Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China; 3School of Computer Science, Shaanxi Normal University, Xian 710062, China;
3、School of Information Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China; 5GuiZhou University, Guizhou Provincial Key Laboratory of Public Big Data, Guiyang 550025, China Received 11 December 2016/Accepted 16 January 2017/Published online 28 September 2017 Abstrac
4、t A novel rational protocol to share two arbitrary qubits among multiple parties is investigated in this paper. First, the protocol is presented, which is learned from Li et al.s protocol. Second, the utility, security, correctness, fairness, Nash equilibrium, and Pareto optimality of our scheme are
5、 discussed in detail, where the utility, correctness, and fairness of rational quantum state sharing protocols are creatively given because the agent who recovers the state plays a dierent and more important role. Another important point is that assumptions about our protocol are more practical and
6、suitable than existing protocols. Keywords rational, quantum state sharing, Nash equilibrium, secure, correct Citation Dou Z, Xu G, Chen X-B, et al. A secure rational quantum state sharing protocol. Sci China Inf Sci, 2018, 61(2): 022501, doi: 10.1007/s11432-016-9151-x 1 Introduction In the secret s
7、haring (SS) problem, there exists a dealer Alice and some agents Bobi. Alice owns a secret or some bits, which are split by her a nd shared by all the agents. Since the secret is fatal, Alice will send part of it to each a gent, instea d of the integrated secret. Only sucient agents can recover the
8、secret with the help of each other. This problem was rst investigated by Shamir 1 and Blakley 2 in 1979. The quantum secret sharing (QSS) protocol is the quantum version so lution of the SS problem. Quantum mechanics was introduced to ensure the unconditional security of the protocol 3, 4. In 1999,
9、Hillery et al. 5 proposed a QSS scheme with the Greenberger-Horne-Zeilinger (GHZ) state. At the same time, Cleve et al. 6 studied how to share quantum information (a quantum secret), instead of classical bits, among dierent agents. This kind of protocol is called quantum state sharing (QSTS). Owing
10、to the quantum no-cloning principle 7, an unknown quantum state cannot be copied as several ones. Only one agent, who is named Bobk or Cha rlie, can obtain the state with the help of the others. In 2004, Li et al. 8 proposed a QSTS protocol to share a n arbitrary unknown qubit via sharing Bell sta t
11、es and multi-particle GHZ basis measurement. Lance et al. 9 investigated a (2, 3) threshold quantum state sharing scheme in the same year. They demonstrated that average delity is equal to 0.730.04. In 2 005, Deng et al. 10 proposed a multi-party controlled scheme to teleport an arbitrary two-partic
12、le state. In this scheme, a three-particle GHZ state were utilized as the quantum resource. Actually, most * Corresponding author (email: ) c Science China Press and Springer-Verlag Berlin Heidelberg 2017 1 1,5 Dou Z, et al. Sci China Inf Sci February 2018 Vol. 61 022501:2 controlled teleportation
13、could be regarded as a QSTS protocol with or without a little mo dication 11. The same is true of Deng et al.s scheme 10. After that, Li et al. 11 simplied the process of this scheme. Pa rticipants in 11, do not need to perform multi-party entanglement measurement or two- qubit joint operation, whic
14、h makes their protocol ea sier to implement. They also expanded the scheme to a multi-particle version to extend its use. Later, Muralidharan and Panigrahi 12 designed a perfect QSTS pro to col to sha re arbitrary single- and two-qubit states via maximally entangled ve-qubit states. To complete the
15、task, multi-particle measurements are needed. Recently, Li et al. 13 investigated how to share an arbitrary two-qubit state by using a cluster sta te and a Bell state. There are two agents in this scheme. Security analysis shows that it is safe. In addition, the deterministic QSTS in cavity quantum
16、electrodynamics was investigated. Halpern and Teague 14 considered a rational classical SS protocol in 2004. Rational players are not supposed to be honest or malicious. On the contrary, they only pay attention to their own benet, and make decisions to maximize it. They will cooperate with others or
17、 not depending on which choice is more advantageous for themselves. Another all-importa nt standpoint is that no rationa l multi-party computa tion proto col ca n be accomplished in a deterministic time 14. In the view of assumption a bout players, we rechecked all the above QSTS protocols 6, 8 12,
18、and found that agents are supposed to accomplish the sharing faithfully even if they are malicious. Indeed, the same assumption also holds in the general case. We do not think this is reasonable enough. Players actually will also have incentive to obstruct the accomplishment of sharing if they can b
19、enet more. Maitra et al. 15 investigated the rational QSTS scheme for the rst time in 2015. The state is enco ded by CSS code. A (3, 7) rational QSTS scheme was investigated rst. In this scheme, the dealer is semi-oine. The generation to a (t, n) version scheme was given second. Correctness, fairnes
20、s, and the existence of Nash equilibrium were analyzed. A (t, n) QSTS proto col with the oine dealer and the corresponding ana lysis were also described. Another important assumption is whether the dealer knows the information about the state or not. In Maitra et al.s protocol 15, the dealer does kn
21、ow, so she can copy the state and distribute the same particles to dierent agents. In addition, t agents can obtain the state simultaneously. However, it makes the protocol more like a remote state preparing (RSP) protocol, instead of the QSTS. In g eneral, the dealer do es not know the state, much
22、less copy it. Only one agent can recover the state accordingly. The general case is more reasonable indeed. The third assumption of a protocol is whether the setting of agents is Byzantine or fail-stop. In the fail- stop setting, a player will o nly fulll his duty or drop out, depending on which cho
23、ice is more benecial. In contrast, a Byzantine agent may deviate from the protocol, such as sending false bits. It is evident that Byzantine agents are more practical and harder to investigate. In this paper, we follow the work of Li et al. 11 and Maitra et al. 15, and design a novel rational QSTS p
24、rotoco l. The processes are learned from 11. Some delicate and necessary mo dications are made, which makes players prefer the strategy Cooperating. The properties of rational multi-party computation are also ensured: correctness, fairness, and the existence o f Nash equilibrium. Furthermore, Cooper
25、ating is also the strateg y which satises the Pareto optimality. In addition, our proto col is also a s safe as Li et al.s 11. Compared with Maitra et al.s protocol 15, on the one hand, we suppose that only one agent can obtain the state, instead of all t agents. On the other hand, the setting is By
26、zantine, rather than fail-stop. Our assumptions are more practical. A detailed discussion is a lso given. The rest of paper is arra nged as follow. Preliminaries, including a random electoral method, quan- tum mechanics, Li et al.s scheme 11, and the basic concepts of the rationa l multi-party proto
27、co l are introduced in Section 2. Our novel rational QSTS proto col and analysis are shown in Sections 3 and 4, respectively. Discussion is described in Section 5 . Fina lly, conclusion is given in Section 6. 2 Preliminaries Dou Z, et al. Sci China Inf Sci February 2018 Vol. 61 022501:3 2.1 A simple
28、 method to randomly elect one player among N Fo r N players Pj (1 j N ), a simple way to elect a representative among themselves randomly could be described as follows. E-1 2 2 N virtual players are added. These players will not do anything. Now, there are 2 players in the election. log N 2 E-2 Each
29、 of N N real players randomly publishes one bit cj. Then they can co mpute the XOR of cj , C= j=1 cj. Here, denotes the addition module 2. 2 E-3If C = 0, then the rst half of 2 will lose, and vice versa. players will still have the right to be elected, the others (1 ) If all the players who have the
30、 rights are virtual, the election will be resta rted. (2 ) If more than one player has the right, and at least one of them is real, they will reperform the Steps E-2 and E-3 until only one exists. (3 ) If only one player has the right, and is real, he will be the chosen o ne. Although this way is no
31、t the true random in the quantum case 16, it is simple and ea sy to perform. More importantly, the result is co-determined by all the players instead of part of them. If this election is considered as a game, we can show that it is fair. The analysis is given in Subsection 4.4. 2.2 Quantum mechanics
32、 Some vital quantum states and bases in our paper are now introduced. (1 ) Bell states and Bell basis. Four Bell states are written as follows: 2 2 They could be denoted as | with dierent V and P , and constructed as the Bell basis. Similarly, |+ and | could be rewritten as |P , and constructed as t
33、he X basis. (2 ) GHZ states. In the three-particle case, there are 8 = 23 GHZ states in total. They are listed as 00 1 01 1 | 10 = 2 1 (|0 00 |111 ), | 11 = 2 1 (|0 01 |110 ), (2) | = 2 (|0 10 |101 ), 00+ 1 | = 2 (|0 11 |100 ). The best known state of these eight is | = 2 (|000 + |111 ). In the (n +
34、 2)-particle case, the counterpart is | = 2 ( i=1 |0 + i=1 |1 ). 2.3 Review of Li et al.s QSTS protocol In 2006, Li et al. 11 proposed a multi-party QSTS protocol. In the protocol, there are n + 1 agents Bobi (1 i n + 1) and a boss Alice. Suppose that the quantum state they want to share is an arbit
35、rary two-particle sta te: | xy = (a|00 + b|01 + c|10 + d|11 )xy, (3) where, |a| + |b| + |c| + |d| = 1. The pro cesses are simply reviewed a s follows. L-1Alice prepares two (n + 2)-particle GHZ states | s 1 = | s 2 = 2 ( i=1 |0 + i=1 |1 ), and shares them with the n + 1 agents. The whole system is s
36、hown as 1 2 N N 1 1 | = (|00 |1 1 ) = (| + + | ), 2 2 (1) 1 1 | = (|01 |1 0 ) = (| + | ). V P follows: n+2 n+2 2 2 2 2 n+2 n+2 | | | | = (a|00 + b|01 + c|10 + d|11 )xy Dou Z, et al. i=1 Sci China Inf Sci n+2 i i i=1 February 2018 Vol. 61 022501:4 i=1 i=1 b i . (4) Alice sends the photons a i and bi
37、to the agent Bobi , respectively. L-2 Alice performs the Bell basis measurement on her photon x and an+2 The rest of the system becomes , y and b n+2 , respectively. | sub = n+1 i=1 |00 aibi + n+1 i=1 |01 aibi + n+1 i=1 |10 aibi + n+1 i=1 |11 aibi . (5) The pa ra meter set , , , is the permutation o
38、f a, b, c, d, and is related to the measurement n+2 n+2 n+2 n+2 L-3 Suppose that Bob1 will recover the state, then Bobi (2 i n + 1) will perform X ba- sis measurement to help him. The measurements they performed can be expressed as follows: M ( +|) ( |) a ( +|) ( |) b, where, ( +|) ( |) a denotes th
39、e measurement operation re- lated to the particles ai , ( +|) ( |) b is related to bi. The symbols t and q are the numbers of agents who obtain the result |, respectively. L-4 After previous measurements, the state collapsed into 1 1 1 1 Bobi will perform lo cal operatio ns to recover the state | ac
40、cording to the public measurement results n+2 n+2 n+2 i=2 i n+2 i=2 i 2.4 Rational multi-party computation protocol i i=1 i i=1 i i=1 i ith player. The strategy set player Pi may perform is Ai. Let A A1 A2 An , then a = i=1 denotes the utility function. If Pi prefers strategy a to a , then we say ui
41、(a) ui(a ). In addition, the outcome of this game is denoted as o(a) = (o1, o2, . . . , on). Further, for a given strategy a = (a1 , a2, . . . , an ), ai is dened as ai (a1 , . . . , ai1 , ai+1 , . . . , an ), i i 1 i1 i i+1 n Nash equilibrium and Pareto optimality are two vita l denitions about the
42、 game. The general descrip- tions are also given below. Denition 1 (Strict Nash equilibrium). A strategy vector a in the game is a strict N ash equilibrium, i i Denition 2 (Pareto optimality). A strategy vector a in the game is a P areto optimality if it is impossible to improve anyones utility with
43、out reducing at least one others. In other word, if ui(a ) ui(a), then there exists at least one player j which has uj(a ) uj(a), we say a is a Pareto improvement of a. Since only one agent can obtain the state in our proto col, the utility, correctness, and fairness of our protocol are dierent from
44、 general protocols. The details are given in Section 4. 3 Our new rational QSTS protocol In this section, we propose a new rational QSTS protocol. The processes follow Li et al.s 11. There is also a boss (dealer) Alice and n + 1 agents. The protocol contains r rounds. The pro cesses Alice and Bobi n
45、eed to perform are described as follows. n+2 1 2 |0 + |1 n+2 n+2 1 |0 + |1 i 2 results V , V , P , and P . nt t nq q nt t nq q q t (q+t) | =(|00 + (1) |01 + (1) |10 + (1) |1 1 ) . n+1 n+1 V , V , P , and P . Here, P = P P , P = P P . n n n n (a , a ,. . . , a ) A is ca lled as a strategy vector of t
46、his game. Here, a is the strategy of P and u if for each player P and his any other strategy a we have u (a , a ) 0) here. (2) Obtaining a false state is disadvanta geous for an agent, hence U e Ups = Upe. (6) For simplicitys sake, we suppose thatonly U f is proportio nal to ri . The other utilities are independent with and r i. Next, we describe the utility of agents when they choose the strategy Cheating or Cooperating in a two-agent version as an example (Ta