Efficient Three-party Quantum Secure Direct Communication with Epr Pairs

In order to get rid of the drawback of information leakage which existed in Chong et al.'s protocol (Opt. Commun., 284, 2011, 515-518), an efficient three-party quantum secure direct communication (3P-QSDC) based on some ideas of quantum dense coding with EPR pairs is proposed, in which each entangled pair can be used to exchange a longer length of secret message between three legal users. By improving the classical channels and the qubit transmissions, our scheme can avoid this kind of drawback. Thus, the secret messages are not leaked out to other people from the public information. Moreover, compared with Chong et al.'s protocol, our protocol can achieve higher efficiency.


Introduction
Quantum secure direct communication (QSDC) is an important branch of quantum cryptography, in which the secret messages are directly transmitted in a quantum channel between two legitimate parties, say Alice and Bob, without creating a private key to encode and decode the messages.Since QSDC has a great advantage of unconditional security based on quantum mechanics for the legal users to communicate, much attention has been focused on this research field and many schemes have been presented [1][2][3][4][5][6][7][8][9][10][11][12].
In 2002, Long and Liu [1] proposed the first QSDC scheme based on EPR pairs.Beige et al. [2] presented a QSDC protocol based on the exchange of single photons.Boström et al. [3] proposed a ping-pong QSDC scheme based on EPR pairs, which was improved by Li et al. [4] in 2011.Deng et al. [5] proposed an efficient QSDC scheme.However, the mode of message transmission in QSDC is one-way.Thus, in 2004, quantum dialogue or the so-called bidirectional QSDC was proposed [7].Recently, many three-party QSDC schemes were proposed, in which a party can obtain the other two parties' messages simultaneously through a quantum channel.Jin et al. [10] presented a 3P-QSDC by using the GHZ states, and Man et al. [11] improved this scheme.Chamoli [12] also presented a 3P-QSDC with GHZ states.In 2007, Wang et al. [13] presented a 3P-QSDC by using EPR pairs.In 2011, Chong et al. [14] proposed an enhancement on Wang et al.' scheme [13].They pointed out that the communication can be paralleled and thus the protocol efficiency is improved.
For simplicity, References [13,14] are shortened as CH protocol and WY protocol, respectively.From CH protocol, we can see that the main features of their work are the paralleled communication and the improved protocol efficiency.However, there are some questions in Chong et al.'s scheme, which can be summarized as follows: 1) The qubit transmissions in WY protocol are thought to be sequential by Chong et al., i.e., Alice → Bob → Charlie → Alice.That is, every party needs to wait for the other's response.So in CH scheme, the qubit transmissions are designed as Alice → (Bob and Charlie) and (Bob and Charlie) → Alice.However, the improvement has the following disadvantages: (a) The goal here is to save the response time throughout the process, but this new way can lead to double workload in Alice's site.Thus, this improvement would be of no great importance or value in practical application; (b) As will be described later, the qubit transmission mode of CH protocol can reduce 3P-QSDC protocol efficiency.
2 3) From step 11 in CH protocol, we can see that there exists a message correlation between three parties.Let us take for example.If . Thus, from the public classical channels, Eve can know the secret bits transmitted by three parties must be one of randomly, which contains 2 bit of information.This insecurity is called information leakage or classical correlation [15,16].In fact, WY protocol also has this kind of drawback.In this paper, we present an efficient 3P-QSDC scheme based on some ideas in quantum dense coding with EPR pairs.Each photon pair can be used to exchange a longer length of secret message and the drawback of information leakage does not exist in our scheme.Moreover, in an ideal quantum channel, the efficiency of CH protocol is 50%, but our 3P-QSDC efficiency can be increased to 60%.Finally, the security of our scheme is analyzed.

Description of the Protocol
Firstly, let us introduce two-qubit entangled states.An EPR pair is one of the four Bell states, i.e., where 0 and 1 are the up and down eigenstates of Pauli operator z  .
  Suppose that Alice, Bob, and Charlie have a secret message to exchange respectively.Their messages can be assumed as the following in sequence: , , , , where .

 
, , , , , 0,1 n n n n n n i j k l p q  Three parties agree that the four Pauli operations rep-resent two-bit classical information, respectively, i.e., An EPR pair can be transformed into another EPR pair by performing the unitary operation .Then the encoding of our 3P-QSDC can be summarized as Table 1.
Now, let us describe the present protocol in detail by the following steps.
Step 1. Alice prepares EPR pairs and each EPR pair is one of the four Bell states randomly.Alice takes one particle from each EPR pair to form two single photon sequences h Q and t , where denotes the first (the second) particle in each pair.She encodes her message into t by performing the operation (9).Alice prepares five sets of decoy photons, A , randomly chosen from D 0 , 1 ,  , and  .Moreover, she generates single photon sequence r , in which the particles is defined a one-to-one correspondence with the initial states prepared by herself, i.e., Q 0 ,1 , , and  (9).After that, Bob asks Alice to send him .
old, this protocol is aborted.Otherwise, Bob picks out h and performs Bell measurements on h and t (encoded sequence), which forms two new sequences , , , 0,1,2,3 Bob's measurement results, where 0, 1, 2, 3 denote , , , respectively.Bob asks Alice to announce the measurement basis of , then he measures r with the same basis to form . Subsequently, Bob randomly inserts the particles in respectively, thus she can generate a corresponding bit string according to the initial states prepared randomly by herself in Step 1, where .
, , , , According to all above steps, Bob and Charlie can get the other two users' messages.Thus three parties can exchange their secret messages successfully.The simple steps can be seen in Figure 1.Decoding rules can be described as: 1) According to Table 1, Alice can know the final states in her site, which are also the initial states in Bob's site, from the initial states prepared by herself and her own operations in Step 1. Combining the final-

Security Analysis
Now, we analyze the security of our protocol in detail below.The transmission security of the particle sequences in the present 3P-QSDC scheme is similar to that of Chong et al.'s scheme which is based on security of Wang et al.'s scheme.In addition, we can see that the entangled photon pairs act as a quantum channel based on the idea of two-step transmission in our protocol.If the sequence is securely transmitted, Eve can not obtain any encoded information because one can not gain the secret messages from one particle of an EPR pair.On the other hand, although contains a sub-sequence r which directly corresponds to the initial states prepared by Alice, Eve can not get any useful information about Alice's message or Bob's.This is because the decoy photons in R .Next, Eve may get a message correlation between three parties by combining with M .However, because has the nature of randomness, Eve also cannot get any secret information.So all the secret bits exchanged between three parties are not leaked out from the classical channels.

Discussion and Conclusion
In the following, let us discuss the efficiency of the present protocol.The efficiency of a quantum communication scheme is defined as   denotes the expected number of secret bits received by the users, t is the number of transmitted qubits, and t is the number of needed classical bits.In CH protocol, we can see that , thus the efficiency is 50%.In our scheme, 3P-QSDC protocol can achieve higher efficiency with   . For clarity, we make a comparison between CH protocol and our protocol, which can be seen in Table 2.
In this paper, we point out that CH protocol has a drawback of information leakage and propose a new protocol to get rid of this kind of drawback.Moreover, our scheme has higher efficiency.In summary, our protocol is efficient and secure in theory.

D
and checks the quantum channel by analyzing the error rate.If the error rate exceeds the thresh inserts the particles in 2 and r into to form Q .Finally, he sends to Charlie.

Step 4 . 2 AQ 1 A 1 A 1 A
After Charlie receives t , Bob announces the positions of and the states of C .Then Charlie measures C and checks the quantum channel by analyzing the error rate.If the error rate exceeds the threshold, this protocol is aborted.Otherwise, Charlie encodes his message into t by performing the unitary operations according to Equation (9).After picking out r , Charlie measures this sequence with the basis announced by Alice.Next, Charlie randomly inserts the particles and verifies if the transmission of is secure by analyzing the error rate.If no, the protocol is aborted.Otherwise, Alice picks out t been encoded by herself, Bob, and Charlie.Finally, Alice asks Bob to send her .b h Step 6.After Alice receives h , Bob announces the positions and the states of .Then Alice checks the quantum channel by measuring .If the transmission of is insecure, the protocol is aborted.Otherwise, after picking out , Alice performs Bell measurement on h and t Q (encoded sequence), and she records the measurement results as .Alice encodes 00, 01,10,11 into b

Figure 1 .
Figure 1.Qubit transmissions.states in Bob' site   B R , Alice can deduce Bob's operations.Thus she obtains B M .From the initial states

R
Alice randomly inserts all particles in z  or x  .He can judge if the quantum channel is secure by analyzing the error rate.If no, Bob aborts the communication.Otherwise, after picking out t , he encodes his message into t by performing the operation

Table 1 ,
Eve can infer the final state in Alice's site and Bob's operation must be Eve can only explore 4 bits of secret information exchanged between Alice and Bob (each user has 2 bits).Thus Eve cannot get any information from B CM ,