An Improved Quantum Information Hiding Protocol Based on Entanglement Swapping of χ-type Quantum States

Transcription

1 Commun. Theor. Phys. 65 (2016) Vol. 65, No. 6, June 1, 2016 An Improved Quantum Information Hiding Protocol Based on Entanglement Swapping of χ-type Quantum States Shu-Jiang Xu (Å ), 1, Xiu-Bo Chen (í ), 2 Lian-Hai Wang ( ), 1 Qing-Yan Ding (òã), 1 and Shu-Hui Zhang ( ) 1 1 Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center (National Supercomputer Center in Jinan), Jinan , China 2 Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing , China (Received November 25, 2015; revised manuscript received April 25, 2016) Abstract In 2011, Qu et al. proposed a quantum information hiding protocol based on the entanglement swapping of χ-type quantum states. Because a χ-type state can be described by the 4-particle cat states which have good symmetry, the possible output results of the entanglement swapping between a given χ-type state and all of the 16 χ-type states are divided into 8 groups instead of 16 groups of different results when the global phase is not considered. So it is difficult to read out the secret messages since each result occurs twice in each line (column) of the secret messages encoding rule for the original protocol. In fact, a 3-bit instead of a 4-bit secret message can be encoded by performing two unitary transformations on 2 particles of a χ-type quantum state in the original protocol. To overcome this defect, we propose an improved quantum information hiding protocol based on the general term formulas of the entanglement swapping among χ-type states. PACS numbers: Dd, a, w, Hk Key words: quantum communication, quantum information hiding, χ-type quantum state, entanglement swapping 1 Introduction With the continuous development of mobile internet and cloud computing, information security is more and more valued. [1 4] Information hiding, which is one of the important protection measures for private information, has made rapid progress in the past few decades. [5 7] As the analog of classical information hiding in the field of quantum information, quantum information hiding becomes one of the popular topics of quantum communication. Since the quantum mechanical effect is employed, quantum information hiding, as well as other quantum communication protocol including quantum key distribution (QKD), [8 10] quantum secure direct communication (QSDC) [11 13] and quantum secret sharing (QSS), [14 16] can achieve unconditional security. Based on the entanglement swapping of quantum states, scholars have proposed several quantum information hiding protocols [17 22] in which covert channels are established to send secret messages. By the entanglement swapping of Bell states and the entanglement swapping of χ-type states, Qu et al. [17 18] respectively proposed a quantum information hiding protocol based on the given QSDC channels. Ye and Jiang [19 20] presented two quantum information hiding protocols respectively based on the entanglement swapping among GHZ states and the entanglement swapping between GHZ states and Bell states in cavity quantum electrodynamics. It is really interesting to build up quantum covert channels inside other quantum communication channels. The type of quantum information hiding protocols can achieve a high capacity and a good imperceptibility. However, these protocols in Refs. [17 20] consume some auxiliary quantum states besides the given QSDC channels. Employing the same secret message encoding rule as the no extended one used in Ref. [17], Fatahi and Naseri [21] proposed a quantum information hiding protocol based on Bell states in In 2013, Mo et al. [22] presented a quantum information hiding protocol based on the entanglement swapping property of Bell dual basis. The two protocols in Refs. [21 22] do not Supported by the National Natural Science Foundation of China under Grant Nos , , , and , the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2013FM025, ZR2013FQ001, ZR2014FM003, and ZY2015YL018, the Shandong Provincial Outstanding Research Award Fund for Young Scientists of China under Grant Nos. BS2015DX006 and BS2014DX007, the National Development Foundation for Cryptological Research, China under Grant No. MMJJ , the Priority Academic Program Development of Jiangsu Higher Education Institutions and Jiangsu Collaborative Innovation Center on Atmospheric Environment and Equipment Technology Funds, and the Shandong Academy of Sciences Youth Fund Project, China under Grant Nos. 2015QN003 and 2013QN007 xsj3917@163.com c 2016 Chinese Physical Society and IOP Publishing Ltd

2 706 Communications in Theoretical Physics Vol. 65 consume any auxiliary quantum state, but their capacity is only 1/2 that of the given QSDC channels, while the capacity of protocols in Refs. [17 18] can be 2/3 that of the given QSDC channels. In Ref. [23], Xu et al. showed the main idea why a covert channel can be established within arbitrary QSDC channel which employs unitary transformations to encode information bits, and proposed a quantum information hiding protocol. In Ref. [24], a quantum information protocol is presented based on the tensor product of Bell states. The two protocols [23 24] can achieve a high efficiency without consuming auxiliary quantum state. This paper will point out that the secret messages are difficult to read out in the protocol [18] because there are 8 groups instead of 16 groups of different results and each result occurs twice in each line (column) of its secret messages encoding rule table. As we know, the χ-type states can be described by the 4-particle cat states. Due to the symmetry of the 4-particle cat states, the possible output results of the entanglement swapping between χ ij and χ i j are the same as that of the entanglement swapping between χ ij and χ (i 1)(j 2) (also χ (i 1)(j 2) and χ i j ). As a result, the possible output results of the entanglement swapping between a given χ-type state and all of the 16 χ-type states are divided into 8 groups instead of 16 groups of different results when the global phase is not considered. So we point out that a 3-bit instead of a 4-bit secret message can be encoded each time by the original secret message encoding rule. According to the general relationship between the input states and the output states of the entanglement swapping for the χ-type states, this paper proposes an improved quantum information hiding protocol so as to overcome the defect without consuming any auxiliary quantum state. The rest of this paper is organized as follows. A defect of the protocol [18] is shown in Sec. 2. Section 3 improves the original protocol. Finally, we conclude this work in Sec Defect of Qu s Protocol [18] This section briefly introduces the properties of χ-type states, and then points out a defect of the protocol. [18] 2.1 The Properties of Entanglement States The four-particle χ-type states are asymmetrical maximum entanglement states. For convenient description of the χ-type states, the four Pauli transformations are given as follows: [ ] 1 0 σ 0 = =, 0 1 [ ] 0 1 σ 1 = =, 1 0 [ ] 0 1 σ 2 = =, 1 0 [ ] 1 0 σ 3 = =. (1) 0 1 We use χ ij 3214 (i, j = 0, 1, 2, 3) to respectively denote the 16 χ-type states: χ = 1 2 ( ) 3214, (2) χ ij 3214 = (σ i 3 σj 1 ) χ , i, j 0, 1, 2, 3, (3) where the subscripts denote the particle order in a χ-type state, σ i k means that the Pauli transformation σi is performed on the k-th particle of the state χ All of the 16 χ-type entangled states form a set of orthonormal basis, FMB, for the 4-qubit space: FMB = χ ij 3214 i, j = 0, 1, 2, 3. (4) Due to the distinctive structure, the χ-type entangled states have some significant properties. For example, the responding reduced density matrices of the particle pairs 1, 3 and 2, 4 are both equal to the complete mixture: [13] ρ = 1 ( ). (5) 4 As a result, only respectively performing measurement on particle pairs 1, 3 and particle pairs 2, 4, the χ-type states are difficult to distinguish from each other. The only effective way to distinguish these χ-type states is to measure them by the FMB basis. To show another property of the χ-type states, four set of two-qubit orthonormal basis are given as follows: AMB 1 = AMB 2 = BMB 1 = BMB 2 = Φ ± 1, Ψ± 1 Φ ± 1 = 1 ( Φ + ± Ψ ), Ψ ± 1 = 1 ( Ψ + ± Φ ), (6) 2 2 Φ ± 2, Ψ± 2 Φ± 2 = 1 ( Φ + ± Ψ + ), Ψ ± 2 = 1 ( Ψ ± Φ ), (7) , 0, 1+, 1 1 ± = 2 ( 0 ± 1 ), (8) +0, 0, +1, 1 1 ± = 2 ( 0 ± 1 ). (9) Measured the particles 2 and 4 of a χ-type state by the basis BMB 1 (BMB 2 ), and the particles 3 and 1 of the state

3 No. 6 Communications in Theoretical Physics 707 will collapse on the basis AMB 1 (AMB 2 ). For instance, the state χ can be rewritten as the following two forms: χ = 1 2 ( Φ Φ Ψ Ψ+ 1 1 ), (10) χ = 1 2 ( Φ Φ Ψ Ψ+ 2 1 ). (11) The property can be employed to design a defense strategy so as to guard against eavesdropping in the transmission. [13] In [18], the χ-type states are rewritten by the 4-particle cat states which have good symmetry. For example, χ = 1 2 (ψ(0) + ψ(6) ψ(13) ψ(11)) 3214, (12) where ψ(i) is a cat state. All the 16 four-particle cat states form a set of measurement basis, CMB = ψ(i) 0 i 15, of the 4-qubit space: ψ(i) = 1 2 ( i 1 i 2 i 3 i 4 + i 1 i 2 i 3 i 4 ), 0 i 7, (13) ψ(i) = 1 2 ( j 1 j 2 j 3 j 4 j 1 j 2 j 3 j 4 ), 8 i 15, (14) where i 1 i 2 i 3 i 4 and j 1 j 2 j 3 j 4 respectively are the binary form of i (0 i 7) and i 8 (8 i 15), i k + ī k = 1, j k + j k = 1 (k = 1, 2, 3, 4) and + denotes modulo-2 addition. 2.2 Defect of Protocol [18] Let us first briefly describe the original protocol. [18] Based on the entanglement swapping of χ-type quantum states which can be illustrated by the 4-particle cat states, the original protocol proposed a secret messages encoding rule which is described in Table 1. Using the secret message encoding rule, Qu et al. presented a quantum information hiding protocol, [18] in which a covert channel is established inside the given QSDC channel. The sender uses the given QSDC channel to transfer normal information bits, while his purpose is to transfer secret messages in the covert channel without being detected. The receiver can read out the secret message with the secret message encoding rule and the normal information bits. For details of the information protocol, please refer to Ref. [18]. Table 1 The result collections of arbitrary two χ-type states, χ ij 3214 and χ i j entanglement swapping on qubits 1 and 4. (The superscript denotes the type of Pauli operations and the subscript denotes the number of qubits, S i is the number of result collections). σ 0 3 σ 0 1 σ 0 3 σ 1 1 σ 1 3 σ 2 1 σ 1 3 σ 3 1 σ 2 3 σ 0 1 σ 2 3 σ 1 1 σ 3 3 σ 2 1 σ 3 3 σ 3 1 σ 0 3 σ0 1 S 0 S 1 S 0 S 1 S 14 S 15 S 14 S 15 σ 0 3 σ1 1x S 4 S 5 S 4 S 5 S 10 S 11 S 10 S 11 σ3 1σ2 1 S 0 S 1 S 0 S 1 S 14 S 15 S 14 S 15 σ3 1σ3 1 S 4 S 5 S 4 S 5 S 10 S 11 S 10 S 11 σ3 2σ0 1 S 11 S 10 S 11 S 10 S 5 S 4 S 5 S 4 σ3 2σ1 1 S 15 S 14 S 15 S 14 S 1 S 0 S 1 S 0 σ3 3σ2 1 S 11 S 10 S 11 S 10 S 5 S 4 S 5 S 4 σ3 3σ3 1 S 15 S 14 S 15 S 14 S 1 S 0 S 1 S 0 The defect can be clearly seen in the encoding rule table of the original protocol. There are 8 groups instead of 16 groups of different results in each line (column) of Table 1 which has 16 lines (columns). And each of the results will occur twice in a line (column). So it is difficult to exact decode the 4-bit secret message, which has been sent over the covert channel. Note that there are 16 groups of different output results of the entanglement swapping between any two χ-type states, but only 8 groups of them appear in each line (column) of Table 1. According to Ref. [18], we can easily find that σ3σ i j 1ψ(k) = σ(3 i) 3 σ (3 j) 1 ψ(k), (15) when the global phase is not considered. As a result, the possible output results of the entanglement swapping between χ ij and χ i j are same as that of the entan-

4 708 Communications in Theoretical Physics Vol. 65 glement swapping between χ ij and χ (i 1)(j 2) (also χ (i 1)(j 2) and χ i j ). Therefore, the possible output results of the entanglement swapping between a given χ- type state and all of the 16 χ-type states are divided into 8 groups instead of 16 groups when the global phase is not considered. So a 3-bit instead of a 4-bit secret message can be encoded by performing the unitary transformation σ3 iσj 1 (i, j = 0, 1, 2, 3) on a χ-type state in the original protocol. Moreover, the protocol [18] consumes some auxiliary quantum state which reduces the embedding capacity. All the protocols [17 24] establish covert channels within the given QSDC channels to transfer secret messages. The difference among them is the secret message encoding methods, the secret message encoding methods of protocols [17 22] employ the entanglement swapping between two quantum states, the secret message encoding method of protocol [23] employs single quantum state, and the secret message encoding method of protocol [24] employs the tenser product of two or more quantum states. In fact, the secret message encoding methods of Refs. [17 22] can be considered as the special case of that in protocol, [24] because the input/output of the entanglement swapping between two quantum states can be seen as a tensor product of two quantum states. The former has a high computation complexity for the entanglement swapping, while the latter not only does not need complicated computation, but also has a good generality. To overcome the weakness of the original protocol [18] and reduce its computational complexity, the next section will improve it based on the general term formula of the entanglement swapping for the χ-type states without complicated computation. 3 An Improved Quantum Information Hiding Protocol for Qu s Protocol [18] In this section, we further research the general relationship between the input states and the output states of the entanglement swapping for the χ-type states, and then give an improved quantum information hiding protocol for Qu s protocol. [18] The improved protocol not only can overcome the defect shown in Sec. 2, but also has a large capacity because it does not consume any auxiliary quantum state. The improved protocol has a given QSDC channel and a covert channel. The information sequence can be sent through the given QSDC channel and the secret message can be sent via the covert channel under the cover of the given QSDC channel. 3.1 The Improved Secret Message Encoding Rule Based on the relationship between the input states and the output states of the entanglement swapping for the χ-type states, a new secret message encoding rule is presented in this subsection. Suppose that the result of entanglement swapping between two χ-type states satisfies: χ i1j χ i2j χi3j χ i4j (16) Qu et al. [18] draw the formula of the entanglement swapping for the χ-type states: (σ i 3σ j 1 σp 3 σq 1 )( χ χ ) = (σ i 3 σq 1 σp 3 σj 1 )( χ χ ), (17) this formula can be simplified as: χ ij 3214 χ pq = χiq χpj (18) According to Eqs. (16) and (18), we can obtain i 1 i 2 i 3 i 4, j 1 j 2 j 3 j 4, (19) where means modulo-4 addition operation. As a result, (i 3 i 4 ) can determine (i 1 i 2 ), and (j 3 j 4 ) can determine (j 1 j 2 ). So we can employ the output result of the entanglement swapping to encode the secret message. In the given QSDC channel, the two χ-type states χ i3j and χ i4j are encoded to carry the quaternary information sequences i 3 j 3 and i 4 j 4 by respectively performing the unitary transformations σ 2k1+k2, σ 2k3+k4, σ 2k5+k6, and σ 2k7+k8 on the particles 3, 1, 3 and 1. Here i 3 = 2k 1 + k 2, j 3 = 2k 3 + k 4, i 4 = 2k 5 + k 6, j 4 = 2k 7 + k 8, k l 0, 1. (20) To encode a 8-bit secret message s, a secret message encoding rule is presented based on the above four unitary transformations: s = (i 3 a 1 )(j 3 a 2 )(i 4 a 3 )(j 4 a 4 ), (21) where means modulo-4 addition operation, and the constants a i 0, 1, 2, 3 (i = 1, 2, 3, 4). Rewrite the quaternary s into a binary form, a new secret message encoding rule can be obtained. Compared with the secret message encoding rule in Refs. [17 20], this one has a low computation complexity since the complex computation for entanglement swapping of the quantum states is no longer needed. Moreover, this rule is more generalized than the rules in Refs. [17 20] because the 4 constants are adopted. 3.2 An improved quantum information hiding protocol based on the entanglement swapping of χ-type quantum states Based on the proposed secret message encoding rule, an improved quantum information hiding protocol is given in this subsection. The step of the improved protocol is illustrated in detail as follows: Step 1 Initializing (a) Before performing the protocol, the sender Alice and the receiver Bob of the communication agree that they utilize the dense encoding rule in the given QSDC

5 No. 6 Communications in Theoretical Physics 709 channel and the proposed secret message encoding rule in the covert channel, respectively. And they determine the value of the 4 constants a i (i = 1, 2, 3, 4). (b) Alice prepares n χ-type entangled state χ , and respectively selects the i-th particle of each entangled state χ to form 4 ordered particles sequences: C i = [Pi 1, P i 2,..., P i n j ] (i = 1, 2, 3, 4). Here, Pi means the i-th particle of the j-th χ-type entangled state. She keeps the particle sequences C 1 and C 3, and sends the particle sequences C 2 and C 4 to Bob. Step 2 Eavesdropping check (a) Bob chooses sufficient sample particles from the sequences C 2 and C 4 randomly, and then measures these corresponding particles using the basis BM 1 (or BM 2 ). (b) Bob announces the position of the sample particles and the corresponding measurement basis in classical channel. (c) Based on the information that Bob announced, Alice measures the corresponding particles in the sequences C 3 and C 1 by the corresponding basis AM 1 (or AM 2 ). (d) Alice and Bob compare their measurement results in the public channels. If the error rate does not exceed a given threshold, they confirm that the quantum channel is secure and turn to the next step. Otherwise, they abort the communication and restart it again. Step 3 Information encoding and secret messages encoding (a) Information encoding: If Alice wants to send the information sequence m 1 m 2 m 3 m 4 to Bob, she can respectively apply the corresponding local unitary operations σ 2m1+m2 and σ 2m3+m4 to the particles 3 and 1, here m i 0, 1. (b) Secret messages encoding: Suppose that Alice wants to send the secret message s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 (s l 0, 1, l = 1, 2,..., 8) via a covert channel under the cover of the given QSDC channel. Alice should choose four proper particles P3 m, P m 1, P m+1 m+1 3 and P1 from the encoded sequences C 3 and C 1, respectively. Here the four particles are employed to transfer the information bits n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 8 by performing the four unitary transformations σ 2n1+n2, σ 2n3+n4, σ 2n5+n6, and σ 2n7+n8 on them in step (a). The tensor product of the two encoded χ-type quantum states, χ (2n1+n2)(2n3+n4) 3214 χ (2n5+n6)(2n7+n8) 3214, is taken as an output result of the entanglement swapping for the secret message encoding rule. According to the secret message encoding rule, s i and n i should satisfy: 2n 1 n 2 a 1 2s 1 s 2, (22) 2n 3 n 4 a 2 2s 3 s 4, (23) 2n 5 n 6 a 3 2s 5 s 6, (24) 2n 7 n 8 a 4 2s 7 s 8, (25) where means modulo-4 addition operation. Since n i, s i 0, 1, n i can be easily determined by s i. After that, Alice sends these encoded particles to Bob. Note that the position information m should be transmitted via the given QSDC channel at the same time. Step 4 Information decoding and secret messages decoding (a) Information decoding: After receiving the particles that Alice has sent, Bob performs measurements on each group of particles [P j 3, P j 2, P j 1, P j 4 ] by the basis FMB. According to his measurement results, he can decode the information that is sent through the QSDC channel. (b) Secret messages decoding: After receiving the hidden position information m, Bob first decodes the information carried by the particles P3 m, P m 1, P m+1 m+1 3, and P1 in the given QSDC channel, and then he can obtain the unitary transformations σ 2n1+n2, σ 2n3+n4, σ 2n5+n6, and σ 2n7+n8 which are respectively performed on the four particle P3 m, P m 1, P m+1 m+1 3, and P1. Finally, the secret message can be decoded based on Eqs. (22) (25). To further illustrate the improved quantum information hiding protocol, a simple example is given as follows. If Alice wants to send the secret message to Bob via the covert channel, she should choose four particles P3 m, P m 1, P m+1 m+1 3, and P1 from the encoded particles sequences, which are encoded to carry the information in the given QSDC channel. After receiving the position information, Bob first obtains the unitary transformations σ 1, σ 0, σ 2, and σ 3 that have been applied to the four particles P m 3, P m 1, P m+1 3, and P m+1 1 by the FMB measurement, and then he can read out the secret message in the covert channel based on Eqs. (22) (25). From the steps of the improved quantum information hiding protocol, we can find that all the sender and the receiver need to do in the covert channel is classical computation. That is to say, there is no quantum operation in the covert channel. So the improved protocol has no effect on the imperceptibility and security of the original protocol. Because the improved protocol does not consume any auxiliary particle, its capacity is larger than that of the original one. In fact, the improved protocol has the same capacity as that of the given QSDC protocol, while the capacity of the protocols in Refs. [17 20] is 2/3 that of the given QSDC protocol at most. As we know, the QSDC protocol is a high-efficiency quantum communication protocol. So the improved quantum information hiding protocol achieves a high efficiency. 4 Conclusion According to the symmetry of the cat states which can be utilized to describe the χ-type states, we point out that the possible output results of the entanglement swapping between a given χ-type state and all of the 16 χ-type

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