Multiparty Controlled Quantum Secure Direct Communication of d-dimensional using GHZ state

Jian Dong1, Jianfu Teng1, Shuyan Wang2

1 School of Electronic Information Engineering, Tianjin University, Tianjin, 300072, China

2 Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin, 300072, China

dongjian@tju.edu.cn

Abstract

Quantum secure direct communication (QSDC) aims to allow two remote parties to communicate directly without creating a private key in advance and then using it to encrypt the secret message. In this paper, a multiparty controlled secure quantum communication of d-dimensional is proposed based on d-dimensional maximally entangled Greenberger–Horne–Zeilinger (GHZ) state and teleportation. After using decoy photons to check eavesdropping efficiently for the purpose of the quantum channel security, the sender encodes the secret message of d-dimensional on a sequence of additional particle states and then fulfills teleportation of the secret through d-dimensional Bell state measurement (DBM) and X-basis or Z-basis measurement. The receiver can recover the secret message by combining his measurement result with the sender’s and controller’s results. If an eavesdropper Eve cannot escape from the communication parties eavesdropping check, our scheme is completely secure because the transmitting particle sequence does not carry the secret message. 1. Introduction

The development of quantum cryptography was motivated by the short comings of classical cryptographic methods, which are basing security on the assumed difficulty of mathematical problems, which means time barrier. Security of classical cryptography is facing the threat of quantum computers. The theoretical ability of quantum computers to parallel process large amounts of information removes the time barrier to factoring large numbers. Since quantum cryptography does not depend on difficulty of mathematical problems for its security, quantum cryptography is under development. The QKD protocol was first published by Charles Bennett and Gilles Brassard in 1984 [1]. From then on, there is a large number of research on this field [2-5]. Recently, a practical quantum cryptosystem prototype has been developed for metro area

applications by NEC Corporation[6]. Besides, there are also some research in wireless field [7,8] and the European project Secoqc QKD network is also proposed [9-10], which aims at developing a global network for unconditionally secure key distribution. Meanwhile, commercial quantum products are also being developed [11], making its deployment a feasible alternative for securing real data networks.

Quantum teleportation plays an important role in the field of quantum cryptography and quantum communication. The quantum teleportation process allows the two remote parties to teleport an unknown quantum state, utilizing the nonlocal correlation of the quantum channel by means of multi-particle joint measurement. The quantum controlled teleportation scheme (QCTS), first published by [12], and is to make an unknown qubit or qudit quantum state be regenerated by a receiver with the help of one or many controllers [13-16]. Recently, a novel concept, quantum secure direct communication (QSDC), has been put forward and studied by some groups [17-19]. Different from QKD, QSDC aims to allow two remote parties to communicate directly without creating a private key in advance and then using it to encrypt the secret message. The works on QSDC attracted a great deal of attention and can be divided into two kinds, one utilizes single photon [20, 21], the other base on entangled state [22]. A multiparty controlled quantum secret direct communication scheme by using GHZ state is presented in [17], based on the idea of dense coding of three-photon entangled state and qubit transmission in blocks. J. Wang et al. [19] present an (n, n) threshold quantum secret sharing scheme of secure direct communication using Greenberger–Horne–Zeilinger state and teleportation. In Ref.[23], a three-party quantum secure direct communication protocol is presented by using GHZ states and entanglement swapping. A scheme for quantum secure direct communication with quantum encryption is presented in [24]. In Ref.[25], a three-party simultaneous quantum secure direct communication (QSDC) scheme is presented using GHZ states. And this

Second International Symposium on Intelligent Information Technology Application

978-0-7695-3497-8/08 $25.00 © 2008 IEEE

DOI 10.1109/IITA.2008.321

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Second International Symposium on Intelligent Information Technology Application

978-0-7695-3497-8/08 $25.00 © 2008 IEEE

DOI 10.1109/IITA.2008.321

551

Second International Symposium on Intelligent Information Technology Application

978-0-7695-3497-8/08 $25.00 © 2008 IEEE

DOI 10.1109/IITA.2008.321

551

scheme can be directly generalized to N-party QSDC by using n-particle GHZ states.

In this paper, we present a multiparty controlled QSDC (MCQSDC) scheme based on GHZ state and teleportation. In present scheme, the sender’s secret message is transmitted to the receiver by teleportation using GHZ state as quantum channel and can only be reconstructed by the receiver with the help of all the controllers. Different from quantum secret sharing (QSS), the sender transmits his/her secret message to the receiver directly and the information of the receiver is asymmetric to that of the controllers. Based on the ideas of controlled teleportation of qudit state[15, 16, 27], a multiparty controlled QSDC (MCQSDC) scheme of d-dimensional is proposed. The communication parties first perform eavesdropping check to detect whether there is eavesdropping in the transmission line. After using decoy photons to check eavesdropping efficiently for the purpose of the quantum channel security, the sender encodes the secret message of d-dimensional on a sequence of additional particle states and then fulfills teleportation of the secret through d-dimensional Bell state measurement (DBM) and X-basis or Z-basis measurement. The receiver can only recover the secret message by combining his measurement result with the sender’s and controller’s results. If an eavesdropper Eve cannot escape from the communication parties eavesdropping check, our scheme is completely secure because the transmitting particle sequence does not carry the secret message. Our scheme is completely secure if the quantum channel is secure.

The rest of the paper is organized as follows. Section 2 describes single-party Controlled teleportation of a qudit state. Multiparty controlled teleportation of a qudit state is listed in Section 3. In Section 4, the proposed QSDC protocol of d-dimensional based on GHZ state is describes in details. Finally, we give a summary in Section 5. 2. Single-party controlled teleportation of a qudit state

For clarity, let us first give the fundamental of quantum teleportation for a qudit by using a three-qudit maximally entangled GHZ state under 1-controller (for any d-dimensional, d≥2). The multiparty case is given in next section.

First of all the related notation and definition are list as follows [15, 16, 26, 27]. For a d-dimensional Hilbert space, its basis has d eigenvectors along z-direction Zd, can be written as

1,...,2,1,0 −d (1)

The d eigenvectors of another unbiased measuring basis (MB) Xd can be described as

∑−

=

=1

0

/21~ d

l

dilu led

u π

(2) A set of maximally d-dimensional Bell states can be

described as follows [31]

vlled

d

l

diluuv ⊕⊗= ∑−

=

1

0

/21 πφ (3)

The invert form of Eq. (3) is gotten as

∑−

=⊕

−=1

0,,

/21 d

vu

uvvij

dilued

ij φδπ

(4)

The vectors set { }uvφ is a set of orthonormal basis of an inner product space d2. So it can be used as a d-dimensional Bell state measurement (DBM). Furthermore,

we define the state uvφ

corresponding to two classical

dits “uv”, u=0,1,…,d-1; v=0,1,…,d-1, i.e. 00φ to “00”,

01φ to “01”, )1(0 −dφ to “0(d-1)”,

)1)(1( −− ddφ to “(d-1)(d-1)”.

Suppose the sender Alice wants to send an quantum

state, for example qudit a, aχ

to the receiver Charlie, where χ=0,1,…,d-1.

a

d

lla

la∑−

=

=1

0χ

(5) And the quantum channel is a three-qudit GHZ shared

by the sender Alice, the controller Bob and the receiver Charlie in advance in the form as

∑−

=

⊗⊗=1

0

1 d

lCBAABC

llld

φ (6)

where the qudit A is with Alice and the qudit B and C are in the hand of Bob and Charlie respectively. Therefore, the composite state of the quantum system composed of the particles in an unknown state and the GHZ state is

ABCaF φχ ⊗=

(7) When a d-dimensional Bell state measurement (DBM)

on particle a and particle A is done.

∑

∑∑−

=

−

=

−

=

=

⊗⊗⊗=

⊗=

1

0,

1

0

1

0

1

1

d

jlCBaAl

d

lCBAa

d

ll

ABCa

jjljad

llld

la

F φχ

(8)

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Suppose the results of the GDBM are α1α2, i.e. 21ααφ ,

and then the state of the quantum system becomes the

total state can be rewritten in terms of aA

uvφ as

According to Eq.(4), we can get

∑−

=

− ⊕⊕=1

0,,

/21 d

vulCBaA

uvdilul vlvlea

dF φπ

(9) And then controller Bob measure the particles B with

the Xd basis. Suppose the result of the measurement on

the particle B isβ~

, and then the state of the quantum system becomes

∑

∑−

=

++−

−

=

−

⊕⎟⎠⎞

⎜⎝⎛=

⊕⊕=

1

0,,,

/])([22/3

1

0,,

/2

1

1

d

mvulCBaA

uvdmvlluil

d

vulCBaA

uvdilul

vlmead

vlvlead

F

φ

φ

π

π

(10) The retained subsystem composed of the particles

controlled by Charlie is in the state

βααπ

===

−

=

++−∑ ⊕= mvu

d

lC

dmvlluil vleaF ,,

1

0

/])([221

|' (11)

Then receiver can perform a corresponding unitary

operation pqUon particle C to reconstruct the original

quantum state according to the measurement of Alice and Bob. The operation is defined as Ref.[15].

lqleUd

l

diplpq ∑

−

=

⊕=1

0

/2π

(12) According Eq.(11) and Eq. (12), we can get

⎩⎨⎧

⊕==⊕

βαα

1

2 0pq

(13)

3. Multiparty controlled teleportation of a qudit state

We now generalize 1-controller to n-controllers case

with an arbitrary n≥ 2. Suppose Alice wants to send an

quantum state aχ

to the receiver Charlie via multi-controllers (Bobi, i=0,1,…,n).

The GHZ state of (n+2) particles is given as

∑−

=

+⊗=1

0

)2(1 d

l

n

CABl

diφ

(14) Similar to the analysis of 1-controller case, the

difference is the particle Bi is held in the hands of Bobi respectively, i=1,2,…,n. And each Bobi measure the particle in his hand with the Xd basis.

The composite state of the quantum system becomes

∑

∑∑−

=

⊗++−+

−

=

+⊗−

=

⊕⎟⎠⎞

⎜⎝⎛=

⊗=

1

0,,,

/])([22/)12(

1

0

)2(1

0

1

1

d

mvulC

n

BiaA

uvdmvlluil

n

d

l

n

a

d

ll

ii

i vmead

ld

laF

χφπ

(15) After the measurement of Alice and Bob, the retained

subsystem composed of the particles controlled by Charlie is in the state

ii

n

ii

mvu

d

lC

dmvllui

l vleaF βαα

π

===

−

=

++−

∑ ⊕∑

= =,,

1

0

/])([2

21

1 |'(16)

Then receiver can perform a corresponding unitary

operation pqUon particle C to reconstruct the original

quantum state according to the measurement of Alice and Bobi.

According Eq.(11) and Eq.(16), we can get

⎪⎩

⎪⎨⎧

⊕=

=⊕

∑=

n

iip

q

11

2 0

βα

α

(17) 4. Proposed QSDC protocol of d-dimensional based on GHZ state

In this section, a multiparty controlled QSDC protocol of d-dimensional based on GHZ state is proposed. Suppose Alice wants to send a dit (d-dimensional bit) information (m, m=0,1,…,d-1) to Charlie via many controllers Bobi using QSDC. Alice first checks eavesdropping in the transmission line by using random Z-basis or X-basis measurement. After ensuring the security of the quantum channel, Alice encodes her secret message on each of the GHZ states and sends it to the receivers directly using teleportation. According to Alice’s measurement results, the receivers can collaborate to acquire Alice’s secret message. The following content describes the proposed QSDC protocol in detail.

Step 1 According to random selection a basis from Z-

basis { 1,...,2,1,0 −d } or X-basis

{

~1,...,2~,1~,0~ −d

} for each time, Alice prepares randomly N (n+2)-particle GHZ states, each of which is in the state as Eq.(14) or following state

∑−

=

+⊗=

1

0

)2(~1~ d

l

n

CABl

di

φ (18)

We denote the ordered N GHZ states by {[P1(A), P1(Bi), P1(C)], [P2(A), P2(Bi), P2(C)], . . ., [PN(A), PN(Bi), PN(C)]}, where the subscript indicates the order of each state in the sequence, and A, B, C represent the three

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particles of each state, i=1,2,…,n. Alice takes one particle from each state to form an ordered partner particle sequence, [P1(A), P2(A), . . ., PN(A)], called A sequence. The remaining partner particles compose Bi sequence, [P1(Bi), P2(Bi), . . ., PN(Bi)], and C sequence, [P1(C), P2(C), . . ., PN(C)].

Step 2 Alice sends Bi sequence and C sequence to Bobi and Charlie, respectively. Bobi and Charlie inform Alice that they each have received N particles.

To prevent the eavesdropper Eve or the controller Bobi from stealing the information about the unknown qudit state in this scheme, the parties can exploit the way in Ref. [13, 15] to set up the quantum channel securely. Alice prepares some decoy photons which are randomly in the

states { 1,...,2,1,0 −d ,

~1,...,2~,1~,0~ −d

}, to insert them into the original sequence of particles before sending them to Bobi and Charlie, respectively. After confirming the receipt of the sequences, Alice picks up the decoy photons to check the eavesdropping in the quantum line by comparing the results of measurement with the MB Zd or the MB Xd. Thus the process for setting up the quantum channel can be made to be secure.

Then if Alice confirms that there is no eavesdropping, they continue to execute the next step. Otherwise, they abort the communication.

Step 3 After ensuring the security of the quantum channel, Alice encodes her secret message on GHZ sequence. If the bit value of Alice’s secret message is “m”, m=0, 1,…, d-1, she prepares a particle a in the state m~ or m for the GHZ state in the sequence according to

which basis is selection in step 1. That is a in the state m~

if Z-basis is chosen, otherwise, a in the state m . We denote particle a as Pi(a), where i=1,2,…,N. If the state of

Pi(a) is m~ , then the state of particles Pi(a), Pi(A), Pi(Bi), and Pi(C) is

∑−

=

+⊗⊗=Ψ1

0

)2(1~ d

l

n

al

dm

(19) where the subscript a denotes particle Pi(a). If the state

of Pi(a) is m , then the state of the four particles becomes

∑−

=

+⊗⊗=Ψ

1

0

)2(~1'd

l

n

al

dm

(20) Step 4 The sender Alice first performs a d-dimensional

Bell state measurement (GDBM) on the particles Pi(a) and Pi(A) using Z-basis or X-basis as in step 1. And then each Bobi measure the particles Pi(Bi) with the X-basis or Z-basis. Here Alice transmits the message of basis selection to Bobi secretly through classical communication.

Step 5 Charlie can perform a unitary operation Upq on each of her particle Pi(C) according Eq.(17) to regenerate

the quantum state with the help of controller Bob and the public result of Bell measurement by Alice. And then Charlie measure the Pi(C) to get secret message “m” on the basis which Bobi through classical communication.

Now let us discuss the security for the present scheme. The proposed scheme is secure if the quantum transmission is secure. That is to say, if an eavesdropper Eve cannot escape from the communication parties eavesdropping check at Step 2, the scheme is secure. Charlie can not recover and measurement correctly without Bobi’ help because of two possible bases(X-basis or Z-basis).

5. Conclusions

In summary, we have presented a multiparty controlled

QSDC protocol of d-dimensional based on GHZ state. If an eavesdropper Eve cannot escape from the communication parties eavesdropping check, our scheme is completely secure because the transmitting particle sequence does not carry the secret message. Our scheme is completely secure if the quantum channel is secure.

Acknowledgement

This paper is supported by Tianjin Nature Science

Fund 06YFJMJC00800.

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