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In wireless quantum networks, nodes communicate by means of pre-distribution for entangled pairs and relay path establishment for quantum teleportation. However, simple point-to-point communication seriously restricts the efficiency of quantum communication. Inspired by sharing idea of quantum secret sharing (QSS), which is based on three collaborative nodes with pre-shared GHZ (Greenberger-Horne-Zeilinger) states, we propose a quantum secret broadcast scheme to improve network performance. In a cluster net-work cored on three parties of QSS, three cluster heads with pre-shared GHZ states are senders, while cluster members are receivers. One cluster head encodes secret messages on auxiliary particles by performing certain operations on them with GHZ particles, then three cluster heads measure their own par-ticles and broadcast measurement results honestly. Based on the specific correlation of measurement results and secret messages, all receivers can re-cover the secret messages. Furthermore, to prevent eavesdropping, cluster heads can update an encoding key periodically. Analysis shows the proposed scheme is more efficient than previous schemes in wireless quantum net-works, especially when the number of receivers is larger. Besides, in the proposed scheme, attacks on quantum channel based on GHZ state can be detected, and eavesdroppers cannot recover messages correctly for lack of suitable decoding key.

Since the first quantum key distribution (QKD) protocol BB84 [

Quantum secret sharing (QSS) is an important branch of quantum crypto- graphy, which is a generalization of the classical secret sharing into the quantum domain. Since the first QSS protocol was presented by Hillery et al. [

Inspired by the feature of QSS that more than one party can receive secret message each time, we introduce the idea of QSS into a wireless quantum network, and propose a quantum secret broadcast scheme to solve the troubling efficiency problem. In a cluster network cored on three parties of QSS, three cluster heads, Alice, Bob, and Charlie will collaborate honestly to broadcast messages to cluster members by using pre-shared GHZ states. The communi- cation mode can be whole-network broadcast or intra-cluster broadcast. Furthermore, to prevent illegal eavesdropping, three cluster heads will perio- dically update a encoding key Y. Consequently, illegal nodes cannot read out the message correctly for lack of suitable decoding key.

Wireless quantum networks (WQN) has been studied by many groups [

Although WQN has been explored further in the aspects of EPR-pair allo- cation [

relay path establishment.

In 2009, Liu et al. [

By convention, the sender is denoted as Alice (A), and the receivers is denoted as Bob (B) and Charlie (C). First, they share three-particle GHZ states, each of which is:

After eavesdropping check to ensure the security of quantum channel (GHZ states), Alice prepares an EPR pair in the state:

Four unitary operators are defined as:

The system state after encoding can be expressed as:

where

Next, Alice applies a controlled-NOT (CNOT) gate on both particle

Then Alice applies a Bell-state measurement on both particles

Their measurement results will be correlated in certain forms according to different encoding operation

In Liu’s QSS scheme [

The main idea of our scheme is that in a cluster network cored on three parties of QSS, Alice (A), Bob (B), and Charlie (C) are cluster heads. In each communication period, one cluster head plays the role of a message sender, other two cluster heads are assistants to help sender broadcast messages. Moreover, we design two types of communication modes, namely whole- network broadcast and intra-cluster broadcast, to meet different requirement of a sender.

The quantum secret broadcast scheme is described as follows:

Step 0: Initializing

where

Before communication, quantum channel of GHZ states should be checked for potential attacks, the GHZ among A, B and C can be rewritten as:

where

performs measurement under basis

Step 1: Encoding key updating. When each communication period begins, three cluster heads generate a two-bit “encoding key”

Assume that it is A’s turn to produce the “encoding key”

qubits for B and C, respectively, in which the state

represented

When B, C both receive the encoding key

After that, the whole network knows the latest encoding key. So the encoding key updating phase is completed.

Step 2: Communication mode selecting. To meet different requirement of cluster heads, the communication mode can be selected to whole-network broadcast or intra-cluster broadcast.

If a cluster head needs to broadcast messages to the whole network, it will announce an application in this step. Then three cluster heads discuss to select one appropriate cluster heads to be the sender of whole-network broadcast. Otherwise, the communication switches into the mode of intra-cluster broadcast if no cluster heads announce an application.

1) Whole-network broadcast

Step 3: Message encoding. Assume that cluster head A is chosen to be the sender. For each shared GHZ state, she prepares an auxiliary EPR state as follows:

state.

We define four unitary operators:

Step (3.1) A encodes message on EPR state with two operators

where

For convenience, we assume the encoding key

Step (3.2) A applies a CNOT gate on both particles

Step (3.3) A performs a Hadamard gate on

System state after encoding is as shown in

Step4: Measurement result broadcasting. Sender A performs Bell-state mea- surement on both particles

Step 5: Message decoding.

Step (5.1) All the cluster members consider three cluster heads’ measurement results and can read out A’s messages

Step (5.2) As

Let us take an example to illustrate how the mode of whole-network broadcast works. At first, we assume that the encoding key

B’s and C’s results | ||||
---|---|---|---|---|

A’s results | ||||

00 | 11 | 11 | 00 | |

01 | 10 | 10 | 01 | |

11 | 00 | 00 | 11 | |

10 | 01 | 01 | 10 |

to read out massages

2) Intra-cluster broadcast

Assume that no cluster heads announce an application, the communication will switch into the mode of intra-cluster broadcast, in which three cluster heads can send messages to their own cluster members. We take A as an example to illustrate this mode.

Step 3: Message encoding. Similarly, based on the idea of QSS, A randomly selects two cluster members (denote as

As is shown in

We define two unitary operators for encoding:

A encodes one bit message on particle 1 with two operators

where

For convenience, we assume “

From Equations (21)-(22), system state is changed as shown in

Step 4: Measurement result Broadcasting. A performs a Bell-state mea- surement on both particles 1 and 3,

Step 5: Message decoding.

Step (5.1) A’s cluster members consider three parties’ results and read out message

Step (5.2)

We take an example of A cluster to illustrate how intra-cluster mode works. Assume that the encoding key

Our scheme aims to achieve message broadcast in WQN, which attempts to extend communication mode and improve network performance. Different from conventional schemes based on quantum teleportation [

For the reason of applying QSS, our scheme has differences with conventional schemes which are based on quantum teleportation in many aspects. Our scheme transmits classical messages by broadcast, while previous ones transmit quantum state by teleportation. Our scheme makes a attempt to achieve message broadcast, the number of receivers can be

A’s results | ||||

0 | 1 | 1 | 0 | |

0 | 1 | 1 | 0 | |

1 | 0 | 0 | 1 | |

1 | 0 | 0 | 1 |

nodes as assistants to build a routing path from source to destination, the farther distance between source and destination is, the more assistant nodes are needed. we denote

The efficiency in our quantum communication protocol can be defined as:

where

We make a comparison between our scheme and Cao’s scheme [

Consider a situation that a cluster head sends messages to his and other cluster heads’ members. In our scheme,

We can see from

Items | Our scheme | Conventional schemes |
---|---|---|

Basic technology | QSS | Quantum teleportation |

Things to transmit | Classical message | Qubit |

Communication mode | Broadcast | Point-to-point |

Assistant nodes | 2 | |

Number of receivers | 1 |

Scheme | Our scheme | Cao’s scheme |
---|---|---|

1 | 1 | |

5 | ||

1 | ||

3 | ||

6 |

In the proposed scheme, no qubits carrying messages are transmitted directly, so quantum channel only exists in the GHZ states. If an eavesdropper Eve cannot escape from the security check at the phase of Step 0: Initializing, our scheme is secure. The security check method of our scheme is the same as QSS schemes in [

The error rate involved in Eve is

Considering that receivers read out messages according to the three cluster head’s measurement results, another secure problem is that if an eavesdropper also knows the correlation between messages and measurement results, it can obtain messages. To solve this problem, our scheme generates an encoding key

In the whole-network mode, by randomly selecting a key, an eavesdropper will recover a message with the error rate 75%, while in the intra-cluster mode, the error rate is 50%. So our scheme uses security check for quantum channel and key updating for secret messages to ensure security.

In this paper, a quantum secret broadcast scheme was proposed to solve efficiency problem in WQN, where each two bits are encoded in an auxiliary EPR states. The proposed scheme constructs a cluster network cored on three- party QSS, three cluster heads share three-party GHZ states, and each cluster head shares EPR pairs with its own cluster members. For different requirement of cluster heads, the scheme can be selected into whole-network broadcast, in which one cluster head is message sender and other two cluster heads are assistants to help broadcast messages to whole network, or intra-cluster broad- cast, in which each cluster head chooses two cluster members as assistants to help broadcast messages to its intra-cluster members. Furthermore, a wireless quantum network with more than three cluster heads needs to be investigated for extensive application.

This project was supported by the National Natural Science Foundation of China (No. 61571024) and the National Key Research and Development Program of China (No. 2016YFC1000307) for valuable helps.

Shang, T., Du, G. and Liu, J.W. (2017) Quantum Secret Broadcast for Wireless Quantum Networks. Int. J. Communications, Network and System Sciences, 10, 7-18. https://doi.org/10.4236/ijcns.2017.108B002