Quantum Risk Assessment Model Based on Two Three-Qubit GHZ States

: With the acceleration of the construction of quantum communication networks, scholars have proposed different quantum communication protocols for different application scenarios. However, few scholars pay attention to the risk assessment process before communication. In this paper, we propose a novel quantum risk assessment model based on quantum teleportation technology with two three-qubit GHZ states. Only by using Bell states measurements (BSMs) and two-qubit projective measurements (PJMs), the communicators can recovery any arbitrary two-qubit state. This protocol can transmit two-dimension risk assessment factors with better security performance. On the one hand, more suf ﬁ cient evaluation factors allow the two communicating parties to more objectively eval-uate the risk level of communication with the other party, and on the other hand, it also improves the qubit ef ﬁ ciency of the protocol. Moreover, we introduce the third party in this scheme can be semi-trusted, which must be full-trusted in our previous work. This change can reduce the dependence of the communication parties on the third-party organization and improve the privacy of communication. The security analysis shows that this scheme can resist internal and external attacks, and the quantum circuit diagrams also prove that our protocol is physi-cally easier to implement.

From the introduction above, we can find that most of the quantum communication protocols are based on QKD system which are proved to be unconditional secure, and researchers have pre-supposed that the communicating parties are honest and reliable, or set some prerequisites so that the protocol is not disturbed by the honesty of the communication participants. However, as classic communication network, we must perform a risk assessment of the communication participants to evaluate their reliability and honesty in a quantum communication network. We have studied quantum trust model based on node evaluation [24]. In our previous work, we used quantum teleportation to transmit single qubit as the trust value of node. However, with the deepening of our research, we find that the risk assessment factor value of a node in quantum communication network should be multi-dimensional, and we should find an efficient method to evaluate the risk assessment of the communication participants.
Recently, Li et al. [2] introduced a practical quantum teleportation scheme with two three-qubit GHZ states. The arbitrary two-qubits w j i AB ¼ a 00 j iþb 10 j iþc 01 j iþg 11 j i ð Þ AB can be teleportation only by using Bell states measurements (BSMs) and two-qubit projective measurements (PJMs) with their protocol.
Inspired by Li et al., we are thinking whether we can transmit more quantum risk assessment factors by using more efficient quantum teleportation technology. We find that the arbitrary two-qubits w j i AB ¼ a 00 j iþb 10 j iþc 01 j iþg 11 j i ð Þ AB can be applied to carry two-dimension risk assessment factors.
In this paper, we introduce a quantum risk assessment model based on two three-qubit GHZ states. The features of our protocol are reflected in the following aspects: 1. Only by using Bell states measurements (BSMs) and two-qubit projective measurements (PJMs), the communicators can recovery any arbitrary two-qubits state. Our protocol can transmit two-dimension risk assessment factors with better particle utilization efficiency and lower quantum operation complexity. 2. The third trust party (TTP) must be honest in our previous work [24], in this scheme, however, the third party can be semi-trusted because it cannot obtain any useful information when it performs the illegal operation.
The organization of this paper is demonstrated as follows. In Section 2, we review the basic theory of quantum risk assessment, and quantum teleportation with two three-qubit GHZ states. Then we express our proposed scheme in Section 3, and the security analysis is discussed in Section 4. Finally, our conclusion of this scheme is drawn in Section 5.

Fuzzy Comprehensive Theory of Quantum Risk Assessment
There are mainly two relationship of trust in the research of trust in QCN [24]: 1) Objective trust. Objective trust refers to the state of trust based on objective facts and not interfered by subjective emotions. 2) Subjective trust. Subjective trust refers to the state of trust based on experience and personal emotion. In the real classic network, when a new node wants to join the network, we usually use subjective trust to make a preliminary evaluation, and determine the specific trust value of a node through objective trust, because objective trust is based on the fuzzy and uncertain real evidence to make a credit evaluation. For a more vivid description of fuzzy comprehensive evaluation theory, we set up a QCN with n user nodes (u 1 ; u 2 ; …; u n ) and a trust third party (TTP) and the communication diagram is shown in Fig. 1. The establishment rules of QCN are as follows: each node must register with the TTP before joining the network, and the submitted registration information contains various risk assessment factors. The TTP conducts fuzzy comprehensive risk assessment for all nodes according to the following steps.
Step 1 TTP determine the factor set. The determination of factor set refers to the set of all factors that evaluate the object to be evaluated (in this case, the user node u i ), and it is the various elements that can affect the evaluation of the object to be evaluated. Factor set refers to the set of factors that can evaluate behavior credibility. Here, it is assumed that the factor set of the evaluation node is E ¼ e 1 ; e 2 ; …; e n f g , where n is the number of factors. When the node u i submits the registration information, TTP will sort out and classify the information to form a set of different factors for risk assessment.
Step 2 TTP determine the comment set. Comment set refers to the different evaluation levels of a single factor, which is a set of various comment levels for a certain factor. For example, to classify scores, you can define a comment set {excellent, good, medium, poor} according to scores, which is a comment set. Here, set the comment set as C ¼ c 1 ; c 2 ; …; c m f g , where m represents the total number of comments, which can also be called comment level.
Step 3 TTP determine the weight vector of evaluation factors. Because the proportion of each factor of the evaluated research object is not the same, TTP needs to introduce the weight vector of evaluation factors.
We set the W ¼ w 1 ; w 2 ; …; w t 0 w i 1 j f g as the weight vector of evaluation factors, P t i¼0 w i ¼ 1 and t is the number of factors set. It is worth to note that the determination of the weight value depends on the subjective evaluation result of TTP. The commonly used methods to determine the weight value include expert experience method, expert investigation method, eigenvalue method and weighted average method. Step 4 The establishment of fuzzy evaluation matrix. Single factor fuzzy evaluation is to determine the membership degree of evaluation object to evaluation set C by evaluating single factor, which is called single factor fuzzy evaluation. The matrix that combines the membership of all single factor fuzzy evaluation to the comment set is called the fuzzy evaluation matrix. Here we set a single fuzzy evaluation vector as R jk ¼ r j1 ; r j2 ; …; r jk È É , then the fuzzy evaluation matrix can be defined as follow: Step 5 Vector synthesis of comprehensive evaluation results. TTP selects the appropriate fuzzy synthesis operator for the corresponding synthesis operation, and finally obtains the fuzzy comprehensive evaluation result vector of all the evaluated objects. In this protocol, we define the fuzzy comprehensive evaluation result as: Step 6 TTP uses weighted average method to obtain fuzzy comprehensive evaluation results. The principle of weighted average is to sum the result vectors of fuzzy comprehensive evaluation by weighting, and calculate the average value. The expression of weighted average principle can be expressed as: where k is the undetermined coefficient, and the value of it is k ¼ 1 or k ¼ 2.
Step 7 TTP prepares the arbitrary two-qubits quantum states w j i AB ¼ a 00 j i þ b 10 j i þ c 01 j i þ g 11 j i ð Þ AB to transmit the risk assessment factors of node u i based on the obtained fuzzy comprehensive evaluation results. In our previous work, we use intuitionistic fuzzy sets theory to describe a node's trust degree when quantum communication occurs. A single quantum qubits ' j i ¼ a 0 j i þ b 1 j i to express the node's trust factor in quantum communication network, where a j j 2 represents the membership degree of trust factor and b j j 2 represents the non-membership degree of trust factor, and a j j 2 þ b j j 2 ¼ 1. However, as research progresses, we think that the trustworthiness of a node in a realistic quantum network environment should be determined by a combination of factors, and assessing the risk of a node cannot be based on a single factor alone. Then we tried to use the arbitrary two-qubits quantum states w Þ AB to encode more risk evaluation factors, a j j 2 þ b j j 2 þ g j j 2 þ c j j 2 ¼ 1 and we can set that a j j 2 to represent the legal operation degree of user node, b j j 2 represents the honest degree of user node, c j j 2 represents the identity credibility degree of user node, and g j j 2 represents the communication reliability degree of user node.

Quantum Teleportation of an Arbitrary Two-Qubit State by Using Two Three Qubit GHZ States
A quantum teleportation protocol by using two three-qubit GHZ states and the six-qubit entangled state has been established [2], and Li et al. have described the steps of the protocol in detail. Before giving the detailed steps of our scheme, the basic theory of quantum teleportation protocol by using two three-qubit GHZ states need to be reviewed. Suppose that Alice, Bob and Charlie share two three-qubit GHZ states. Alice possesses qubits pairs (1, 6), Bob possesses qubits pairs (2,4) and Charlie possesses qubits pairs (3,5) respectively. The two three-qubit cluster state is described as follows: Alice holds the arbitrary two-qubits w Þ AB , which is the qubits to be transmitted. Now the whole system can be written as: Alice performs BSMs on her qubit pairs (A, 1) and (B, 6). In more detail, Alice perform the tensor product operation on w j i AB and S 16 ( w j i AB S 16 ), then she uses Bell measurement basis 00 j i; 11 j i; 10 j i; 01 j i ð Þ to obtain the BSMs results. After that, she can get 16 kinds of measurement results with equal probability of 1/16. After Alice's measurement operations, the qubit pairs (2, 3, 4, 5) will also have 16 kinds of collapse results to the corresponding states w j i 2345 . Alice informs the measurement results to Bob and Charlie through a classical communication channel. Then Charlie performs a two-qubit projective measurement (PJMs) on the qubit pairs (3,5), and Charlie sends the measurement outcomes to Bob. After that, Bob performs an appropriate unitary on his qubit pairs (2,4) and he can reconstruct the state w j i AB , which means that the qubit w j i AB holds by Alice has been transmitted to Bob with quantum teleportation scheme. The detailed quantum teleportation process is shown in Fig. 2. It should be noted that this scheme uses two three-qubit GHZ states to construct quantum teleportation channels for information transmitting. Compared with other multi-particle states, GHZ state particles have better particle state stability and are easier to prepare. Moreover, this new quantum teleportation method is much more secure and efficient than our previous protocol.

Protocol
In this section, we describe the process of our quantum trust model in detail. And we assume that STP is the semi-trust third party, node u i wants to communicate with node v i , the detail steps of our scheme are as follow.

Initial Phase
On assumption that the information used to describe the fuzzy comprehensive risk assessment value of u i is stored in STP previously by means of registering, which is expressed in the form of quantum state as follows (prepared by STP before sending): Here a i 2 þ b i 2 þ c i 2 þ g i 2 ¼ 1 and i ¼ 1; 2; …; n. STP prepares the two three-qubit GHZ states, each of which is in the state w

Distribute the Quantum Key
After registration, STP prepares and shares the quantum key Key uT with node u i , and the quantum key Key vT with node v i . This process can be implemented by QKD system. More Specifically, Wang et al. [45] proposed a novel Measurement Device Independent Quantum Key Distribution (MDI-QKD) scheme with two-mode state source. Their protocol is relatively simple to operate and enables the quantum key to be transmitted over long distances with a small number of quantum resources. These features meet the requirements for quantum resources in quantum communication networks, so we choose this protocol to generate the quantum key.
In our scheme, node u i and node v i are the two senders who control the light sources separately to generate a two-mode state. In general, the T-mode is detected by u i or v i , and the S-mode is sent to the Semi-Trust Party (which named STP in our protocol). Therefore, our scheme needs an additional signal state and single quantum qubits as light source (when the MDI-QKD system uses the BB84 protocol as original protocol for quantum key generation).

Establish Communication Connection
Suppose that node u i wants to communicate with node v i , u i sends a request to STP. When STP receives the request and confirms which is sent from u i , STP prepares quantum state w j i AB ¼ a 00 j i þ b 10 j i þ c 01 j i þ g 11 j i ð Þ AB based on the registration information of u i , and informs v i that u i wants to communicate with v i . STP encrypts S 35 with key Key uT and sends it to node u i . Then STP encrypts S 24 with Key vT and sends it to node v i .

Transmit Risk Assessment Factors by Quantum Teleportation
After both two nodes are sure to receive the quantum sequences, the two nodes decrypt the quantum sequence with their secret quantum key. STP performs BSMs on qubit pairs A; 1 ð Þ and B; 6 ð Þ, and the measurement result is M B and the 2; 3; 4; 5 ð Þwill collapse to the corresponding entangled states, which is shown in Tab. 1.
STP informs node u i to perform a two-qubit projective measurements (PJMs) on 3; 5 ð Þ, and the measurement result is M P . Node u i encrypts M P with Key uT and sends it to STP with classic communication channel, then STP decrypts and obtains M P . STP encrypts the M B and M P with Key vT , and sends the result to node v i with classic communication channel. After decryption, based on the information of M B and M P , node v i can recovery the quantum state w j i AB which represents comprehensive trust factor of node u i by using appropriate unitary operation, and the detailed operation is shown in Tab. 2, and U (2, 4) means node v i performs Unitary operation on qubits pairs (2,4). STP sends the detail information about w j i AB to node v i , node v i calculates the risk assessment factors value of u i based on the quantum state w j i AB he reconstructed and compares the two quantum states. According to the above comparison, he can judge whether node u i is credible or not.

Security Analysis
Before starting security analysis, we discuss that third party can be semi-honest: The risk assessment factors information is stored in STP, so STP must be reliable to some extent. But we can use obfuscation function or quantum random numbers to encrypt trust information, and use quantum one-time pad (OTP) to update the node's trust information. The two quantum keys and the three qubit pairs are prepared by STP, which means STP may insert some decoy qubits to eavesdrop the communication information. In our proposed scheme, if STP performs the illegal operation, the entanglement relationship between STP and two nodes will be disrupted, and both two nodes can find this illegal behavior by qubit measurement. Therefore, we can introduce a semi-trust third party to reduce the dependence of a third-party center.

The Internal Attack
Suppose node u i wants to cheat node v i , then node u i must cooperate with Eve to intercept the original qubits sent by STP. After that, node u i will send v i the entanglement qubits that are prepared beforehand. In this case, u i will find it tough to get the appropriate quantum information when he needs to reconstruct the original qubits. In another case, if node v i wants to cheat node u i , with the help of STP, both two nodes can obtain the risk assessment factors value, which means that neither node can cheat the other with internal attack.

The External Attack
If the external attacker Eve wants to forge the trust factor state w j i AB , Eve must know the information about key Key uT and Key vT . The QKD and QOTP guarantee the unconditional security of the two keys and the communication channels. Even if Eve obtains the two keys, the forge is impossible without STP's BSMs outcomes and the qubit pairs S 24 hold by node v i . We can quantify the Eve and node v i 's (u i 's) mutual information of the key Key vT (Key uT ) by Shannon entropy theory. Suppose the probability for Eve is detected by v i (u i ) is P d , and the mutual information can be defined as: The relationship between the amount of mutual information Eve obtains and the probability that he is detected is shown in Fig. 3. It shows that if Eve want to obtain the full information (H B : E ð Þ¼ 1), the probability of the eavesdropping detection is P d ¼ 50%, When the length of the two keys (n) is large enough, Eve's eavesdropping behavior must be detected.

Analysis of the Two Pre-Shared Keys
Due to the unconditional security of quantum key distribution, only STP and node u i know the secret key Key uT . According to the analysis of the protocol flow, Key uT is used to encrypt the quantum sequence S 35 . After that, node v i or external attackers are unable to obtain any information related to Key uT . The analysis of Figure 3: Relationship between the amount of mutual information eve obtains and his detected probability key Key vT is the same as that of Key uT . When a failure occurs in the eavesdropping check or when the secret keys are used for a long period of time does, the new secret keys have to be shared again.

The Qubit Efficiency
Quantum protocol's qubit efficiency can be calculated with equation . Parameter c represents the number of secret bits, q is the total number of transmitted qubits, and b is the total number of classical bits. In our protocol, we assume that the length of w j i AB is n, and the length of measurement results (M B and M P ) is 14n: The length of M P is 2n which is equals to the length of qubit sequence S 35 . The length of M B is 4n which is equals to the length of qubit sequence S 16 and w j i AB . Analysis of the protocol process reveals that the measurement result M B will be transmitted two times. Based on the above analysis, we can obtain that c ¼ 2n, q ¼ 2 Â 2n þ 4n ¼ 8n and b ¼ 4n þ 2n ¼ 6n. In our previous scheme [24], the trust factor values are encoded in a single qubit ' j i a , and TTP performs a Bell states measurement on ' j i a with Bell Þ , and measurement results are transmitted via a classical channel, then node u will perform unitary operation corresponding to TTP's announce classical information 01, 10, 11, and 00. Therefore, we can obtain that c ¼ n, q ¼ 2n þ 3n ¼ 5n and b ¼ 3n. Tab. 3 shows the comparison of this scheme's qubit efficiency with our previous work.

The Implementation of Quantum Circuit
Based on the above analysis, our scheme needs to perform the Bell states measurements (BSMs) operation, two-qubit projective measurements (PJMs) and unitary operation. The quantum qubit prepared in this protocol are two three-qubit GHZ states w j i 123 ¼ 1 2 000 j iþ 111 j i ð Þ 123 and w j i 456 ¼ 1 2 000 j iþ 111 j i ð Þ 456 . Moreover, an arbitrary two-qubits states w j i AB ¼ a 00 j i þ b 10 j i þ c 01 j i þ g 11 j i ð Þ AB is also needed in this protocol.
The quantum circuit implementation diagram of our scheme is shown in Fig. 4. In the Fig. 4, "H" stands for Hadamard-gate operation, which turns a single photon into an entangled state and its transformation process of the Hadamard-gate operation can be represented by Eq. (9).
"þ" represents the range of entangled qubits under the Hadamard-gate operation. And "BSMs m; n ð Þ" implies performing Bell States Measurement on qubits pairs (m, n). "PJMs m ð Þ" means to perform two qubits Projective Measurement on qubit m and "U m; n ð Þ" implies performing Unitary operation on qubits pairs (m, n).

Conclusion
In this paper, we introduce a quantum risk assessment model based on two three-qubit GHZ states. By using a new way of quantum teleportation, our scheme can carry more risk assessment factors, which is more in line with the characteristics of classic networks. The security analysis show that our scheme has a better performance in communication security and qubit efficiency. The quantum circuit diagram also shows that the implementation of our scheme is easier to implement physically. As a future work, we shall research ways to eliminate the influence of various noises, and how to use the advantages of quantum blockchain [46] to reduce dependence on third parties. Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.