Private comparison is the basis of many encryption technologies, and several related Quantum Private Comparison (QPC) protocols have been published in recent years. In these existing protocols, secret information is encoded by using conjugate coding or orthogonal states, and all users are quantum participants. In this paper, a novel semi-quantum private comparison scheme is proposed, which employs Bell entangled states as quantum resources. Two semi-quantum participants compare the equivalence of their private information with the help of a semi-honest third party (TP). Compared with the previous classical protocols, these two semi-quantum users can only make some particular action, such as to measure, prepare and reflect quantum qubits only in the classical basis

Since Bennett and Brassard published the initial Quantum Key Distribution (QKD) protocol [

Secure Multiparty Computing (SMC), also known as secure function evaluation, is a primitive basic form of distributed computation. It can correctly distribute computing to outputs when inputs are given by a group of distrustful users. As a subfield of QSMC, Quantum Private Comparison (QPC) was first established as a computing task by Yao [

Boyer et al. [

In order to improve the qubits’ efficiency and make classical users be involved in quantum private comparison, we propose an SQPC protocol based on Bell state. Two semi-quantum users can compare their private information with the help of a semi-honest TP. Nevertheless, both of them can only make specific actions, such as measuring, preparing and reflecting the quantum qubits on the classical basis

The rest of this paper is arranged as follows. The detailed description of the SQPC protocol is described in Section 2, and the security analysis of the protocol is explained in Section 3. In Section 4, the discussion and conclusion of this protocol are provided, and the following semi-quantum research work is analyzed and arranged.

In the following, the detailed description of an SQPC scheme is provided step by step. Two semi-quantum participants, Alice and Bob, are involved. Both of them have the same length of secret information. _{AB} ( _{AB} is used for indicating Alice and Bob to choose the operation of MEASURE or REFLECT. When

The description of the scheme is the following steps.

_{1} and _{2}, consisting of sequences _{1} and _{2}. Then TP sends the qubits _{1} and _{2} to Alice and Bob one by one, respectively.

_{AB}, Alice (Bob) performs the operational rules of semi-quantum, MEASURE or REFLECT, on each qubit of _{1} (_{2}) sequence. When _{i} for calculating _{A}^{i} (if

_{1}_{2}

_{AB} to TP through the public channel. If these two _{AB} are not the same, TP terminates the protocol. Otherwise, TP proceeds to the next step.

_{AB}, TP divides the result of measurement into MEASURE (

Then TP takes the next two steps:

Verifying the equivalence. Assume that TP prepares the initial Bell state to be

Publishing the result of comparison. Assume that TP prepares the initial Bell state to be

For clarity, we describe the flowchart of the proposed protocol in ^{rd}, 4^{th}, 5^{th} and 7^{th} qubits are used for security detection. If the results of Bell basis measurement are not ^{nd} and 3^{rd} positions. Thus TP announces 1. It can be concluded that TP has finished the comparison and cannot obtain any secret information from both sides.

In this section, the security of the proposed protocol is analyzed from two aspects: (1) The secret information of participants is plagued by external eavesdroppers, and (2) Dishonest users or the semi-honest TP may steal the secret information in the procedure of the scheme. Then, the efficiency analysis of the scheme with some previous SQPC protocols are provided.

We will give out the eavesdropping detection that Eve may take at every step of the proposed protocol.

In Step 1, When TP sends _{1} and _{2} to Alice and Bob, respectively, Eve may launch an attack on the quantum channel. The attack is titled the Trojan horse attack [

The external eavesdropper Eve intercepts the Bell states sent from TP to Alice (Bob) and prepares two-particle states according to measurement results, then she sends these qubits to Alice and Bob. Eve will be inevitably detected for two reasons: (1) Two-particle states can only be prepared randomly because this is the closest method to simulating the original sequence, and (2) Alice and Bob’s operation are still random to Eve, even though Alice and Bob publish the sequence _{AB} in Step 4. For example, the initial Bell state TP prepared is

When Alice and Bob choose REFLECT operation, TP makes Bell basis measurement on

The initial Bell state | Fake particles | Alice(Bob)’s choice | The result | Probability of being detected or M/R | |
---|---|---|---|---|---|

REFLECT | secret Inf | 1/2 |
100% | ||

MEASURE | Different | 1/2 |
Right | ||

Same | 1/2 |
Mistake | |||

REFLECT | 1/2 |
0 | |||

MEASURE | Different | 1/2 |
Mistake | ||

Same | 1/2 |
100% | |||

REFLECT | 1/2 |
100% | |||

MEASURE | Different | 1/2 |
Mistake | ||

Same | 1/2 |
Mistake |

It should also be pointed out that Even Eve cannot obtain any secret information by performing intercept-resend attacks. She can still affect the comparison of secret information in some cases. The protocol can avoid Eve’s mistake by performing the detection firstly (Step 5).

The measure-resend attack refers to that Eve intercepts the particles sent from TP to Alice (Bob), measures them, then sends the measured states to Alice (Bob). She inevitably causes the original Bell state to collapse into two-particle states. When Alice and Bob choose REFLECT operation, TP only has 50% possibility to obtain the initial Bell state. For the MEASURE operation, Eve cannot be detected and does not cause any interfere with the comparison result. In

The initial Bell state | The measurement result | Alice(Bob)’s choice | The result of TP’s measurement | Probability of being detected or M/R | |
---|---|---|---|---|---|

REFLECT | secret Inf | 1/2 |
50% | ||

MEASURE | Same | 1/2 |
Right | ||

Different | 1/2 |
Right | |||

REFLECT | 1/2 |
50% | |||

MEASURE | Same | 1/2 |
Right | ||

Different | 1/2 |
Right |

During the flip attack, Eve interferes with the correctness of the comparison by modifying the intercepted particles’ information. This scheme can use the entanglement correlation of the Bell states to avoid this attack. Assuming that TP prepares the initial Bell state to be

The entangle-measure attack means that Eve performs attack (UE,UF) on the Bell states among TP, Alice and Bob. UE and UF share a common probe space with initial state

Thus

In the proposed protocol, dishonest users and semi-honest TP may try to obtain secret information. We analyze them in two ways.

In Step 1, TP sends S1 to Alice and S2 to Bob. Firstly, both Alice and Bob can never perform certain operations on the other sequence. TP performs all joint measurements. This is the reason why Alice or Bob cannot obtain other’s secret information. Besides, if Alice or Bob deliberately choose different KAB sequence, it will be checked out in Step 4. In the last step, TP only uses 1 qubit to stand for the equivalence of their private information. They have no way to know the different of secret information.

The Semi-honesty determines that TP must implement the protocol base on the rules. Therefore, TP has only one way to obtain the private information of participants through M sequence (the sequence are all qubits that participants encode with their private information). For example, if the M sequence is 00 11 01 10, Eve only has the probability of 1/2 to obtain the initial state. When n is large enough, the probability of obtaining the private information of Alice is

In this subsection, we aim to compare the efficiency of the proposed protocol with an SQPC protocols from References [

In terms of the quantity of the preparation of initial states and workload of the participants, this protocol is better than Reference [

In addition, the qubits’ efficiency of the proposed protocol is highest among these three protocols. The qubits efficiency [

As for the proposed protocol, in order to compare n-bit secret information of Alice and Bob (

The protocol of Reference [ | The protocol of Reference [ | The present protocol | |
---|---|---|---|

Characteristic | Measure-resend | Measure-resend | Measure-resend |

Quantum resource | Bell entangled states (8 |
Two-particle product states (8 |
Bell entangled states (2 |

TP | Semi-honest | Semi-honest | Semi-honest |

Quantum measurementfor TP | Bell basis measurements for case 1 and case 4 | Single-photon measurements (4 kinds of situations) | Bell basis measurements for all returned particles |

Whether TP know the comparison result or not | Yes | Yes | Yes |

Pre-shared SQKD/SQKA key | Yes | Yes | No (K_{AB} can be considered as classical pre-shared key) |

Qubit efficiency |

In this paper, we have proposed a novel SQPC protocol with detailed procedures based on Bell entangled states. As the only quantum participant, TP can calculate the equivalence of private information of Alice and Bob, but he cannot obtain any private information of them. In addition, TP only needs to release 1 qubit through public channel to announce whether their private information is same. In addition, the paper has shown the detail of security against some eavesdropping attacks, and the qubit efficiency of the proposed scheme is higher than two other protocols.

Meanwhile, the quantum participants need several techniques in the scheme, such as the generation of Bell states in Reference [

As for the decoherence noise channel, the coupling of the quantum system to the environment will cause the decay of quantum information. It can be described as:

where

Further, future studies will focus on analyzing the impact of the noise channel to quantum cryptography protocols and preventing the classical users’ operations from the influence of noise channels. Our studies also continue to track the possibilities between block-chain and quantum secure communication in Reference [