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Efficient quantum private comparison employing single photons and

collective detection

Bin Liu, Fei Gao, Heng-yue Jia, Wei Huang, Wei-wei Zhang, Qiao-yan Wen

abstract

Two efficient quantum private comparison (QPC) protocols are proposed, employing single

photons and collective detection. In the proposed protocols, two distrustful parties (Alice and Bob)

compare the equivalence of information with the help of a semi-honest third party (TP). Utilizing

collective detection, the cost of practical realization is reduced greatly. In the first protocol, TP

gains the result of the comparison. While in the second protocol, TP cannot get the comparison

result. In both of our protocols, Alice and Bob only need be equipped with unitary operation

machines, such as phase plates. So Alice and Bob need not to have the expensive quantum devices,

such as qubit generating machine, quantum memory machine and quantum measuring machine.

Security of the protocols is ensured by theorems on quantum operation discrimination.

Keywords: quantum private comparison, single photons, collective detection

1. Introduction

Along with the rapid development of quantum mechanics in recent years, more and more

attentions are paid to the applications of quantum mechanics in communication, such as quantum

information and quantum cryptography. Since Bennett and Brassard designed the first quantum

cryptographic protocol [1] in 1984, various quantum cryptographic protocols have been proposed,

such as quantum key distribution [2-10], quantum coin flipping [11-13], quantum secure

multiparty computation (QSMC) [14-21] and so on.

Secure multiparty computing (SMC), also called secure function evaluation, has been

extensively studied in classical cryptography. Private comparison is an important branch of SMC,

which allows two participants to determine whether they own the same secrets without disclosing

their secrets. Based on the millionaires' problem presented by Yao in 1982 [22], in which two

millionaires determine who is richer without knowing each others' actual property, Boudot

proposed a private comparison protocol [23] used to determine whether two millionaires are

equally rich . However, the security of classical private comparison protocols relies on the

computation complexity, which is susceptible to the strong ability of quantum computation. In fact,

Lo pointed out that it is impossible to construct a secure equality function in a two-party scheme

[24].

Quantum private comparison (QPC) is an attractive application of quantum mechanics. With

the help of a third party (TP), QPC allows two parties (Alice and Bob) to check whether their

secrets are equal without disclosing any information about their secrets. As opposed to classical

private comparison, the security of QPC does not rely on computational assumptions but simply

on the laws of physics such as Heisenberg uncertainty principle and quantum no-cloning theorem.

Yang et al. designed the first QPC protocol [25]. Since then many QPC protocols have been

proposed [26-30]. Most of them employ entangled states [25, 27-30]. In all of the previous QPC

protocols, TP is limited to be semi-honest. A semi-honest TP is a party who always follows the

procedure of the protocol. He/she will take a record of all intermediate computations, and will not

be corrupted by an outside eavesdropper. However, he/she might try to steal the information from

the record. What's more, in all these protocols, detection is taken in every step of the particles'

transmission. However, this detection strategy which is called as the step-by-step detection makes

these protocols intricate and inefficient. On the other hand, the action of detection always requires

the participant to have quantum devices, e.g., the qubit generating machine, or the quantum

memory, or the quantum measuring machine. Considering that these devices are expensive in the

practical situation, it is unrealistic that every participant is equipped with all these quantum

devices. Therefore, QPC protocols would be more efficient and easier to achieve if the detection is

taken after the whole process of the particle transmission, i.e. the collective detection.

In this paper, we propose two QPC protocols employing single photons and collective

detection, which is more efficient and easier to realize than the previous QPC protocols. More

details on the proposed protocols and the security analysis are discussed in later sections. Section

2 describes the two protocols. The security is analyzed in Section 3. Finally, a short conclusion is

given in Section 4.

2. The proposed QPC protocols employing single photons and

collective detection

In this section, two efficient QPC protocols are described in steps. In the first protocol, TP

learns the result of the comparison. While in the second protocol, TP has no idea of the

comparison result. In both the two protocols, Alice and Bob only need be equipped with phase

plates. Here is our first QPC protocol (See Fig 1.).

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