Multi-party quantum private comparison of size relationship with two third parties based on d-

2025-05-02 0 0 412.66KB 15 页 10玖币

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Multi-party quantum private comparison of size

relationship with two third parties based on

dimensional Bell states

Jiang-Yuan Lian, Xia Li, Tian-Yu Ye*

College of Information & Electronic Engineering, Zhejiang Gongshang University, Hangzhou 310018, P.R. China

E-mail: yetianyu@zjgsu.edu.cn (T.Y. Ye)

Abstract: In this paper, we put forward a multi-party quantum private comparison (MQPC) protocol with two

semi-honest third parties (TPs) by adopting

-dimensional Bell states, which can judge the size relationship of

private integers from more than two users within one execution of protocol. Each TP is permitted to misbehave on

her own but cannot collude with others. In the proposed MQPC protocol, TPs are only required to apply

dimensional single-particle measurements rather than

-dimensional Bell state measurements. There are no

quantum entanglement swapping and unitary operations required in the proposed MQPC protocol. The security

analysis validates that the proposed MQPC protocol can resist both the outside attacks and the participant attacks.

The proposed MQPC protocol is adaptive for the case that

users want to compare the size relationship of their

private integers under the control of two supervisors. Furthermore, the proposed MQPC protocol can be used in the

strange user environment, because there are not any communication and pre-shared key between each pair of users.

Keywords: Multi-party quantum private comparison; size relationship;

-dimensional Bell states; semi-honest

third party.

1 Introduction

Classical private comparison is one of the most important branches of classical secure

multiparty computation (SMC). In 1982, the first classical private comparison protocol, called as

the millionaire problem, was proposed by Yao [1], which aims to determine who is the richer one

within the mentioned two millionaires without leaking their actual wealth. However, the security of

classical private comparison relies on the computation complexity, which means that the

corresponding protocol cannot guarantee the security as long as the computer has enough ability to

deal with extremely complex data. In order to overcome this problem, quantum private comparison

(QPC) was put forward in 2009 by combining quantum mechanics and classical private comparison

[2]. Subsequently, lots of two-party QPC protocols of equality, which can compare the equality of

private inputs from two users, have been proposed [3-19]. In 2013, the first multi-party quantum

private comparison (MQPC) protocol of equality, which can compare the equality of private inputs

from more than two users within one execution of protocol, was put forward by Chang et al. [20].

Later, numerous MQPC protocols of equality have been constructed with different quantum

technologies [21-26]. However, none of the QPC protocols in Refs.[2-26] have the function of

judging the size relationship (i.e., greater than, equal to or smaller than) of private inputs from more

than two users within one execution of protocol, which restricts their applications in practice.

In 2013, the first QPC protocol of size relationship was put forward by Lin et al. [27] by using

-dimensional Bell states, which can compare the size relationship of private inputs from two users.

Then, Yu et al. [28] designed a two-party QPC protocol of size relationship with

- dimensional

single-particle states in 2013; Guo et al. [29] proposed a two-party QPC protocol of size relationship

based on entanglement swapping of

- dimensional Bell states in 2014; Chen et al. [30] put forward

a two-party QPC protocol of size relationship via quantum walks on circle in 2021. However, the

QPC protocols of size relationship in Refs.[27-30] are not feasible for the multi-user circumstance.

Fortunately, in 2014, Luo et al. [31] put forward the first MQPC protocol of size relationship by

using

-dimensional entangled states, which can judge the size relationship of private inputs from

more than two users within one execution of protocol. Since then, a series of MQPC protocols of

size relationship have been proposed, such as the ones with single-particle states [32-33], the one

with

-dimensional GHZ states [34] and the one with

-dimensional Bell states [35]. At present,

the number of MQPC protocols of size relationship is still few. Moreover, the first protocol of

Ref.[32] is the only MQPC protocol of size relationship which has two supervisors.

Based on the above analysis, in this paper, we are devoted to considering the case that

strange users want to compare the size relationship of their private integers under the control of two

supervisors. We use

-dimensional Bell states to design a novel MQPC protocol of size relationship

with two semi-honest third parties (TPs). Two TPs only need to perform

- dimensional single-

particle measurements rather than

-dimensional Bell state measurements. The proposed MQPC

protocol requires neither quantum entanglement swapping nor unitary operations. The proposed

MQPC protocol can be adaptive for the strange user environment.

2 Protocol description

-dimensional Bell state can be depicted as

vjje

iju

vu = 

−





, (1)

where

 

, 0,1, , 1u v d−

, and



denotes the modulo

addition. Two common conjugate bases in

the

-dimensional quantum system can be described as

 

10 , 1 , , 1Td=−

, (2)

 

20 , 1 , , 1T F F F d=−

, (3)

where

1it

F t e





−

=

with

0,1, , 1td=−

represents the

- dimensional discrete quantum

Fourier transform. It is apparent that, when the two qudits of the

-dimensional Bell state in Eq.(1)

are measured with the

basis, they are collapsed into

and

jv

, respectively. As a result, we

have

jv

⊝

jv=

, where ⊝ denotes the modulo

subtraction. We will use this property of the

dimensional Bell state in Eq.(1) to design the proposed MQPC protocol.

Suppose that there are

strange users,

, , , n

P P P

, whose private integers can be represented

ppp ,,, 21 

, respectively;

and

are two TPs, each of whom is permitted to misbehave on

her own but cannot collude with others. Here,

 

0,1, ,

ph

and

1,2, ,in=

. When

( )

,2 0mod d =

we set

; otherwise, we set

h−

and

pre-share the private key

through a secure

quantum key distribution (QKD) protocol,

 

0,1, , 1

kd−

and

1,2, ,in=

, , , n

P P P

want to

compare the size relationship of their private integers under the control of

and

on the condition

that there are not any communication and pre-shared key between any two of them, which implies

that only when both

and

agree can

, , , n

P P P

obtain the size relationship of their private

integers. In the following, we put forward a novel MQPC protocol suitable for the strange user

environment to accomplish this goal. Here, the quantum channels are assumed to be noiseless, while

the classical channels are supposed to be authenticated.

Step 1:

prepares

-dimensional Bell states in Eq.(1), and picks out all of the first and the

second particles of these

Bell states to form sequences

and

, respectively. Here,

and

can be

represented by

1 1 1 1

, ,..., n

S m m m



=

(4)

and

2 2 2 2

, ,..., n

S m m m



=

, (5)

respectively. Then,

records the second label of the

- dimensional Bell state as

. Here,

 

, 0,1, , 1

m m d−

and

1,2, ,in=

Step 2:

prepares

decoy photons which are randomly selected from the sets

and

. Then

she inserts them into

to form a new sequence

and sends it to

via a quantum channel. After all

particles of

are received by

and

start the security check on the quantum channel between

them. To be specific,

announces the positions and the preparation bases of all decoy photons in

; then,

acquires the measurement results of these decoy photons by measuring them with

the correct measuring bases and returns these measurement results to

; afterward,

judges

whether there is an eavesdropper or not by comparing the received measurement results with decoy

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Multi-partyquantumprivatecomparisonofsizerelationshipwithtwothirdpartiesbasedon-dimensionalBellstatesJiang-YuanLian,XiaLi,Tian-YuYe*CollegeofInformation&ElectronicEngineering,ZhejiangGongshangUniversity,Hangzhou310018,P.R.ChinaE-mail:yetianyu@zjgsu.edu.cn(T.Y.Ye)Abstract:Inthispaper,weputforwardamul...

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