Quantum Private Comparison Protocol with Cluster States

Abstract

In this paper, we introduce a quantum private comparison (QPC) protocol designed for two players to securely and privately assess the equality of their private information. The protocol utilizes four-particle cluster states prepared by a semi-honest third party (TP), who strictly adheres to the protocol without deviation or collusion with any participant. The TP facilitates the private comparison by enabling users to encode their information through bit-flip or phase-shift operators applied to the received quantum sequences. Once the information is encoded, the sequences are returned to the TP, who can derive the comparison results without accessing any details of the private information. This design ensures correctness, privacy, and fairness throughout the process. The QPC protocol is robust against both external threats and participant attacks due to the incorporation of the decoy-state method and quantum key distribution techniques. Additionally, the protocol employs unitary operations and Bell-basis measurements, enhancing its technical feasibility for practical implementation. Notably, the proposed protocol achieves a qubit efficiency of up to 50%. This efficiency, combined with its strong security features, establishes the QPC protocol as a promising solution for private information comparisons within the realm of quantum cryptography.

1. Introduction

The rise of the Internet has spurred the growth of cooperative computation applications, where participants collaborate to perform tasks based on their private inputs. A significant challenge in this context is conducting joint computations without compromising the privacy of those inputs, known as the secure multi-party computation (SMC) problem [1]. This area has become a crucial focus of research in modern cryptography. SMC has a wide array of applications, including privacy-preserving data mining, secure auctions, and bidding. Goldreich [2] notes that general results in SMC may not be universally applicable, leading to inefficiencies when directly implemented across diverse contexts. Consequently, specialized protocols tailored to the specific needs of particular applications are often developed to enhance both efficiency and practicality. Colbeck showed that unconditional security cannot be achieved in two-party classical computation for numerous classes of functions [3]. In contrast, quantum mechanics provides a framework that allows for the potential of achieving unconditional security. This paradigm shift opens new avenues for designing secure protocols that leverage the unique properties of quantum systems. The integration of quantum mechanics into secure computation has led to significant advancements that were previously unattainable within classical frameworks. These developments not only bolster security but also facilitate innovative applications in quantum information processing and cryptography. Notable examples include quantum key distribution [4,5,6], quantum key agreement [7,8,9], quantum secure direct communication [10,11,12,13], and quantum private set intersection [14,15,16].

Despite the advantages offered by quantum mechanics, the impossibility of unconditionally secure quantum multi-party computation protocols, as noted by Lo [17], underscores the challenges in achieving absolute security. However, by introducing specific assumptions or conditions, it is possible to enhance security in quantum multi-party computation scenarios. In 2009, Yang and Wen [18] were the first to utilize decoy particles and EPR (Einstein–Podolsky–Rosen) pairs to facilitate a secure private comparison known as the quantum private comparison (QPC) protocol. This protocol enables participants to assess the equality of their inputs while ensuring that any dishonest party cannot access sensitive information. Subsequently, Chen et al. [19] introduced an improved version of the QPC protocol, which leverages triplet entangled states and implements straightforward single-particle measurements, thereby increasing efficiency and practicality in secure comparisons. Since then, researchers have explored a variety of quantum states as resources for QPC protocols, including single photons [20,21,22,23,24,25], Bell states [26,27,28,29,30], multi-qubit entangled states [31,32,33,34,35,36], and d-level quantum states [37,38,39,40]. Additionally, various quantum technologies, such as entanglement swapping and unitary operations, have been integrated into these protocols to facilitate private comparisons.

In the context of cluster states as quantum resources, the QPC protocol proposed by Xu et al. [41] utilizes a four-qubit cluster state to provide a secure and efficient method for comparing private information. By capitalizing on the robustness and symmetry inherent in cluster states, this protocol effectively upholds participant privacy while producing reliable comparison results. Similarly, Sun and Long [42] introduce a novel QPC protocol that employs four-particle cluster states as information carriers, supported by a semi-honest third party (TP). This design allows all participants, including the TP, to perform only single-particle measurements, significantly simplifying technical implementation and enhancing the protocol’s accessibility for practical applications. Chang et al. [43] propose another innovative QPC protocol that utilizes a five-particle cluster state, also with the assistance of a semi-honest TP. This protocol enables the simultaneous comparison of private information from two groups, each consisting of two users, thereby enhancing scalability. Li et al. [44] present an efficient QPC protocol that incorporates entanglement swapping between four-qubit cluster states and extended Bell states. This approach allows for the comparison of three bits of secret inputs during each comparison, thus improving the protocol’s efficiency. Collectively, these advancements in QPC protocols illustrate the potential of cluster states to serve as effective resources for secure and efficient private information comparisons in quantum cryptography.

The four-particle cluster state demonstrates considerable robustness against decoherence. Once a four-qubit cluster state is generated, universal quantum computation can be performed using only measurements and simple local operations. Given these advantageous properties, the application of four-qubit cluster states in quantum private comparison (QPC) protocols is particularly significant. In this paper, we propose a QPC protocol that employs four-particle cluster states, involving a semi-honest third party (TP). The protocol consists of three parties: TP, Alice, and Bob. This framework enables Alice and Bob to securely determine the equality of their private information while ensuring the confidentiality of their inputs. Each participant encodes their information using bit-flip or phase-shift operators on the quantum sequences they receive. After encoding, these sequences are sent back to TP, which can ascertain the comparison results without gaining any insight into the private information. This design ensures correctness, privacy, and fairness throughout the process. It is resilient against both external threats and participant attacks, due to the integration of the decoy-state method and quantum key distribution techniques. Additionally, the protocol employs unitary operations and Bell-basis measurements, enhancing its technical feasibility for implementation. Importantly, our proposed QPC protocol achieves a qubit efficiency of up to 50%.

The remainder of this paper is structured as follows: In Section 2, we provide a detailed description of the QPC protocol. Section 3 presents an analysis of the proposed protocol. Following that, Section 4 offers a comparison. Finally, Section 5 concludes this paper with a summary of our findings.

2. The QPC Protocol

The QPC protocol involves three parties: a third party (TP), Alice, and Bob. A critical assumption is that the TP is semi-honest, meaning it follows the protocol correctly and does not deviate from its specified operations, but may attempt to extract additional information about the participants’ private inputs during the execution of the protocol. Furthermore, the TP cannot collude with either participant.

The protocol we present guarantees the following properties:

Privacy: No participant should gain any information beyond their designated output. Additionally, any external eavesdropping on private inputs will fail.

Correctness: Each participant obtains a correct output.

Fairness: No player can gain an advantage over the other during the comparison process.

A four-particle cluster state can be defined as follows:

$$\begin{array}{l}\left|C_4\right.〉={\displaystyle \frac{1}{2}}\left({\left|0000\right.〉}_{1234}+{\left|0011\right.〉}_{1234}+{\left|1100\right.〉}_{1234}+{\left|1111\right.〉}_{1234}\right)\\ ={\displaystyle \frac{1}{2}}\left({\left|00\right.〉}_{23}{\left|00\right.〉}_{14}+{\left|01\right.〉}_{23}{\left|01\right.〉}_{14}+{\left|10\right.〉}_{23}{\left|10\right.〉}_{14}+{\left|11\right.〉}_{23}{\left|11\right.〉}_{14}\right)\\ ={\displaystyle \frac{1}{2}}\left({\left|{\phi }^{+}\right.〉}_{23}{\left|{\phi }^{+}\right.〉}_{14}+{\left|{\phi }^{-}\right.〉}_{23}{\left|{\phi }^{-}\right.〉}_{14}+{\left|{\psi }^{+}\right.〉}_{23}{\left|{\psi }^{+}\right.〉}_{14}+{\left|{\psi }^{-}\right.〉}_{23}{\left|{\psi }^{-}\right.〉}_{14}\right)\end{array}$$

(1)

where four Bell states $\left|{\phi }^{+}\right.⟩,\left|{\phi }^{-}\right.⟩,\left|{\psi }^{+}\right.⟩$ and $\left|{\psi }^{+}\right.⟩$ are defined as follows:

$$\left|{\phi }^{+}\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left(\left|0\right.〉\left|0\right.〉+\left|1\right.〉\left|1\right.〉\right)$$

(2)

$$\left|{\phi }^{-}\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left(\left|0\right.〉\left|0\right.〉-\left|1\right.〉\left|1\right.〉\right)$$

(3)

$$\left|{\psi }^{+}\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left(\left|0\right.〉\left|1\right.〉+\left|1\right.〉\left|0\right.〉\right)$$

(4)

$$\left|{\psi }^{-}\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left(\left|0\right.〉\left|1\right.〉-\left|1\right.〉\left|0\right.〉\right)$$

(5)

Meanwhile, the four-particle cluster state consists of maximally entangled states, meaning it exhibits a high degree of entanglement among its constituent qubits.

The bit-flip and phase-shift operations are defined as follows:

$$X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right), Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

(6)

When applying the operations $X^{a_1}{Z}^{a_2}$ and $X^{b_1}{Z}^{b_2}$ (where $a_1,a_2,b_1,b_2\in \left\{\mathrm{0,1}\right\}$) to qubits 2 and 3 of the four-particle cluster state, and then performing a Bell measurement on qubits 2 and 3, we can conclude that the measurement results of qubits 2 and 3 will be identical to those of qubits 1 and 4 if, and only if, $a_1{a}_2=b_1{b}_2$.

The detailed steps of the proposed QPC protocol can be outlined as follows:

Alice and Bob each possess a private integer: Alice has X and Bob has Y. The binary representations of these integers in $F_{2^L}$ can be expressed as follows:

$$X=\left(x_{L-1},\cdots ,x_1,x_0\right)$$

(7)

$$Y=\left(y_{L-1},\cdots ,y_1,y_0\right)$$

(8)

where $x_i,y_i\in \left\{\mathrm{0,1}\right\}$ for $i=\mathrm{0,1},\cdots ,L-1$. The objective of this protocol is to determine whether X is equal to Y without revealing any additional information about X or Y.

2.1. Preparing Step

(1)

Alice and Bob split their binary representations of X and Y into $⌈L/2⌉$ groups, respectively, with each group containing two-bit classical information. If L is odd (i.e., $L\mathit{mod}2=1$), Alice and Bob will add a zero to the last group to ensure it contains two bits. After grouping, X and Y can be rewritten as $X^{\prime }=\left(x_{⌈L/2⌉-1}^{\prime },x_{⌈L/2⌉-2}^{\prime },\cdots ,x_0^{\prime }\right)$ and $Y^{\prime }=\left(y_{⌈L/2⌉-1}^{\prime },y_{⌈L/2⌉-2}^{\prime },\cdots ,y_0^{\prime }\right)$, respectively. Where $x_j^{\prime }={(x}_{2j+1},x_{2j)}$ and $y_j^{\prime }={(y}_{2j+1}{,y}_{2j})$ for $j=\mathrm{0,1},\cdots ,⌈L/2⌉-2,⌈L/2⌉-1.$

(2)

Alice and Bob share a secret key sequence $K_{AB}=\left(k_{L-1}{k}_{L-2}\cdots k_0\right)$=$\left(k_{⌈L/2⌉-1}^{\prime },k_{⌈L/2⌉-2}^{\prime },\cdots ,k_0^{\prime }\right)$ via a QKD protocol, where each $k_j^{\prime }=k_{2j+1}{k}_{2j}\in \left\{\mathrm{00,01,10,11}\right\}$for $j=\mathrm{0,1},\cdots ,⌈L/2⌉-2,⌈L/2⌉-1.$

(3)

TP prepares an ordered sequence of four-qubit cluster states, which can be represented as follows:

$$\left[\left(p_2^0,p_3^0,p_1^0,p_4^0\right),\left(p_2^1,p_3^1,p_1^1,p_4^1\right),\cdots ,\left(p_2^{⌈L/2⌉-1},p_3^{⌈L/2⌉-1},p_1^{⌈L/2⌉-1},p_4^{⌈L/2⌉-1}\right)\right]$$

(9)

(4)

TP picks up particles 2 and 3 from each cluster state to form two ordered photon sequences: $T_A=\left\{p_2^0,p_2^1,\cdots ,p_2^{⌈L/2⌉-1}\right\}$ and $T_B=\left\{p_3^0,p_3^1,\cdots ,p_3^{⌈L/2⌉-1}\right\}$. The remaining particles from each cluster state form another ordered sequence $T_{TP}=\left\{p_{14}^0,p_{14}^1,\cdots ,p_{14}^{⌈L/2⌉-1}\right\}$.

(5)

To enhance the security of the protocol and prevent eavesdropping, TP prepares two sets of decoy-photon sequences $D_A$ and $D_B$ randomly chosen from four nonorthogonal states $\left\{\left|0\right.⟩,\left|1\right.⟩,\left|+\right.⟩,\left|-\right.⟩\right\}$, and inserts $D_A$ into $T_A$ and $D_B$ into $T_B$ at random positions, After the insertion, the new sequences are denoted as $T_A^{\prime }$ and $T_B^{\prime }$, respectively. Then, TP sends $T_A^{\prime }$ and $T_B^{\prime }$ to Alice and Bob, respectively.

2.2. Checking Step

After Bob and Charlie confirm the receipt of their respective photon sequences $T_A^{\prime }$ and $T_B^{\prime }$, TP, along with Bob and Charlie, conducts a series of checks to detect any potential eavesdropping during the transmission. The procedures are as follows:

(1)

The TP publicly announces the positions and the measuring bases (Z-basis and X-basis) of the decoy-photon sequences $D_A$ and $D_B$.

(2)

Alice and Bob measure the states of the decoy-photon sequence $D_A$ and $D_B$ according to the bases published by the TP, and announce their measurement outcomes to the TP.

(3)

The TP calculates the error rate based on the announced outcomes. Specifically, it compares the results obtained by Bob and Charlie with the prepared decoy-photon sequence $D_A$ and $D_B$. If the error rate exceeds a predetermined threshold, this indicates a potential eavesdropping attempt or other transmission issues. In such a case, the TP will abort the protocol and return to the preparation step. Otherwise, the protocol can proceed to the next steps.

2.3. Coding Step

(1)

Alice (Bob) discards all decoy photons $D_A$ ($D_B)$ from $T_A^{\prime }$($T_B^{\prime }$) to recover $T_A$ ($T_B$).

(2)

Alice (Bob) performs $X^{a_{2j+1}}{Z}^{a_{2j}}$ ($X^{b_{2j+1}}{Z}^{b_{2j}}$) on $T_A$ ($T_B$) to generate a new sequence $S_A$ ($S_B$), where $a_{2j+1}=x_{2j+1}\oplus k_{2j+1}$,$a_{2j}=x_{2j}\oplus k_{2j}$, $b_{2j+1}=y_{2j+1}\oplus k_{2j+1}$ and $b_{2j}=y_{2j}\oplus k_{2j}$ for $j=\mathrm{0,1},\cdots ,⌈L/2⌉-2,⌈L/2⌉-1.$

(3)

Alice (Bob) prepares a decoy-photon sequence $D_A^{\prime } \left(D_B^{\prime }\right)$ whose states are chosen from four nonorthogonal states $\left\{\left|0\right.⟩,\left|1\right.⟩,\left|+\right.⟩,\left|-\right.⟩\right\}$ and inserts $D_A^{\prime } \left(D_B^{\prime }\right)$ into $S_A$ ($S_B$) to generate a new sequence denoted as $S_A^{\prime } \left(S_B^{\prime }\right)$.

(4)

Alice (Bob) sends $S_A^{\prime } \left(S_B^{\prime }\right)$ to the TP.

2.4. Decoding Step

(1)

Upon receiving their sequences $S_A^{\prime }$ (from Alice) and $S_B^{\prime }$(from Bob), the TP interacts with Alice and Bob to verify the presence of any eavesdroppers in the same manner as previously described. If the checks confirm that there is no eavesdropper (i.e., the error rates are acceptable), the TP discards all decoy photons $D_A^{\prime }$ from $S_A^{\prime }$ and $D_B^{\prime }$ from $S_B^{\prime }$ to recover $S_A$ and $S_B$. Otherwise, they abort the protocol and return to the preparation step.

(2)

The TP performs Bell measurements on $S_A$ and $S_B$ to get a result sequence $R_{AB}$ and performs Bell measurements on $T_{TP}$ to get a result sequence $R_T$.

(3)

The TP compares $R_{AB}$ and $R_T$ to obtain the comparison result. If $R_{AB}$ is completely consistent with $R_T$, the TP can conclude that $X=Y$. Otherwise, $X\ne Y$.

(4)

The TP informs the comparison result to both Alice and Bob.

3. Analysis

3.1. Correctness

In fact, we can know that qubits 2 and 3 can be considered a Bell state in one of four forms $\left|{\phi }^{+}\right.⟩,\left|{\phi }^{-}\right.⟩,\left|{\psi }^{+}\right.⟩$ and $\left|{\psi }^{+}\right.⟩$ from Equation (1). When applying the operations $X^{a_{2j+1}}{Z}^{a_{2j}}$ ($X^{b_{2j+1}}{Z}^{b_{2j}}$) to qubits 2 and 3 of the four-particle cluster state, followed by performing a Bell measurement on these qubits, we obtain the resulting states, which are summarized in

Table 1. The resulting states.

Operations	$\left|{\mathit{\phi }}^{+}\right.⟩$	$\left|{\mathit{\phi }}^{-}\right.⟩$	$\left|{\mathit{\psi }}^{+}\right.⟩$	$\left|{\mathit{\psi }}^{-}\right.⟩$
$X^0{Z}^0\otimes X^0{Z}^0$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$
$X^0{Z}^0\otimes X^0{Z}^1$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$
$X^0{Z}^0\otimes X^1{Z}^0$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$
$X^0{Z}^0\otimes X^1{Z}^1$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$
$X^0{Z}^1\otimes X^0{Z}^0$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{-}\right.⟩$
$X^0{Z}^1\otimes X^0{Z}^1$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{+}\right.⟩$
$X^0{Z}^1\otimes X^1{Z}^0$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$
$X^0{Z}^1\otimes X^1{Z}^1$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$
$X^1{Z}^0\otimes X^0{Z}^0$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$
$X^1{Z}^0\otimes X^0{Z}^1$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$
$X^1{Z}^0\otimes X^1{Z}^0$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$
$X^1{Z}^0\otimes X^1{Z}^1$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$
$X^1{Z}^1\otimes X^0{Z}^0$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$
$X^1{Z}^1\otimes X^0{Z}^1$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$
$X^1{Z}^1\otimes X^1{Z}^0$	$\left|{\phi }^{-}\right.⟩$	$\left|{\phi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$
$X^1{Z}^1\otimes X^1{Z}^1$	$\left|{\phi }^{+}\right.⟩$	$\left|{\phi }^{-}\right.⟩$	$\left|{\psi }^{+}\right.⟩$	$\left|{\psi }^{-}\right.⟩$

.

From Table 1, we observe that the resulting states remain unchanged only when the same operations are performed on the first and second particles of the Bell states. From Equation (1), we note that, when performing Bell measurements on qubits (2, 3) and (1, 4), the measurement results are identical. Therefore, we can deduce that if, and only if, the same operations are applied to qubits 2 and 3 of the four-particle cluster state, the measurement results from performing Bell measurements on qubits (2, 3) and (1, 4) will be identical. The same operation is equivalent to encoding using the same secret, allowing us to further determine whether the secret is indeed identical.

3.2. Privacy

In our protocol, an external eavesdropper, Eve, may perform quantum attacks such as intercept–resend, entangle–measure, and Trojan horse attacks to gain information about the private inputs of the participants. Since our protocol is designed within a semi-honest model, all participants, including the TP, adhere strictly to the protocol’s rules. They perform their roles correctly without deviating from the prescribed operations. Despite their compliance, participants may still seek to infer additional information about others’ secret inputs. This could involve analyzing the results of their own computations and any public announcements made during the protocol. We will consider the following attacks: (1) an external adversary attempting to intercept communication between the participants to gain access to their private inputs, X or Y; (2) one participant (either Alice or Bob) behaving dishonestly with the intent to learn the other participant’s private input; and (3) the TP attempting to deduce the private inputs, X or Y.

Case I An external adversary attempts to intercept communication between the participants to gain access to their private inputs, X or Y.

If Eve, the external adversary, attempts to intercept the particles sent from the TP to either Alice or Bob and replace them with fake particles, she will inadvertently introduce an increased error rate in the communication. This is due to her lack of knowledge regarding the precise positions and original states of the decoy photons. If m decoy photons used for eavesdropping detection, the probability of detecting Eve can be calculated as follows [45]:

$$P=1-{\left({\displaystyle \frac{3}{4}}\right)}^m$$

(10)

Here, ${\left({\displaystyle \frac{3}{4}}\right)}^m$ represents the probability that Eve successfully avoids detection by randomly guessing the states of the decoy photons. As m (the number of decoy photons) increases, the probability of detecting Eve increases. Given this framework, if Eve attempts to eavesdrop by intercepting and replacing particles, the protocol is designed to reveal her presence through increased error rates. Therefore, this attack is unlikely to succeed, as the participants can effectively detect her interference through the eavesdropping detection mechanism built into the protocol.

If Eve intercepts the qubits transmitted in the TP–Alice or TP–Bob quantum channel, entangles intercepted qubits with ancillary particles prepared, and performs measurements on the ancillary particles to gain access to the participant’s secret data, she will fail. Eve’s unitary operation U acts on both states $\left\{\left|0\right.⟩,\left|1\right.⟩\right\}$ and her ancillary particles can be expressed as follows:

$$U\left|a\right.〉\left|0\right.〉={\delta }_{00}\left|a_{00}\right.〉\left|0\right.〉+{\delta }_{01}\left|a_{01}\right.〉\left|1\right.〉$$

(11)

$$U\left|a\right.〉\left|1\right.〉={\delta }_{10}\left|a_{10}\right.〉\left|0\right.〉+{\delta }_{11}\left|a_{11}\right.〉\left|1\right.〉$$

(12)

where $\left|a\right.⟩$ is the ancillary particle, $\left|a_{00}\right.⟩,\left|a_{01}\right.⟩,\left|a_{10}\right.⟩,\mathrm{a}\mathrm{n}\mathrm{d}\left|a_{11}\right.⟩$ are four pure states determined by unitary operation U, and the parameters ${\delta }_{00},{\delta }_{01},{\delta }_{10},\mathrm{a}\mathrm{n}\mathrm{d} {\delta }_{11}$ satisfy ${\left|{\delta }_{00}\right|}^2+{\left|{\delta }_{01}\right|}^2=1$, ${\left|{\delta }_{10}\right|}^2+{\left|{\delta }_{11}\right|}^2=1$. When Eve’s unitary operation U acts on both states $\left\{\left|+\right.⟩,\left|-\right.⟩\right\}$ and her ancillary particles, we can obtain the following:

$$\begin{array}{l}U\left|+\right.〉\left|a\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left({\delta }_{00}\left|a_{00}\right.〉\left|0\right.〉+{\delta }_{01}\left|a_{01}\right.〉\left|1\right.〉+{\delta }_{10}\left|a_{10}\right.〉\left|0\right.〉+{\delta }_{11}\left|a_{11}\right.〉\left|1\right.〉\right)\\ ={\displaystyle \frac{1}{2}}\left|+\right.〉\left({\delta }_{00}\left|a_{00}\right.〉+{\delta }_{01}\left|a_{01}\right.〉+{\delta }_{10}\left|a_{10}\right.〉+{\delta }_{11}\left|a_{11}\right.〉\right)\\ +{\displaystyle \frac{1}{2}}\left|-\right.〉\left({\delta }_{00}\left|a_{00}\right.〉-{\delta }_{01}\left|a_{01}\right.〉+{\delta }_{10}\left|a_{10}\right.〉-{\delta }_{11}\left|a_{11}\right.〉\right)\end{array}$$

(13)

$$\begin{array}{l}U\left|-\right.〉\left|a\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left({\delta }_{00}\left|a_{00}\right.〉\left|0\right.〉+{\delta }_{01}\left|a_{01}\right.〉\left|1\right.〉-{\delta }_{10}\left|a_{10}\right.〉\left|0\right.〉-{\delta }_{11}\left|a_{11}\right.〉\left|1\right.〉\right)\\ ={\displaystyle \frac{1}{2}}\left|+\right.〉\left({\delta }_{00}\left|a_{00}\right.〉+{\delta }_{01}\left|a_{01}\right.〉-{\delta }_{10}\left|a_{10}\right.〉-{\delta }_{11}\left|a_{11}\right.〉\right)\\ +{\displaystyle \frac{1}{2}}\left|-\right.〉\left({\delta }_{00}\left|a_{00}\right.〉-{\delta }_{01}\left|a_{01}\right.〉-{\delta }_{10}\left|a_{10}\right.〉+{\delta }_{11}\left|a_{11}\right.〉\right)\end{array}$$

(14)

If Eve aims to conduct her eavesdropping without introducing any detectable errors during the eavesdropping check, her unitary operation U must satisfy ${\delta }_{01}={\delta }_{10}=0$ and ${\delta }_{00}\left|a_{00}\right.⟩={\delta }_{11}\left|a_{11}\right.⟩$. If Eve’s unitary operation U acts on Alice’s qubits of $\left|C_4\right.⟩$ and her ancillary particles, we can obtain the following:

$$\begin{array}{l}U{\left|C_4\right.〉}_A\left|a\right.〉={\displaystyle \frac{1}{2}}\left(\begin{array}{l}\left({\delta }_{00}\left|a_{00}\right.〉{\left|0\right.〉}_2+{\delta }_{01}\left|a_{01}\right.〉{\left|1\right.〉}_2\right){\left|0\right.〉}_3{\left|00\right.〉}_{14}\\ +\left({\delta }_{00}\left|a_{00}\right.〉{\left|0\right.〉}_2+{\delta }_{01}\left|a_{01}\right.〉{\left|1\right.〉}_2\right){\left|1\right.〉}_3{\left|01\right.〉}_{14}\\ +\left({\delta }_{10}\left|a_{10}\right.〉{\left|0\right.〉}_2+{\delta }_{11}\left|a_{11}\right.〉{\left|1\right.〉}_2\right){\left|0\right.〉}_3{\left|10\right.〉}_{14}\\ +\left({\delta }_{10}\left|a_{10}\right.〉{\left|0\right.〉}_2+{\delta }_{11}\left|a_{11}\right.〉{\left|1\right.〉}_2\right){\left|1\right.〉}_3{\left|11\right.〉}_{14}\end{array}\right)\\ ={\displaystyle \frac{1}{2}}\left(\begin{array}{l}{\delta }_{00}\left|a_{00}\right.〉{\left|00\right.〉}_{23}{\left|00\right.〉}_{14}+{\delta }_{00}\left|a_{00}\right.〉{\left|01\right.〉}_{23}{\left|01\right.〉}_{14}\\ +{\delta }_{00}\left|a_{00}\right.〉{\left|10\right.〉}_{23}{\left|10\right.〉}_{14}+{\delta }_{00}\left|a_{00}\right.〉{\left|11\right.〉}_3{\left|11\right.〉}_{14}\end{array}\right)\\ ={\delta }_{00}\left|a_{00}\right.〉{\left|00\right.〉}_{23}{\left|00\right.〉}_{14}\end{array}$$

(15)

Similarly, if Eve’s unitary operation U acts on Bob’s qubits of $\left|C_4\right.⟩$ and her ancillary particles, we can obtain the following:

$$\begin{array}{l}U{\left|C_4\right.〉}_B\left|a\right.〉={\displaystyle \frac{1}{2}}\left(\begin{array}{l}{\left|0\right.〉}_2\left({\delta }_{00}\left|a_{00}\right.〉{\left|0\right.〉}_3+{\delta }_{01}\left|a_{01}\right.〉{\left|1\right.〉}_3\right){\left|00\right.〉}_{14}\\ +{\left|0\right.〉}_2\left({\delta }_{10}\left|a_{10}\right.〉{\left|0\right.〉}_3+{\delta }_{11}\left|a_{11}\right.〉{\left|1\right.〉}_3\right){\left|01\right.〉}_{14}\\ +{\left|1\right.〉}_2\left({\delta }_{00}\left|a_{00}\right.〉{\left|0\right.〉}_3+{\delta }_{01}\left|a_{01}\right.〉{\left|1\right.〉}_3\right){\left|10\right.〉}_{14}\\ +{\left|1\right.〉}_2\left({\delta }_{10}\left|a_{10}\right.〉{\left|0\right.〉}_3+{\delta }_{11}\left|a_{11}\right.〉{\left|1\right.〉}_3\right){\left|1\right.〉}_3{\left|11\right.〉}_{14}\end{array}\right)\\ ={\displaystyle \frac{1}{2}}\left(\begin{array}{l}{\delta }_{00}\left|a_{00}\right.〉{\left|00\right.〉}_{23}{\left|00\right.〉}_{14}+{\delta }_{00}\left|a_{00}\right.〉{\left|01\right.〉}_{23}{\left|01\right.〉}_{14}\\ +{\delta }_{00}\left|a_{00}\right.〉{\left|10\right.〉}_{23}{\left|10\right.〉}_{14}+{\delta }_{00}\left|a_{00}\right.〉{\left|11\right.〉}_3{\left|11\right.〉}_{14}\end{array}\right)\\ ={\delta }_{00}\left|a_{00}\right.〉{\left|00\right.〉}_{23}{\left|00\right.〉}_{14}\end{array}$$

(16)

If the intercepted particles and the ancillary particles are entangled instead of being in a product state, the interaction could introduce no error. However, from Equations (15) and (16), we observe that the intercepted particles and the ancillary particles are in a product state form. This indicates that Eve’s strategy is ineffective.

Case II One participant (either Alice or Bob) behaves dishonestly with the intent to learn the other participant’s private input.

In this quantum protocol, both Alice and Bob are positioned as legitimate participants. However, each participant may behave dishonestly with the intent to learn the other participant’s private input. The protocol treats Alice and Bob as having equivalent roles; we assume that Bob may try to uncover Alice’s secrets encoded in the quantum sequence $T_A$ through quantum operations such as bit flip and phase shift operations. For Bob to access Alice’s secrets, he would need both the transformed quantum sequence $T_A$ and $S_A$. However, since Alice and Bob do not communicate directly, Bob’s only option to obtain $T_A$ is by intercepting the quantum communication between the TP and Alice. If Bob intercepts and modifies the quantum signals, deviations from the expected outcomes will occur, and this interception behavior will be undoubtedly detected, resulting in the failure of Bob’s eavesdropping attempt. Similarly, if Alice attempts to infer information regarding Bob’s private inputs, an analysis can be carried out in the same manner. Therefore, the secrets of both participants are safeguarded, even in the presence of potential attacks from either side.

Case III The TP attempts to deduce the private inputs, X or Y.

The TP follows the protocol correctly and does not deviate from its specified operations but may attempt to glean additional information about the participants’ private inputs during the execution of the protocol. Its primary tasks include preparing the four-particle cluster states and performing Bell-basis measurements. Although the TP knows the initial prepared four-particle cluster states and the measurement results $R_{AB}$ and $R_T$, she still cannot derive the users’ secrets due to different combinations of bit flip and phase shift operations can yield the same result. The TP may prepare single photons instead of four-particle cluster states, measure the received single photons, and analyze the measurement results to infer some operations performed by Alice and Bob. However, the TP cannot derive the users’ secrets without access to the private key $K_{AB}$, which is vital for encrypting the participants’ private messages $X^{\prime }$ and $Y^{\prime }$. Therefore, the actual secret inputs of Alice and Bob remain confidential for the TP.

3.3. Fairness

Fairness is effectively ensured through the involvement of the TP. Once the TP publishes the comparison result, Alice and Bob can receive this information simultaneously, ensuring that each party has equal access to the result, preventing any participant from gaining an undue advantage.

4. Comparison

The qubit efficiency [46,47,48,49] is crucial for evaluating the performance of quantum protocols, which can be defined as follows:

$${\eta }_e={\displaystyle \frac{q_c}{q_t}}$$

(17)

where $q_c$ denotes the number of classical bits that can be compared and $q_t$ represents the total number of photons used. In the proposed protocol, L four-qubit cluster states serve as quantum information carriers that can be utilized to compare 2L classical bits of information. Therefore, the qubit efficiency of the proposed protocol is ${\eta }_e={\displaystyle \frac{2L}{4L}}=50\%$. The comparison results between our protocol and those based on cluster states in Refs. [41,42,43,44] are summarized in

Table 2. Comparison among similar QPC protocols.

	Ref. [41]	Ref. [42]	Ref. [43]	Ref. [44]	Ours
Consumption of quantum resources	(L + k) Four-qubit cluster states for comparing 2L bits	L Four-particle cluster states for comparing L bits	(L + k) Five-particle cluster states for comparing 2L bits	L Four-qubit cluster states and L extended Bell states for comparing 3L bits	$L$Four-particle cluster state for comparing 2L bits
Whether needs the entanglement swapping	No	No	No	Yes	No
Unitary operation for TP	No	No	$I \mathrm{and} i{\delta }_y$	No	No
Unitary operation for users	I and X	Four Pauli operators	No	No	Z and X
Quantum Measurement for TP	No	Z basis	Z basis and X basis	Bell basis	Bell-basis
Quantum measurement for users	Single-particle, Bell basis and $\left\{\left|00\right.⟩,\left|01\right.⟩,\left|10\right.⟩,\left|11\right.⟩\right\}$ basis	Z basis	Z basis and X basis	Bell-basis and extend Bell basis	No
QKD method between two users	Yes	No	No	Yes	Yes
QKD method between users and TP	Yes	No	No	Yes	No
Qubit efficiency	<50%	25%	<40%	50%	50%

.

Ref. [41] requires L + k four-qubit cluster states as quantum resources to compare 2L classical-bit information, where k four-qubit cluster states are used to verify whether Bob shares the cluster state with Alice. Thus, the qubit efficiency is ${\eta }_e={\displaystyle \frac{2L}{4(L+k)}}<50\%$. Additionally, the users will measure their qubits using the bases $\left\{\left|00\right.⟩,\left|01\right.⟩,\left|10\right.⟩,\left|11\right.⟩\right\}$, single particle, and the Bell basis and $\left\{\left|0\right.⟩,\left|1\right.⟩\right\}$. Alice and the TP, Bob and the TP, and Alice and Bob will share a secret key. These processes increase the consumption of quantum resources, quantum measurements, and additional particles in key distribution.

Ref. [42] utilized L four-qubit cluster states as quantum resources to compare L classical-bit information, resulting in a qubit efficiency of ${\eta }_e={\displaystyle \frac{L}{4L}}=25\%$. This efficiency is relatively lower than many QPC protocols.

Ref. [43] utilized L + k five-qubit cluster states as quantum resources to compare 2L classical-bit information, where k five-qubit cluster states are used for resisting the TP’s malicious behavior by executing the second eavesdropping check. Thus, the qubit efficiency is ${\eta }_e={\displaystyle \frac{2L}{5(L+k)}}<40\%$.

Ref. [44] utilized the L extended Bell state and four-qubit cluster states as quantum resources to compare 3L classical-bit information, achieving a qubit efficiency of ${\eta }_e={\displaystyle \frac{3L}{6L}}=50\%$. Quantum measurements need to be performed by users and the TP. Alice and Bob, Alice and the TP, and Bob and the TP all need to negotiate a key with each other, which increases consumption. Importantly, the entanglement swapping of extended Bell states and four-qubit cluster states is more challenging to implement compared to unitary operations.

Our protocol also has a qubit efficiency of 50%, and the users perform Z and X operations; only Alice and Bob need to share a secret key. This further reduces additional resource consumption.

5. Conclusions

In this paper, we propose a quantum private comparison (QPC) protocol that utilizes four-particle cluster states as quantum resources. In this framework, a semi-honest third party (TP) facilitates the comparison of private information between two participants without revealing any private data. Each user encodes their information using bit-flip or phase-shift operators applied to the received quantum sequences. These modified sequences are then returned to the TP, who can derive and announce the comparison results while maintaining the confidentiality of the users’ information. Our protocol ensures correctness, privacy, and fairness throughout the entire process. It is specifically designed to withstand both external threats and attacks from participants, owing to the integration of the decoy-state method and quantum key distribution (QKD) techniques. Additionally, the use of unitary operations and Bell-basis measurements enhances the technical feasibility of the protocol, making it suitable for real-world applications. A comparative analysis demonstrates that our scheme outperforms the existing QPC protocols based on the four-qubit cluster state in terms of efficiency. In the future, we will focus on developing methods to compare numerical sizes rather than simply determining equality, and we will design semi-quantum private comparison protocols to reduce users’ demand for quantum capabilities.