# A New Quantum Private Protocol for Set Intersection Cardinality Based on a Quantum Homomorphic Encryption Scheme for Toffoli Gate

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## Abstract

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## 1. Introduction

## 2. Preliminary

#### 2.1. Pauli Gates

#### 2.2. Quantum Toffoli Gate

#### 2.3. Information-Theoretic Security

## 3. Quantum Private Computation Protocol for Set Intersection Cardinality

## 4. Analysis and Comparison

#### 4.1. Correctness Analysis

#### 4.2. Implementation of Quantum PSI-CA Protocols on IBM Quantum Experience Platform

#### 4.3. Security Analysis

#### 4.3.1. Outside Attacks

#### 4.3.2. Participant Attack

#### 4.4. Comparison

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 3.**The circuit U used to privately calculate $T\left|{\psi}_{k}^{A}\right.\u232a\left|{\psi}_{k}^{B}\right.\u232a\left|{\psi}_{k}^{C}\right.\u232a$.

**Figure 4.**The circuit used to privately calculate $T\left|1\right.\u232a\left|1\right.\u232a\left|1\right.\u232a$ and the experiment results.

**Figure 5.**The circuit used to privately calculate $T\left|0\right.\u232a\left|0\right.\u232a\left|0\right.\u232a$ and the experiment results.

**Figure 6.**The circuit used to privately calculate $T\left|1\right.\u232a\left|1\right.\u232a\left|0\right.\u232a$ and the experiment results.

**Figure 7.**The circuit used to privately calculate $T\left|1\right.\u232a\left|0\right.\u232a\left|1\right.\u232a$ and the experiment results.

**Figure 8.**The circuit used to privately calculate $T\left|0\right.\u232a\left|1\right.\u232a\left|1\right.\u232a$ and the experiment results.

**Figure 9.**The circuit used to privately calculate $T\left|0\right.\u232a\left|1\right.\u232a\left|0\right.\u232a$ and the experiment results.

**Figure 10.**The circuit used to privately calculate $T\left|0\right.\u232a\left|0\right.\u232a\left|1\right.\u232a$ and the experiment results.

**Figure 11.**The circuit used to privately calculate $T\left|1\right.\u232a\left|0\right.\u232a\left|0\right.\u232a$ and the experiment results.

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**MDPI and ACS Style**

Liu, W.; Li, Y.; Wang, Z.; Li, Y.
A New Quantum Private Protocol for Set Intersection Cardinality Based on a Quantum Homomorphic Encryption Scheme for Toffoli Gate. *Entropy* **2023**, *25*, 516.
https://doi.org/10.3390/e25030516

**AMA Style**

Liu W, Li Y, Wang Z, Li Y.
A New Quantum Private Protocol for Set Intersection Cardinality Based on a Quantum Homomorphic Encryption Scheme for Toffoli Gate. *Entropy*. 2023; 25(3):516.
https://doi.org/10.3390/e25030516

**Chicago/Turabian Style**

Liu, Wen, Yangzhi Li, Zhirao Wang, and Yugang Li.
2023. "A New Quantum Private Protocol for Set Intersection Cardinality Based on a Quantum Homomorphic Encryption Scheme for Toffoli Gate" *Entropy* 25, no. 3: 516.
https://doi.org/10.3390/e25030516