# Quantum Secure Multi-Party Summation Using Single Photons

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## Abstract

**:**

## 1. Introduction

## 2. The Three-Party Quantum Summation Protocol

#### 2.1. Proposed Protocol

**Correctness:**The result of modulo-2 summation of all participants’ secret inputs should be correct.

**Fairness:**All participants receive the summation result simultaneously.

**Privacy:**Participants’ secret inputs are private. In other words, no participant can learn about other participants’ secret inputs, even though the participant can launch various quantum attacks and up to n-2 participants are allowed to conspire but not with TP and an outside eavesdropper (here, n is the number of participants in the protocol).

**Security:**An outside eavesdropper cannot learn any information about each participant’s secret input without being detected.

**Step 1:**TP sends a secret key sequence ${l}_{A}$(${l}_{B}$, ${l}_{C}$) to Alice (Bob, Charlie) via the quantum key distribution protocol [11] beforehand, where ${l}_{A}=({l}_{A1},\dots ,{l}_{Am})({l}_{B}=({l}_{B1},\dots ,{l}_{Bm}),$${l}_{C}=({l}_{C1},\dots ,{l}_{Cm})),{l}_{At}({l}_{Bt},{l}_{Ct})\in \{0,1\},t=1,2,\cdots ,m.$

**Step 2:**According to the secret key sequence ${l}_{A}$ previously shared with Alice, TP generates m copies of single photon states and uses these particles to construct a sequence

**Step 3:**After confirming that Alice has received all the particles from TP, Alice checks the security of the transmission of $\overline{{S}_{A}}$ with TP. Specifically, TP announces the insertion positions and the bases of the decoy photons in $\overline{{S}_{A}}$ to Alice. Then, based on the announced information, Alice measures these decoy states on the correct bases and publishes the measurement results to TP. Subsequently, according to Alice’s measurements, TP checks for the presence of eavesdroppers in the quantum channel. If the error rate is higher than the threshold determined by the channel noise, TP cancels this protocol and restarts it. Otherwise, TP proceeds to the next steps.

**Step 4:**After determining that the transmission has not eavesdropped, Alice obtains ${S}_{A}$ by extracting decoy photons from $\overline{{S}_{A}}$. Then, Alice encodes his secret input ${x}_{A}$ on the sequence ${S}_{A}$. Concretely, Alice performs the unitary operation ${U}_{Y}^{{x}_{At}}$ on the t th particle of ${S}_{A}$. Here, the operators are defined by [37],

**Step 5:**After receiving the sequence $\overline{{S}_{TP}}$, TP and Charlie jointly check the security of the transmission channel. TP obtains the sequence ${S}_{TP}$ by extracting decoy photons after confirming that the channel is safe. Then, TP computing ${L}_{1}={l}_{A1}\oplus {l}_{B1}\oplus {l}_{C1},$${L}_{2}={l}_{A2}\oplus {l}_{B2}\oplus {l}_{C2},\dots ,{L}_{m}={l}_{Am}\oplus {l}_{Bm}\oplus {l}_{Cm}$. If ${L}_{t}=1(t=1,2,\dots ,m)$, TP will perform the Hadamard operation on the t th particle of ${S}_{TP}$. Otherwise, the particles in ${S}_{TP}$ will remain the same. After the above operation, TP measures the particles with the Z basis. Then, TP can acquire the summation ${x}_{A1}\oplus {x}_{B1}\oplus {x}_{C1},{x}_{A2}\oplus {x}_{B2}\oplus {x}_{C2},\dots ,{x}_{Am}\oplus {x}_{Bm}\oplus {x}_{Cm}$. Finally, TP announces the summation result to Alice, Bob, and Charlie via a public channel.

**Sample 1:**We assume the secret input of Alice Bob and Charlie are ${x}_{A}$ = (010), ${x}_{B}$ = (011) and ${x}_{C}$ = (011), respectively.

**Channel loss of the cited QKD protocol:**Although we assumed that the quantum channel is noiseless, channel loss is a major issue in the construction of QKD, so it is discussed here. The QKD protocol [11] used in our protocol utilizes coherent states to against realistic flawed sources and ensure security by adopting the reference technique. A proof-of-principle experiment in Ref. [11] demonstrates the feasibility of the QKD protocol in terms of resistance to channel loss.

#### 2.2. Correctness

#### 2.3. Security Analysis

#### 2.3.1. Outside Attack

**1**)

**The Trojan horse attacks**

**2**)

**The entangle-measure attack**

**0**donates a column zero vector. In the same way, we can infer

**3**)

**The intercept-resend attack**

**4**)

**The measurement-resend attack**

#### 2.3.2. Participant Attack

## 3. The Multi-Party Quantum Summation Protocol

#### 3.1. Proposed Protocol

**Step 1:**TP sends a secret key sequence ${l}_{i}$ to ${P}_{i}$ via the quantum key distribution protocol [11] in advance, where ${l}_{i}=({l}_{i1},{l}_{i2},\dots ,{l}_{im}),{l}_{it}\u03f5\{0,1\}$, t = 1, 2, …, m.

**Step 2:**TP encodes the secret key sequence ${l}_{1}$ previously shared with ${P}_{1}$ according to the above agreement, and he can obtain m copies of single photon states

**Step 3:**After confirming that ${P}_{1}$ received all the particles $\overline{{S}_{1}}$, TP and ${P}_{1}$ check the transmissions between them for eavesdroppers. Concretely, TP first announces the insertion positions and the bases of the decoy photons to ${P}_{1}$. Then, based on the announced information, ${P}_{1}$ uses the correct basis to measure these decoy photons and publishes the measurement results to TP. Subsequently, according to their measuring results, TP checks whether eavesdroppers exist in the quantum channels. If the error rate is higher than the threshold determined by the channel noise, TP cancels this protocol and restarts it. Otherwise, TP proceeds to the next step.

**Step 4:**By extracting all the decoy photons from $\overline{{S}_{1}}$ and discarding them, ${P}_{1}$ can acquire the sequence ${S}_{1}$. Then, ${P}_{1}$ performs the unitary operation ${U}^{{x}_{1t}}$ on the t-th particle of ${S}_{1}$ according to his secret input ${x}_{1}$. When ${P}_{1}$ completes the encoding operation, the quantum states in ${S}_{1}$ would change to $\left(\right)$. Define the changed sequence as ${S}_{2}$. ${P}_{1}$ mixes ${S}_{2}$ with m decoy states randomly in $\left(\right)$ to form a new sequence $\overline{{S}_{2}}$. Finally, ${P}_{1}$ sends $\overline{{S}_{2}}$ to ${P}_{2}$.

**Step 5:**For j = 2, 3, …, n: when ${P}_{j}$ received the sequence $\overline{{S}_{j}}$ from ${P}_{j-1}$, ${P}_{j-1}$ checks the security of transmission with ${P}_{j}$, which similar to Step 3. After determining that the channel is secure, ${P}_{j}$ removes the decoy states and encodes his secret input ${X}_{j}$ similar to Step 4. Furthermore, ${P}_{j}$ encodes the information according to the secret key sequence ${l}_{j}$ sent by TP. To be clear, ${P}_{j}$ performs the unitary operation ${H}^{{l}_{jt}}$ on the t th particle of ${S}_{j}$. Then, ${P}_{j}$ mixes sequence ${S}_{j}$ and decoy photons randomly to form a new sequence $\overline{{S}_{j}}$, and sends it to ${P}_{j+1}$. Of note, the last participant ${P}_{n}$ sends the particle sequence $\overline{{S}_{n+1}}$ to TP.

**Step 6:**When TP has received the sequence $\overline{{S}_{n+1}}$, TP checks the security of transmission channel with ${P}_{n}$. TP obtains ${S}_{n+1}$ by extracts and discards decoy photons from $\overline{{S}_{n+1}}$. Then, TP computing ${L}_{1}$ = $\sum _{i=1}^{n}}{l}_{i1$, ${L}_{2}$ = $\sum _{i=1}^{n}}{l}_{i2$, …, ${L}_{m}$ = $\sum _{i=1}^{n}}{l}_{im$. If the result ${L}_{t}$ is 1, TP will perform the Hadamard operation on the t th particle of ${S}_{n}$. Otherwise, the particles in ${S}_{n}$ will not change. After the above steps, TP measures the particles with Z basis. Then, TP can obtain the summation ${\sum}_{i=1}^{n}{x}_{i1}$, ${\sum}_{i=1}^{n}{x}_{i2}$, …, ${\sum}_{i=1}^{n}{x}_{im}$. Finally, TP announces the summation result to ${P}_{1}$, ${P}_{2}$, …, ${P}_{n}$.

#### 3.2. Correctness

#### 3.3. Security Analyse

#### 3.3.1. Outside Attack

#### 3.3.2. Participant Attack

**Case 1:****The participant attack from one or more dishonest parties**

**a**)

**The participant attack from one dishonest party**

**b**)

**The participant attack from more than one dishonest party.**

**Case 2:****The participant attack from TP**

## 4. Comparisons

Ref. [32] | Ref. [35] | Ref. [34] | Ref. [27] | Ref. [44] | Ref. [28] | Ref. [30] | Ref. [31] | Our Protocol | |
---|---|---|---|---|---|---|---|---|---|

Quantum resource | single photons in both polarization and spatial-mode degree of freedom | d-level single-photon state | d-level single-photon state | three-particle entangled state | d-level N-particle entangled state | N-particle entangled state | Bell state | phase GHZ state | single-photon state |

Quantum operations | Single-photon operators | Unitary operations | Two unitary operations (${X}_{d}$ and ${Y}_{d}$) | CNOT and Hadamard operators | Quantum Fourier transformand and Pauli operators | Pauli and Hadamard operators | NOT and identity operators | No | Pauli and Hadamard operators |

Particle transmission mode | circle-type | circle-type | circle-type | tree-type | tree-type | tree-type | circle-type | star-type | circle-type |

Quantum measurements for TP | Single-photon projective measurements | single qudit measurements | Single qudit measurement | No | No | Single-photon projective measurements | Single-photon projective measurements | No | Single-photon projective measurements |

Quantum measurement for participants | No | No | No | Single-photon projective measurements | Single qudit measurements | No | Single-photon projective measurements | Computational basis measurements | No |

Qubit efficiency | $\frac{2}{2N+3}$ | $\frac{1}{2N+3}$ | $\frac{1}{2N+3}$ | $\frac{1}{4N-1}$ | $\frac{1}{3N-2}$ | $\frac{1}{3N-2}$ | $\frac{1}{5N}$ | $\frac{1}{4{N}^{4}+3N}$ | $\frac{1}{2N+2}$ |

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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${\mathit{x}}_{1}$ = (0, 1, 0) | ${\mathit{x}}_{2}$ = (0, 1, 0) | ${\mathit{x}}_{3}$ = (0, 1, 1) |
---|---|---|

${|0\rangle}_{0}\stackrel{{U}_{{x}_{A1}}}{\u27f6}{|0\rangle}_{0}$ | $\stackrel{{U}_{{l}_{B1}}{U}_{{x}_{B1}}}{\to}{|0\rangle}_{0}$ | $\stackrel{{U}_{{l}_{C1}}{U}_{{x}_{C1}}}{\to}{|0\rangle}_{0}$ |

${|0\rangle}_{1}\stackrel{{U}_{{x}_{A2}}}{\u27f6}{|1\rangle}_{0}$ | $\stackrel{{U}_{{l}_{B2}}{U}_{{x}_{B2}}}{\to}{|0\rangle}_{1}$ | $\stackrel{{U}_{{l}_{C2}}{U}_{{x}_{C2}}}{\to}{|1\rangle}_{0}$ |

${|0\rangle}_{0}\stackrel{{U}_{{x}_{A3}}}{\u27f6}{|1\rangle}_{1}$ | $\stackrel{{U}_{{l}_{B3}}{U}_{{x}_{B3}}}{\to}{|1\rangle}_{1}$ | $\stackrel{{U}_{{l}_{C3}}{U}_{{x}_{C3}}}{\to}{|0\rangle}_{0}$ |

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**MDPI and ACS Style**

Wu, W.-Q.; Xie, M.-Z.
Quantum Secure Multi-Party Summation Using Single Photons. *Entropy* **2023**, *25*, 590.
https://doi.org/10.3390/e25040590

**AMA Style**

Wu W-Q, Xie M-Z.
Quantum Secure Multi-Party Summation Using Single Photons. *Entropy*. 2023; 25(4):590.
https://doi.org/10.3390/e25040590

**Chicago/Turabian Style**

Wu, Wan-Qing, and Ming-Zhe Xie.
2023. "Quantum Secure Multi-Party Summation Using Single Photons" *Entropy* 25, no. 4: 590.
https://doi.org/10.3390/e25040590