# Authenticated Semi-Quantum Key Distribution Protocol Based on W States

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- It reduces the quantum hardware equipment when compared to other ASQKD protocols.
- (2)
- It does not require pre-share keys.
- (3)
- Wen et al. demonstrated that the proposed ASQKD protocol is robust against typical attacks.
- (4)
- It is highly efficient than some of the existing ASQKD protocols.

- The proposed ASQKD protocol ensures the procedure is functional.
- The proposed ASQKD protocol does not require quantum memory and legally fulfills a semi-quantum environment [17].
- The proposed ASQKD protocol, based on W states, only reduces the quantum hardware requirements.
- The qubit efficiency of the proposed ASQKD protocol is 1.6 times higher than that of Wen et al.’s ASQKD protocol.
- The proposed ASQKD protocol does not require classical cryptography (i.e., the hash function), which does not show the potential menace of the advance quantum computing.

## 2. Review of Wen et al.’s ASQKD Protocol

- Step W1.
- Alice prepares n GHZ-like states as shown in Equation (1) and divides these states into three sequences: ${S}_{a},{S}_{b},{S}_{c}$. Every photon in ${S}_{a}$ represents all the first particles in GHZ-like states, and ${S}_{b}$ and ${S}_{c}$ represent all the second and third particles in GHZ-like states. Then, she inserts random decoy states $\left\{|0\rangle ,|1\rangle ,|+\rangle ,|-\rangle \right\}$ into ${S}_{a}$ and obtains ${{S}^{\prime}}_{a}$ as follows:$$\begin{array}{ll}|G{\rangle}_{abc}& =\frac{1}{2}{\left(|001\rangle +|010\rangle +|100\rangle +|111\rangle \right)}_{abc}\\ & =\frac{1}{2}\left({|0\rangle}_{a}{|{\psi}^{+}\rangle}_{bc}+{|1\rangle}_{a}{|{\varphi}^{+}\rangle}_{bc}\right)\end{array}$$

- Step W2.
- Alice encodes state string $|Y\rangle $ according to the specific coding rule (see also Table 1), generates a binary message string $L=\{{L}_{i}|i=1,2,\dots ,2n.\}$, then recodes the binary string $L$ according to the following rules: binary message {0,1} recodes into the Z-basis $\{|0\rangle ,|1\rangle \}$. Eventually, Alice obtains the new particle string |${\phi}_{L}\rangle $ = {$|{\phi}_{{L}_{j}}\rangle \text{}|\text{}j=1,\text{}2,\text{}\dots ,2n.$}. Alice performs W-basis measurement on $(|Y\rangle {,\text{}|S}_{1}\rangle {,\text{}|S}_{2}\rangle )$ and $(|{\phi}_{L}\rangle ,{\text{}|S}_{1}\rangle {,\text{}|S}_{2}\rangle )$ and performs Bell measurement on $(|{S}_{b}\rangle ,|{S}_{c}\rangle ).$ Then, Alice can obtain the measurement results $M{R}_{Y},M{R}_{L},M{R}_{Bell},$ respectively. Alice informs Bob of the measurement results via a classical channel. It should be noted that $M{R}_{Y},M{R}_{L}$ is presented in notation format $\{|{\kappa}^{+}\rangle ,|{\kappa}^{-}\rangle ,|{\gamma}^{+}\rangle ,|{\gamma}^{-}\rangle \}.$

- Step W3.
- Bob receives the measurement results, $M{R}_{Y},M{R}_{L}$ from Alice. He can perform the unitary operation on ${S}_{3}$ to recover the states $|Y\rangle \text{}\mathrm{and}\text{}|{\phi}_{L}\rangle $ based on the measurement result $M{R}_{Y},M{R}_{L}$. Bob measures $|{\phi}_{L}\rangle $ in ${S}_{3}$ using a Z-basis. Then, he records the measurement results as ${M}_{3}=\left\{{m}_{3}^{1},{m}_{3}^{2},\dots \dots ,{m}_{3}^{2n}\right\}$. According to the notation and the measurement result of ${M}_{3}$, Bob applies the corresponding coding rule and obtains ${M}_{3}^{\prime}=\left\{{{m}^{\prime}}_{3}^{1},{{m}^{\prime}}_{3}^{2},\dots ,{{m}^{\prime}}_{3}^{2n}\right\}$ as shown in Table 2 below.

- Step W4.
- Bob obtains $|Y\rangle \text{}\mathrm{and}$ ${M}_{3}^{\prime}$ and returns $|{Y}^{\prime}\rangle $ based on ${M}_{3}^{\prime}.$ If ${M}_{3}^{\prime}=00$ or ${M}_{3}^{\prime}=01$, then he measures $|Y\rangle $ in the Z-basis, prepares the same photon as $|{Y}^{\prime}\rangle $, and resends it back to Alice. If ${M}_{3}^{\prime}=10$ or ${M}_{3}^{\prime}=11$, then Bob returns $|Y\rangle $ as $|{Y}^{\prime}\rangle $ directly to Alice. Furthermore, Alice checks the received decoy states using the correct corresponding basis.

- Step W5.
- Alice measures $|{Y}^{\prime}\rangle $ on the correct basis and checks if $|{Y}^{\prime}\rangle $ equals to the original $|Y\rangle .$ Alice then announces the position of $|{\phi}_{L}\rangle $ and decoy photons in ${S}_{a}^{\prime}$ to Bob via the classical channel. According to this announcement, Bob removes the decoy state in ${S}_{a}^{\prime}$ and recovers sequence ${S}_{a}.$ Then, Bob measures ${S}_{a}$ on the Z-basis to check its correlation with $M{R}_{Bell}$.

- Step W6.
- Based on the measurement results of ${S}_{a}$, Bob generates binary string ${L}_{B}=\{{L}_{{B}_{j}}\text{}|j=1,\text{}2,\text{}\dots ,\text{}2n\}$ according to the following coding rules: if the measurement result is $|0\rangle $, then he encodes ${L}_{{B}_{j}}=00$. Furthermore, while the measurement result is $|1\rangle $, he encodes ${L}_{{B}_{j}}=01$. Bob hashes ${L}_{B}$ to obtain the hash value H(${L}_{B}$). Bob then sends H(${L}_{B}$) to Alice.

- Step W7.
- Alice calculates H($L$) and checks if H(${L}_{B}$) equals H($L$).

## 3. Security Issues in Wen et al.’s ASQKD Protocol

#### 3.1. Teleportation of W States in Wen et al.’s ASQKD Protocol

#### 3.2. Quantum Environment Issue in Wen et al.’s ASQKD Protocol

## 4. Proposed Measure-Resend ASQKD Protocol

**.**Specifically, ${K}_{1}$ determines the initial state of the prepared W state and ${K}_{2}$ represents measure-resending or reflecting photons. ${K}_{3}$ determines the photon to be the check sequence or key sequence. Figure 1 illustrates the proposed scheme. The steps involved in the proposed ASQKD protocol are as follows:

- Step 1.
- Alice prepares the initial W states based on ${K}_{1}.$ If ${K}_{1}=0$, then Alice prepares $|{\kappa}^{+}\rangle $, and while ${K}_{1}=1$, she prepares $|{\gamma}^{+}\rangle $. Alice then divides the W states into three sequences: ${W}_{1}$, ${W}_{2}$
**,**and ${W}_{3}.$ ${W}_{1}$ represents all the first particles of W states. Similarly, ${W}_{2}$ and ${W}_{3}$ represent all the second and third particles of W states, respectively. Alice sends ${W}_{2}$ and ${W}_{3}$ to Bob one photon at a time.

- Step 2.
- For every received photon, Bob performs measure-resending or reflects photons based on ${K}_{2}$.
- If ${K}_{2}=0$, then Bob measures the received photon on a Z-basis, prepares the same photon as the measurement result, and resends it to Alice. For the measured sequence at ${K}_{2}=0$, if ${K}_{3}=0$, then Bob records the measurement results to the check sequence $M{R}_{CB}$; if ${K}_{3}=1$, then Bob records the measurement results to the key sequence $M{R}_{KB}$.
- If ${K}_{2}=1$, then Bob measures the received photon on a Z-basis, prepares the same photon as the measurement result, and resends it to Alice.
- If ${K}_{2}=2$, then Bob reflects the received photon back to Alice without any influence.

- Step 3.
- Alice receives ${W}_{2}^{\prime}$ and ${W}_{3}^{\prime}$ from Bob. She performs Z-basis or W-basis measurements based on ${K}_{2}$.
- If ${K}_{2}=0$, then Alice performs a Z-basis measurement and classifies it into two measured sequences based on ${K}_{3}$. If ${K}_{3}=0$, then Alice records the measurement results as a check sequence $M{R}_{CA}$, whereas if ${K}_{3}=1$, Alice records the measurement results to the key sequence $M{R}_{KA}$.
- If ${K}_{2}=1$, then Alice performs a W-basis measurement to check the entanglement correlation of the W states. Hence, according to the uncertainty principle, if the initial state is $|{\kappa}^{+}\rangle $ or $|{\gamma}^{+}\rangle $, then the measured states should collapse into $\{|{\kappa}^{+}\rangle ,|{\kappa}^{-}\rangle $} or $\{|{\gamma}^{+}\rangle ,|{\gamma}^{-}\rangle $}. Otherwise, if the states remain the same as the initial state, then it is inferred that Bob does not measure the photons and Alice will detect that the protocol may suffer from the reflecting attack.
- If ${K}_{2}=2$, then Alice performs a W-basis measurement for the eavesdropping check. Alice compares the measurement results with the initial states. This implies that if the states remain the same as the original states, neither Bob nor Eve measure the photons, proving the security of the transmission.

After the eavesdropping check, Alice announces $M{R}_{CA}$ to Bob.

- Step 4.
- Bob checks if $M{R}_{CA}=M{R}_{CB}$ to secure the channel. Eventually, Alice and Bob share a raw key as the measurement result of $M{R}_{KA},M{R}_{KB}$ (i.e., if one measures $|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle $, represents classical bits “00”, “01”, “10”, “11”, respectively.). Then, they perform a privacy amplification process [67,68] on the raw key to distill the private key.

## 5. Security Analysis

#### 5.1. Impersonation Attack

#### 5.1.1. Assume Eve Essayed to Impersonate Alice

_{CB}. Bob detects eavesdropping because $M{R}_{CB}\ne M{R}_{CA}.$ Eve cannot impersonate any check sequence MR

_{CA}because it was generated based on ${K}_{3}$, which was previously pre-shared privately. Hence, Eve cannot successfully impersonate Alice in the proposed ASQKD protocol.

#### 5.1.2. Assume Eve Essayed to Impersonate Bob

#### 5.2. Reflecting Attack

#### 5.3. Man-in-the-Middle Attack

#### 5.4. Collective Attack

**Theorem**

**1.**

**Proof**

**of**

**Theorem 1.**

_{1}operation, the possibilities are described as follows:

#### 5.5. Key Leakage Problem

## 6. Efficiency Analysis

- The proposed ASQKD protocol ensures the procedure is functional.
- The proposed ASQKD protocol does not require quantum memory and legally fulfills a semi-quantum environment [17].
- The proposed ASQKD protocol, based on W states, only reduces the quantum hardware requirements.
- The qubit efficiency of the proposed ASQKD protocol is 1.6 times higher than that of Wen et al.’s ASQKD protocol.
- The proposed ASQKD protocol does not require classical cryptography (i.e., the hash function), which does not show the potential menace of the advance quantum computing.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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$|\mathit{Y}\rangle =|0\rangle $ | $|\mathit{Y}\rangle =|1\rangle $ | $|\mathit{Y}\rangle =|+\rangle $ | $|\mathit{Y}\rangle =|-\rangle $ |
---|---|---|---|

$L=00$ | $L=01$ | $L=10$ | $L=11$ |

${\mathit{m}}_{3}=0$ | ${\mathit{m}}_{3}=1$ | |
---|---|---|

${\kappa}^{+}$ | ${m}_{3}^{\prime}=0$ | ${m}_{3}^{\prime}=1$ |

${\kappa}^{-}$ | ${m}_{3}^{\prime}=0$ | ${m}_{3}^{\prime}=1$ |

${\gamma}^{+}$ | ${m}_{3}^{\prime}=1$ | ${m}_{3}^{\prime}=0$ |

${\gamma}^{-}$ | ${m}_{3}^{\prime}=1$ | ${m}_{3}^{\prime}=0$ |

Yu et al.’s ASQKD Protocol [59] | Li et al.’s ASQKD Protocol [60] | Zebboudj et al.’s ASQKD Protocol [63] | Chang et al.’s ASQKD Protocol [64] | Wang et al.’s ASQKD Protocol [65] | Wen et al.’s ASQKD Protocol [66] | The Proposed ASQKD Protocol | |
---|---|---|---|---|---|---|---|

Quantum resource | Bell states | Bell states, Single photons | Single photons | Single photons | Single photons | GHZ-like states W states | W states |

Qubit efficiency | 10% | 11% | 14% | 17% | 14% | $7\%$ | $11\%$ |

Required pre-shared keys (in bits) | 6n | 4n | 3n | 3n | 3n | 4n | 5n |

Classical participant’s quantum capabilities | Generate Reflect Measure | Generate Reflect Measure | Generate Reflect Measure | Generate Reflect Measure | Generate Reflect Measure | Generate Reflect Measure Quantum memory | Generate Reflect Measure |

Classical participant does not require quantum memory | Yes | Yes | Yes | Yes | Yes | No | Yes |

Legal semi-quantum environment | Yes | Yes | Yes | Yes | Yes | No | Yes |

The protocol does not require the hash function | Yes | No | No | No | No | No | Yes |

Runnable protocol | Yes | Yes | Yes | Yes | Yes | No | Yes |

Required classical channel | Yes | Yes | Yes | Yes | No | Yes | Yes |

Robustness of the reflecting attack | Yes | Yes | Yes | No | Yes | Yes | Yes |

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**MDPI and ACS Style**

Wang, H.-W.; Tsai, C.-W.; Lin, J.; Yang, C.-W. Authenticated Semi-Quantum Key Distribution Protocol Based on W States. *Sensors* **2022**, *22*, 4998.
https://doi.org/10.3390/s22134998

**AMA Style**

Wang H-W, Tsai C-W, Lin J, Yang C-W. Authenticated Semi-Quantum Key Distribution Protocol Based on W States. *Sensors*. 2022; 22(13):4998.
https://doi.org/10.3390/s22134998

**Chicago/Turabian Style**

Wang, Hung-Wen, Chia-Wei Tsai, Jason Lin, and Chun-Wei Yang. 2022. "Authenticated Semi-Quantum Key Distribution Protocol Based on W States" *Sensors* 22, no. 13: 4998.
https://doi.org/10.3390/s22134998