#
Secure Physical Layer Network Coding versus Secure Network Coding^{ †}

^{1}

^{2}

^{3}

^{4}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. CAF and Secure CAF

#### 2.1. CAF

#### 2.2. Secure CAF

#### 2.3. Concrete Expressions for Mutual Information

## 3. Butterfly Network

#### 3.1. Conventional Protocol

#### 3.2. Secure NC

**(A1)**- Two source nodes ${V}_{1}$ and ${V}_{2}$ share a secret number L,

**(S1)**- When the eavesdropper attacks only one of the edges, she obtains no information for each message ${M}_{i}$.
**(S2)**- When the nodes do not collude, each node obtains no information for the unintended messages.

#### 3.3. Secure PLNC

#### 3.3.1. Use of Secure CAF

**(A2)**- The pairs $({e}_{1},{e}_{2})$, $({e}_{4},{e}_{5})$, and $({e}_{6},{e}_{7})$ are given as Gaussian MACs such as (2).

#### 3.3.2. Use of CAF

#### 3.4. Comparison

## 4. Network with Three Source Nodes

#### 4.1. Secure NC

**(S3)**- When Eve eavesdrops only one edge among three edges (channels) between the intermediate nodes and the destination node, she obtains no information about each message.
**(S4)**- When Eve eavesdrops only one intermediate (untrusted) node among three intermediate (untrusted) nodes, she obtains no information for each message. Here, no node colludes with another node.

#### 4.1.1. Security (S3)

#### 4.1.2. Security (S4)

#### 4.2. Secure PLNC

#### 4.2.1. Use of Secure CAF

**(A3)**- The channels over the pairs $({e}_{1},{e}_{6})$, $({e}_{2},{e}_{4})$, and $({e}_{3},{e}_{5})$ are Gaussian MACs as in (2).

#### 4.2.2. Use of CAF

#### 4.3. Comparison

## 5. Conclusions and Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Transmission time for four schemes when $GT=1$ and the base of the logarithm is 2. The upper solid line (black) expresses the time $\frac{2GT}{2I{(Y;{A}_{1}\oplus {A}_{2})}_{\mathrm{Equation}(2)}-I{(Y;{A}_{1},{A}_{2})}_{\mathrm{Equation}(2)}}+\frac{GT}{I{(Y;A)}_{\mathrm{Equation}(16)}}$ of the secure PLNC protocol given in Section 3.3.1. The upper dashed line (blue) expresses the time $\frac{4GT}{I{(Y;A)}_{\mathrm{Equation}(16)}}$ of the secure NC protocol given in Section 3.2 without the MAC channel. The lower dashed line (red) expresses the time $\frac{2GT}{I{(Y;A)}_{\mathrm{Equation}(16)}}+\frac{2GT}{I{(Y;{A}_{1},{A}_{2})}_{\mathrm{Equation}(2)}}$ of the secure NC protocol given in Section 3.2 with the MAC channel. The lower solid line (green) expresses the time $\frac{2GT}{I{(Y;{A}_{1}\oplus {A}_{2})}_{\mathrm{Equation}(2)}}+\frac{GT}{I{(Y;A)}_{\mathrm{Equation}(16)}}$ of the secure PLNC protocol given in Section 3.3.2.

**Figure 5.**Transmission Time for four schemes when $GT=1$ and the base of the logarithm is 2. The upper dashed line (blue) expresses the time $\frac{8GT}{I{(Y;A)}_{\mathrm{Equation}(16)}}$ of secure NC protocol given in Section 4.1.2 without the MAC channel. The lower dashed line (red) expresses the time $\frac{6GT}{I{(Y;{A}_{1}{A}_{2}{A}_{3})}_{\mathrm{Equation}(21)}}+\frac{2GT}{I{(Y;{A}_{1},{A}_{2})}_{\mathrm{Equation}(2)}}$ of the secure NC protocol given in Section 4.1.2 with the MAC channel. The solid line (green) expresses the time $\frac{3GT}{I{(Y;{A}_{1}\oplus {A}_{2})}_{\mathrm{Equation}(2)}}+\frac{2GT}{I{(Y;{A}_{1},{A}_{2})}_{\mathrm{Equation}(2)}}$ of the secure PLNC protocol given in Section 4.2.2. The solid line (black) expresses the time $\frac{3GT}{I{(Y;{A}_{1}{A}_{2}{A}_{3})}_{\mathrm{Equation}(21)}}+\frac{GT}{2I{(Y;{A}_{1}\oplus {A}_{2})}_{\mathrm{Equation}(2)}-I{(Y;{A}_{1},{A}_{2})}_{\mathrm{Equation}(2)}}$ of the secure PLNC protocol given in Section 4.2.1. The solid line (black), the solid line (green), and the lower dashed line (red) intersect around $h=1.7$.

Protocol | Assumption | Security | Type |
---|---|---|---|

Section 3.2 | (A1) | (S1) (S2) | Secure NC |

Section 3.3.1 | (A2) | (S2) | Secure PLNC with secure CAF |

Section 3.3.2 | (A1) (A2) | (S1) (S2) | Secure PLNC with CAF and secure NC |

Time Slot | Time i | Time ii | Time iii | Time iv | Time v |
---|---|---|---|---|---|

Channel | ${e}_{1}$, ${e}_{4}$ | ${e}_{2}$, ${e}_{7}$ | ${e}_{3}$ | ${e}_{5}$ | ${e}_{6}$ |

Time Slot | Time i | Time ii | Time iii |
---|---|---|---|

Channel | $({e}_{1},{e}_{2})$ | ${e}_{3}$, ${e}_{4}$, ${e}_{7}$ | ${e}_{5}$, ${e}_{6}$ |

Time Slot | Time i | Time ii | Time iii |
---|---|---|---|

Channel | $({e}_{1},{e}_{2})$ | ${e}_{3}$ | $({e}_{4},{e}_{5})$,$({e}_{6},{e}_{7})$ |

Protocol | Assumption | Security | Type |
---|---|---|---|

Section 4.1.1 | – | (S3) | Secure NC |

Section 4.1.2 | – | (S3) (S4) | Secure NC |

Section 4.2.1 | (A3) | (S4) | Secure PLNC with secure CAF |

Section 4.2.2 | (A3) | (S3) (S4) | Secure PLNC with CAF and secure NC |

Time Span | Time i | Time ii | Time iii | Time iv | Time v |
---|---|---|---|---|---|

Channel | ${e}_{1},{e}_{2},{e}_{3}$ | ${e}_{4},{e}_{5},{e}_{6}$ | ${e}_{7}$ | ${e}_{8}$ | ${e}_{9}$ |

Time Span | Time i | Time ii |
---|---|---|

Channel | $({e}_{1},{e}_{6})$, $({e}_{2},{e}_{4})$,$({e}_{3},{e}_{5})$ | $({e}_{7},{e}_{8},{e}_{9})$ |

Time Span | Time i | Time ii |
---|---|---|

Channel | $({e}_{1},{e}_{6})$, $({e}_{2},{e}_{4})$, $({e}_{3},{e}_{5})$ | $({e}_{7},{e}_{8},{e}_{9})$ |

Time Span | Time i | Time ii | Time iii | Time iv |
---|---|---|---|---|

Channel | $({e}_{1},{e}_{6})$, $({e}_{2},{e}_{4})$,$({e}_{3},{e}_{5})$ | $({e}_{8},{e}_{9})$ | $({e}_{7},{e}_{9})$ | $({e}_{7},{e}_{8})$ |

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**MDPI and ACS Style**

Hayashi, M.
Secure Physical Layer Network Coding versus Secure Network Coding. *Entropy* **2022**, *24*, 47.
https://doi.org/10.3390/e24010047

**AMA Style**

Hayashi M.
Secure Physical Layer Network Coding versus Secure Network Coding. *Entropy*. 2022; 24(1):47.
https://doi.org/10.3390/e24010047

**Chicago/Turabian Style**

Hayashi, Masahito.
2022. "Secure Physical Layer Network Coding versus Secure Network Coding" *Entropy* 24, no. 1: 47.
https://doi.org/10.3390/e24010047