# Confidential Cooperative Communication with the Trust Degree of Jammer

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Trust Degree and System Model

#### 2.1. Trust Degree

#### 2.2. System Model

## 3. Problem Formulation and Solution

**Theorem**

**1.**

**Proof.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

- If ${\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}}{{g}_{tr}{g}_{je}}\right]}^{+}\le {\rho}_{J}\le \frac{{g}_{te}-{g}_{tr}}{{g}_{tr}{g}_{je}}$, ${\rho}_{c}^{o}\in [{\rho}_{c}^{l},\phantom{\rule{3.33333pt}{0ex}}{\rho}_{c}^{u})$, the optimal transmit SNR of the confidential message that maximizes (A1) is obtained by ${\rho}_{c}^{\star}=\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{\rho}_{J}{g}_{tr}{g}_{je}+{g}_{tr}-{g}_{te}}{2{g}_{tr}{g}_{te}}$;
- If ${\rho}_{J}\le min\left\{{\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}}{{g}_{tr}{g}_{je}}\right]}^{+},\frac{{g}_{te}-{g}_{tr}}{{g}_{tr}{g}_{je}}\right\}$, ${\rho}_{c}^{o}\in [0,\phantom{\rule{3.33333pt}{0ex}}{\rho}_{c}^{l}]$, the optimal transmit SNR of the confidential message is obtained by ${\rho}_{c}^{\star}={\rho}_{c}^{l}$ due to the concavity of (A1).

- If $\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}-{g}_{tr}}{{g}_{tr}{g}_{je}}<{\rho}_{J}$, ${\rho}_{c}^{o}>{\rho}_{T}$, the optimal transmit SNR of the confidential message is obtained by ${\rho}_{c}^{\star}={\rho}_{T}$ due to the concavity of (A1);
- If $max\left\{\frac{{g}_{te}-{g}_{tr}}{{g}_{tr}{g}_{je}},\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}}{{g}_{tr}{g}_{je}}\right\}<{\rho}_{J}\le \frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}-{g}_{tr}}{{g}_{tr}{g}_{je}}$, ${\rho}_{c}^{o}\in ({\rho}_{c}^{l},\phantom{\rule{3.33333pt}{0ex}}{\rho}_{T}]$, the optimal transmit SNR of the confidential message is obtained by ${\rho}_{c}^{\star}=\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{\rho}_{J}{g}_{tr}{g}_{je}+{g}_{tr}-{g}_{te}}{2{g}_{tr}{g}_{te}}$;
- If $\frac{{g}_{te}-{g}_{tr}}{{g}_{tr}{g}_{je}}<{\rho}_{J}\le \frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}}{{g}_{tr}{g}_{je}}$, ${\rho}_{c}^{o}\in [0,\phantom{\rule{3.33333pt}{0ex}}{\rho}_{c}^{l}]$, the optimal transmit SNR of the confidential message is ${\rho}_{c}^{\star}={\rho}_{c}^{l}$, and this result can be categorized into (3) in this Appendix.

## Appendix B. Proof of the Concavity of (A2)

## Appendix C. Proof of Corollary 1

- If $0\le \alpha <{\left[\frac{{g}_{te}-{g}_{tr}-{\rho}_{T}{g}_{tr}{g}_{te}}{{g}_{te}}\right]}^{+}$, ${\rho}_{c}^{o}<0$, the optimal transmit SNR of the confidential message is ${\rho}_{c}^{\star}=0$ due to the concavity of (A5);
- If ${\left[\frac{{g}_{te}-{g}_{tr}-{\rho}_{T}{g}_{tr}{g}_{te}}{{g}_{te}}\right]}^{+}\le \alpha <{\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}}{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}}\right]}^{+}$, ${\rho}_{c}^{o}\in [0,\phantom{\rule{3.33333pt}{0ex}}{\rho}_{c}^{l})$, the optimal transmit SNR of the confidential message that maximizes (A5) is ${\rho}_{c}^{\star}=\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}+\alpha {g}_{te}}{(2-\alpha ){g}_{tr}{g}_{te}}$;
- If ${\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}}{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}}\right]}^{+}\le \alpha <{\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}-{g}_{tr}}{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}}\right]}^{+}$, ${\rho}_{c}^{o}\in [{\rho}_{c}^{l},\phantom{\rule{3.33333pt}{0ex}}{\rho}_{T}]$, the optimal transmit SNR of the confidential message is ${\rho}_{c}^{\star}={\rho}_{c}^{l}$ due to the concavity of (A5).

- If $0\le \alpha <{\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}-{g}_{tr}}{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}}\right]}^{+}$, ${\rho}_{c}^{o}\in [0,\phantom{\rule{3.33333pt}{0ex}}{\rho}_{T}]$, the optimal transmit SNR of the confidential message is ${\rho}_{c}^{\star}=\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{tr}-{g}_{te}+\alpha {g}_{te}}{(2-\alpha ){g}_{tr}{g}_{te}}$;
- If ${\left[\frac{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}-{g}_{tr}}{{\rho}_{T}{g}_{tr}{g}_{te}+{g}_{te}}\right]}^{+}\le \alpha \le 1$, ${\rho}_{c}^{o}>{\rho}_{T}$, the optimal transmit SNR of the confidential message is ${\rho}_{c}^{\star}={\rho}_{T}$ due to the concavity of (A5).

## References

- Zhang, M.; Chen, X.; Zhang, J. Social-aware relay selection for cooperative networking: An optimal stopping approach. In Proceedings of the 2014 IEEE International Conference on Communications (ICC), Sydney, Australia, 10–14 June 2014; pp. 2257–2262. [Google Scholar]
- Mao, H.; Feng, W.; Zhao, Y.; Ge, N. Joint social-position relationship based cooperation among mobile terminals. IEEE Commun. Lett.
**2014**, 18, 2165–2168. [Google Scholar] [CrossRef] - Ryu, J.Y.; Lee, J.; Quek, T.Q. Trust degree based beamforming for MISO cooperative communication system. IEEE Commun. Lett.
**2015**, 19, 1957–1960. [Google Scholar] [CrossRef] - Ryu, J.Y.; Lee, J.H. Trust Degree-Based MISO Cooperative Communications with Two Relay Nodes. Wirel. Commun. Mob. Comput.
**2019**. [Google Scholar] [CrossRef] - Zhao, M.; Ryu, J.Y.; Lee, J.; Quek, T.Q.; Feng, S. Exploiting trust degree for multiple-antenna user cooperation. IEEE Trans. Wirel. Commun.
**2017**, 16, 4908–4923. [Google Scholar] [CrossRef] - Sun, Y.L.; Han, Z.; Yu, W.; Liu, K.R. A Trust Evaluation Framework in Distributed Networks: Vulnerability Analysis and Defense Against Attacks. In Proceedings of the 25th IEEE International Conference on Computer Communications (INFOCOM 2006), Barcelona, Spain, 23–29 April 2006; pp. 1–13. [Google Scholar]
- Zhao, M.; Wang, X.; Feng, S. Joint Power Splitting and Secure Beamforming Design in the Multiple Non-Regenerative Wireless-Powered Relay Networks. IEEE Commun. Lett.
**2015**, 19, 1540–1543. [Google Scholar] [CrossRef] [Green Version] - Mavoungou, S.; Kaddoum, G.; Taha, M.; Matar, G. Survey on threats and attacks on mobile networks. IEEE Access
**2016**, 4, 4543–4572. [Google Scholar] [CrossRef] - Jameel, F.; Wyne, S.; Kaddoum, G.; Duong, T.Q. A comprehensive survey on cooperative relaying and jamming strategies for physical layer security. IEEE Commun. Surv. Tutor.
**2018**. [Google Scholar] [CrossRef] - Vuppala, S.; Tolossa, Y.J.; Kaddoum, G.; Abreu, G. On the physical layer security analysis of hybrid millimeter wave networks. IEEE Trans. Commun.
**2017**, 66, 1139–1152. [Google Scholar] [CrossRef] - Atallah, M.; Kaddoum, G. Secrecy Analysis in Wireless Network with Passive Eavesdroppers by Using Partial Cooperation. IEEE Trans. Veh. Technol.
**2019**. [Google Scholar] [CrossRef] - Ryu, J.Y.; Lee, J.; Quek, T.Q. Confidential Cooperative Communication with Trust Degree of Potential Eavesdroppers. IEEE Trans. Wirel. Commun.
**2016**, 15, 3823–3836. [Google Scholar] [CrossRef] - Tang, L.; Chen, H.; Li, Q. Social tie based cooperative jamming for physical layer security. IEEE Commun. Lett.
**2015**, 19, 1790–1793. [Google Scholar] [CrossRef] - Wang, H.M.; Xu, Y.; Huang, K.W.; Han, Z.; Tsiftsis, T.A. Cooperative secure transmission by exploiting social ties in random networks. IEEE Trans. Commun.
**2018**, 66, 3610–3622. [Google Scholar] [CrossRef] - Wang, L.; Wu, H.; Stüber, G.L. Cooperative jamming-aided secrecy enhancement in P2P communications with social interaction constraints. IEEE Trans. Veh. Technol.
**2017**, 66, 1144–1158. [Google Scholar] [CrossRef] - Gong, X.; Chen, X.; Zhang, J. Social group utility maximization game with applications in mobile social networks. In Proceedings of the 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, 2–4 October 2013; pp. 1496–1500. [Google Scholar]
- Gong, X.; Chen, X.; Zhang, J. Social group utility maximization in mobile networks: From altruistic to malicious behavior. In Proceedings of the 2014 48th Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA, 19–21 March 2014; pp. 1–6. [Google Scholar]
- Chen, X.; Proulx, B.; Gong, X.; Zhang, J. Exploiting social ties for cooperative D2D communications: A mobile social networking case. IEEE Trans. Wirel. Commun.
**2015**, 23, 1471–1484. [Google Scholar] [CrossRef] - Coon, J.P. Modelling trust in random wireless networks. In Proceedings of the 2014 11th International Symposium on Wireless Communications Systems (ISWCS), Barcelona, Spain, 26–29 August 2014; pp. 976–981. [Google Scholar]
- Li, J.; Li, R.; Kato, J. Future trust management framework for mobile ad hoc networks. IEEE Commun. Mag.
**2008**, 46, 108–114. [Google Scholar] - Theodorakopoulos, G.; Baras, J.S. On trust models and trust evaluation metrics for ad hoc networks. IEEE J. Sel. Areas Commun.
**2006**, 24, 318–328. [Google Scholar] [CrossRef] [Green Version] - Zouridaki, C.; Mark, B.L.; Hejmo, M.; Thomans, R.K. A quantitative trust establishment framework for reliable data packet delivery in MANETs. In Proceedings of the Milcom 2010 Military Communications Conference, San Jose, CA, USA, 31 October–3 November 2005; pp. 1–10. [Google Scholar]
- Changiz, R.; Halabian, H.; Yu, F.R.; Lambadaris, I.; Tang, H.; Mason, P.C. Trust establishment in cooperative wireless networks. In Proceedings of the 3rd ACM Workshop on Security of Ad Hoc and Sensor Networks, Lexandria, VA, USA, 7 November 2005; pp. 1074–1079. [Google Scholar]
- Jorgensen, M.L.; Yanakiev, B.R.; Kirkelund, G.E.; Popovski, P.; Yomo, H.; Larsen, T. Shout to secure: Physical-layer wireless security with known interference. In Proceedings of the IEEE GLOBECOM 2007—IEEE Global Telecommunications Conference, Washington, DC, USA, 26–30 November 2007; pp. 33–38. [Google Scholar]
- Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]

**Figure 2.**The expected secrecy rate versus trust degree $\alpha $, where ${\rho}_{T}={\rho}_{J}=30$ dB.

**Figure 3.**The expected secrecy rate versus the transmit SNR of jamming signal ${\rho}_{J}$, where ${\rho}_{T}=30$ dB.

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**MDPI and ACS Style**

Zhao, M.; Liu, D.; Gao, H.; Feng, W.
Confidential Cooperative Communication with the Trust Degree of Jammer. *Entropy* **2019**, *21*, 595.
https://doi.org/10.3390/e21060595

**AMA Style**

Zhao M, Liu D, Gao H, Feng W.
Confidential Cooperative Communication with the Trust Degree of Jammer. *Entropy*. 2019; 21(6):595.
https://doi.org/10.3390/e21060595

**Chicago/Turabian Style**

Zhao, Mingxiong, Di Liu, Hui Gao, and Wei Feng.
2019. "Confidential Cooperative Communication with the Trust Degree of Jammer" *Entropy* 21, no. 6: 595.
https://doi.org/10.3390/e21060595