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SCIENCE CHINA Information Sciences, Volume 62, Issue 4: 042304(2019) https://doi.org/10.1007/s11432-018-9660-0

On proactive eavesdropping using anti-relay-selection jamming in multi-relay communication systems

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  • ReceivedJul 6, 2018
  • AcceptedOct 22, 2018
  • PublishedFeb 26, 2019

Abstract

This paper considers a multi-relay legitimate surveillance system, where a suspicious link may transmit illegal information with the help of relays and a legitimate monitor aims at intercepting the information. We assume that the relay selection method is adopted by the suspicious link to improve the efficiency of the information transmission. We propose anti-relay-selection (ARS) jamming strategies for two different information transmission scenarios, namely the event-based information and content-based information. Beamforming vector of jamming is optimized and the iterative algorithms are used to improve the jamming efficiency. We demonstrate the effectiveness and the superiority of the ARS jamming strategies through numerical results and simulations.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61871241, 61771263, 61801248) and Nature Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 18KJB510036).


Supplement

Appendix

Proof of Proposition 1

For (P4), let us investigate the following KKT conditions: \begin{align}\mathbf{I} + \mathrm{Tr}({\boldsymbol H}_{s,r_i})\sum\limits_{i=1}^{M}\mu_{i}{\boldsymbol h}_{r_i^{*},e}{\boldsymbol h}_{r_i^{*},e}^{\dagger}-\mathrm{Tr}({\boldsymbol H}_{s,r_i^{*}})\sum\limits_{i=1}^{M}\mu_{i}{\boldsymbol h}_{r_i,e}{\boldsymbol h}_{r_i,e}^{\dagger}-{\boldsymbol V} = 0, \tag{24} \end{align} \begin{equation}{\boldsymbol W}{\boldsymbol V} = 0, \tag{25}\end{equation} \begin{align}\mathrm{Tr}({\boldsymbol H}_{s,r_i})(\mathbf{Tr}({\boldsymbol h}_{r_i^{*},e}{\boldsymbol h}_{r_i^{*}锛?e}^{\dagger}{\boldsymbol W})+1) < \mathrm{Tr}({\boldsymbol H}_{s,r_i^{*}})(\mathbf{Tr}({\boldsymbol h}_{r_i,e}{\boldsymbol h}_{r_i,e}^{\dagger}{\boldsymbol W})+1), \tag{26} \end{align} \begin{equation}r_i \in \mathcal{I}_{\rm out}, {\boldsymbol W} \succeq 0, \tag{27}\end{equation} where ${\boldsymbol~W}$ denotes the optimal primal variable, and $\mu_{i}~\geq~0$, $i~=~1,2,\ldots,M$ and ${\boldsymbol~V}~\succeq~0$ are the optimal dual variables associated with constraints (A1)–(A4), respectively. It can be inferred from (A1) and (A2) that $\mathrm{rank}({\boldsymbol~V})~\ge~N_{{\boldsymbol~W}}~-~1$ and $\mathrm{rank}({\boldsymbol~W})~\leq~1$. Since $\mathrm{rank}({\boldsymbol~W})~=~0$ violates (A3) owing to adoption conditions of ARS jamming discussed before, we have $\mathrm{rank}({\boldsymbol~W})~=~1$.


References

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  • Figure 1

    (Color online) System model.

  • Figure 2

    (Color online) Performance of achievable surveillance rate, $P_e$ = 20 dB.

  • Figure 3

    (Color online) Performance of non-outage probability, $P_e$ = 20 dB.

  • Figure 4

    (Color online) Average iterations of ARS jamming strategies.

  • Figure 5

    (Color online) Achievable surveillance rate with different transmit power of the monitor, the number of relays is 4.

  • Figure 6

    (Color online) Outage probability of surveillance with different transmit power of the monitor, the number of relays is 4.

  •   

    Algorithm 1 ARS jamming for event-based information

    Sort $\mathcal{I}_{\rm~svl}$ according to $\gamma_{s,r_i}$ and set $k$ express the number of elements.

    while $k~\ne~0$ do

    Set $r_i^*~=~r_k$ and solve P4 with relay $r_i^*$;

    if P4 is feasible then

    if $P_{\rm~min}~\le~P_e$ then

    $\chi~=~1$;

    goto final;

    else

    $k~=~k~-~1$;

    end if

    else

    $k~=~k~-~1$;

    end if

    end while

    if $k~=~0$ then

    $\chi~=~0$;

    end if

    final

    Return $\chi$.

  •   

    Algorithm 2 ARS jamming for content-based information

    Sort $I_{\rm~svl}$ according to $\textrm{min}\{\gamma_{s,{r_i}},\gamma_{{r_i},d}\}$ and set $k$ express the number of elements.

    while $k~\ne~0$ do

    Set $r_i^*~=~r_k$ and solve P6 with relay $r_i^*$;

    if P6 is feasible then

    if $P_{\rm~min}~\le~P_e$ then

    $R_{\rm~svl}~=~R_{{\rm~svl},r_i^*}$;

    goto final;

    else

    $k~=~k~-~1$;

    end if

    else

    $k~=~k~-~1$;

    end if

    end while

    if $k~=~0$ then

    $R_{\rm~svl}~=~0$;

    end if

    final

    Return $R_{\rm~svl}$;

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