Distributed beamforming designs to improve physical layer security in wireless relay networks
© Qian et al.; licensee Springer. 2014
Received: 13 March 2014
Accepted: 8 April 2014
Published: 28 April 2014
The Erratum to this article has been published in EURASIP Journal on Advances in Signal Processing 2014 2014:129
This paper investigates security-oriented beamforming designs in a relay network composed of a source-destination pair, multiple relays, and a passive eavesdropper. Unlike most of the earlier works, we assume that only statistical information of the relay-eavesdropper channels is known to the relays. We propose beamforming solutions for amplify-and-forward (AF) and decode-and-forward (DF) relay networks to improve secrecy capacity. In an AF network, the beamforming design is obtained by approximating a product of two correlated Rayleigh quotients to a single Rayleigh quotient using the Taylor series expansion. Our study reveals that in an AF network, the secrecy capacity does not always grow as the eavesdropper moves away from the relays or as total relay transmit power increases. Moreover, if the destination is nearer to the relays than the eavesdropper is, a suboptimal power is derived in closed form through monotonicity analysis of secrecy capacity. While in a DF network, secrecy capacity is a single Rayleigh quotient problem which can be easily solved. We also found that if the relay-eavesdropper distances are about the same, it is unnecessary to consider the eavesdropper in a DF network. Numerical results show that for either AF or DF relaying protocol, the proposed beamforming scheme provides higher secrecy capacity than traditional approaches.
Cooperative communications, in which multiple nodes help each other transmit messages, has been widely acknowledged as an effective way to improve system performance [1–3]. However, due to the broadcast property of radio transmission, wireless communication is vulnerable to eavesdropping which consequently makes security schemes of great importance as a promising approach to communicate confidential messages.
The traditional secure communication schemes rely on encryption techniques where secret keys are used. However, as the high-layer secure protocols have attracted growing attacks in recent years, the implementation of security schemes at physical layer becomes a hotspot. It was first proved by Wyner that it is possible to communicate perfectly at a non-zero rate without a secret key if the eavesdropper has a worse channel than the destination . This work was extended to Gaussian channels in  and to fading channels in . Recently, there has been considerable work on secure communication in wireless relay networks (WRNs) [7–15]. A widely acknowledged measurement of system security in WRNs is the maximal rate of secret information exchange between source and destination which is defined as secrecy capacity. A decode-and-forward (DF)-based cooperative beamforming scheme which completely nulls out source signal at eavesdropper(s) was proposed in , and this work was extended to the amplify-and-forward (AF) protocol and cooperative jamming in . Hybrid beamforming and jamming was investigated in  where one relay was selected to cooperate and the other to make intentional interference in a DF network. Combined relay selection and cooperative beamforming schemes for DF networks were proposed in  where two best relays were selected to cooperate. The authors of [11, 12] considered the scenario where the relay(s) could not be trusted in cooperative MIMO networks. Additionally, a new metric of system security is brought up in  as intercept probability and optimal relay selection schemes for AF and DF protocols based on the minimization of intercept probability were proposed.
In earlier works, it is widely assumed that the relays have access to instantaneous channel state information (CSI) of relay-eavesdropper (RE) channels [7, 8, 13–15]. This assumption is ideal but unpractical in a real-life wiretap attack since the malicious eavesdropper would not be willing to share its instantaneous CSI. Thus, security schemes using instantaneous CSI of the eavesdropper cannot be adopted anymore. However, the instantaneous CSI of relay-destination (RD) channels is available since the destination is positive. The statistical information of the RE channels is also available through long-term supervision of the eavesdropper's transmission . It is worth mentioning that even if the relays do not have access to the perfect CSI of RD channels, they can still estimate these channels by training sequences and perform beamforming based on the estimated CSI .
Our focus is on secrecy capacity, and we are interested in maximizing it with appropriate weight designs of relays. The remainder of this paper is organized as follows. Section 2 introduces system model under AF and DF protocols using relay beamforming. The optimization problem in an AF network is addressed and solved in Section 3 along with some analyses of secrecy capacity. Section 4 provides the optimal beamforming design for a DF network along with a surprising finding that considering the eavesdropper sometimes may not be necessary. Numerical results are given in Section 5 to compare the performances of different designs, and Section 6 provides some concluding remarks.
2. System model
where v i is the additive noise at R i .
where w = (w1, …, w M )T, ρ fg = (ρ1f1g1, …, ρ M f M g M )T, ρ g = (ρ1g1, …, ρ M g M )T, v = (v1, …, v M )T, and vD represents additive white Gaussian noise (AWGN) at the destination. The total relay transmit power is wHw = P.
where ρ fh = (ρ1f1h1, …, ρ M f M h M )T, ρ h = (ρ1h1, …, ρ M h M )T, and vE represents AWGN at the eavesdropper.
where g = (g1, …, g M )T and h = (h1, …, h M )T.
3. Distributed beamforming design for AF
where , , and γD and γE are received signal-to-noise ratios (SNRs) at the destination and the eavesdropper, respectively. We aim to improve CS by exploiting appropriate beamforming designs. The following subsection describes the proposed beamforming design for an AF network.
3.1 Proposed design for AF (P-AF)
where Γ g = diag(ρ12|g1|2, …, ρ M 2|g M |2), , and Now we discuss how to design w to maximize CS, and the proposed solution is denoted by . It is obvious that maximizing CS is equivalent to maximizing . Hence, in what follows, the objective function will be .
where . This is a product of two correlated Rayleigh quotients  which is generally difficult to maximize. However, it would be much easier to get a suboptimal solution if we approximate the objective function to a single Rayleigh quotient.
Denote the matrices D h + P- 1I, Γ g + P- 1I, and Γ h + P- 1I as A, B, and C, respectively. For simplicity, we also let a i , b i , and c i represent the i th diagonal entry of A, B, and C, respectively, and define p = (P1, …, P M )T.
where Φ = A- 1B- 1C(Psρ fg ρ fg H + Γ g + P- 1I).
Channel coefficients used in Figure 2
Channel coefficients realizations
M = 6
We also address the traditional design for AF (T-AF) where the eavesdropper is ignored and the goal is to maximize CD. It can be easily proved that the optimal solution is where cAF is a constant chosen to satisfy .
3.2 Discussion about secrecy capacity in AF networks
It is natural to conjecture that secrecy capacity would grow as the eavesdropper moved away or as the total relay transmit power increased. However, we find that this conjecture is not always right.
For simplicity, we assume the distances between relays are much smaller than those between the relays and the source, so the path losses of the SR channels are almost the same. The same assumption is also made to the destination/eavesdropper. Denote the SR, RD, and RE distances as dSR, dRD, and dRE, respectively, and the corresponding channel variances as , , and , respectively.
Due to the difficulty of calculating the eigenvalues of Φ, we replace the non-diagonal elements in Φ with their mean value 0 and the i th diagonal element with where . Thus, Φ becomes λ(P)I after replacement.
By setting , we obtain the positive stationary point of CS(P) as described in (18).
If dRE > dRD (), ∀P∈(0, Psubopt), we have ; ∀P∈(Psubopt, + ∞), we have . Hence, if the destination is much nearer than the eavesdropper is, CS(P) is an increasing function over (0, Psubopt) and a decreasing function over (Psubopt, + ∞), which means that CS(Psubopt) is the maximum of CS(P).
This monotonicity of CS and the accuracy of Psubopt under the case of dRE > dRD will be verified in the next section. It needs to be pointed out that the above analysis is not for any certain design, so the optimal value of P for a certain design would be different from but around Psubopt. It also needs to be pointed out that the replacement of the channel coefficients in Φ with their mean values may result in the loss of the security benefit that is supposed to be achieved by exploiting the perfect CSI of SR and RD channels. This loss does not affect the monotonicity of CS greatly under the case of dRE > dRD because the destination is much nearer and therefore much more advantageous in communication than the eavesdropper is. However, when dRE < dRD (or dRE = dRD), such replacement becomes inappropriate, since the instantaneous CSI of f i and g i improves the system security significantly.
It can be observed from (22) that the positivity of depends on the value of P. Thus, CS(P) is neither convex nor concave.
This equation does not involve , which implies CS is a constant in this case wherever the eavesdropper is.
4. Distributed beamforming design for DF
This section focuses on the security-oriented beamforming design for DF protocol. Similar to the design for AF protocol, the mission is to find the optimal design under a total relay transmit power constraint to maximize secrecy capacity.
4.1 Proposed design for DF (P-DF)
For comparison purpose, we also address the traditional design for DF (T-DF) and denote the solution by . The optimization problem is formulated as , and the optimal solution is obviously where cDF is a constant chosen to satisfy the power constraint .
4.2 Discussion about secrecy capacity in DF networks
It is a natural thought that no matter under what channel assumption, secrecy capacity achieved by security-oriented designs would be higher than that achieved by traditional designs. However, the fact is that these designs may have the same performance which means that sometimes we can just ignore the eavesdropper.
Remark 2. In a DF network, if the RE distances are about the same (which is widely assumed), it is unnecessary to consider the eavesdropper as the security-oriented design and the traditional design are indeed the same.
we have umax(I + P ggH) = umax(P ggH). Thus, which is the same as .
5. Numerical results
In this section, we investigate the performance of the above beamforming designs numerically. The simulation environment follows the model of Section 2. We perform Monte Carlo experiments consisting of 10,000 independent trials to obtain the average results.
Assume the number of relays is M = 6, and the source transmit power is Ps = 10 dB. In order to show the influence of the RE distance in AF protocol, we fix the source at (0,0), the destination at (2,0), and the relays at (1,0) and move the eavesdropper from (1.25,0) to (5,0). We assume that the distances between relays are much smaller than SR/RD/RE distances. Therefore, the SR channels and RD channels follow a distribution, and the RE distance dRE varies from 0.25 to 4.
To verify Remark 2, we now examine the P-DF design and T-DF design under different variance assumptions of the RE channels.
In this paper, we focused on security-oriented distributed beamforming designs for relay networks in the presence of a passive eavesdropper. We provided two beamforming designs under a total relay transmit power constraint, one of which is for AF and the other is for DF. Each design is to maximize secrecy capacity by exploiting information of SR, RD, and RE channels. To derive the beamforming solution for AF requires approximating the optimization objective by using the Taylor series expansion, while the solution for DF is obtained much more easily. We also found that secrecy capacity does not always grow if the relays use more power to transmit or if the eavesdropper gets farther from the relays, and that taking the eavesdropper into consideration is not always necessary. Moreover, for AF, we derived a suboptimal value of the total relay transmit power if the destination is nearer than the eavesdropper is. Numerical results showed the efficiency of the proposed designs.
Thus, K ≈ 0 with large P.
This work is supported by the Natural Science Foundation of China under Grants 61372126 and 61302101, and the open research fund of National Mobile Communications Research Laboratory in Southeast University under Grant 2012D11.
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