Joint power allocation and relay selection for decode-and-forward cooperative relay in secure communication

The Journal of China Universities of Posts and Telecommunications April 2013, 20(2): 79–85 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Joint power allocation and relay selection for decode-and-forward cooperative relay in secure communication YANG Yun-chuan1 (), ZHAO Hui1, SUN Cong2, WANG Wen-bo1 1. Key Laboratory of Universal Wireless Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. ICMSEC, AMSS, Chinese Academy of Sciences, Beijing 100190, China

Abstract In this paper, we investigate the cooperative strategy with total power constraint in decode-and-forward (DF) relaying scenario, in the presence of an eavesdropper. Due to the difference of channel for each source-relay link, not all relay nodes have constructive impacts on the achievable secrecy rate. Besides, the final achieved secrecy rate depends on both source-relay and relay-destination links in DF relaying scenario. Therefore, the principal question here is how to select cooperative strategy among relays with proper power allocation to maximize the secrecy rate. Three strategies are considered in this paper. First, we investigate the cooperative jamming (CJ) strategy, where one relay with achieved target transmission rate is selected as a conventional relay forwarding signal, and remaining relays generate artificial noise via CJ strategy to disrupt the eavesdropper. Two CJ schemes with closed-form solutions, optimal cooperative jamming (OCJ) and null space cooperative jamming (NSCJ), are proposed. With these solutions, the corresponding power allocation is formulated as a geometric programming (GP) problem and solved efficiently by convex programming technique. Then, to exploit the cooperative diversity, we investigate the cooperative relaying (CR) strategy. An iterative algorithm using semi-definite programming (SDP) and GP together with bisection search method is proposed to optimize the cooperative relaying weight and power allocated to the source and relays. Furthermore, to exploit the advantages of both CR and CJ, we propose an adaptive strategy to enhance the security. Simulation results demonstrate that the efficiency of the proposed cooperative strategies in terms of secrecy rate. Keywords

physical layer security, CJ, CR, power allocation, SDP, GP

1 Introduction Privacy and security issues play important roles in wireless networks. Physical layer security technique, which exploits the physical characteristic of wireless channel to guarantee the message being transmitted securely, has attracted significant attentions. In Ref. [1], Wyner developed the concept of wire-tap channel and established the possibility of creating secrecy links without relying on the privacy cryptograph, where the notion of ‘secrecy capacity’ was proposed. Node cooperation via single-antenna relays can be regarded as an effective way to enhance the secrecy

Received date: 23-07-2012 Corresponding author: YANG Yun-chuan, E-mail: [email protected] DOI: 10.1016/S1005-8885(13)60032-1

rate [2]. Two cooperative strategies, CR and CJ based on DF or amplify-and-forward (AF) relaying protocol, were proposed for the secure communication in one-way relay channel [3]. In Ref. [4], several relay selection criterions based on DF protocol were proposed to enhance the security in one-way relay channel. In this scenario, two relay nodes were selected, where one relay acted as conventional relay to forward signal and another as a jammer to disrupt eavesdropper. Although relay selection can choose the ‘best’ nodes to improve the secrecy rate, without cooperation the cooperative diversity is not exploited well, which will result in the limitation of the achieved secrecy rate limited. In Ref. [5], a relay chatting scheme based on AF protocol was proposed, where the remaining relays transmitted a random message via distributed beamforming.

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In this paper, we study cooperative strategy with power allocation for secure communication in DF relaying networks. Three strategies are considered. Our work is different from the aforementioned researches in following aspects: 1) We consider a DF relaying protocol with total transmit power constraint, which results in that the achievable secrecy rate depends on the performance of two phases. 2) Comparing with the strategy to select both relaying and jamming nodes strategy [4], we consider one relaying node selection and let the remaining relays acted as the jamming nodes, or all relays act as forwarding nodes via cooperative relaying. Then the relays can exploit not only selection diversity but also the cooperative diversity to enhance the security. Notations Superscripts (⋅)* , (⋅)T and (⋅) H stand for complex conjugate, transpose, and conjugate transpose operations, respectively. I N is the identity matrix with size N × N . Tr A denotes the trace of matrix A . E {} ⋅ denotes the statistical expectation. ⋅ denotes the absolute value of a scalar, and ⋅ 2 denotes the norm of a vector. A0 means that A is a positive semi-definite matrix.

( x)

+

denotes max(0, x) .

= {1, 2,..., N } is the index

2013

yRi = PS hSR i x + nRi . Here, hSR i stands for the channel between source S and the ith relay. nR i is the additive zero-mean noise with variance σ R2 i at the ith relay.

Fig. 1 Joint relay selection and CJ model for secure communication

Considering that the fading characteristic of each source-to-relay link is different with each other, we define a decoding set Cd ⊆ ( i ∈ Cd if CR i > R0 , where CR i is the instantaneous capacity of link between source S and ith relay, and R0 is the target transmission rate), where relays can decode the received signal successfully with a given transmit power PS . Then, in the second slot, the ‘best relay’ is chosen from the decoding set Cd , and is denoted as R * , which forwards the signal with power PR . Meanwhile, the remaining relays operate as jammer nodes and cooperatively transmit artificial noise z with power PJ

2 System model and problem formulation

aiming to disrupt the performance of the link between the selected relay and the eavesdropper. So the received signal at the legitimate receiver is expressed as: H yD = PR hR* D x + PJ hJD wJ z + nD (1)

2.1

where hR*D ∈ C and hJD ∈ C ( N −1)×1 stand for the channel

set of N relays.

System model

Consider a wireless communication system consisting of a source S, a legitimate receiver D, an eavesdropper E and N relay nodes

{R } i

N

i=1

, as depicted in Fig. 1. Assume that

the two direct links: S-D and S-E, are not available, due to some practical constraints, such as limited transmit power and shadow fading effects. Assume that the global channel state information (CSI) is available [6]. In this model, the source transmits messages with the help of relays while keeping the eavesdropper ignorant of information. Relays not only decode and forward signal but also generate artificial noise to disrupt the interception of eavesdropper. In the first slot, the source broadcasts signal x to the relays with transmit power PS , where

{ } = 1 . The received signal

E x

2

yRi at the ith relay is

coefficients from the selected relay and jammers to the legitimate receiver, respectively. wJ ∈ C ( N −1)×1 is a jamming vector with wJH wJ = 1 . nD is the additive zero-mean noise with variance σ D2 at legitimate receiver. The signal received at the eavesdropper is given by: yE = PR hR* E x + PJ hJEH wJ z + nE (2) where

hR*E ∈ C

and

hJE ∈ C ( N −1)×1

are the channel

coefficients from the selected relay and jammers to the eavesdropper. nE is the additive zero-mean noise with variance σ E2 at the eavesdropper. The instantaneous capacity at the legitimate receiver and eavesdropper, denoted as CD and CE , are written as,: 2 ⎛ PR hR* D 1 ⎜ CD = lb 1 + 2 ⎜ P hH w 2 + σ 2 J JD J D ⎝

⎞ ⎟ ⎟ ⎠

(3)

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2 ⎛ ⎞ PR hR* E 1 ⎜ ⎟ CE = lb 1 + (4) 2 ⎜ P hH w 2 + σ 2 ⎟ J JE J E ⎝ ⎠ The achieved rate at the selected relay in the first slot can be expressed as: 2 ⎛ ⎞ 1 ⎜ PS hSR* ⎟ CR* = lb 1 + (5) σ R2 * ⎟ 2 ⎜ ⎝ ⎠ Thus, the final secrecy rate in the DF relaying scenario can be defined as [7]:

CS = ( min(CD , CR* ) − CE ) 2.2

+

(6)

Problem formulation

To improve the achievable secrecy rate, we need to balance the quality of two links via optimizing the cooperative jamming weights as well as allocating appropriate power for each node. Since a joint relay selection and power allocation strategy is considered in this paper, at least one relay can be chosen from the decoding set. Therefore, the transmit power of source should be constrained to meet the target transmission rate R0 as follows: PSPS0 =

(22 R0 − 1)σ R2 * hSR*

(7)

2

Mathematically, the corresponding problem can be formulated as: ⎛ 1 + γ D 1 + γ R* ⎞ ⎫ max min ⎜ , ⎟⎪ PS , PR , PJ , wJ ⎝ 1 + γ E 1 + γ E ⎠⎪ ⎪ s.t. ⎪ ⎬ PS + PR + PJ = P ⎪ ⎪ wJH wJ = 1 ⎪ PSPS0 ⎪⎭

(

where γ D = PR hR*D

(P h J

H JE

2

)

2

)(

2

)

optimization

(8)

(

H PJ hJD wJ + σ D2 , γ E = PR hR*E

(

wJ + σ E2 , γ R* = PS hSR*

2

)σ

2 R*

2

)

, and P is the

total power constraint over source and all relay nodes.

3 Joint power allocation and CJ with relay selection strategy 3.1

Relay selection criterion

The relay selection criterion is based on the instantaneous signal noise ration (SNR) of link from

source S to relays for simplicity as: 2 ⎛ ⎞ PS hh i ⎟ ⎜ SR * R = arg max ⎜ ⎟ 2 i∈ ⎜ σ Ri ⎟ ⎝ ⎠

81

(9)

3.2 Power allocation and CJ strategy Noticing that Eq. (8) is a non-convex optimization problem, we maximize the lower bound of the objective function in Eq. (8) as: γ * ⎞⎫ ⎛ γ max min ⎜ D , R ⎟⎪ PS , PR , PJ , w J ⎝ 1 + γ E 1 + γ E ⎠⎪ ⎪ s.t. ⎪ (10) ⎬ PS + PR + PJ = P ⎪ ⎪ wJH wJ = 1 ⎪ PSPS0 ⎪⎭ In this subsection, we propose a two stage approach to solve Eq. (10). We first solve the jamming weight by fixing the power allocation, where two schemes are applied. 1) OCJ scheme For a given jamming vector wJ , we maximize the jamming signal power hJEH wJ

2

at the eavesdropper while

allowing a small amount of leakage interference at legitimate receiver. The corresponding problem is formulated as follows: 2 max w2H wJ ⎫ wJ ⎪ ⎪ s.t. ⎪ (11) ⎬ 2 H w1 wJ = z ⎪ ⎪ wJH wJ = 1 ⎪⎭ Here, w1 = hJD hJD 2 and w2 = hJE hJE 2 . Define r = w1H w2 , we obtain the optimal objective value as

(

G ( z ) = 1 − r 1 − r − (1 − r 2 ) z

)

2

[7]. Thus, substituting

G ( z ) into Eq. (10), the problem can be written as: 2 ⎛ P h * 2 f ( P ) −1 ⎞⎫ PS hSR* σ R−2* R 1 J R D ⎜ ⎟⎪ max min , PS , PR , PJ ⎜ 1 + P h 2 f ( P )−1 1 + P h 2 f ( P )−1 ⎟⎪ R 2 J R 2 J R* E R *E ⎝ ⎠⎪⎪ ⎬ s.t. ⎪ PS + PR + PJ = P ⎪ ⎪ PSPS0 ⎪⎭ (12)

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Here f1 ( PJ ) = PJ hJD

2 2

2

z + σ D2 , f 2 ( PJ ) = PJ hJE 2 G ( z ) +

σ E2 . They are the functions over jamming power PJ for a

given z ∈ [ 0,1] . Although Eq. (12) is not a convex

optimization problem in general, it can be transformed to a class of standard generalized GP [8]. With simple transformation, we obtain the equivalent form as: min t ⎫ PS , PR , PJ ⎪ ⎪ s.t. ⎪ − − 2 2 2 PR−1 f1 ( PJ ) hR* D + hR* E hR* D f1 ( PJ ) f 2 ( PJ ) −1t ⎪⎪ ⎬ −2 −2 2 PS−1σ R2 * hSR* + PS−1 PR σ R2* hR* E hSR* f 2 ( PJ )−1t ⎪ ⎪ ⎪ PS + PR + PJ = P ⎪ PSPS0 ⎪⎭ (13) Eq. (13) can be solved efficiently by the convex software, where the global optimum is guaranteed [8–9]. With the power solution p = [ PS , PR , PJ ] from Eq. (13), Eq. (12) is reduced as a single variable optimization over z: 2 2 ⎛ ⎞ PR hR* D PS hSR* ⎜ ⎟ 2 ⎜ ⎟ σ R2 * PJ hJD 2 z + σ D2 ⎟ max min ⎜ , 2 2 z∈[0,1] ⎜ ⎟ PR hR* E PR hR* E ⎜ 1+ ⎟ 1 + 2 2 2 2 ⎟ ⎜ h + h + P G z P G z σ σ ( ) ( ) J JE E J JE E ⎝ ⎠ 2 2 (14) We will show that one dimensional optimization technique can be used to search for the solution of Eq. (14) [10]. 2) Suboptimal scheme: NSCJ scheme Although the optimal solution of Eq. (14) can be found by one dimensional search method, OCJ scheme suffers from high computational complexity. To reduce the complexity, we require that the CJ vector wJ should lie in the null space of the channel between the selected relay and legitimate receiver, i.e. w1H wJ = 0 . Thus, we have a simple choice of jammer as: wJ = Gc , where G is the column-orthogonal matrix corresponding to zero singular value of w1H and c is the combination coefficient vector with dimension of ( N − 2) ×1 . As a result, the optimization problem of Eq. (11) is transformed into: max c H G H w2 w2H Gc ⎫ c∈C ( N −1)×1 ⎪⎪ (15) ⎬ s.t. ⎪ cHc = 1 ⎪⎭

It has closed-form solution as: c = (G H hJE )

2013

( h G ). H JE

2

Taking the solution into Eq. (10), we obtain the following problem as Eq. (16). This problem also belongs to the generalized GP. min t ⎫ PS , PR , PJ ⎪ ⎪ s.t. ⎪ −2 −2 2 −1 2 −1 2 ⎪⎪ PR σ D hR* D + σ D hR* E hR* D f 3 ( PJ ) t ⎬ −2 −2 2 PS−1σ R2 * hSR* + PS−1 PR σ R2* hR* E hSR* f 3 ( PJ ) −1t ⎪ ⎪ ⎪ PS + PR + PJ = P ⎪ PSPS0 ⎪⎭ (16) 2

where f3 ( PJ ) = PJ hJEH Gc + σ E2 . Proposition 1 In the high SNR regime, null space jamming strategy is asymptotically the optimal jamming. Proof For a given power allocation p = [ PS , PR , PJ ] , the instantaneous rate between the source and relay link will approach infinity as SNR increases. Thus, the corresponding optimization problem is reduced as: 1+ max

z∈[ 0,1]

1+

PR hR* D PJ hRD PR hR* E

2 2

2

z 2

(17)

2

PJ hRE 2 G ( z )

To guarantee a non-zero secrecy rate achieved, solving Eq. (17) is equivalent to solving the following problem as: 2

max

z∈[ 0,1]

hRE 2 G ( z ) hRD

2

(18)

z 2

Obviously, the sufficient and necessary condition to maximize the objective function is z = 0 .

4 CR strategy and adaptive strategy 4.1 CR with power allocation strategy Different from CJ strategy, each relay transmits a weighted version of the decoded signal in CR strategy. Thus, the received signals at legitimate receiver and eavesdropper are expressed as, respectively: H yD = PR hRD wR x + nD ⎪⎫ (19) ⎬ H yE = PE hRE wR x + nE ⎪⎭ where hRD ∈ C N ×1 and hRE ∈ C N ×1 are channel

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coefficient vectors from all relays to the legitimate receiver and the eavesdropper. wR ∈ C N ×1 is the beamforming

max t

W ∈C N × N

s.t.

weight vector of relays, and PR is the power allocated to relays. Similarly to Eq. (8), Mathematically, the optimization problem is formulated as: 2 2 ⎛ ⎞⎫ H P h ⎜ 1 + PR hRD wR 1 + min S SR i ⎟ ⎪ i∈ ⎜ σ R2 i ⎟ ⎪ σ D2 max min ⎜ , ⎟⎪ 2 2 PS , PR , wR H H ⎜ wR wR ⎟ ⎪ PR hRE PR hRE 1+ ⎜1+ ⎟⎪ ⎜ ⎟⎪ σ E2 σ E2 ⎝ ⎠ ⎪ (20) ⎬ s.t. ⎪ 2 ⎪ PS hSR i 2 R0 ⎪  min 2 1 − i∈ σ R2 i ⎪ ⎪ PR + PS = P ⎪ ⎪ wRH wR = 1 ⎭ This problem is also a non-convex optimization problem. So we propose an iterative algorithm to solve it. 1) Optimization of relay weight In the first stage, we calculate the beamforming weight wR with fixed PS and PR . Here, we define W = wR wRH , using the semi-definite relaxation (SDR) technique in Ref. [11], the relaxed problem is obtained as Eq. (21). Obviously, for any t , the feasible set in Eq. (21) is quasi-convex. We can use the bisection algorithm to solve Eq. (21) by solving a convex problem at each step [11]. For each step, problem Eq. (21) is reformulated as a standard SDP problem, and can be solved by interior point method. In simulation, we use SeDuMi to implement modern interior point methods for SDP problem [12]. The detail of algorithm is stated as follows: Algorithm 1 Bisection algorithm Define an interval [0, t ] known to contain the optimal value t * . Step 1 Initialize t min = 0 , t max = t . Step 2 Set t = (tmin + tmax ) 2 . Step 3 Solve the problem with given t. Step 4 Update t by the bisection algorithm a) If problem Eq. (21) is feasible: t min = t . b) If problem Eq. (21) is infeasible: t max = t . Step 5 Until t max − tmin < δ , where δ is the given threshold. Then the converged t min is the optimal solution of problem Eq. (21).

⎛ ⎛ hH h hH h Tr ⎜⎜ PR ⎜ RD 2 RD − t RE 2RE σE ⎝ ⎝ σD

⎞ ⎞ ⎟W ⎟⎟ +(1 − t )0 ⎠ ⎠ ⎛P h i ⎛ ⎛ hH h ⎞ ⎞ Tr ⎜⎜ t ⎜ I N + PR RE 2RE ⎟ W ⎟⎟1 + min ⎜ S SR 2 i∈ ⎜ σE ⎠ ⎠ ⎝ ⎝ ⎝ σ Ri Tr W = 1 W 0

83

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ 2 ⎞⎪ ⎟⎪ ⎟⎪ ⎠ ⎪ ⎪ ⎪⎭ (21)

2) Optimization of source and relay powers The second stage is to optimize PS and PR with wR fixed. Thus, the relaxed optimization problem is stated as: min t% ⎫ PS , PR ,t% ⎪ ⎪ s.t. ⎪ 2 H ⎪ hRE wR σ D2 PR−1σ D2 t% + ⎪ 2 2 H H ⎪ hRD wR hRD wR σ E2 ⎪ (22) ⎬ 2 H 2 −1 −1 2 PS PR hRE wR σ R i PS σ R i ⎪ t% ⎪ + 2 2 min hSRi min hSR i σ E2 ⎪ i∈ i∈ ⎪ 2 ⎪ min PS hSRi σ R2 i (22 R0 − 1) i∈ ⎪ ⎪⎭ PS + PR = P The problem also belongs to the class of GP problem. According to the above two stages, we can summary the framework of iterative algorithm for CR with power allocation strategy as follows: Algorithm 2 Iterative optimization algorithm Step 1 Set the initial value of PS and PR . Step 2 Solve the problem Eq. (21) with fixed PS and PR through Algorithm 1, and obtain the updated wR . Step 3 Solve the problem Eq. (22) with fixed wR through GP, and obtain the updated PS and PR . Step 4 Continue until converge. 4.2

Adaptive strategy

To guarantee all relays decoding signal successfully, most power is allocated to the source, which will result in little fraction of power allocated to relays. In order to compensate the insufficiency of power allocated to relays, all relays should cooperate together to exploit diversity to enhance the secrecy rate.

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In CJ strategy, the power allocated to source is only required that the instantaneous capacity at the selected relay satisfies the target transmission rate, so that more power can be allocated to the relays forwarding signal and jamming eavesdropper. Obviously, as the number of cooperative relays increases, the power allocated to jamming reduces due to that the available null space in Eq. (6) increases, which will be shown in the simulation results. Based on the above analysis, we propose an adaptive cooperative strategy to guarantee secrecy. In particular, when in the low SNR regime, both the legitimate receiver and eavesdropper only receive small information signal. In this situation, CR strategy should be preferable for the reason that it exploits cooperative diversity. While in the high SNR, the eavesdropper receives much more information signal from the relay. Thus, the final achievable secrecy rate can be determined as: CA = max(CJ , CR ) , where CJ and CR are the secrecy

Fig. 2 N= 4

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Secrecy rate achieved by CJ strategies vs. SNR with

In Fig. 3, we plot the power allocation of the proposed OCJ scheme versus SNR with N=4, 6 and 8. As observed from the figure, more power is allocated to jamming than forwarding the desired signal in low SNR regime, while the conclusion is reversed in high SNR regime.

rate achieved with CJ and CR strategies, calculated according to Eq. (8) and Eq. (21), respectively.

5 Performance analysis In our simulation, we assume he noise power at relays, legitimate receiver and eavesdropper are assumed equal and set as σ 2 . 5.1 Joint power allocation and CJ strategy Fig. 3

In this subsection, we focus on the performance of the OCJ and NSCJ schemes. We also evaluate the performances of two CJ strategies explained as follows: 1) Strategy 1: CJ strategy with relay selection, no power allocation. 2) Strategy 2: CJ strategy with power allocation, random relay selection. As reflected in Fig. 2, both two proposed strategies provide significant improvements in secrecy rate compared with Strategy 1 and Strategy 2. Relay selection creates the opportunity for using “the best relay” to forward desired signal, which brings in the selection gain. Compared the secrecy rate achieved by two CJ schemes, there is a small gap between OCJ and NSCJ as shown in the enlarged box in the low SNR regime. While in the high SNR regime, OCJ and NSCJ schemes have the same performance, which further verifies Corollary 1.

Optimal power of sources transmit power PS , relay

power PR and jammer power PJ vs. SNR with

N= 4

Furthermore, we investigate the impact of the number of relays on the secrecy rate. As depicted in Fig. 4.

Fig. 4

Secrecy rate achieved by OCJ scheme with N= 4, 6 and 8

Obviously, inviting more relays in CJ strategy increases

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the achievable secrecy rate. Increasing the number of relays can increases the opportunity to use ‘the best relay’ to forward desired signal. Fig. 5 illustrates the power allocation of CR strategy versus SNR with N=4, 6 and 8. As reflected in this figure, to guarantee that all relay can decode signal successfully, we need to allocate large amount of power to the source. And as the number of relays increases, the ratio is also increasing.

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We first showed that CJ with relay selection strategy can help to improve the secrecy rate due to an opportunistic relay selection and create intentional interference at eavesdropper. CR strategy exploits the cooperative diversity to enhance security. However, with little amount of power, the improvement on secrecy rate is limited. As the number of relays increases, the benefit due to cooperative diversity reduces, and CJ with relay selection strategy outperforms CR in terms of the secrecy rate. Acknowledgements This work was supported by the National Key Technology R&D Program of China (2012ZX03004005), and the Foundational Research Funds for the Central Universities.

References

Fig. 5

Optimal power of source transmit power PS , relay

power PR with CR strategy vs. SNR with N= 4, 6 and 8

5.2

CR and adaptive strategies

To exploit the advantage of both CJ and CR strategies, we compare the secrecy rate achieved by CR, CJ and adaptive strategies. As observed from Fig. 6, the adaptive cooperative strategy outperforms both CR and CJ strategies over the entire range of SNR values.

Fig. 6 Comparison with adaptive, OCJ with relay selection and CR strategies vs. SNR with N= 6

6 Conclusions This paper has investigated the three cooperative strategies with power allocation for secure communication.

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(Editor: WANG Xu-ying)