IEEE ICC 2015 - Wireless Communications Symposium
Energy Efficiency Maximization for Secure Data Transmission over DF Relay Networks Dong Wang∗† , Bo Bai∗ , Member, IEEE, Wei Chen∗ , Senior Member, IEEE, and Zhu Han‡ , Fellow, IEEE ∗ State
Key Laboratory on Microwave and Digital Communication Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Electronic Engineering, Tsinghua University, Beijing 100084, China † New Star Research Institute of Applied Technology, Hefei 230031, China ‡ Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77004, USA E-mail:
[email protected] [email protected] [email protected] [email protected] Abstract—The security requirements of data transmission over wireless networks are energy-limited in many situations. In this paper, the secure energy efficiency (EE), is defined as the ratio of the secrecy rate to the total power, and is investigated in a systematic way considering a decode-and-forward (DF) relay network with a potential eavesdropper. We maximize the secure EE subject to the individual power constraint and the minimum decoding rate constraint of the relay. To deal with the nonconvexity of the formulated problem, a fractional programming approach embedded with DC (difference of convex functions) programming is proposed to solve the problem by twolayer iterations. The key point of the proposed algorithm is to translate the primal problem into a series of convex subproblems, which can be solved by convex programming. It is verified by simulation that the proposed algorithm achieves much better secure EE than the conventional secrecy rate maximization yet with a minor performance loss measured by average secrecy rate or secrecy outage probability.
I. I NTRODUCTION Most recently, many works have focused on improving the spectral efficiency of collaborative relay networks, such as [1] and [2]. However, another key problem of collaborative relay networks is security issue. Information security is a critical issue when confidential data are transmitted over wireless networks. Recently, physical layer security has attracted increasing attentions with the purpose of transmitting the confidential data securely while keeping eavesdroppers from intercepting the data. However, the secure strategies at the physical layer are confined by limited power and energy in many situations. As a result, energy-efficient secure communication is an interesting topic but has not yet been considered in a systematic way. For implementing secure communication at the physical layer, most works focused either on improving secrecy rate with the limited power constraint or saving the power with the minimum secrecy rate requirement, such as [3], [4], and [5]. In [6], the aforementioned optimization objectives and constraints were employed to improve the transmission efficiency This paper is partially supported by NSFC under grant No. 61322111 and No. 61401249, the National Basic Research Program of China (973 Program) No. 2013CB336600, Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant No. 20130002120001, Chuanxin Funding, Beijing nova program No. Z121101002512051, and NSFC under grant No. 61428101.
978-1-4673-6432-4/15/$31.00 ©2015 IEEE
and reliability in MIMO relay networks. A jamming-resistant broadcast scheme was proposed in [7] to enhance both the efficiency and the security by using the node cooperation. However, these schemes mentioned above cannot ensure to achieve the global optimization of secure energy efficiency (EE). Thus, some works mentioned in [8] and [9] have raised concerns on EE for the physical layer security. In [8], the secure communication in a low-SNR regime was studied with the metric of minimum energy per secret bit. In [9], the partial secrecy scenario in which the equivocation rate at the eavesdropper may be smaller than the transmission rate was explored, and the fact that the level of secrecy can be traded off for energy efficiency was revealed. The previous works solved the secure EE problem in part due to the specific scenarios. In addition, these works only considered the direct transmission systems without relays. In contrast to the literature, we investigate the secure EE maximization for a decode-and-forward (DF) relay network with the presence of an eavesdropper. The EE is maximized considering the physical layer security with the given individual power constraints and the minimum decoding rate constraint of relay. The problem is formulated as a nonconvex optimization which is challenging to solve. To overcome this difficulty, an algorithm characterized by fractional programming and DC (difference of convex functions) programming is developed, which solves the optimization problem by twolayer embedded iterations. The inner iterative process is based on the DC programming while the outer iterative process is based on the fractional programming. Simulation results illustrate that our proposed algorithm, compared with the conventional secrecy rate maximization, can bring a significant improvement of secure EE while with negligible degeneration of average secrecy rate and secrecy outage probability. II. S YSTEM M ODEL A ND P ROBLEM F ORMULATION A. System Model We consider a DF relay network as shown in Fig. 1. Each node with single antenna is assumed to be half-duplex. The source transmits a confidential message to the destination with the help of the relay while preventing the eavesdropper from overhearing the message. Assume that all channels are
2313
IEEE ICC 2015 - Wireless Communications Symposium
We can interpret the source message by using the received signals in both phases. The rates in Eq. (7) and (8) can be achieved by maximal ratio combining. The factor 1/2 is because of the fact that two time units required in two phases. The achievable secrecy rate is defined as
ĚĞƐƚŝŶĂƚŝŽŶ ^
Z
ƐŽƵƌĐĞ ƌĞůĂLJ
ĞĂǀĞƐĚƌŽƉƉĞƌ
Rsec = [Rd − Re ]+ ,
Fig. 1. A relay network with an eavesdropper.
quasi-static flat fading. Let hsd and hrd denote the legitimate channels from the source and the relay to the destination, respectively. The wiretap channels from the source and the relay to the eavesdropper are denoted by gse and gre , respectively. The channel from the source to the relay is hsr . It is assumed that the full channel state information (CSI) of all channels are known at each node. The CSI of the eavesdropper can be obtained in the case where the eavesdropper is active in the network [3]. In the first phase, the source transmits symbol x (E{|x|2 }= 1) to the relay and destination. The transmitted power of source ps should ensure that the relay can decode correctly. Then the received signal at the relay, destination and eavesdropper are, respectively, given by √ (1) yr = ps hsr x+zr , √ (1) (2) yd = ps hsd x + zd , √ ye(1) = ps gse x + ze . (3) The notations zr , zd , ze are the independent additive white Gaussian noise with the same distribution CN (0, σz2 ). In the second phase, the relay forwards the re-encoded symbol x with power pr using the same codewords as the source. Then at the destination and eavesdropper, the received signal are, respectively, expressed as √ (2) (4) yd = pr hrd x + zd , √ ye(2) = pr gre x + ze . (5) Then, the receiving rate at the relay, destination, and eavesdropper can be, respectively, obtained by 2 1 ps |hsr | Rr = log2 1 + , (6) 2 σz2 2 ps |hsd | 1 pr |hrd |2 Rd = log2 1 + , (7) + 2 σz2 σz2 2 ps |gse | 1 pr |gre |2 Re = log2 1 + . (8) + 2 σz2 σz2 ξ(ps , pr ) =
Rsec = Ptot
log2 1 +
ps |hsd |2 σz2
(9)
where [x]+ denotes max{x, 0}. During two-phase transmission, the total power consumption is the sum power of the both phases, i.e., Ptot = pph1 + pph2 1 pr 1 ps + ptcs + prcr + + ptcr = 2 η 2 η 1 ps + pr t r t + pc s + pc r + p c r , = 2 η
(10)
where ptcs and ptcr are the transmitted circuit power of the source and relay excluding the power consumed by the power amplifier, respectively. prcr is the power consumed by the relay for reception. The power amplifier efficiency is denoted by η. Therefore, the secure EE (unit: bits/joule) is defined as the ratio of the secrecy rate to the total power shown in Eq. (11). B. Problem Formulation In order to use energy efficiently, we intend to maximize the secure EE of the DF relay network. By optimizing power allocation, the secure EE can be maximized subject to the individual power constraints and the minimum decoding rate constraint of relay. The problem can be formulated as max ξ(ps , pr ) ps ,pr⎧ , ⎨ 0 ≤ ps ≤ pmax s , 0 ≤ pr ≤ pmax s.t. r ⎩ Rr ≥ R 0 .
(12)
The problem is formulated with the more practical consideration of the individual power constraints rather than the sum and pmax are the maximum power constraints, where pmax s r power of the source and relay, respectively. The notation R0 is the predefined minimum decoding rate of relay, i.e., the minimum target secrecy rate. The constraint Rr ≥ R0 ensures that the relay is able to decode [10]. III. S ECURE EE M AXIMIZATION A LGORITHM The problem as Eq. (12) is challenging to solve due to the nonconvexity of the objective function. To overcome this difficulty, a fractional programming approach embedded with DC programming is explored in this section. The proposed algorithm includes two-layer iterative processes: the inner iteration of DC programming and the outer iteration of fractional programming.
pr |hrd |2 σz2 ps +pr + η
+
2314
− log2 1 +
ps |gse |2 σz2
ptcs + prcr + ptcr
+
pr |gre |2 σz2
+ .
(11)
IEEE ICC 2015 - Wireless Communications Symposium
A. Problem Transformation in the Outer Iteration In order to decode correctly, the rate at the relay should satisfy Rr ≥ R0 . By Eq. (6), we can obtain
2R 2 0 − 1 σz2 ps ≥ . (13) |hsr |2 Then the feasible domain of Eq. (12) can be rewritten as 22R0 − 1 σz2 max max S= ≤ ps ≤ p s , 0 ≤ pr ≤ p r . |hsr |2 (14) We can see that the objective function in Eq. (12) is a nonconvex fractional form. Our strategy is to convert the problem into equivalent parametric programming based on the following Proposition 1. Then the problem can be solved by the Dinkelbach’s method [11], which is adopted to solve the general EE maximization problem without consideration of security in [12]. R(x) Proposition 1: The optimization problem max{ P (x) : x ∈ S} is defined as the fractional programming, which associates with the parametric programming max {R(x) − ξP (x) : x ∈ S} , parameter ξ ∈ R,
(15)
where S is nonempty and compact, and P (x) > 0. R(x) and P (x) are continuous. The maximum value R(x) R(x∗ ) ∗ = max :x ∈ S (16) ξ = P (x∗ ) P (x) can be achieved if and only if ξ ∗ and x∗ satisfy ∗
max {R(x) − ξ P (x) : x ∈ S} = 0.
(17)
Based on Proposition 1, the problem in Eq. (12) can be equivalently translated into finding ξ ∗ and p∗ to satisfy max{F (ξ, p) = Rsec (p) − ξPtot (p) : p ∈ S} = 0, ∗
(18)
∗
where p (ps , pr ), ξ and p are the maximum secure EE and the optimal power vector, respectively. In Eq. (18), the maximization problem is a parametric programming associated with parameter ξ. By the Dinkelbach’s method with given initial value ξ0 < ξ ∗ of ξ, the optimal solution ξ ∗ and p∗ can be found by solving the following secondary problem at each iteration, i.e., p∗ (ξi ) = arg max{F (ξi , p) = Rsec (p)−ξi Ptot (p) : p ∈ S}, p
(19) where i is the iteration index and ξi is the secure EE at the ith iteration. p∗ (ξi ) is the local optimal power vector with regard to the fixed ξi . ξi can be updated by Rsec (p∗ (ξi )) . (20) Ptot (p∗ (ξi )) In fact, the problem in Eq. (18) is solved by solving Eq. (19) at each iteration to generate an increasing sequence {ξi }. Referring to [11], the sequence converges to ξ ∗ at least linearly, even super-linearly when ξ0 < ξ ∗ . Therefore, given some threshold ε > 0, the iteration process terminates when |F (ξi , p∗ (ξi ))| ≤ ε. Then the optimal solution is obtained. ξi+1 =
B. Solution of the Inner Iteration with Fixed ξi It is still difficulty to solve the parameterized secondary problems due to the nonconvexity of the objective function in Eq. (19). Therefore, the DC programming [13] [14] described in the following Proposition 2 is introduced. Proposition 2: If the functions B(x) and D(x) are differentiable convex functions on the convex set S, the optimization problem min {F (x) = B(x)−D(x) : x ∈ S} is defined as the DC programming, which can be solved iteratively by min{B(x) − D(xk ) − ∇D(xk ), x − xk : x ∈ S}. (21) In Eq. (21), ∇D(xk ) is the gradient of D(x) at xk , and a, b stands for the dot product of a and b. In Eq. (19), for the fixed ξi , the maximization problem is equivalent to min{−F (ξi , p) = −Rsec (p)+ξi Ptot (p) : p ∈ S}.
(22)
The objective function in Eq. (22) can be decomposed as −F (ξi , p) = B(p) − D(p), where B(p) and D(p) are respectively written as p s + pr t r t B(p) = ξi + pc s + p c r + p c r η 2 pr |hrd |2 ps |hsd | −log2 1 + + , σz2 σz2 2 ps |gse | pr |gre |2 . + D(p) = −log2 1 + σz2 σz2
(23)
(24)
(25)
It is obvious that both B(p) and D(p) are convex, and feasible domain S is a convex set. Therefore, the optimization problem as Eq. (22) is the canonical DC programming. According to Proposition 2, by solving the convex programming min{B(p)−D(pk )−∇D(pk ), p−pk : p ∈ S},
(26)
the problem shown in Eq. (22) can be solved iteratively, where k is the iteration index. The gradient ∇D(p) is determined by ⎞ ⎛ 2 2 − σ|g2reln| 2 − σ|g2seln| 2 z z ⎠. , ∇D(p) = ⎝ ps |gse |2 pr |gre |2 ps |gse |2 |2 1+ σ2 + σ2 1+ σ2 + pr |gσre 2 z z z z (27) Because D(p) is convex, ∀p ∈ S, it follows that D(p) ≥ D(pk )+∇D(pk ), p−pk . We can derive D(pk+1 ) ≥ D(pk ) + ∇D(pk ), pk+1 − pk .
(28)
Moreover, pk+1 is the optimal solution of Eq.(26) while pk is only feasible solution. Accordingly, we can deduce B(pk+1 ) − D(pk+1 ) ≤ B(pk+1 ) − D(pk ) −∇D(pk ), pk+1 − pk ≤ B(pk ) − D(pk ).
(29)
Eq. (29) indicates that the next solution pk+1 is always better than the previous solution pk , i.e., the iteration is
2315
IEEE ICC 2015 - Wireless Communications Symposium
50 45 Secure EE maximization Secrecy rate maximization
40 Average secure EE
Algorithm 1 Secure EE maximization algorithm for DF relay networks Input: ε, τ , hsr , hsd , hrd , gse , gre . Output: p∗ , ξ ∗ 1: Give initial value ξ0 , calculate p∗ (ξ0 ), i := 0; 2: while |F (ξi , p∗ (ξi ))| > ε do 3: i := i + 1; 4: Update ξi by Eq. (20); 5: For ξi , give initial value p0 , compute p1 , −F (ξi , p0 ), and −F (ξi , p1 ), k := 1; 6: while |−F (ξi , pk ) + F (ξi , pk−1 )| > τ do 7: k := k + 1; 8: For pk−1 , solve Eq. (26) to obtain pk by convex programming; 9: Calculate −F (ξi , pk ); 10: end while 11: p∗ (ξi ) = pk , 12: end while 13: return ξ ∗ = ξi and p∗ = p∗ (ξi ).
35 30 25 20 15 10 5 0 50
100
150 200 250 300 350 400 Distance between the source and relay
450
Fig. 2. Average secure EE versus the distance between the source and relay: R0 = 1 bit/s/Hz. 140 120
Secure EE maximization Secrecy rate maximization
100 Average secure EE
convergent. The convergence has been proved in [13]. The iterative process is repeated until the convergence condition |−F (ξi , pk ) + F (ξi , pk−1 )| ≤ τ is satisfied, where τ > 0 is the convergence tolerance. Then the local optimal power vector with respect to ξi is p∗ (ξi ) = pk .
80 60 40
C. Summary of Secure EE Maximization Algorithm
20 0 250
300
350 400 450 500 550 600 650 Distance between the source and destination
700
750
Fig. 3. Average secure EE versus the distance between the source and destination: R0 = 1 bit/s/Hz. 25 Secure EE maximization Secrecy rate maximization
20 Average secrecy rate
For providing a comprehensive understanding, the secure EE maximization algorithm is detailed in Algorithm 1. Firstly, the primal problem is associated with a parametric programming with respect to parameter ξ, which can be solved by solving a sequence of parameterized secondary problems. Then, the DC programming is employed for converting the secondary problem into a series of convex subproblems (i.e., Eq. (26)). Finally, the subproblems are solved by convex programming. Algorithm 1 includes two layers of loop. The inner loop is the DC programming where the parameterized secondary problem (i.e., Eq. (19)) is solved with given ξi . The outer loop is the fractional programming where parameter ξi is updated and the equivalent problem (i.e., Eq. (18)) of the primal problem is solved.
15
10
5
IV. S IMULATION R ESULTS In this section, we present simulation results to demonstrate the performance of the proposed algorithm. For simplicity and fairness, all nodes are placed along a horizontal line. We fix the source at point (0, 0). The simulation parameters are given as: σz2 = −100 dBm, η = 0.38, pmax = pmax = 500 mW, s r ptcs = prcr = ptcr = 25 mW. The log-distance model is adopted with path loss exponent α = 3.5. The simulation is performed over 1000 independent Rayleigh channel realizations to obtain the average results. Firstly, the average secure EE of the secure EE maximization and the conventional secrecy rate maximization are compared in Fig. 2, where the destination and eavesdropper
0 250
300
350 400 450 500 550 600 650 Distance between the source and destination
700
750
Fig. 4. Average secrecy rate versus the distance between the source and destination: R0 = 1 bit/s/Hz.
are fixed at (500, 0) and (550, 0) (units: m), respectively. The location of the relay is moved from (50, 0) to (450, 0). We can see that the secure EE of our proposed scheme far outperforms the secure EE of the conventional secrecy rate maximization over the different locations of relay. In Fig. 3, we place the relay and eavesdropper at (250, 0)
2316
IEEE ICC 2015 - Wireless Communications Symposium
are sufficiently stronger than the channels to the eavesdropper. Fig. 6 shows the secrecy outage probability of the different target secrecy rate under the similar simulation conditions as Fig. 5. The secrecy outage probability is defined as the probability of secrecy outage events, i.e., Pr(Rsec < R0 ). Compared with the secrecy rate maximization, the secure EE maximization may be bring little degeneration of the outage performance.
50 45
Average secure EE
40
Secure EE maximization Secrecy rate maximization
35 30 25 20
V. C ONCLUSION
15 10 5 0
300 400 500 600 700 Distance between the source and eavesdropper
Fig. 5. Average secure EE versus the distance between the source and eavesdropper: R0 = 1 bit/s/Hz.
0
10
Secrecy outage probability
R0= 2 −1
10
R = 1.5 0
R =0
A secure EE maximization algorithm for physical layer security is developed for a DF relay network considering a potential eavesdropper. We maximize the secure EE by optimizing power allocation, which is implemented by a fractional programming approach embedded with the DC programming. The proposed scheme is constituted by two-layer iterations including an inner loop characterized by the DC programming and an outer loop characterized by the fractional programming. The simulation results demonstrate that our proposed scheme can achieve much better secure EE than that of the conventional secrecy rate maximization yet with negligible degeneration of security performance.
0
R EFERENCES −2
10
R = 0.5 0
−3
R0= 1
10
Secure EE maximization Secrecy rate maximization −4
10
200
300 400 500 600 700 Distance between the source and eavesdropper
800
Fig. 6. Secrecy outage probability of different target secrecy rate.
and (500, 0), respectively. Then the average secure EE is illustrated versus the different positions of the destination from (250, 0) to (750, 0). As expected, the secure EE of our proposed algorithm is much better than that of the secrecy rate maximization over all the positions of the destination. It is worth noting that, with the increase of the distance between the source and destination, the secure EE of the both schemes declines, since the channels to the destination weaken gradually compared with the channels to the eavesdropper. In order to observe the effect of the secure EE maximization for the secrecy rate, we investigate the secrecy rate of the two schemes in Fig. 4 with the same configurations as Fig. 3. It can be seen that the secrecy rate of the proposed algorithm, compared with that of the secrecy rate maximization, has a minor decrease due to the tradeoff between EE and spectral efficiency. In Fig. 5, the relay and destination are placed at (250, 0) and (500, 0), respectively, and the eavesdropper is moved from (250, 0) to (750, 0). It is obvious that our proposed algorithm can achieve much better secure EE than the conventional secrecy rate maximization when the channels to the destination
[1] W. Chen, “CAO-SIR: channel aware ordered successive relaying,” IEEE Trans. Wireless Commun., vol.13, no.12, pp.6513-6527, Dec. 2014. [2] B. Bai, W. Chen, K. B. Letaief, and Z. Cao, “A unified matching framework for multi-flow decode-and-forward cooperative networks,” IEEE J. Sel. Areas Commun., vol.30, no.2, pp.397-406, Feb. 2012. [3] J. Li, A. P. Petropulu, and S. Weber, “On cooperative relaying schemes for wireless physical layer security,” IEEE Trans. Signal Process., vol. 59, no. 10, pp. 4985-4996, Oct. 2011. [4] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Secure wireless communications via cooperation,” in Proc. 46th Annu. Allerton Conf. Commun., Control, Computing, Monticello, IL, Sep.-Oct. 2008. [5] J. Mo, M. Tao, Y. Liu, and R. Wang, “Secure beamforming for MIMO two-way communications with an untrusted relay,” IEEE Trans. Signal Process., vol. 62, no. 9, pp. 2185-2199, May 2014. [6] J. Huang and A. L. Swindlehurst, “Cooperative jamming for secure communications in MIMO relay networks,” IEEE Trans. Signal Process., vol. 59, no. 10, pp. 4871-4884, Oct. 2011. [7] L. Xiao, H. Dai, and P. Ning, “Jamming-resistant collaborative broadcast using uncoordinated frequency hopping,” IEEE Trans. Inf. Forensics Security, vol. 7, no. 1, pp. 297-309, Feb. 2012. [8] M. C. Gursoy, “Secure communication in the low-SNR regime,” IEEE Trans. Commun., vol. 60, no. 4, pp. 1114-1123, Apr. 2012. [9] C. Comaniciu and H. V. Poor, “On energy-secrecy trade-offs for Gaussian wiretap channels,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 2, pp. 314-323, Feb. 2013. [10] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. [11] W. Dinkelbach, “On nonlinear fractional programming,” Managment Science, vol. 13, no. 7, pp. 492-498, Mar. 1967. [12] K. T. K. Cheung, S. Yang, and L. Hanzo, “Achieving maximum energy-efficiency in multi-relay OFDMA cellular networks: a fractional programming approach,” IEEE Trans. Commun., vol. 61, no. 7, pp. 27462757, Jul. 2013. [13] T. P. Dinh and H. A. L. Thi, “Recent advances in DC programming and DCA,” Transactions on Computational Intelligence XIII. Springer, pp. 1-37, 2014. [14] H. H. Kha, H. D. Tuan, and H. H. Nguyen, “Fast global optimal power allocation in wireless networks by local D.C. programming,” IEEE Trans. Wireless Commun., vol. 11, no. 2, pp. 510-515, Feb. 2012.
2317
