IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 8, AUGUST 2014
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On Transmission Secrecy Outage of a Multi-Antenna System With Randomly Located Eavesdroppers Tong-Xing Zheng, Hui-Ming Wang, Member, IEEE, and Qinye Yin
Abstract—This letter studies the physical-layer security of a multi-antenna transmission system in the presence of Poisson distributed eavesdroppers. The transmission secrecy outage probability (TSOP) is adopted to evaluate the security. We derive an accurate integral expression as well as a closed-form upper bound on TSOP for the noncolluding eavesdroppers’ case and a closed– form solution for the colluding eavesdroppers’ case, respectively. Based on these, we define a novel concept of security region to intuitively illustrate the security from a spatial perspective. We further analyze the impacts of various factors on the security, such as the number of transmit antennas, the node intensity, and the target secrecy rate. Index Terms—Physical layer security, secrecy outage, random network, fading.
I. I NTRODUCTION
S
ECURITY issue of wireless communications has received considerable attention due to the broadcast characteristic of the wireless medium. During the past decades, physical layer security, or, information-theoretic security has been widely investigated since Wyner’s pioneering work [1]. Various wiretap models have been studied, such as fading channels, multi-antenna channels and broadcast channels. The secrecy capacities/rates of these channels have been investigated and secrecy transmission schemes have been proposed (see [2] and its references). However, in most of these works, only smallscale fading, say Rayleigh fading, has been considered in the channel models, and large-scale path loss has been ignored. In this case, the secrecy transmission becomes irrelative to the relative physical locations of the legitimate terminals and the eavesdroppers. However, loosely speaking, since a positive secrecy capacity/rate can only be achieved when the equivalent legitimate link is superior to the wiretap link, the impacts of the locations of the terminals and the propagation path losses are very critical to the secrecy transmission. The difficulty of taking the path loss into consideration is that the locations of the eavesdroppers are unknown since they work in a passive way. Recently, stochastic geometry theory provides a powerful tool to model the random locations distribution
Manuscript received April 9, 2014; revised May 18, 2014; accepted June 16, 2014. Date of publication June 20, 2014; date of current version August 8, 2014. This work was supported in part by the NSFC under Grants 61102081 and 61221063, by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant 20110201120013, by the New Century Excellent Talents Support Fund under Grant NCET-13-0458, by the Industrial Research Fund of Shaanxi Province under Grant 2012GY2-28, by the Fok Ying Tong Education Foundation under Grant 141063, and by the Fundamental Research Funds for the Central University of China under Grant 2013jdgz11. The associate editor coordinating the review of this paper and approving it for publication was X. Zhou. (Corresponding author: Hui-Ming Wang.) The authors are with the MOE Key Laboratory for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China (e-mail:
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LCOMM.2014.2332172
of the nodes in a wireless network [3], and facilitates us to investigate the impact of eavesdroppers located randomly to the security, e.g., [4]–[7]. Specially, the Poisson point process (PPP) is an efficient model to describe the locations distribution of unknown eavesdroppers. In [4], the authors modeled the locations of multiple legitimate pairs with fixed distances and those of the eavesdroppers as PPP, and studied the average secrecy throughput in such a wireless network. In [5], the authors considered Nakagami fading channels, and derived the secrecy outage probability (SOP) from a transmitter to a randomly distributed receiver, exposed to PPP distributed eavesdroppers. Both [4] and [5] considered the single-antenna node case. In [6], the authors studied securing a multiple input multiple output (MIMO) transmission subjected to Rayleigh fading using beamforming along with artificial noise, and derived the probability of a positive secrecy rate. In [7], the authors analyzed the secrecy rate and the SOP under a regularized channel inversion precoding in broadcast channels with PPP located eavesdroppers. In this letter, we investigate the transmission secrecy outage probability (TSOP) of a multi-antenna transmission system subjected to Rayleigh fading, coexisting with PPP distributed eavesdroppers. Here, the TSOP is defined as the probability that the secrecy capacity is below a target threshold conditioned on the non-outage transmission of the legitimate channel, which is different from the SOP adopted in [5]–[7].1 We consider both the non-colluding and colluding eavesdroppers (NCE/CE) cases, and derive a closed-form upper bound and a rigorous expression for the TSOP, respectively. Based on these, we introduce a novel concept of security region (SR) to intuitively describe the security. The SR is a geometry region in which the TSOP is below a prescribed threshold. Therefore, roughly speaking, some level of secrecy can be guaranteed once the legitimate receiver locates in the SR. Note that, our concept of security region is significantly distinguished from the secrecy region defined in [10]–[13], where only a single eavesdropper is present in the network and the secrecy region is defined as the region where this eavesdropper has a low probability of interception, while our SR is defined as the set of the legitimate receiver’s locations where it has a guaranteed level of secrecy exposed to numerous randomly located eavesdroppers. Besides, the so-called secrecy guard zone in [4] is defined as the region where any eavesdropper is not allowed to exist, and is also obviously different from ours. II. S YSTEM M ODEL AND P ROBLEM D ESCRIPTION We consider a secure transmission from a transmitter Alice to a legitimate receiver Bob, in the presence of randomly located 1 The definition of the TSOP has been firstly introduced in [8] and is different from the original SOP defined in [9], which does not distinguish between the reliability and security of the transmission.
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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 8, AUGUST 2014
eavesdroppers Eves. Alice uses an M -antenna uniform linear array (ULA) with element spacing half-wavelength. Bob and Eves are equipped with single antenna. Without loss of generality, we let Alice locate at the origin of the polar coordinate with the array placed along the horizontal line, and Bob locate at (rb , θb ).2 We assume Eves’ positions {ek : (rk , θk ) ∈ R2 } obey a stationary PPP Φ of intensity λe . The wireless link is subjected to flat Rayleigh fading together with a path loss governed by the exponent α ≥ 2, thus the channel coefficient vector with respect to (w.r.t.) node i is −(α/2) hi ri ai , where hi is complex Gaussian fading coefficient with zero mean and unit variance, i.e., hi ∼ CN (0, 1) and Δ ai = [1, ejπ cos θi , . . . , ejπ(M −1) cos θi ]T is the antenna manifold vector [16] of the ULA. The signals received at Bob and the k-th Eve can be expressed by [16] √ −α yb = hb rb 2 wH ab P s + nb , (1) √ −α yk = hk rk 2 wH ak P s + nk , ∀ek ∈ Φ, (2) where w is the normalized transmit weight vector, s is the confidential message with E[|s|2 ] = 1 and P is its power, ni ∼ CN (0, σn2 ) are additive white Gaussian noises (AWGN). We assume the channel state information (CSI) related to Bob is known to Alice, while that of Eves is not available, as they work as passive listeners. With the knowledge of the steering vector ab w.r.t. Bob, Alice adopts maximum ratio combining (MRC) beamforming to Bob, i.e., the weight vector √ should be w = ab / M . From (1) and (2), the channel gains Δ for Bob and the k-th Eve can be denoted by ηb = M |hb |2 rb−α 2 and ηk = |hk |2 ψk rk−α , where ψk = (1/M )|aH b ak | . We consider both the non-colluding Eves (NCE) and colluding Eves (CE) cases. For NCE case, Eves individually decode the messages, while for CE case, Eves jointly decode the messages with MRC reception. The secrecy capacity (SC) is the difference between the capacities of the legitimate channel CB and the equivalent wiretap channel CE , which can be uniformly expressed as below in both cases Δ
Δ
CS = [log2 (1 + ηb ρ) − log2 (1 + ηe ρ)]+ ,
(3)
Δ
where [x]+ = max(0, x), ρ = P/σn2 is the transmit signal to noise ratio (SNR), and ηe is the equivalent wiretap channel gain. For NCE case, it is a compound wiretap channel [14] with ηe = maxek ∈Φ (ηk ). Hence, the SC herein is impacted by the Eve with the best channel. For CE case, ηce = ek ∈Φ ηk , and the SC is influenced by all Eves. Since the CSI of Bob is available, Alice can decide whether to transmit or not so that the outage due to reliability failure and security failure can be distinguished. Therefore, to describe the security more explicitly, we investigate the transmission secrecy outage probability (TSOP) proposed in [8], which is defined as3 Δ
Ptso = Pr {CS < Rs |CB > Rs } .
transmission (the capacity of the main channel CB is larger than the target secrecy rate Rs ). To intuitively describe the security w.r.t. Bob’s position, we propose a concept of security region (SR), which is defined as the geometry region where the TSOP is below a prescribed outage threshold ε ∈ [0, 1], i.e., Δ 2 R(ε) s = (rb , θb ) : ∀(rb , θb ) ∈ R , Ptso ≤ ε .
The SR is instructive to guide the legitimate receiver’s position. Once Bob is located in the SR, some level of secrecy (TSOP) can be guaranteed. Conversely, for a given position of Bob, we know how secure the transmission is. III. S ECRECY O UTAGE AND S ECURITY R EGION In this section, we derive the TSOP for both NCE and CE cases, and depict the SR based on it. The TSOP in (4) is a conditional probability which can be calculated by Ptso = Pso /Pt , Δ Δ where Pso = Pr{CS < Rs , CB > Rs } and Pt = Pr{CB > Rs }. Since hb ∼ CN (0, 1), we have the probability distribution α function (PDF) of ηb as fηb (x) = (rbα /M )e−rb x/M , and then α Pt = e−(βrb /M ) , where β = (2Rs − 1)/ρ. Next, we calculate Pso as follows 1 + ηb ρ Pso = Pr log2 < Rs , log2 (1 + ηb ρ) > Rs 1 + ηe ρ 1 = Pr ηe > (ηb − β), ηb > β (6a) T ∞ ∞ (a) = fηe (x)fηb (y) dx dy β y−β T ∞
1 − Fη e
=
we use the notation (ri , θi ) to denote the position of node i with the distance ri from the origin and the direction θi , see Fig. 1. 3 Traditional SOP is defined as Pr{C < R }, where R is the target s s S secrecy rate.
fηb (y) dy,
(6b)
where T = 2Rs , fηe (x) and Fηe (x) are the PDF and the cumulative distribution function (CDF) of ηe , respectively. Eq. (a) is derived due to the independence between ηb and ηe . A. NCE Case For NCE case, we have ηe = maxek ∈Φ (ηk ). Consider the statistic characteristics of hk and the randomness of Eves’ positions, the CDF Fηe (x) can be calculated by
Fηe (x) = Pr{ηe < x} = EΦ Pr max(ηk ) < x|Φ ek ∈Φ
= EΦ Pr{ηk < x|Φ} = EΦ
ek ∈Φ
ek ∈Φ
⎛
= exp ⎝−λe
(b) 2 Hereafter
y−β T
β
(4)
We can see the TSOP is the probability that the secrecy capacity CS is below a target rate Rs conditioned on the message reliable
(5)
1−e
2π ∞ 0
0
λe Ψ Γ = exp − α
(c)
α −rk x/ψk
⎞
re−r 2 α
α
x
x/ψ
2 −α
dr dθ⎠
,
(7)
ZHENG et al.: ON TRANSMISSION SECRECY OUTAGE OF A MULTI-ANTENNA SYSTEM
1301
where Ex [f (x)] is the mathematical expectation w.r.t. x, Eq. (b) holds for the probability generating functional lemma (PGFL) over PPP [15], (c) holds for the integration formula ∞ α Δ [17, 3.326.1] 0 e−r x dr = (1/α)Γ(1/α)x−(1/α) , and Ψ = 2π 2/α dθ with ψ calculated by 0 ψ 2 2 M −1 1 −jmπδ 1 1 − e−jM πδ 1 H 2 e ψ = |ab a| = = M M m=0 M 1 − e−jπδ 2 M Sinc2 (M δ/2) 1 e−jM πδ/2 sin(M πδ/2) = = , M e−jπδ/2 sin(πδ/2) Sinc2 (δ/2) (8) Δ
Δ
where δ = cos θb − cos θ and Sinc(x) = sin(πx)/πx. Substitute fηb (x) and Fηe (x) into (6b), and transform the integral variable y to z = (1/T )(y − β), we can obtain Pso . Then the TSOP Ptso = Pso /Pt can be derived by Ptso
T rbα =1− M
∞
λe Ψ Γ exp − α
2 2 T rbα −α z dz. − z α M
0
(9) From (9), we find Ptso is a function of various factors, e.g., M , λe , Rs , α as well as Bob’s position (rb , θb ). For any given M , λe , Rs , and α, Ptso solely depends on Bob’s position. In some regions, certain level of secrecy can be ensured, which is just the basic idea of the SR. Detailed analysis on the impacts of these factors to the security can be seen in Section IV. Note that Ptso is independent to the transmit power P , i.e., increase P will not enlarge the SR. This is because Ptso is determined by the rate difference of the main and wiretap channel, which is influenced by the relative strength between the two channels but not the absolute value. It is complicated to obtain a closed-form expression for Ptso in (9). However, for the case α = 2, (9) can be simplified as λe Ψ T rb2 − z dz exp − 2z M 0 = 1 − 2λe T Ψ/M rb K1 2λe T Ψ/M rb , (10) ∞
Ptso = 1 − T rb2
where K1 (z) is the first-order modified Bessel function of the second kind [17, 8.407]. Eq. (10) holds due to the ∞ integral formula [17, 3.324.1] 0 exp((−μ/4x) − νx)dx = √ (μ/ν)K1 ( μν) with μ = 2λe Ψ and ν = T rb2 /M . For α > 2, we now derive its analytical upper bound. The upper bound presents a conservative estimation of the TSOP. If the upper bound is below the outage threshold ε, the exact TSOP can be definitely guaranteed. We rewrite (9) as
2 2 λe Ψ −α Ptso = 1 − Ez exp − Γ , (11) z α α ∞ α where Ez [g(z)] = (T rbα /M ) 0 g(z)e−T rb z/M dz is the mathematical expectation w.r.t. g(z) with the PDF of variable z α is fz (z) = (T rbα /M )e−T rb z/M . With Jensen’s inequality, we have 2 2 λe Ψ −α Γ Ptso ≤ 1 − exp − . (12) Ez z α α
Fig. 1. Security Region in a 120 m × 120 m area. α = 2, M = 4, λe = 0.0001/ m2 and Rs = 0.1 bits/s/Hz.
From [17, 3.478.1], we have Ez [z −(2/α) ] = (T /M )2/α rb2 Γ(1 − (2/α)). Substitute it into (12), we finally obtain the ub closed-form expression on the upper bound Ptso , as 2 2 T α λe Ψrb2 2 ub Γ Ptso = 1−exp − Γ 1− . (13) α M α α Remark 1: From the numerical results in Section IV, we find ub is asymptotically tight as α increases. In practice, the range Ptso ub of α is about 2 ∼ 7. We see that Ptso gives a considerable ub approximation on Ptso at α = 6. In that case, Ptso is a low complexity alternative to Ptso . Fig. 1 depicts some SRs according to the TSOP in (10) under several ε’s. Each SR is a geometry area bounded by the contour line corresponding to a certain ε. We find the SR stretched along the normal direction of the array. This is due to the adoption of ULA and MRC beamforming, the main-beam at the normal direction is narrowest, leading to the smallest probability that the “best” Eve appears in this beam. It indicates that the normal direction is of the most secure guarantee. Therefore, with the knowledge of Bob’s direction θb , if the direction of the ULA could be adjusted, the normal direction could be aligned to θb to enhance the security. B. CE Case the equivalent wiretap channel gain is ηce = For CE case, −α 2 ek ∈Φ |hk | ψk rk , which is a shot noise process [3]. Similarly to the NCE case, we rewrite (6a) as ce Pso = Pr {β < ηb < T ηce + β} ∞ T y+β fηb (x)fηce (y) dx dy = 0
=e
β
−βrbα /M
⎛ ⎝1 −
⎞
∞ e
−T rbα y/M
fηce (y) dy ⎠ , (14)
0
where the integral item in (14) is the Laplace transform of ηce , denoted by Lηce (s) at s = T rbα /M . According to [3], we have 2 2 λe 2 α Lηce (s) = exp − Γ , (15) Γ 1− Ψs α α α
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Fig. 2. TSOP vs. M with different λe ’s and Rs ’s.
IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 8, AUGUST 2014
monotonically increase w.r.t. rb , and tend to 1 as rb → ∞, which means secrecy outage always occurs for a far enough Bob even conditioned on the reliable transmission. This is because the main channel capacity CB is used to meet the target secrecy rate Rs , leaving little margin against the eavesdroppers. The gap between Ptso ’s for α = 4 and α = 6 becomes wider with an increase of rb , which indicates the degradation on the security caused by the path loss becomes more significe cant when Bob is farther away. The same is true for Ptso . ce We also find Ptso is always higher than Ptso due to the collusion of Eves and becomes asymptotically close to Ptso as α increases. In other words, the collusion provides less additional degradation on the security compared with the NCE case when α becomes larger. The reason is that, more severe path loss makes the influence of the “best” Eve more significant compared with other Eves, which has weakened the impact of Eves’ collusion. R EFERENCES
Fig. 3. TSOP vs. Bob’s distance rb with different α’s.
where Ψ is defined in (8), and α > 2. Substitute (15) into (14), ce ce and by Ptso = Pso /Pt , we derive the TSOP 2 2 λe Ψrb2 T α 2 ce Ptso = 1−exp − Γ Γ 1− . (16) α M α α Eq. (16) and (13) are identical, which implies that the upper bound in NCE case actually represents the CE scenario. IV. N UMERICAL R ESULTS AND D ISCUSSION In this section, we investigate the factors that impact the security. Fig. 2 presents the TSOP in (10) versus the number of transmit antennas M with different intensity λe ’s and secrecy rate Rs ’s. We find a slight increase of M effectively decreases the TSOP, since more antennas at transmitter form a narrower beam that covers fewer Eves, which consequently reduces the probability of message leakage. λe has dramatically lowered the TSOP, as a smaller λe decreases the average number of Eves in a certain area, which naturally reduces the probability of interception. We see that, the larger Rs , the higher TSOP, which indicates there should be a tradeoff between the transmission efficiency and the security. We conclude that, a larger M , a smaller λe and a lower Rs all lead to a lower TSOP. According to the definition of the SR, these factors are obviously beneficial for expanding the SR, thus enhancing the security of the transmission. ce Fig. 3 illustrates Ptso in (9) and Ptso in (16) versus Bob’s distance rb with different path loss exponent α’s. Both TSOPs
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