On the Secrecy of Interference-Limited Networks under Composite Fading Channels
Hirley Alves, Carlos H.M. de Lima, Pedro. H.J. Nardelli, Richard Demo Souza, Matti Latva-aho
11 On the Secrecy of Interference-Limited Networksunder Composite Fading Channels
Hirley Alves
Student Member, IEEE , Carlos H. M. de Lima
Member, IEEE , Pedro. H. J. Nardelli
Member, IEEE ,Richard Demo Souza
Senior Member, IEEE and Matti Latva-aho
Senior Member, IEEE
Abstract —This paper deals with the secrecy capacity of theradio channel in interference-limited regime. We assume thatinterferers are uniformly scattered over the network area accord-ing to a Point Poisson Process and the channel model consists ofpath-loss, log-normal shadowing and Nakagami-m fading. Boththe probability of non-zero secrecy capacity and the secrecyoutage probability are then derived in closed-form expressionsusing tools of stochastic geometry and higher-order statistics. Ournumerical results show how the secrecy metrics are affected bythe disposition of the desired receiver, the eavesdropper and thelegitimate transmitter.
Index Terms —Composite channel, secrecy capacity, secrecyoutage probability, stochastic geometry
I. I
NTRODUCTION
Due to their broadcast nature, wireless communications aresusceptible to security issues since non-intended nodes withinthe communication range of a given transmitter can overhearthe transmission and possibly extract private information [3].To ensure confidentiality, cryptographic techniques (usuallyimplemented in higher layers) depend on secret keys and alsorely on the limited computational power of eavesdroppersand on the reliability guaranteed by channel coding at thePhysical Layer (PHY). However, future wireless systems tendto be deployed in large scale with ubiquitous coverage, dy-namic operation and computational powerful devices, makingencrypted communication through secret keys expensive anddifficult to achieve. PHY security has reemerged [1]–[4] asa viable alternative to enhance the robustness and reduce thecomplexity of conventional cryptography systems since PHYoffers unbreakable and quantifiable secrecy in confidentialbps/Hz, regardless of the eavesdropper’s computational power.PHY security dates back to 1975, when Wyner in hispioneering work [5] introduced the wire-tap channel whichis composed of a pair of legitimate users, known as Al-ice (transmitter) and Bob (receiver), communicating in thepresence of an eavesdropper known as Eve. Alice and Bobcommunicate through the main channel in the presence ofEve, who perceives a degraded version of the message sentto Bob through the eavesdropper channel. In this context it is
H. Alves, C. H. M. de Lima, P. H. J. Nardelli and M. Latva-aho are withthe Centre for Wireless Communications (CWC), University of Oulu, Finland.Contact: [email protected].fi.R. D. Souza and H. Alves are with Federal University of Technology -Paraná (UTFPR), Curitiba, Brazil.C. H. M. de Lima is also with São Paulo State University (UNESP), SãoJoão da Boa Vista, Brazil.The authors also would like to thank Aka and Infotech Oulu GraduateSchool from Finland, and CNPq and Special Visiting Researcher fellowshipCAPES 076/2012 from Brazil. proved that there exist codes which guarantee both low errorprobabilities and a certain degree of confidentiality.Later in [6], it was demonstrated that the secrecy capacity ofthe Gaussian wire-tap channel can be defined as the differencebetween the capacity of the main channel and the eavesdropperchannel considering that the eavesdropper channel is noisierthan the main channel. The wire-tap channel is extendedto a fading scenario and secrecy outage probability is thencharacterized in [1], [2]. Cooperative and jamming techniquesare reviewed in [4] and diversity schemes are assessed in [7].Such recent works, however, focus on a small number ofnodes, whose results provide few insights how PHY securityperforms in large-scale networks [2, Sec. VIII-C]. Differentfrom the point-to-point communication wherein secrecy isguaranteed by keys or tokens, large-scale deployments imposenew challenges to ensure secure communication, due to thecomplexity of distributing and maintaining secret keys [8].Security in large-scale networks are also affected by thespatial distribution of the interferer nodes and eavesdroppers(for more details, refer to [2, Sec. VIII-C] and the referencestherein). Early contributions aim at characterizing the scalinglaws of secrecy capacity [9] and the network connectivity [10]of randomly scattered nodes, or at computing the secure areaspectral efficiency for predefined quality requirements. Recentpapers have also shown that PHY-security can compensate forthe vulnerability of wireless communications and reduce itsimplementation complexity by exploiting the spatial-temporalcharacteristics of the wireless medium [11].In contrast, we characterize here the secrecy capacity oflarge-scale networks in the presence of uncoordinated interfer-ence by applying a general framework that jointly considersnodes spatial distribution and composite-fading channel, amodel introduced in [12], [13]. Besides, as most of existingworks consider thermal noise alone to compute the secrecy ca-pacity, this paper also contributes to the literature by providinga framework that allows for evaluating the aggregate interfer-ence and hence computing the non-zero secrecy capacity andthe secrecy outage probability in closed-form.Built upon the analytic results from [12], [13], we as-sess the joint effects of transmitter-receiver and transmitter-eavesdropper relative distances and density interfering nodeson such capacity metrics. Considering a scenario where inter-ferers follow a Point Poisson Process and the channel modelconsists in path-loss, log-normal shadowing and Nakagami-mfading, we show under which circumstances non-zero secrecycapacity is possible and how the secrecy outage probabilitybehaves in terms of such dynamics. a r X i v : . [ c s . I T ] N ov II. S
YSTEM M ODEL
This section describes the system model used here, follow-ing the basic concepts introduced in [12, Sec.II]. To beginwith, the tagged legitimate pair is defined as the referencelink (transmitter-receiver) so as to compute the aggregate CCIand performance metrics for the scenario under consideration.Specifically the mutual information of the tagged legitimatepair and its related eavesdropper are determined based on theirinstantaneous Signal-to-Interference Ratio (SIR) values. Weadopt the notion for secrecy capacity, probability of existenceof non-zero secrecy capacity and secrecy outage probabilityas in [2, Sec. II].We consider large-scale wireless networks where legitimatetransmitters (interferer nodes) constitute a homogeneous Pois-son Point Processs (PPPs) Φ with density λ [TXs/m ] in R [12], [13]. Transmitters communicate using antennas withomni-directional radiation pattern and fixed transmit power.Let us now consider the set of legitimate transmittersin an arbitrary region R of area A which thus follows aPoisson distribution with parameter λA . We then assumethe (composite) fading effect as a random mark associatedwith each point of Φ . Using Marking theorem, the resultingprocess r Φ “ tp ϕ, x q ; ϕ P Φ u corresponds to a Marked PointProcess (MPP) on the product space R ˆ R ` , whose points(transmitter locations) ϕ belong to the process Φ and therandom variable x refers to the corresponding squared-envelopof the composite fading, as presented later.The scenarios under study are interference-limited andhence the thermal noise is negligible in comparison to theresulting Co-Channel Interference (CCI) [12], [13]. We recallthat as pointed out by [14] the aggregate interference domi-nates over AWGN noise and the distribution of the aggregateinterference is positively skewed and heavy tailed, whichsuggests a Log-Normal distribution [14]. We assume the highmobility random walk model, so each observation period canbe analyzed as an independent realization of the MPP [15].Fig. 1 depicts the network model in a 2D grid. Noticethat the legitimate pair – the tagged transmitter and receiver– is represented in black (star and square, respectively) andare separated by a distance of d l . The field of interferers isdenoted by red circles while the eavesdropper is representedby the blue diamond. We assume that the eavesdropper is ata known distance d e from the tagged transmitter. Through theframework introduced in [12], [13], we are able to computethe aggregate interference seen by the receivers.Radio links are subject to distance-dependent path-loss andshadowed fading, which is assumed to be independent overdistinct network entities and positions. A signal strength decayfunction describes the average power attenuation (unboundedmodel) as l p d i q “ d ´ αi , where α is the path loss exponent and d i represents the distance between a transmitter-receiver pairwith i P t l , e u , which can be either a legitimate receiver orthe eavesdropper.Each interferer then disrupts the communication of thetagged receiver with a component given by p l p d q x , where p represents the interferer’s transmit power, d is the separationdistance from an interferer to the tagged receiver or eavesdrop- − − − − −
25 0 25 50 75 100 d (m) − − − − − d ( m ) Fig. 1. Example of the network model. Legitimate pair is represented in black(square and star) and are separated by a distance d l , while Eve is depicted inblue (diamond) and at distance d e from the tagged transmitter. Note that theinterferers are in red (circles) and that the aggregate power of the interferersdisrupt the communication of the tagged receiver and the eavesdropper. per. From this assumption, we can compute an approximationto the distribution of the aggregate interference caused by allactive transmitters, defined by the MPP, as presented in thenext section.The received squared-envelop due to multipath fading andshadowing is represented by a Random Variable (RV) X P R ` with Cumulative Distribution Function (CDF), F X p x q , andProbability Distribution Function (PDF), f X p x q . Then, thecomposite distribution of the received squared-envelop due toLog-Normal (LN) shadowing and Nakagami- m fading has aGamma-LN distribution, whose PDF is given as [12], [13]: f X p x q “ ż ´ mω ¯ m x m ´ Γ p m q exp ´ ´ mω x ¯ (1) ˆ ξ ? πσω exp « ´ ` ξ ln ω ´ µ Ω p ˘ σ p ff d ω . In this case, m is the shape parameter of the Gamma(Nakagami- m ) distribution, Γ p¨q is the gamma function [17,Eq. 8.310-1], ξ “ ln p q { , Ω p is the mean squared-envelop, µ Ω p and σ Ω p is the mean and standard deviation of Ω p ,respectively.Moreover, Ho and Stüber show in [16] that a compositeGamma–LN distribution can be approximated by a single LNdistribution with mean and variance (in logarithmic scale)given by µ dB “ ξ r ψ p m q ´ ln p m qs ` µ Ω p and σ “ ξ ζ p , m q ` σ p , where ψ p m q is the Euler psi function [17,Eq. 8.360-1] and ζ p , m q is the generalized Riemann zetafunction [17, Eq. 9.551]. III. S
ECRECY C APACITY A NALYSIS
The cumulant-based framework from [12, Secs. II, IV andVII] and [13, Sec.VI] is now used to asses achievable levelsof secrecy and the resulting performance of legitimate link.Suppose that the legitimate transmitter has only CSI of thedesired receiver, which is known as passive eavesdropping[3], [7]. In such case, we resort to a probabilistic view ofsecurity in order to characterize the probability of informationleakage to the eavesdropper. Then, in order to protect thetransmission from an inimical attack, we consider the useof a wiretap code with nR codewords, where R is madeequal to the instantaneous capacity of the legitimate channel,namely C l [1]. Then, the number of codewords per bin isset equal to nR e , where R e represents the eavesdropper’sequivocation rate. Thus, a fixed secure transmission rate isattained as R s “ R ´ R e “ C l ´ R e , which implies that R e “ C l ´ R s varies according to the legitimate channel con-dition. Therefore, as introduced in [1], an outage event occurswhen R s exceeds the difference between the instantaneouscapacities of the legitimate and the eavesdropper channels,thus Pr rr C l ´ C e s ` ă R s s , with r¨s ` fi max t¨ , u .Let us first characterize the probability of existence of non-zero secrecy capacity ( Pr r C l ą C e s ) when the legitimate linkexperiences interference from concurrent transmissions. Weuse Γ l and Γ e to denote the SIR of the legitimate and eaves-dropper links, respectively. Then, the following expressionshows the secrecy capacity of shadowed fading channels C s “ r C l ´ C e s ` “ r log p ` Γ l q ´ log p ` Γ e qs ` , (2)The distributions of Γ l and Γ e can be recovered using [13,Th.1]. Theorem 1:
For the system model described in this paperand the distances d l (from the receiver) and d e (from the eaves-dropper) to the tagged legitimate transmitter, the probabilityof existence of non-zero secrecy capacity is Pr r C s ą s “ Q « µ e ´ µ l a σ l ` σ e ff (3) Proof:
The non-zero secrecy capacity probability is Pr r C s ą s “ Pr r Γ l ą Γ e s “ ż γ l ż f l p γ l q f e p γ e q d γ e d γ l , (4)where C s is defined in (2), the SIR PDFs f l p γ l q and f e p γ e q of the legitimate and eavesdropper nodes follow Lognormal ` µ l , σ l ˘ and Lognormal ` µ e , σ e ˘ . Integrating interms of γ e , we have Pr r Γ l ą Γ e s “ ż
12 Erfc „ µ e ´ log p γ e q? σ e f l p γ l q d γ l . (5)Thereafter, we substitute η “ p µ e ´ log γ e q{? σ e in (5)and adjust the limits of integration accordingly to obtain Pr r Γ l ą Γ e s “ ż ´8 e ´ η ? π Erf „ ´ µ l ` µ e ` ? ασ l ? σ e d η. (6) Eq. (3) is obtained from (6) by using [17, Eq. 8.259-1].We now need to identify the circumstances whereby secrecyis compromised by defining the secrecy outage probability. Definition 1:
Secrecy Outage Probability is defined as theprobability that the instantaneous secrecy capacity C s does notmatch the target secrecy rate R s ą and is expressed as [2]: Pr r C s ă R s s “ Pr r C s ă R s | Γ l ą Γ e s Pr r Γ l ą Γ e s ` Pr r C s ă R s | Γ l ď Γ e s Pr r Γ l ď Γ e s . (7) Theorem 2:
For the system model described in this paper,the secrecy outage probability with respect to the legitimatelink and an arbitrary eavesdropper is given by Pr r C s ă R s s “ ´ N ÿ n “ ω n ? π (8) ˆ Erf « µ l ´ log “ ´ ` R s ` R s exp ` µ e ´ ? η n σ e ˘‰ ? σ l ff . Proof:
Let us start evaluating each summand in (7)separately. Recall from Theorem 1 that Pr r Γ l ą Γ e s “ Q ” p µ e ´ µ l q { a σ l ` σ e ı and since R s ą , it follows that Pr r C s ă R s | Γ l ď Γ e s “ .Then, we rewrite the secrecy capacity in terms of its SIRdistribution and then proceed as Pr r C s ă R s | Γ l ą Γ e s ““ Pr r log p ` Γ l q ´ log p ` Γ e q ă R s | Γ l ą Γ e s“ Pr “ Γ l ă R s p ` Γ e q ´ | Γ l ą Γ e ‰ . (9)Under the assumption that legitimate and eavesdropperchannels are independent, one computes (9) as follows. Pr r C s ă R s | Γ l ą Γ e s ““ r Γ l ą Γ e s ż (cid:15) p ` γ e q´ ż f l p γ l q f e p γ e q d γ l d γ e , (10)where (cid:15) “ R s . After integrating over γ l , we attain Pr r C s ă R s | Γ l ą Γ e s “ r Γ l ą Γ e s p Ξ ´ Ξ q . (11)Employing the same method used in the previous proof [17,Eq. 8.259-1], we have Ξ “ ż f e p γ e q Erf „ µ l ´ log γ e ? σ l d γ e “ Q « µ l ´ µ e a σ l ´ σ e ff , (12)where the second term is integrated using Gauss-Hermitequadrature [18] and η “ p µ e ´ log γ e q {? σ e , so that Ξ “ ż f e p γ e q Erf „ µ l ´ log p´ ` (cid:15) ` (cid:15)γ e q? σ l d γ e “ N ÿ n “ ω n ? π Erf „ µ l ´ log r´ ` (cid:15) ` (cid:15) ζ s? σ l , (13)where ζ “ exp ` µ e ´ ? η n σ e ˘ . Inserting (3), (12) and (13)into (11), yields the secrecy outage probability as in (8). . . . . . . . . . − −
20 0 20 40 P r [ C s > ] Γ l (dB) λ (Node/ m )1 × − × − d e “ d e “
15 m d e “
25 m
Fig. 2. Probability of existence of non-zero secrecy capacity as a functionof the perceived SIR at the legitimate receiver for different distance betweenlegitimate transmitter and eavesdropper d e , and density of interfering nodes λ , considering d l “
15 m . IV. N
UMERICAL R ESULTS
In this section, we apply the framework to evaluate thefeasibility of PHY security in terms of the existence of secrecycapacity and the outage secrecy probability. Eavesdropper andlegitimate nodes are affected by shadowed fading with theLN shadowing following a zero-mean Gaussian distributionwith variance σ “ , Rician fading factor of K “ . ,and Hermite polynomial order of N “ to evaluate ourperformance metrics. We consider that active nodes operate ata fixed transmit power of
20 dBm .Fig. 2 depicts the probability of existence of non-zerosecrecy capacity of the legitimate tagged link by varying itsaverage SIR, eavesdropper distance to the tagged transmitterand density of interfering nodes for d l “ . We observethat the probability of non-zero secrecy capacity increaseswhen the average SIR of the legitimate receiver increases,whereas it decreases by positioning the eavesdropper close thelegitimate transmitter, namely, d e “ away. In other words,the closer the eavesdropper is to the legitimate transmitter, thehigher should be the link quality of the legitimate pair so asto guarantee its secrecy.We can also see from Fig. 2 that the increase of the inter-ferers’ density decreases the non-zero secrecy capacity proba-bility for the same average SIR, regardless of the distance d e .This fact evinces that the increase of co-channel interferencecaused by increasing λ equally decreases the channel capacityfor both legitimate and eavesdropper links and therefore thegap between the curves with different densities tends to beconstant when the same distance d e is set.Fig. 3 shows the secrecy outage probability for increasingsecrecy rate R s , considering that the eavesdropper is at dis- . . . . . . . . .
91 0 2 4 6 8 10 P r [ C s < R s ] R s (bps / Hz) d e ( m)1015 d l “ , , , ,
25 m
Fig. 3. Secrecy outage probability as a function of the secrecy rate for d l “ , , , ,
25 m , while the eavesdropper is positioned d e “
10 m and d e “
15 m away from the transmitter. tance d e “ and d e “
15 m . Note that the distance betweenthe legitimate pair varies with increments of . We caninfer that secrecy outage probability worsens with the increaseof the distance between the legitimate nodes and also withthe proximity of the eavesdropper to the transmitter. However,higher secrecy rates can be achieved with shorter legitimatelinks once the transmitted signal is much stronger than theinterference seen at the receivers.V. F
INAL REMARKS AND CONCLUSION
We investigated the secrecy capacity of the channel in largescale, interference-limited networks. Using a more generalmodel that captures randomness due to interferers’ position,shadowing and fast fading, we derive closed form expressionsfor the probability of non-zero secrecy capacity and secrecyoutage probability. For the proposed scenario, our numericalresults show under which conditions secrecy can be achievedfor different network configurations, evincing the effects of theproximity of the eavesdroppers to the legitimate transmitterand the interferer density on the secrecy outage probability.We plan to continue the investigation introduced in thispaper in scenarios legitimate transmitters use random accessprotocols and links are subject to quality constraints. In thisway, we plan to evaluate both the effective secrecy through-put and the network spatial secrecy throughput, as in [19].Another extension is to assess the secrecy capacity of Poissondistributed networks where multi-hop links are allowed [20]such that the relative positions between relays (defined bysome specific hopping strategy) and eavesdroppers are expectto impact the system performance. R EFERENCES[1] M. Bloch et al., “Wireless information-theoretic security,”
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