2764 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 8, AUGUST 2011

On the Throughput Cost of Physical Layer Securityin Decentralized Wireless Networks

Xiangyun Zhou, Member, IEEE, Radha Krishna Ganti, Member, IEEE,Jeffrey G. Andrews, Senior Member, IEEE, and Are Hjørungnes, Senior Member, IEEE

Abstract—This paper studies the throughput of large-scaledecentralized wireless networks with physical layer securityconstraints. In particular, we are interested in the question ofhow much throughput needs to be sacrificed for achieving acertain level of security. We consider random networks where thelegitimate nodes and the eavesdroppers are distributed accordingto independent two-dimensional Poisson point processes. Thetransmission capacity framework is used to characterize the areaspectral efficiency of secure transmissions with constraints onboth the quality of service (QoS) and the level of security. Thisframework illustrates the dependence of the network throughputon key system parameters, such as the densities of legitimatenodes and eavesdroppers, as well as the QoS and securityconstraints. One important finding is that the throughput costof achieving a moderate level of security is quite low, whilethroughput must be significantly sacrificed to realize a highlysecure network. We also study the use of a secrecy guardzone, which is shown to give a significant improvement on thethroughput of networks with high security requirements.

Index Terms—Physical layer security, decentralized wirelessnetworks, transmission capacity, guard zone.

I. INTRODUCTION

THE problem of securing wireless communications atthe physical layer has recently drawn considerable at-

tention. In the pioneering works on physical layer security,Wyner [1] introduced the wiretap channel for single point-to-point communication, which was extended to broadcastchannels with both common and confidential messages byCsiszar and Korner [2]. Their results showed that perfectsecrecy can be achieved if the intended receiver has a strongerchannel than the eavesdropper. Recent studies on physicallayer security primarily focused on communications involvinga small number of nodes with multi-antenna transmission [3–5], cooperative transmission in relay channels [6, 7] or multipleaccess channels [8, 9]. However, few studies have been carriedout for large-scale wireless networks. Unlike point-to-pointcommunications, where it is reasonably easy to establish

Manuscript received December 21, 2010; revised March 30, 2011; acceptedMay 9, 2011. The associate editor coordinating the review of this paper andapproving it for publication was I.-M. Kim.

X. Zhou and A. Hjørungnes are with UNIK - University GraduateCenter, University of Oslo, Kjeller, NO-2027, Norway (e-mail: {xiangyun,arehj}@unik.no).

R. K. Ganti and J. G. Andrews are with the Department of Electrical andComputer Engineering, the University of Texas at Austin, Austin, TX 78712,USA (e-mail: rganti@austin.utexas.edu, jandrews@ece.utexas.edu).

This work was supported by the Research Council of Norway throughthe FRITEK project 197565/V30 entitled “Theoretical Foundations of MobileFlexible Networks - THEFONE,” and the DARPA IT-MANET Project.

Digital Object Identifier 10.1109/TWC.2011.061511.102257

secret keys and have encrypted transmissions, security is moreexpensive and difficult to achieve in large-scale decentralizednetworks. Therefore, physical layer security may be impor-tant for exchanging secret keys and adding another layer ofprotection in such networks.

The communication between any pair of nodes in large-scale networks strongly depends on the locations of othernodes and how the nodes interact with each other. Whensecure communication is required in the presence of eaves-droppers, the locations and channel state information of theeavesdroppers, which are usually unknown, become extraparameters affecting the network throughput. Initial workson network security from an information-theoretic viewpointmainly considered networks where the legitimate nodes andthe eavesdroppers are randomly distributed, and studied theconnectivity [10–14], coverage [15], and capacity scalinglaws [16–18]. Specifically, various statistical characterizationsof the existence of secure connections were given in [10–12,14]. Using tools from percolation theory, the existence of asecrecy graph was analyzed in [10, 12, 13]. These connectivityresults are concerned with the possibility of having securecommunication, while they do not give insight on the networkthroughput. The authors in [16–18] derived secrecy capacityscaling laws in static and mobile ad hoc networks, i.e., theorder-of-growth of the secrecy capacity as the number of nodesincreases. Although the scaling laws may provide insights intothe information-theoretic performance of large-scale networks,a finer view of throughput is necessary to better understand theimpact of key system parameters and transmission protocols,since most of these design choices affect the throughput butnot the scaling behaviors [19].

A. Approach and Contributions

In this work, we aim to characterize the throughput of securecommunications in large wireless networks and to understandhow the security requirements affect the network throughput.Our approach uses a metric termed the transmission capac-ity [20], which provides the area spectral efficiency (ASE)of decentralized networks with random topology, identicalnodes, and a constraint on outage probability. A tutorial ontransmission capacity can be found in [21], which showedhow analytical results can often be derived in simple forms.We extend this capacity framework to study the impact ofphysical layer security requirements on the network ASE.

The networks considered have both legitimate nodes andeavesdroppers, whose locations follow homogeneous Poisson

1536-1276/11$25.00 c⃝ 2011 IEEE

ZHOU et al.: ON THE THROUGHPUT COST OF PHYSICAL LAYER SECURITY IN DECENTRALIZED WIRELESS NETWORKS 2765

point processes (PPPs). We define the secrecy transmissioncapacity as the achievable rate of successful transmission ofconfidential messages per unit area for given constraints on thequality of service (QoS) and the level of security. The QoSconstraint is given by the outage probability of the transmis-sion between a legitimate transmitter-receiver pair, while thesecurity constraint is given by the probability of a transmissionfailing to achieve perfect secrecy. The secrecy transmissioncapacity shows the dependence of the network ASE on thekey system parameters, i.e., the densities of legitimate nodesand eavesdroppers, as well as the QoS and security constraints.

To illustrate the use of the general capacity formulation,we derive an accurate closed-form lower bound on the se-crecy transmission capacity for Rayleigh fading channels. Thissimple capacity bound gives a quantitative characterization ofthe throughput cost of physical layer security. Specifically, thethroughput reduction for achieving a moderate level of securityis relatively small, while a significant amount of throughputneeds to be sacrificed to realize a highly secure network. Wealso give a condition for the existence of positive secrecytransmission capacity. It turns out that the QoS and securityconstraints as well as the density of eavesdroppers are crucialin determining the existence of positive secrecy transmissioncapacity.

In order to minimize the throughput cost of achieving highnetwork security, it is worthwhile to consider transmissionprotocols that are robust against eavesdropping and can beimplemented in a decentralized manner. Since insecure trans-mission is mainly due to the presence of an eavesdropper closeto the transmitter, we consider the use of a secrecy guard zonefor networks in which the legitimate transmitters are able todetect the existence of eavesdroppers in their vicinities [11,16].1 Transmission of confidential messages take place onlyif no eavesdroppers are found inside the guard zone of thecorresponding transmitter. We consider two transmission pro-tocols when eavesdropper(s) are found inside the guard zone,i.e., the transmitter either remains silent or produces artificialnoise to help the other transmitters. The secrecy transmissioncapacity is studied for both protocols. Numerical results showthat a significant throughput improvement can be achievedfrom the use of a guard zone for networks with high securityrequirements.

The rest of the paper is organized as follows: Section IIpresents the system model and the secrecy transmission capac-ity formulation. In Section III, we obtain analytical results onthe secrecy transmission capacity in Rayleigh fading channels.In Section IV, we investigate the secrecy guard zone withtwo different transmission protocols. Numerical results arepresented in Section V and concluding remarks in Section VI.A summary of the notation used in this paper is given inTable I.

II. SYSTEM MODEL AND CAPACITY FORMULATION

We consider an ad hoc network consisting of both legitimatenodes and eavesdroppers over a large two-dimensional space.

1The application of secrecy guard zone is not always possible. It isapplicable in the scenarios where, for example, the legitimate transmittersare able to physically inspect their surrounding areas [11].

TABLE ILIST OF NOTATION

Φ𝑙 Poisson point process (PPP) of legitimate transmitter locationsΦ𝑒 PPP of eavesdropper locations𝜆𝑙 Density of Φ𝑙

𝜆𝑒 Density of Φ𝑒

𝑅𝑡 Rate of the transmitted codewords𝑅𝑠 Rate of the confidential messages𝑅𝑒 rate loss for securing the messages against eavesdroppingPco Connection outage probabilityPso Secrecy outage probability𝜎 Constraint on Pco

𝜖 Constraint on Pso

𝜏(𝑟) Secrecy transmission capacity with transmission distance 𝑟𝛽𝑡 Threshold signal to interference ratio (SIR) for connection outage𝛽𝑒 Threshold SIR for secrecy outage𝑆 Rayleigh fading gain of the wireless channel𝐷 Radius of secrecy guard zonesℙ(.) Probability operator𝔼{.} Expectation operator

For each snapshot in time, we have a set of legitimatetransmitter locations, denoted by Φ𝑙.2 Each transmitter hasa unique associated intended receiver. The set of receiversis disjoint with the set of transmitters. In addition, we havea set of eavesdropper locations in each snapshot, denotedby Φ𝑒. We model Φ𝑙 and Φ𝑒 as independent homogeneousPPPs with densities 𝜆𝑙 and 𝜆𝑒, respectively. This is a suitablemodel for decentralized networks with nodes having substan-tial mobility [21]. Note that the eavesdroppers need to havesimilar mobility and other behaviors as the legitimate nodessince they can be easily identified otherwise [17]. Furthermore,we assume that the eavesdroppers do not collude with eachother and, hence, must decode the confidential messagesindividually. An example of a network snapshot is shown inFig. 1.

Consider only one active transmitter that wants to sendconfidential messages to its intended receiver in the presenceof the eavesdroppers. Secure encoding schemes, such as theWyner code [1], were found in point-to-point systems withthe notion of weak secrecy. The notion of strong secrecy andthe corresponding encoding scheme were studied in [22]. Ac-cording to Wyner’s encoding scheme, the transmitter choosestwo rates, namely, the rate of the transmitted codewords𝑅𝑡 and the rate of the confidential messages 𝑅𝑠. The ratedifference 𝑅𝑒 = 𝑅𝑡 − 𝑅𝑠 reflects the cost of securing themessages against eavesdropping. If 𝑅𝑡 is less than the mutualinformation between the channel input and output of thelegitimate link, the receiver is able to decode the messagewith an arbitrarily small error. At the same time, if 𝑅𝑒 islarger than the mutual information between the channel inputand output of every eavesdropper link (i.e., links from thetransmitter to every eavesdropper), perfect secrecy is achievedas the mutual information between the confidential messageand every eavesdropper’s received signal approaches zeroratewise. The detailed description of the Wyner code can befound in [1, 23, 24].

In an ad hoc network with simultaneous transmissions frominfinitely many legitimate transmitters, it is difficult to study

2For networks employing a slotted Aloha protocol, Φ𝑙 can be viewed asthe locations of the actual transmitters (out of all potential transmitters) ineach time slot.

2766 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 8, AUGUST 2011

(𝑎) (𝑏)

Fig. 1. An example of a part of a network snapshot. Each legitimate transmitter, T𝑖, 𝑖 = 0, 1, 2, 3, has an intended receiver R𝑖, 𝑖 = 0, 1, 2, 3, locatedat a distance 𝑟. In addition, there are eavesdroppers, E𝑖, 𝑖 = 1, 2, presented in the network. Consider the confidential message transmission from T0. Theintended receiver R0 as well as each eavesdropper E𝑖, 𝑖 = 1, 2 all try to individually decode the transmitted message. The signal reception at R0 and (anarbitrary eavesdropper) E1 are shown in figures (a) and (b), respectively. In both cases, the concurrent transmissions from T1, T2 and T3 act as interference.

the mutual information between any pair of nodes. To makethe design and analysis mathematically tractable, we assumethat the transmitted signal (i.e., channel input) has a Gaussiandistribution and both the intended receivers and the eavesdrop-pers treat the interference from concurrent transmissions asnoise. In addition, we assume that the network is interference-limited, hence, the receiver noise is negligible. With theseassumptions, the mutual information or capacity of either alegitimate link or an eavesdropper link is now determined bythe instantaneous signal to interference ratio (SIR). For anygiven choices of 𝑅𝑡 and 𝑅𝑠 in Wyner’s encoding scheme, thefollowing outage events can result from any transmission [24]:

∙ Connection Outage: The capacity of the channel fromthe transmitter to the intended receiver is below the trans-mission rate 𝑅𝑡. Hence, the message cannot be correctlydecoded by the intended receiver. The probability of thisevent happening is referred to as the connection outageprobability, denoted as Pco.

∙ Secrecy Outage: The capacity of the channel from thetransmitter to one or more eavesdroppers is above the rate𝑅𝑒. Hence, the message is not perfectly secure againsteavesdropping. The probability of this event happeningis referred to as the secrecy outage probability, denotedas Pso.

The connection outage probability can be regarded as thecommunication QoS while the secrecy outage probabilitygives a measure of the security level.

The primary goal of this work is to characterize thethroughput of secure transmissions in decentralized wirelessnetworks. Although it is extremely difficult to find the networkcapacity region, the idea of transmission capacity proposedin [20] often gives useful insights on the network ASE andthe impacts of the key system parameters. Building on the ex-isting transmission capacity framework, we define the secrecytransmission capacity as the achievable rate of successfultransmission of confidential messages per unit area, for agiven connection outage constraint and a given secrecy outage

constraint. Mathematically, the secrecy transmission capacity,with a connection outage probability of Pco = 𝜎 and a secrecyoutage probability of Pso = 𝜖, is defined as

𝜏 = ��𝑠(1 − 𝜎)𝜆𝑙, (1)

where ��𝑠 is the average rate of confidential messages. Inthis work, we focus on a simple scenario where the transmitpower and the distances to the intended receivers have fixedvalues which are the same for all transmitters. Therefore,the confidential message rates also take the same value forall transmitters. Denote the distance between the legitimatetransmitter-receiver pairs as 𝑟, the secrecy transmission ca-pacity can be written as

𝜏(𝑟) = 𝑅𝑠(1 − 𝜎)𝜆𝑙. (2)

The connection outage constraint 𝜎 determines the value of𝑅𝑡, while the secrecy outage constraint 𝜖 determines the valueof 𝑅𝑒. Therefore, the rate of confidential messages 𝑅𝑠 in(2), given by 𝑅𝑡 − 𝑅𝑒, is a function of 𝜎 and 𝜖. With thesechoices of rates for the Wyner code, the probability that amessage transmission can be successfully decoded by theintended receiver is 1−𝜎, while the probability that a messagetransmission is perfectly secure against eavesdropping is 1−𝜖,under the assumption of treating interference as noise.

If we allow the distance between the legitimate transmitter-receiver pair varies over time and/or space but follows someknown distribution 𝑓(𝑟), the secrecy transmission capacity iscomputed as

𝜏 =

∫𝜏(𝑟)𝑓(𝑟)d𝑟. (3)

In practice, the distribution of 𝑟 depends on specific scenarios.Hence, we do not consider the variation in 𝑟 and focus on 𝜏(𝑟)in (2).

ZHOU et al.: ON THE THROUGHPUT COST OF PHYSICAL LAYER SECURITY IN DECENTRALIZED WIRELESS NETWORKS 2767

III. SECRECY TRANSMISSION CAPACITY IN RAYLEIGH

FADING CHANNELS

In this section, we derive analytical results on the secrecytransmission capacity for Rayleigh fading channels. We as-sume that each node has a single antenna for transmissionor reception, and the fading channel states are known at thereceiver side (including the eavesdroppers) but not at thetransmitter side. The derivation of the secrecy transmissioncapacity involves two main steps: 1) Use the connectionoutage constraint 𝜎 to find the value of 𝑅𝑡. 2) Use the secrecyoutage constraint 𝜖 to find the value of 𝑅𝑒.

Our analysis is based on an arbitrarily chosen transmitter-receiver pair, which are named the typical transmitter andreceiver. For confidential message transmission from the typ-ical transmitter, the other transmitters act as interferers to thetypical receiver or any eavesdropper. From Slivnyak’s Theo-rem [25], the spatial distribution of the interferers, given thelocation of the typical transmitter, still follows a homogeneousPPP with density 𝜆𝑙. By slight abuse of notation (since wehave used Φ𝑙 to denote the set of all transmitter locations),we will also refer to Φ𝑙 as the set of interferer locations inthe rest of this paper.

For the typical receiver, a connection outage occurs iflog2(1 + SIR0) < 𝑅𝑡, where SIR0 denotes the SIR at thetypical receiver given by

SIR0 =𝑆0𝑟

−𝛼∑𝑙∈Φ𝑙

𝑆𝑙∣𝑋𝑙∣−𝛼, (4)

where 𝑆0 and 𝑟 are the channel fading gain and the distancebetween the typical transmitter and receiver, respectively, 𝛼 isthe path loss exponent, 𝑆𝑙 and ∣𝑋𝑙∣ are the channel fading gainand the distance between the interferer (at position) 𝑙 in Φ𝑙 andthe typical receiver, respectively. We assume 𝛼 > 2 throughoutthis paper. The fading gains are modeled as independent andidentically distributed (i.i.d.) exponential random variableswith unit mean.

Define a threshold SIR value for connection outage as

𝛽𝑡 = 2𝑅𝑡 − 1. (5)

Hence, the connection outage probability can be written as

Pco = ℙ

(SIR0 < 𝛽𝑡

)= ℙ

(𝑆0𝑟

−𝛼∑𝑙∈Φ𝑙

𝑆𝑙∣𝑋𝑙∣−𝛼< 𝛽𝑡

). (6)

The summation term∑

𝑙∈Φ𝑙𝑆𝑙∣𝑋𝑙∣−𝛼 is a shot noise pro-

cess [26] in two-dimensional space whose Laplace transformis known in a closed form and was used to compute theconnection outage probability in [27] as

Pco = 1 − exp

[−𝜆𝑙𝜋𝑟

2𝛽2/𝛼𝑡 Γ

(1 − 2

𝛼

)Γ(

1 +2

𝛼

)]. (7)

With the connection outage constraint given by Pco = 𝜎,the transmission rate 𝑅𝑡 can be found using (5) and (7) as

𝑅𝑡 = log2

⎛⎝1 +

[ln 1

1−𝜎

𝜆𝑙𝜋𝑟2Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)]𝛼

2

⎞⎠ . (8)

It is clear that a lower connection outage probability (i.e., ahigher QoS) requires a lower 𝑅𝑡.

On the other hand, the confidential message transmission isnot perfectly secure against the eavesdropper (at position) 𝑒in Φ𝑒 if log2(1 + SIR𝑒) > 𝑅𝑒, where SIR𝑒 denotes the SIRat 𝑒 given by

SIR𝑒 =𝑆𝑒∣𝑋𝑒∣−𝛼∑

𝑙∈Φ𝑙𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼

, (9)

where 𝑆𝑒 and ∣𝑋𝑒∣ are the channel fading gain and the distancebetween the typical transmitter and eavesdropper 𝑒 in Φ𝑒,respectively, 𝑆𝑙𝑒 and ∣𝑋𝑙𝑒∣ are the channel fading gain andthe distance between node 𝑙 in Φ𝑙 and eavesdropper 𝑒 in Φ𝑒,respectively. The fading gains are modeled as i.i.d. exponentialrandom variables with unit mean.

Define a threshold SIR value for secrecy outage as

𝛽𝑒 = 2𝑅𝑒 − 1. (10)

Let 𝐴 = {𝑦 ∈ Φ𝑒 : SIR𝑦 > 𝛽𝑒}, i.e., the set of eavesdroppersthat can cause secrecy outage. Hence, we can define thefollowing indicator function: 1𝐴(𝑒), which equals 1 when theeavesdropper 𝑒 is in the set 𝐴. The secrecy outage probabilityequals the probability that at least one of the eavesdroppersin Φ𝑒 causes a secrecy outage, which can be written as

Pso = 1 − 𝔼Φ𝑙

{𝔼Φ𝑒

{𝔼𝑆

{ ∏𝑒∈Φ𝑒

(1 − 1𝐴(𝑒)

)}}},

= 1 − 𝔼Φ𝑙

{𝔼Φ𝑒

{

∏𝑒∈Φ𝑒

(1 − ℙ

( 𝑆𝑒∣𝑋𝑒∣−𝛼∑𝑙∈Φ𝑙

𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼> 𝛽𝑒

∣∣∣Φ𝑒,Φ𝑙

))}}.

(11)

where the independence in the fading gains among differenteavesdroppers is used to move the expectation over 𝑆 ={𝑆𝑒, 𝑆𝑙𝑒} inside the product over Φ𝑒 in (11). Since it isdifficult to express Pso in a closed form, we look for analyticalbounds on the secrecy outage probability. The results aresummarized in the following lemma:

Lemma 1: The secrecy outage probability is bounded fromabove by

PUBso = 1 − exp

⎡⎣− 𝜆𝑒

𝜆𝑙𝛽2/𝛼𝑒 Γ

(1 − 2

𝛼

)Γ(

1 + 2𝛼

)⎤⎦ , (12)

and bounded from below by

PLBso =

1

1 + 𝜆𝑙

𝜆𝑒𝛽2/𝛼𝑒 Γ

(1 − 2

𝛼

)Γ(

1 + 2𝛼

) . (13)

Proof: Using the generating functional of the PPP Φ𝑒 [25],we can express the secrecy outage probability in (11) as

Pso = 1 − 𝔼Φ𝑙

{

exp

[− 𝜆𝑒

∫ℝ2

ℙ

( 𝑆𝑒∣𝑋𝑒∣−𝛼∑𝑙∈Φ𝑙

𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼> 𝛽𝑒

∣∣∣Φ𝑙

)d𝑒

]}.

(14)

2768 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 8, AUGUST 2011

Jensen’s inequality gives an upper bound on Pso

Pso ≤ 1 − exp

[− 𝜆𝑒

∫ℝ2

ℙ

( 𝑆𝑒∣𝑋𝑒∣−𝛼∑𝑙∈Φ𝑙

𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼> 𝛽𝑒

)d𝑒

]

= 1 − exp

[− 2𝜋𝜆𝑒

⋅∫ ∞

0

exp

[−𝜆𝑙𝜋𝑟

2𝑒𝛽

2/𝛼𝑒 Γ

(1 − 2

𝛼

)Γ(

1 +2

𝛼

)]𝑟𝑒d𝑟𝑒

],

(15)

where 𝑟𝑒 denotes the distance between the typical transmitterand eavesdropper 𝑒, (15) is arrived in the same way as (7)followed by changing to polar coordinates. The upper boundin (12) is then obtained by directly evaluating the integrationin (15).

The lower bound on Pso is obtained by considering onlythe eavesdropper nearest to the typical transmitter. Denote theeavesdropper (location) in Φ𝑒 that is nearest to the typicaltransmitter as 𝑒′ and denote the distance between 𝑒′ and thetypical transmitter as 𝑟𝑒′ . The probability distribution of 𝑟𝑒′ isgiven by [28]

𝑓(𝑟𝑒′ ) = 2𝜆𝑒𝜋𝑟𝑒′ exp(−𝜆𝑒𝜋𝑟2𝑒′ ). (16)

The secrecy outage probability is bounded from below bythe probability that the nearest eavesdropper causes a secrecyoutage, i.e.,

Pso ≥∫ ∞

0

ℙ

( 𝑆𝑒′𝑟−𝛼𝑒′∑

𝑙∈Φ𝑙𝑆𝑙𝑒′ ∣𝑋𝑙𝑒′ ∣−𝛼

> 𝛽𝑒

)𝑓(𝑟𝑒′ )d𝑟𝑒′

=

∫ ∞

0

exp

[−𝜆𝑙𝜋𝑟

2𝑒′𝛽

2/𝛼𝑒 Γ

(1 − 2

𝛼

)Γ(

1 +2

𝛼

)]⋅2𝜆𝑒𝜋𝑟𝑒′ exp(−𝜆𝑒𝜋𝑟

2𝑒′ )d𝑟𝑒′ . (17)

The lower bound in (13) is then obtained by directly evaluatingthe integration in (17). ■

Note that the authors in [29] used the same boundingtechniques to derive analytical bounds on the probability ofconnectivity in a different network scenario and numericallystudied the accuracy of the derived bounds. From the numer-ical illustration in [29, Fig. 5], we know that the upper boundPUBso in (12) gives an accurate approximation of the exact

secrecy outage probability over the entire range of Pso ∈ [0, 1],while the lower bound PLB

so in (13) is usually very differentfrom the exact value of Pso. Moreover, both PUB

so and PLBso

are asymptotically tight in the low probability regime. To seethis, we consider PUB

so ≈ 0 and PLBso ≈ 0, in which case the

bounds in (12) and (13) can be approximated by

PUBso ≈ 𝜆𝑒

𝜆𝑙𝛽2/𝛼𝑒 Γ

(1 − 2

𝛼

)Γ(

1 + 2𝛼

) ≈ PLBso . (18)

Hence, both PUBso and PLB

so approach the exact value of Pso

in the low probability regime.Recall that the goal here is to determine the value of 𝑅𝑒

from the secrecy outage constraint of Pso = 𝜖. Using the upperbound on the secrecy outage probability in (12), the value of𝑅𝑒 that guarantees the required security level can be found as

𝑅𝑒 = log2

(1+

[𝜆𝑙

𝜆𝑒Γ(

1− 2

𝛼

)Γ(

1+2

𝛼

)ln

1

1−𝜖

]−𝛼2

). (19)

It is clear that a lower secrecy outage probability (i.e., a highersecurity level) requires a higher 𝑅𝑒.

Having 𝑅𝑡 in (8) and 𝑅𝑒 in (19), a lower bound on thesecrecy transmission capacity is obtained as 𝜏LB(𝑟) = (𝑅𝑡 −𝑅𝑒)(1 − 𝜎)𝜆𝑙. Its expression is presented in the followingtheorem:

Theorem 1: A lower bound on the secrecy transmissioncapacity with a connection outage constraint of 𝜎 and asecrecy outage constraint of 𝜖 is given by

𝜏LB(𝑟) = (1 − 𝜎)𝜆𝑙

⋅ log2

⎛⎜⎝ 1 +

[ln 1

1−𝜎

𝜆𝑙𝜋𝑟2Γ(1− 2𝛼 )Γ(1+ 2

𝛼 )

]𝛼2

1+[𝜆𝑙

𝜆𝑒Γ(

1− 2𝛼

)Γ(

1+ 2𝛼

)ln 1

1−𝜖

]−𝛼2

⎞⎟⎠ .

(20)

From our discussion on the accuracy of PUBso , we know

that the lower bound on the secrecy transmission capacity in(20) is generally accurate for any values of 𝜎 and 𝜖, andis asymptotically tight as 𝜖 → 0. Therefore, we will forsimplicity refer to 𝜏LB(𝑟) in (20) as the secrecy transmissioncapacity in the rest of this paper. It is clear from (20) that𝜏LB(𝑟) reduces as 𝜖 decreases. The reduction in 𝜏LB(𝑟) as 𝜖decreases can be viewed as the throughput cost of improvingphysical layer security.

In practical network design, the connection outage con-straint and the spatial transmission intensity3 may be underthe control of the system designer. The derived closed-formcharacterization of the secrecy transmission capacity allowsthe designer to optimize these system parameters to maximizethe throughput of secure transmissions with a target securitylevel.

A. Existence of Positive Secrecy Transmission Capacity

A fundamental question to ask is the condition underwhich positive secrecy transmission capacity exists. From theexpression in (20), one can find this condition by solving𝜏LB(𝑟) > 0.

Corollary 1: The condition for positive secrecy transmis-sion capacity is given by

ln1

1 − 𝜎ln

1

1 − 𝜖> 𝜋𝑟2𝜆𝑒. (21)

In other words, positive secrecy transmission capacity isachieved if the average number of eavesdroppers within adistance 𝑟 from the transmitter (i.e., having shorter distancesthan the intended receiver) is less than ln 1

1−𝜎 ln 11−𝜖 .

Remark 1: The condition in (21) clearly gives a trade-offbetween the QoS and the security level of a network: The QoSneeds to be compromised (i.e., allowing a larger value of 𝜎)in order to achieve a higher security level (i.e., a smaller valueof 𝜖). Therefore, a moderate connection outage probability isusually desirable for highly secure networks. Furthermore, the

3In networks employing an Aloha protocol, the spatial transmission inten-sity equals the density of potential transmitters multiplied by the probabilityof transmission. In this case, the system designer may control the probabilityof transmission to vary the spatial transmission intensity.

ZHOU et al.: ON THE THROUGHPUT COST OF PHYSICAL LAYER SECURITY IN DECENTRALIZED WIRELESS NETWORKS 2769

𝜏LB(𝑟) = (1 − 𝜎)𝜆𝑙 log2

⎛⎜⎝1 +

[ln 1

1−𝜎

𝜆𝑙𝜋𝑟2Γ(1− 2𝛼 )Γ(1+ 2

𝛼 )

]𝛼2 −

[𝜆𝑙

𝜆𝑒Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)ln 1

1−𝜖

]−𝛼2

1 +[𝜆𝑙

𝜆𝑒Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)ln 1

1−𝜖

]−𝛼2

⎞⎟⎠

≈ (1 − 𝜎)𝜆𝑙

ln 2

[ln 1

1−𝜎

𝜆𝑙𝜋𝑟2Γ(1− 2𝛼 )Γ(1+ 2

𝛼 )

]𝛼2 −

[𝜆𝑙

𝜆𝑒Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)ln 1

1−𝜖

]−𝛼2

1 +[𝜆𝑙

𝜆𝑒Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)ln 1

1−𝜖

]−𝛼2

=1−𝜎

ln 2

[Γ(

1− 2

𝛼

)Γ(

1+2

𝛼

)]−𝛼2

⎡⎣( ln 1

1−𝜎

𝜋𝑟2

)𝛼2

−(

ln 11−𝜖

𝜆𝑒

)−𝛼2

⎤⎦ 1

𝜆𝛼2 −1

𝑙 +[Γ(

1− 2𝛼

)Γ(

1+ 2𝛼

)ln 1

1−𝜖

𝜆𝑒

]−𝛼2

𝜆−1𝑙

. (23)

feasible range of 𝜎 can be found from (21) as

𝜎 ∈(

1 − exp

[−𝜋𝑟2𝜆𝑒

ln 11−𝜖

], 1

). (22)

Remark 2: The condition in (21) does not depend onthe spatial transmission intensity 𝜆𝑙. That is to say, positivesecrecy transmission capacity cannot be achieved simply bybringing in additional legitimate users or deactivating existinglegitimate users, if the connection outage and secrecy outageperformances of the network do not meet the condition in(21). Once this condition is met and the network is operatingwith some positive secrecy transmission capacity, there existsan optimal value of 𝜆𝑙 which can be found numerically using(20). To obtain some analytical insights into the optimal 𝜆𝑙,we consider the low secrecy transmission capacity regime byletting ln 1

1−𝜎 ln 11−𝜖 ≈ 𝜋𝑟2𝜆𝑒 which implies 𝜏LB(𝑟) ≈ 0. We

can approximate (20) as (23) shown on the top of this page.Assuming 𝜏LB(𝑟) in (23) is positive, the optimal value of 𝜆𝑙

that maximizes 𝜏LB(𝑟) is given by

𝜆opt𝑙 =

( 2

𝛼− 2

) 2𝛼 𝜆𝑒

Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)ln 1

1−𝜖

. (24)

From (24), we see that the optimal spatial transmission inten-sity increases as the required security level increases (i.e., as 𝜖decreases). In addition, the optimal spatial transmission inten-sity is usually much higher than the density of eavesdroppersfor highly secure networks. For example, 𝜆opt

𝑙 /𝜆𝑒 ≈ 63 for𝜖 = 0.01 and 𝛼 = 4. Although these observations are made inthe regime of arbitrarily low secrecy transmission capacity, aswe will see in Section V, they are also valid for more generalscenarios.

B. Optimal Connection Outage Probability in Sparse Net-works

When the system designer has control over the connectionoutage constraint, the expression of secrecy transmission ca-pacity in (20) can be used to numerically find the value of𝜎 that maximizes 𝜏LB(𝑟). Here, we present a closed-formsolution of the optimal connection outage probability in sparsenetworks, i.e.,𝜆𝑙𝜋𝑟

2 ≪ 1. Note that our analysis is basedon the assumption of interference-limited networks, which isvalid if the transmit power of the legitimate users is sufficientlyhigh such that the receiver noise is much weaker than theaggregate interference.

From the discussion in Subsection III-A, we know that thevalue of 𝜎 should not be chosen very close to 0. When thenetwork is sparse, i.e.,𝜆𝑙𝜋𝑟

2 ≪ 1, the secrecy transmissioncapacity in (20) can be approximated as

𝜏LB(𝑟) ≈ (1 − 𝜎)𝜆𝑙

⋅ log2

⎛⎜⎝

[ln 1

1−𝜎

𝜆𝑙𝜋𝑟2Γ(1− 2𝛼 )Γ(1+ 2

𝛼 )

]𝛼2

1+[𝜆𝑙

𝜆𝑒Γ(

1− 2𝛼

)Γ(

1+ 2𝛼

)ln 1

1−𝜖

]−𝛼2

⎞⎟⎠

(25)

= (1 − 𝜎)𝜆𝑙 log2

([𝜅 ln

1

1 − 𝜎

]𝛼2

), (26)

where we have assumed in (25) that the path loss exponent 𝛼is not close to 2 (which happens in most outdoor scenarios)and the connection outage probability is not close to 0, and

𝜅 =

(1 +

[𝜆𝑙

𝜆𝑒Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

)ln 1

1−𝜖

]−𝛼2

)− 2𝛼

𝜆𝑙𝜋𝑟2Γ(

1 − 2𝛼

)Γ(

1 + 2𝛼

) . (27)

The optimal connection outage probability that maximizesthe secrecy transmission capacity in (26) is given by

𝜎opt = 1 − 1

exp[

1W0(𝜅)

] , (28)

where W0(⋅) is the real-valued principal branch of Lambert’sW function. This result is obtained by directly taking thederivative of 𝜏LB(𝑟) in (26) w.r.t.𝜎 and solving for the root.Furthermore, one can show that the optimal connection outageprobability increases when a higher security level (i.e., a lower𝜖) is required.

IV. SECRECY GUARD ZONE

In this section, we consider simple protocols for improv-ing the secrecy transmission capacity. We assume that thelegitimate transmitters are able to detect the existence ofeavesdroppers within a finite range. We model this rangeas a disk with radius 𝐷 centered at each transmitter andcall it the secrecy guard zone. Transmission of confidentialmessages only happens when there is no eavesdropper insidethe secrecy guard zone. As we are concerned with decentral-ized networks, it is assumed that each transmitter individually

2770 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 8, AUGUST 2011

Fig. 2. An example of a part of a network snapshot with a secrecy guardzone around each transmitter. Transmitters T0, T1, and T2 do not findany eavesdropper inside their individual guard zone, and hence, transmitconfidential messages to their intended receivers. However, transmitter T3

detects an eavesdropper, E2, insides its guard zone. If the non-cooperativeprotocol is used, T3 remains silent. If the cooperative protocol is used, T3

transmits artificial noise.

decides whether or not to transmit based on the existence ofeavesdroppers inside its own guard zone.

The general idea of guard zone is not new and has beenapplied in wireless ad hoc networks without or with securityconsiderations: In [30], the authors used a guard zone aroundeach receiver such that the receiver is active when there is nointerferers inside the guard zone. The authors in [11] and [16]studied the secure connectivity and secrecy capacity scalinglaw of ad hoc networks in the presence of eavesdroppers,respectively, and applied a secrecy guard zone around eachlegitimate node. In this work, we apply a secrecy guard zonearound each legitimate transmitter and consider the followingtwo transmission protocols:

1) Non-Cooperative Transmitters: The transmitter re-mains silent when eavesdropper(s) are found inside itssecrecy guard zone.

2) Cooperative Transmitters: The transmitter producesartificial noise when eavesdropper(s) are found insideits secrecy guard zone.

The idea of using artificial noise for secrecy was firstproposed for multi-antenna transmissions in [31], which isalso related to the idea of cooperative jamming studied in [6,9, 18, 32, 33]. An example of a network snapshot with secrecyguard zones is shown in Fig. 2. In the following, we study thesecrecy transmission capacity with each transmission protocol.

A. Secrecy Guard Zone with Non-Cooperative Transmitters

The set of actual transmitter locations, denoted as Φ𝑙′ , hasdensity of

𝜆′𝑙 = 𝜆𝑙 exp[−𝜋𝜆𝑒𝐷

2], (29)

where the exponential term in (29) is the probability of noeavesdropper located inside the secrecy guard zone of an arbi-trary transmitter. With the secrecy guard zone, the distributionof the actual transmitters does not still follow a homogeneousPPP. The non-homogeneous nature resulted from the intro-duction of guard zone was discussed in [30]. In particular, theauthors in [30] applied standard Poisson tests to show thatthe distribution of the actual transmitters can still be well-approximated by a homogeneous PPP outside ℬ(𝑏,𝐷) fromthe viewpoint of a receiver at location 𝑏, where the notationℬ(𝑏,𝐷) stands for a disk of radius 𝐷 centered at 𝑏. Based onthis result, we apply the following two approximations: Fromthe viewpoint of eavesdropper 𝑒, the actual transmitter Φ𝑙′

follows a homogeneous PPP with density 𝜆′𝑙 outside ℬ(𝑒,𝐷).

From the viewpoint of any legitimate receiver, the actualtransmitter Φ𝑙′ follows a homogeneous PPP with density 𝜆′

𝑙.4

For the typical receiver, the connection outage5 probabilityis given by

Pco = 1−exp

[−𝜆′

𝑙𝜋𝑟2𝛽

2/𝛼𝑡 Γ

(1− 2

𝛼

)Γ(

1+2

𝛼

)]. (30)

With the connection outage constraint Pco = 𝜎 and 𝛽𝑡 =2𝑅𝑡 − 1, the transmission rate 𝑅𝑡 is found as

𝑅𝑡 = log2

⎛⎝1+

[ln 1

1−𝜎

𝜆′𝑙𝜋𝑟

2Γ(

1− 2𝛼

)Γ(

1+ 2𝛼

)]𝛼

2

⎞⎠ . (31)

From the viewpoint of the typical transmitter located atthe origin 𝑜, the eavesdroppers Φ𝑒 follows a homogeneousPPP with density 𝜆𝑒 outside ℬ(𝑜,𝐷). Similar to the proof ofLemma 1, an upper bound on the secrecy outage probability isobtained by using the generating functional of Φ𝑒 and applyingJensen’s inequality as

PUBso = 1 − exp

[− 𝜆𝑒

⋅∫ℝ2∖ℬ(𝑜,𝐷)

ℙ

( 𝑆𝑒∣𝑋𝑒∣−𝛼∑𝑙∈Φ𝑙′

𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼> 𝛽𝑒

)d𝑒

]

= 1 − exp

[−𝜆𝑒

∫ℝ2∖ℬ(𝑜,𝐷)

𝔼𝑍

{exp[−𝛽𝑒∣𝑋𝑒∣𝛼𝑍]

}d𝑒

]

= 1 − exp

[−2𝜋𝜆𝑒

∫ ∞

𝐷

ℒ𝑍(𝛽𝑒𝑟𝛼𝑒 )𝑟𝑒d𝑟𝑒

], (32)

where 𝑍 =∑

𝑙∈Φ𝑙′𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼 is the sum of interference

power at eavesdropper 𝑒 and ℒ𝑍(⋅) denotes the Laplacetransform of 𝑍 . Note that we have assumed that Φ𝑙′ is ahomogeneous PPP outside ℬ(𝑒,𝐷) from the viewpoint of

4Note that the second approximation usually underestimates the interfer-ence power at the typical receiver, since the potential interferers near thetypical (active) transmitter is more likely to be active than the ones far awayfrom the typical transmitter. However, this underestimation is marginal aslong as 𝜆′

𝑙 is reasonably close to 𝜆𝑙, such as the scenarios to be consideredin Fig. 5.

5The connection outage event is defined as the transmitted message beingundecodable at the intended receiver. Hence, it does not include the event ofno transmission due to the existence of eavesdropper(s) inside the guard zone.A similar note applies to the secrecy outage event. The effect of transmissionprobability on the secrecy transmission capacity is incorporated in 𝜆′

𝑙 .

ZHOU et al.: ON THE THROUGHPUT COST OF PHYSICAL LAYER SECURITY IN DECENTRALIZED WIRELESS NETWORKS 2771

eavesdropper 𝑒. Hence, ℒ𝑍(⋅) is given by [26]

ℒ𝑍(𝑥) = exp

[𝜋𝜆′

𝑙

(𝐷2

𝔼𝑆

{1 − exp[−𝑥𝑆𝐷−𝛼]

}

−𝑥2/𝛼𝔼𝑆

{𝑆2/𝛼

}Γ(

1 − 2

𝛼

)

+𝑥2/𝛼𝔼𝑆

{𝑆2/𝛼Γ

(1 − 2

𝛼, 𝑥𝑆𝐷−𝛼

)})],

(33)

where 𝑆 is an exponentially distributed random variable withunit mean and Γ(⋅, ⋅) is the upper incomplete Gamma function.Using integral identities from [34] to evaluate the expectationsin (33), we have

ℒ𝑍(𝑥) = exp

[𝜋𝜆′

𝑙

(𝑥𝐷2−𝛼

1+𝑥𝐷−𝛼−𝑥2/𝛼Γ

(1+

2

𝛼

)Γ(

1− 2

𝛼

)

+𝑥𝐷2−𝛼

(1− 2𝛼 )(1+𝑥𝐷−𝛼)2

2F1

(1, 2; 2+

2

𝛼;

1

1+𝑥𝐷−𝛼

))],

(34)

where 2F1(⋅) is the Gauss hypergeometric function.Substituting (34) into (32), PUB

so is expressed in an integralform, hence, 𝑅𝑒 can be solved numerically with the secrecyoutage constraint PUB

so = 𝜖 and 𝛽𝑒 = 2𝑅𝑒 − 1. A lower boundon the secrecy transmission capacity is found as

𝜏LB(𝑟) = (𝑅𝑡 −𝑅𝑒)(1 − 𝜎)𝜆′𝑙. (35)

B. Secrecy Guard Zone with Cooperative Transmitters

In this scenario, the legitimate transmitters cooperative witheach other. When eavesdropper(s) are found inside the secrecyguard zone, the transmitter produces artificial noise to helpmasking the confidential message transmissions from others.The artificial noise is assumed to be statistically identical tothe confidential messages and hence, it cannot be distinguishedfrom message transmissions by the eavesdroppers. It is notedthat this cooperative protocol is entirely distributive as itdoes not require any coordination between the legitimatetransmitters.

For any legitimate receiver or eavesdropper, the set ofinterferers is still Φ𝑙 with density 𝜆𝑙. On the other hand, theset of actual transmitters is Φ𝑙′ with density 𝜆′

𝑙 given by

𝜆′𝑙 = 𝜆𝑙 exp[−𝜋𝜆𝑒𝐷

2]. (36)

Since the interferers remain the same as if no secrecy guardzone is applied, the connection outage probability Pco and thetransmission rate 𝑅𝑡 are still given by (7) and (8), respectively.

From the viewpoint of the typical transmitter located at theorigin 𝑜, the eavesdroppers Φ𝑒 follows a homogeneous PPPwith density 𝜆𝑒 outside ℬ(𝑜,𝐷). Again, an upper bound onthe secrecy outage probability is found using the generating

functional of Φ𝑒 and applying Jensen’s inequality as

PUBso

= 1−exp

[−𝜆𝑒

∫ℝ2∖ℬ(𝑜,𝐷)

ℙ

( 𝑆𝑒∣𝑋𝑒∣−𝛼∑𝑙∈Φ𝑙

𝑆𝑙𝑒∣𝑋𝑙𝑒∣−𝛼>𝛽𝑒

)d𝑒

]

= 1−exp

⎡⎣−𝜆𝑒 exp

[−𝜆𝑙𝜋𝛽

2/𝛼𝑒 Γ

(1− 2

𝛼

)Γ(

1+ 2𝛼

)𝐷2]

𝜆𝑙𝛽2/𝛼𝑒 Γ

(1− 2

𝛼

)Γ(

1+ 2𝛼

)⎤⎦ .

(37)

With the secrecy outage constraint of PUBso = 𝜖, we find 𝑅𝑒

as

𝑅𝑒 = log2

⎛⎜⎜⎝1 +

⎡⎢⎣ W0

(𝜆𝑒𝜋𝐷

2[

ln 11−𝜖

]−1)𝜆𝑙𝜋𝐷2Γ

(1 − 2

𝛼

)Γ(

1 + 2𝛼

)⎤⎥⎦

𝛼2

⎞⎟⎟⎠ . (38)

Theorem 2: A lower bound on the secrecy transmissioncapacity for networks having cooperative transmitters withsecrecy guard zones is given by

𝜏LB(𝑟) = (𝑅𝑡 −𝑅𝑒)(1 − 𝜎)𝜆′𝑙

= (1 − 𝜎)𝜆𝑙 exp[−𝜋𝜆𝑒𝐷2]

⋅ log2

⎛⎜⎝ 1 +

[ln 1

1−𝜎

𝜆𝑙𝜋𝑟2Γ(1− 2𝛼 )Γ(1+ 2

𝛼 )

]𝛼2

1 +[W0(𝜆𝑒𝜋𝐷2[ln 1

1−𝜖 ]−1)

𝜆𝑙𝜋𝐷2Γ(1− 2𝛼 )Γ(1+ 2

𝛼 )

]𝛼2

⎞⎟⎠ . (39)

From the closed-form expression of 𝜏LB(𝑟) in (39), we canderive the condition for positive secrecy transmission capacityby solving 𝜏LB(𝑟) > 0.

Corollary 2: The condition for positive secrecy transmis-sion capacity is given by[ 1

1 − 𝜎

](𝐷𝑟 )2

ln1

1 − 𝜎ln

1

1 − 𝜖> 𝜋𝑟2𝜆𝑒. (40)

When the system designer has control over the connectionoutage constraint and the guard zone size, positive secrecytransmission capacity can be achieved by carefully choosingthe values of 𝜎 and/or 𝐷 to meet the condition in (40).Similar to the networks without guard zones, the density ofthe legitimate transmitters has no say for the existence ofpositive secrecy transmission capacity for fixed connectionoutage probability and guard zone size. Comparing (21) with(40), we see that the improvement from having the secrecyguard zone is given by the factor of [ 1

1−𝜎 ](𝐷/𝑟)2 . For a fixed 𝑟,the minimum required value of 𝐷 increases as 𝜎 or 𝜖 decreases(below certain threshold value). Hence, one may also expectthat the optimal value of 𝐷, which maximizes the secrecytransmission capacity in (39), increases as 𝜎 or 𝜖 decreases.

To further improve the network throughput, the secrecyguard zone could in the future be used in combination withother types of guard zone, such as carrier sense multiple access(CSMA) at the transmitters [35] and the interference guardzone at the receivers [30].

V. NUMERICAL RESULTS AND DISCUSSION

In this section, we present numerical results on the secrecytransmission capacity. We first show the interplay of different

2772 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 8, AUGUST 2011

10−3

10−2

10−1

100

0

0.01

0.02

0.03

0.04

0.05

0.06

Spatial Transmission Intensity, λl

Secr

ecy

Tra

nsm

issi

on C

apac

ity

ε = 0.05

ε = 0.02

ε = 0.01

ε = 1

Fig. 3. The secrecy transmission capacity 𝜏LB(𝑟) in (20) versus the densityof legitimate transmitters 𝜆𝑙. Results are shown for networks with differentsecrecy outage constraints, i.e., 𝜖 = 0.01, 0.02, 0.05, as well as no secrecyconstraint, i.e., 𝜖 = 1. The other system parameters are 𝑟 = 1, 𝛼 = 4,𝜎 = 0.3, and 𝜆𝑒 = 0.001.

system parameters and their effects on the secrecy trans-mission capacity for networks without secrecy guard zones.Fig. 3 shows the secrecy transmission capacity 𝜏LB(𝑟) in(20) versus the spatial transmission intensity 𝜆𝑙 with differentsecurity requirements. Comparing between the four curves,we see that the gap in 𝜏LB(𝑟) between 𝜖 = 1 and 𝜖 = 0.05is relatively small over a wide range of 𝜆𝑙. This suggeststhat the throughput cost of achieving a moderate securityrequirement is relatively low. On the other hand, 𝜏LB(𝑟) dropsdramatically as 𝜖 decreases towards 0. For example, there isa 84% reduction in 𝜏LB(𝑟) for improving the security levelfrom 𝜖 = 0.02 to 𝜖 = 0.01 at 𝜆𝑙 = 0.01. This reflects asignificant increase in the throughput cost of achieving highlysecure networks.

For each curve in Fig. 3, we see that the optimal value of𝜆𝑙 is generally much larger than 𝜆𝑒. This suggests that it isdesirable to have a significantly larger number of legitimatenodes than the number of eavesdroppers in the network, whichcreates a high level of interference to mask the confidentialmessage transmissions against eavesdropping. Furthermore,the optimal value of 𝜆𝑙 increases as 𝜖 decreases. For example,the optimal 𝜆𝑙 is 0.04 for 𝜖 = 0.05, while it increases to 0.051for 𝜖 = 0.02 and to 0.068 for 𝜖 = 0.01. Note that the optimalvalue of 𝜆𝑙 computed from (24) is 0.063 for 𝜖 = 0.01, whichis very close to the numerical result.

Fig. 4 shows the secrecy transmission capacity 𝜏LB(𝑟) in(20) versus the connection outage probability 𝜎 with differentsecurity requirements. The feasible range of 𝜎 for positivesecrecy transmission capacity never reaches 0, which agreeswith the result in (22). We see that a moderate connectionoutage probability is desirable for achieving high secrecytransmission capacity. Furthermore, the optimal value of 𝜎increases as 𝜖 reduces. This is because that a larger 𝑅𝑒 isneeded for a stronger security requirement, in which caselarger 𝑅𝑡 and (hence) 𝜎 are desirable for maximizing thesecrecy transmission capacity. For example, the optimal 𝜎 is0.4 for 𝜖 = 0.05 while it increases to 0.5 for 𝜖 = 0.02 andto 0.6 for 𝜖 = 0.01. Note that the optimal value of 𝜎 can be

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Connection Outage Probability, σ

Secr

ecy

Tra

nsm

issi

on C

apac

ity

ε = 1

ε = 0.05

ε = 0.02

ε = 0.01

Fig. 4. The secrecy transmission capacity 𝜏LB(𝑟) in (20) versus theconnection outage probability 𝜎. Results are shown for networks with differentsecrecy outage constraints, i.e., 𝜖 = 0.01, 0.02, 0.05, as well as no secrecyconstraint, i.e., 𝜖 = 1. The other system parameters are 𝑟 = 1, 𝛼 = 4,𝜆𝑙 = 0.01, and 𝜆𝑒 = 0.001.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Secrecy Outage Probability, ε

Secr

ecy

Tra

nsm

issi

on C

apac

ity

cooperativenon−cooperative

D = 0 (i.e. no guard zone)

D = 6

D = 3

Fig. 5. The secrecy transmission capacity 𝜏LB(𝑟) with guard zone in(35) and (39) versus the secrecy outage probability 𝜖. Results are shown fornetworks with different guard zone radii, i.e.,𝐷 = 0, 3, 6. The other systemparameters are 𝑟 = 1, 𝛼 = 4, 𝜎 = 0.3, 𝜆𝑙 = 0.01, and 𝜆𝑒 = 0.001.

accurately computed from the closed-form expression in (28)for sparse networks.

Now, we present the results on the use of a guard zonefor improving the secrecy transmission capacity. Fig. 5 shows𝜏LB(𝑟) in (35) and (39) versus the secrecy outage probability 𝜖for both the non-cooperative and cooperative protocols.6 Thisfigure clearly shows the remarkable benefit of guard zone fornetworks with high security requirements. For example, thesecrecy transmission capacity at 𝜖 = 0.01 increases from 0.003at 𝐷 = 0 (i.e., no guard zone) to 0.018 with non-cooperativeprotocol and 0.021 with cooperative protocol at 𝐷 = 3. On theother hand, the benefit of guard zone reduces as the securityrequirement reduces. We also see that the cooperative protocoloutperforms the non-cooperative protocol and the difference

6Although we have chosen 𝑟 = 1 in Fig. 5, the benefit of using a secrecyguard zone demonstrated in Fig. 5 is also observed for other values of 𝑟 whichcan be either less than or greater than 𝐷 (plots omitted for brevity).

ZHOU et al.: ON THE THROUGHPUT COST OF PHYSICAL LAYER SECURITY IN DECENTRALIZED WIRELESS NETWORKS 2773

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Secrecy Outage Probability, ε

Con

nect

ion

Out

age

Pro

babi

lity,

σ

D = 0D = 3D = 6

Region of Positive SecrecyTransmission Capacity

Fig. 6. The region of positive secrecy transmission capacity with andwithout secrecy guard zone. The case of cooperative transmitters is shownwith different guard zone radii, i.e.,𝐷 = 0, 3, 6. The curves are plottedbased on the relationship between the connection outage probability 𝜎 andthe secrecy outage probability 𝜖 given in (40). The other system parametersare 𝑟 = 1 and 𝜆𝑒 = 0.001.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8

9

10

Secrecy Outage Probability, ε

Opt

imal

Gua

rd Z

one

Rad

ius

λ

l = 0.01

λl = 0.05

σ = 0.1

σ = 0.3

Fig. 7. The optimal secrecy guard zone radius versus the secrecy outageprobability 𝜖 for the case of cooperative transmitters. Results are shown fornetworks with different connection outage constraints, i.e.,𝜎 = 0.1, 0.3, anddifferent densities of legitimate transmitters, i.e.,𝜆𝑙 = 0.01, 0.05. The othersystem parameters are 𝑟 = 1, 𝛼 = 4, and 𝜆𝑒 = 0.001.

in secrecy transmission capacity is significant for networkswith high security requirement. For example, the increase in𝜏LB(𝑟) from the non-cooperative case to the cooperative caseat 𝜖 = 0.01 is 17% when 𝐷 = 3 and 14% when 𝐷 = 6.

The feasible regions of the connection outage probability𝜎 and the secrecy outage probability 𝜖 for positive secrecytransmission capacity are illustrated in Fig. 6. The case ofcooperative transmitters is considered when the guard zoneis used. In general, it is impossible to have arbitrarily lowoutage probabilities while still operating at some positivesecrecy transmission capacity. Nevertheless, the use of a guardzone greatly enlarges the feasible ranges of both outageprobabilities.

Fig. 7 shows the optimal guard zone radius for cooperativetransmitters. We see that the optimal value of 𝐷 reduces asthe acceptable secrecy outage probability 𝜖 increases and it

reaches zero at moderate to high values of 𝜖. We can also seethe dependence of the optimal 𝐷 on 𝜎 and 𝜆𝑙. The generaltrend is that the optimal 𝐷 reduces as 𝜎 or 𝜆𝑙 increases.The dependence of the optimal 𝐷 on 𝜎 agrees with ourearlier observation from (40). The dependence on 𝜆𝑙 canbe understood as follows: As the interference level (i.e.,𝜆𝑙)increases, the signal power received at the eavesdroppers isallowed to be higher for maintaining the same SIR. This inturn allows us to reduce the guard zone radius, which increasesthe spatial intensity of message transmissions. In practice, alegitimate transmitter may only be able to detect the existenceof eavesdroppers in its vicinity, hence, the guard zone isusually small. Nevertheless, we have seen from Fig. 5 that asignificant improvement in the secrecy transmission capacitycan be achieved even with a small guard zone. Furthermore,by allowing a moderate connection outage probability, it ispossible for the network to operate with the optimal guardzone radius which is within the maximum detection range ofthe transmitters.

VI. CONCLUSIONS

In this work, we defined a performance metric named thesecrecy transmission capacity, which was used to study theimpact of physical layer security requirements on the through-put of large-scale decentralized wireless networks. Using toolsand existing results from stochastic geometry, the secrecytransmission capacity can usually be characterized in simpleanalytical forms, as shown in this paper for Rayleigh fadingchannels. One important finding is that the throughput cost ofachieving a moderate security level is relatively low, while itbecomes very expensive to realize a highly secure network.In addition, we showed that the application of secrecy guardzone with artificial noise is a simple technique that can beused to dramatically reduce the throughput cost of achievinghighly secure networks.

This model of secrecy transmission capacity can be ex-tended to analyze and design networks with other transmissiontechniques, medium access control protocols, and eavesdrop-ping strategies in future work. Similar to other transmissioncapacity formulations, the main limitation of this model isthat it only considers single-hop transmissions, while thecommunication between an arbitrary source-destination pairusually requires multiple hops. End-to-end throughput analysisof wireless networks with physical layer security requirementsis still an open problem. Another limitation of the currentmodel is the homogeneous Poisson distribution of nodes. Theimpact of eavesdropper distribution on secrecy throughput isan interesting problem to investigate.

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Xiangyun Zhou (S’08, M’11) received the B.E.(hons.) degree in electronics and telecommunica-tions engineering and the Ph.D. degree in telecom-munications engineering from the Australian Na-tional University, Australia, in 2007 and 2010, re-spectively. He is currently a postdoctoral researchfellow at UNIK - University Graduate Center, Uni-versity of Oslo, Norway. His research interests arein the fields of wireless communications and signalprocessing, including MIMO systems, ad hoc net-works, relay and cooperative communications, and

physical layer security. He is a recipient of the Best Paper Award at the 2011IEEE International Conference on Communications.

Radha Krishna Ganti (M’10) is a Postdoctoralresearcher in the Wireless Networking and Com-munications Group at UT Austin. He received hisB.Tech. and M.Tech. in EE from Indian Instituteof Technology, Madras, and Masters in AppliedMath and Ph.D. in EE from University of NotreDame in 2009. His doctoral work focused on thespatial analysis of interference networks using toolsfrom stochastic geometry. He is co-author of themonograph Interference in Large Wireless Networks.

Jeffrey Andrews (S’98, M’02, SM’06) receivedthe B.S. in Engineering with High Distinction fromHarvey Mudd College in 1995, and the M.S. andPh.D. in Electrical Engineering from Stanford Uni-versity in 1999 and 2002, respectively. He is anAssociate Professor in the Department of Electricaland Computer Engineering at the University ofTexas at Austin, and the Director of the WirelessNetworking and Communications Group (WNCG),a research center comprising 17 faculty and 13industrial affiliates. He developed Code Division

Multiple Access systems at Qualcomm from 1995-97, and has consulted forentities including the WiMAX Forum, Microsoft, Apple, Clearwire, Palm,Sprint, ADC, and NASA.

Dr. Andrews is co-author of two books, Fundamentals of WiMAX (Prentice-Hall, 2007) and Fundamentals of LTE (Prentice-Hall, 2010), and holds theEarl and Margaret Brasfield Endowed Fellowship in Engineering at UTAustin, where he received the ECE department’s first annual High Gain awardfor excellence in research. He is a Senior Member of the IEEE, served as anassociate editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICA-TIONS from 2004-08, was the Chair of the 2010 IEEE Communication TheoryWorkshop, and is the Technical Program co-Chair of ICC 2012 (Comm.Theory Symposium) and Globecom 2014. He has also been a guest editor fortwo recent IEEE JSAC special issues on stochastic geometry and femtocellnetworks.

Dr. Andrews received the National Science Foundation CAREER award in2007 and is the Principal Investigator of a 9 university team of 12 faculty inDARPA’s Information Theory for Mobile Ad Hoc Networks program. He hasbeen co-author of five best paper award recipients, two at Globecom (2006and 2009) Asilomar (2008), the 2010 IEEE Communications Society BestTutorial Paper Award, and the 2011 Communications Society Heinrich HertzPrize. His research interests are in communication theory, information theory,and stochastic geometry applied to wireless ad hoc and heterogeneous cellularnetworks.

ZHOU et al.: ON THE THROUGHPUT COST OF PHYSICAL LAYER SECURITY IN DECENTRALIZED WIRELESS NETWORKS 2775

Are Hjørungnes works as a Professor at the Fac-ulty of Mathematics and Natural Sciences at theUniversity of Oslo, Norway with office located atUNIK - University Graduate Center. He obtained hisSivilingeniør (M.Sc.) degree (with honors) in 1995from the Department of Telecommunications at theNorwegian Institute of Technology in Trondheim,Norway, and his Doktor ingeniør (Ph.D.) degree in2000 from the Signal Processing Group at the Nor-wegian University of Science and Technology. Hiscurrent main research areas are signal processing,

communications, and wireless networks. He authored the book Complex-Valued Matrix Derivatives: With Applications in Signal Processing andCommunications (Cambridge University Press, 2011).

From August 2000 to December 2000, he worked as a researcher at Tam-pere University of Technology, in Finland, within the Tampere InternationalCenter for Signal Processing. From March 2001 to July 2002, he workedas a postdoctoral fellow at the Federal University of Rio de Janeiro inBrazil, within the Signal Processing Laboratory. From September 2002 toAugust 2003, he worked as a postdoctoral fellow at the Helsinki Universityof Technology in Finland, within the Signal Processing Laboratory. FromSeptember 2003 to August 2004, he was working as a postdoctoral fellow atthe University of Oslo in Norway, at the Department of Informatics, within

the Digital Signal Processing and Image Analysis Group.He has held visiting appointments at the Image and Signal Processing

Laboratory at the University of California, Santa Barbara, the Signal Pro-cessing Laboratory of the Federal University of Rio de Janeiro, the MobileCommunications Department at Eurecom Institute in France, the Universityof Manitoba in Canada, the Alcatel-Lucent Chair at SUPELEC in France,the Department of Electrical and Computer Engineering at the University ofHouston in USA, the Electrical and Computer Engineering Department atUniversity of California, San Diego, USA, and the Department of ElectricalEngineering, University of Hawai’i at Manoa, USA.

Since March 2007, he has been serving as an Editor for IEEE TRANSAC-TIONS ON WIRELESS COMMUNICATIONS. In 2010 and 2011, he was a GuestEditor for IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING

and IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, in thespecial issues on “Model Order Selection in Signal Processing Systems” and“Cooperative Networking - Challenges and Applications,” He co-authoredthe papers winning the best paper awards at IEEE International Conferenceon Wireless Communications, Networking and Mobile Computing (WiCOM2007), 7th International Symposium on Modeling and Optimization in Mo-bile, Ad Hoc, and Wireless Networks (WiOpt 2009), and 5th InternationalConference on Internet Monitoring and Protection (ICIMP 2010).