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Energy-Efficient Secrecy in Wireless NetworksBased on Random Jamming

Azadeh Sheikholeslami, Majid Ghaderi, Member, IEEE, Hossein Pishro-Nik, Member, IEEE,Dennis Goeckel, Fellow, IEEE

Abstract—Secure energy-efficient routing in the presenceof multiple passive eavesdroppers is considered. Previouswork in this area has considered secure routing assumingprobabilistic or exact knowledge of the location andchannel-state-information (CSI) of each eavesdropper. Inwireless networks, however, the locations and CSIs of pas-sive eavesdroppers are not known, making it challengingto guarantee secrecy for any routing algorithm.

We develop an efficient (in terms of energy consumptionand computational complexity) routing algorithm that doesnot rely on any information about the locations and CSIsof the eavesdroppers, and moreover, guarantees secrecyeven in disadvantaged wireless environments, where mul-tiple eavesdroppers try to eavesdrop each message, areequipped with directional antennas, or can get arbitrarilyclose to the transmitter. The key to achieving this isusing additive random jamming to exploit inherent non-idealities of the eavesdropper’s receiver, which makes theeavesdroppers incapable of recording the messages. Wehave simulated our proposed algorithm and compared itwith existing secrecy routing algorithms in both single-hop and multi-hop networks. Our results indicate thatwhen the uncertainty in the locations of eavesdroppersis high and/or in disadvantaged wireless environments,our algorithm outperforms existing algorithms in termsof energy consumption and secrecy.

I. INTRODUCTION

Information secrecy has traditionally been achievedby cryptography, which is based on assumptions oncurrent and future computational capabilities of the ad-versary. However, there are numerous examples of cryp-tographic schemes being broken that were supposedlysecure [1]. This motivates the consideration of physicallayer schemes which are based on information-theoreticsecrecy [2]. In a scenario where an adversary tries toeavesdrop on the main channel between a transmitterand a receiver, Wyner showed that, if the eavesdropper’schannel is degraded with respect to the main channel,a positive secrecy rate can be achieved. This idea waslater extended to Gaussian channels [3], and to the more

This work was supported by the National Foundation under GrantCIF-1421957.

general case of a wiretap channel with a "more noisy"or "less capable" eavesdropper channel [4]. Thus, thekey to obtain information-theoretic secrecy is having anadvantage for the main channel against the eavesdropperchannel. However, such an advantage cannot always beguaranteed. In particular, the locations of eavesdroppersare not known and an eavesdropper might be much closerto the transmitter than the intended receiver. To overcomethis problem, one must design algorithms to obtain therequired advantage for the intended recipient over theeavesdroppers.

The idea of adding artificial noise to the signal bymeans of multiple antennas at the transmitter or somehelper nodes was introduced in [5], [6]. The artificialnoise is placed in the null space of the channel from thetransmitter to the intended recipient and thus does notaffect it. But, it degrades the eavesdropper’s channel withhigh probability. Subsequently, cooperative jamming forphysical layer secrecy has been extensively studied, e.g.[7]–[12]. These works mainly focus on one-hop networksconsisting of one transmitter, one receiver, one eaves-dropper and maybe a few helper nodes that generate theartificial noise. The case of two-hop networks consistingof one transmitter, one receiver, one relay, one eaves-dropper and a few noise generating helper nodes has alsobeen considered extensively in the literature [13]–[16]. Inthe case of multi-hop networks with multiple transmittersand receivers and in the presence of many eavesdroppers,often the asymptotic results for large networks have beeninvestigated [17]–[25].

However, whereas one-hop, two-hop and asymptoti-cally large networks are most amenable to analysis anddo provide insight into wireless network operation, mostad hoc networks in practice operate with a number ofnodes and a number of hops that is between these twoextremes. Hence, the design of algorithms to providesecrecy in networks of arbitrary "moderate" size is ofinterest, which is considered here. We consider a networkwith multiple system nodes where a source node com-municates with a destination node in a multi-hop fashionand in the presence of multiple passive eavesdroppers.We define the cost of communication to be the total

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energy spent by the system nodes to securely and reliablytransmit a message from the source to the destination.Thus, our goal is to find routes that minimize thecost of transmission between the source and destinationnodes. Energy efficiency is an important considerationin designing the routing algorithms, and energy efficientrouting has been extensively studied in the literature, e.g.[26]–[34]. However, only a few works have consideredenergy-aware routing with secrecy considerations [35],[36].

In [35], [36], the authors use a general probabilisticmodel for the location of each eavesdropper, and intro-duce a routing algorithm called SMER (secure minimumenergy routing) which employs cooperative jamming toprovide secrecy at each hop such that the end-to-endsecrecy of the multi-hop source-destination path is guar-anteed. When the density of eavesdroppers is low suchthat there is only one eavesdropper per hop, the locationof the eavesdroppers are known, and the eavesdroppersare restricted to use omni-directional antennas, this ap-proach is promising. However, since we are consideringpassive eavesdroppers, their location and channel-state-information (CSI) are not known to the legitimate nodes.Further, in a disadvantaged wireless environment, manypassive eavesdroppers might try to intercept the messageat each hop, with large uncertainty in the locationsof the eavesdroppers, and the eavesdroppers might getarbitrarily close to the transmitters. In such a situation,the energy consumption of any cooperative jammingapproach including the scheme of [35], [36] can becomevery high. Further, if we plan for the wrong number ofeavesdroppers or do not correctly anticipate the qualityof the eavesdroppers’ channels, the secrecy will becompromised. Hence, in this paper we seek methods thatdo not rely on the quality of eavesdroppers’ channels andtheir locations and can provide secrecy in disadvantagedenvironments at a reasonable cost.

In [37], [38], the authors propose adding randomjamming with large variations and based on an ephemeralkey to the signal to obtain everlasting secrecy in disad-vantaged wireless environments. In contrast to previousmethods based on a key to facilitate secrecy in wirelessnetworks, the work in [37], [38] does not presume thatthe key is kept secret from the eavesdropper indefinitely;rather, the jamming is used to build an advantage forthe intended receiver over the eavesdropper by inhibitingthe eavesdropper’s ability to even record a reasonableversion of the message for later decoding. Here, we usethe approach of [37], [38] in a network setting, where thesource and intermediate relays at each hop add a randomjamming signal to the message to protect it against theeavesdroppers. We design a fast (polynomial time) and

efficient routing algorithm such that the aggregate energyspent to convey the message and to generate the randomjamming signal is minimized. Our contributions in thispaper are:• We develop a routing algorithm that minimizes

the end-to-end energy consumption of sending amessage and jamming signal, while satisfying anend-to-end secrecy requirement.

• Our routing algorithm is independent of the number,location and channel-state-information of the eaves-droppers. It can guarantee secrecy in worst casescenarios: when many eavesdroppers are present,the eavesdroppers can get very close to the trans-mitters, and they use directional antennas pointedat the transmitters.

• We conduct detailed simulations to evaluate the per-formance of our proposed algorithm and compare itwith the algorithm proposed in [35], [36]. Our sim-ulation results show that our algorithm can achievea significant improvement in terms of the averageenergy expended for various network simulationscenarios in disadvantaged wireless environmentsand/or when the uncertainty in the locations of theeavesdroppers is high.

The rest of the paper is organized as follows. SectionII describes the system model, the approach which isused in this paper, and the metric. The analysis of theproblem and the algorithm for minimum energy routingwith secrecy constraints is presented in Section III. InSection IV, the results of numerical examples for var-ious realizations of one-hop and multi-hop systems areprovided, and the comparison of the proposed method toSMER algorithm is presented. Conclusions are discussedin Section V.

II. SYSTEM MODEL AND APPROACH

A. System Model

We consider a wireless network with nodes that aredistributed arbitrarily. A source node generates the mes-sage and conveys it to a destination node in a multi-hop fashion. An H-hop path from the source to thedestination is denoted by Π = 〈l1, . . . , lH〉, where li isthe link that connects two nodes Si and Di along the pathΠ. There are also non-colluding eavesdroppers present inthe network such that the message transmission of eachlink is prone to be overheard by multiple eavesdroppers.We denote the set of eavesdroppers by E . The eavesdrop-pers are assumed to be passive, and thus their locationsand their channel-state informations are not known tothe legitimate nodes. We assume that the system nodesare equipped with omni-directional antennas while the

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eavesdroppers can be equipped with more sophisticateddirectional antennas.

For the channel, we consider transmission in a quasi-static Raleigh fading environment. Let hS,D be thefading coefficient between node S and node D (Thisassumption is relaxed for eavesdroppers’ channels, asdiscussed later.). Without loss of generality, we assumeE[|hS,D|2] = 1. Suppose the transmitter S transmits thesignal xS at power level PS . The signal that the receiverD (analogously, eavesdropper E) receives is:

yD =xShS,D

dα

2

S,D

+ nD

where dS,D is the distance between S and D, α is thepath-loss exponent, and nD ∼ N

(0, σ2

D

)is additive

white Gaussian noise (AWGN) at the receiver D.Because compression of a receiver’s front-end dy-

namic range is the biggest challenge when operatingin the presence of strong jamming, we also considerthe effect of the analog-to-digital converter (A/D) onthe received signal, which consists of the quantizationnoise and the quantizer’s overflow. The quantizationnoise is a result of the limited resolution of the A/D,and the quantizer’s overflow happens when the rangeof the received signal is larger than the span of theA/D. We assume that the quantization noise is uniformlydistributed [39, Section 5]. The resolution of a b-bit A/Dwith full dynamic range [−r, r] is

δ =2r

2b.

Since the power of the received signal at the eavesdrop-per E (analogously, receiver D) is PS |hS,E |2

dαS,E, we set the

range of the A/D as,

r = l

√PS |hS,E |dα/2S,E

,

where l is a constant that maximizes the mutual infor-mation between the transmitted signal and the receivedsignal [40]. Hence, the resolution of the A/D of theeavesdropper E (receiver D) is:

δ =2l√PS |hS,E |

2bdα/2S,E

.

B. Approach: Random Jamming for Secrecy

We use the random jamming scheme of [37], [38] toprovide everlasting secrecy. In this scheme, based on acryptographic key that is shared between the legitimatenodes, a jamming signal with large variation is addedto the transmitted signal. It is assumed that the crypto-graphic key should be kept secret just for the time of

transmission, and can be revealed to the eavesdropperright after transmission without compromising secrecy.The legitimate receiver can use its key to cancel theeffect of the jamming before analog-to-digital-conversion(A/D), while the eavesdropper must record the signal andjamming, and cancel the effect of jamming later fromthe recorded signal (after analog-to-digital-conversion).Hence, the signal that the legitimate receiver receives iswell-matched to its A/D converter. On the other hand,the large variation of the random jamming signal causesoverflow of the eavesdropper’s A/D. The eavesdroppermay enlarge the span of her A/D to prevent overflows;however, it degrades the resolution of its A/D, thusincreasing the A/D noise. In [37], [38] it is shown that,although increasing the span of the A/D causes theeavesdropper to suffer from more quantization noise, theoverflows are more harmful, and thus the best strategythat the eavesdropper can employ is to enlarge the spanof its A/D such that it captures all of the signal and thusno overflow occurs.

The random jamming signal J that the transmitteradds to its signal follows a uniform distribution with2K jamming levels. Hence, K bits of the cryptographickey to generate each jamming symbol are needed. Thedistance between two consecutive jamming levels is2l√PS . Thus, the average energy that is spent on the

random jamming signal is,

PJ = E[J2]

=1

2K

2K−1∑j=0

(2l√PSj

)2

=4l2PS

2K

2K−1∑j=0

j2

=4l2PS

2K× 23K+1 − 3× 22K + 2K

6

=2l2(22K+1 − 3× 2K + 1

)3

PS

= βPS (1)

where β is a constant that depends on K.Suppose that the eavesdropper uses a bE-bit A/D.

Since the power of the signal at the eavesdropper’sreceiver is PS |hS,E |2

dαS,E, and considering the automatic-gain-

control of the eavesdropper’s receiver, the resolution ofthe eavesdropper’s A/D before jamming is:

δE =2l√PS |hS,E |2

2bEdα/2S,E

(2)

Now suppose that the transmitter adds the jamming toits signal. Since the eavesdropper does not know the key,

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it should enlarge the span of its A/D to capture all thesignal plus jamming. The maximum amplitude of thesignal plus jamming can be written as,√PS |hS,E |2

dα/2S,E

+(2K−1)

√PS |hS,E |2

dα/2S,E

= 2K√PS |hS,E |2

dα/2S,E

Thus, the resolution of eavesdropper’s A/D is:

δ′E =2l√PS |hS,E |2

2bEdα/2S,E

× 2K =2l√PS |hS,E |2

2bE−Kdα/2S,E

(3)

The random jamming scheme of [37], [38] relies onthe limited resolution of the eavesdropper’s A/D. Hence,we should assume that the legitimate nodes either knowa bound on the quality of the eavesdroppers’ A/Ds, orplan for the case that all eavesdroppers use the bestA/D technology available at the time. The realizationof this assumption is facilitated by the fact that A/Dtechnology progresses very slowly1. Hence, throughoutthis paper we assume that the resolution of the A/D ofeach eavesdropper is equal to or less than bE bits.

C. Jamming Cancellation at the Legitimate Receiver

Nearly all techniques that exploit jamming for secrecyignore the effects of channel estimation error (e.g. [5]–[12], [35], [37], [38]), yet it is important since in realsystems the jamming power is high, and thus the residualjamming due to imperfections in channel estimationcan be considerable. Note that from [41], [42], thechannel estimation error might be very small, but, sincewe have high-power jammers, the residual interferenceis still important and can have an impact on systemperformance. Hence, we consider the residual jammingat the receiver due to errors in the channel estimates.Given a pilot-based approach for channel estimation,the channel estimate is conditionally Gaussian, wherethe mean of this Gaussian distribution is the minimummean-squared estimate (MMSE) channel estimate. Theestimation error of this MMSE estimate is a zero meanGaussian random variable N (0, σ2

J) with variance (e.g.see [43]),

σ2J = θ2PJ |hS,D|2

dαS,D

where θ is a constant coefficient describing the normal-ized prediction error of the legitimate receiver. Hence,the estimation error acts like zero mean additive whiteGaussian noise with variance proportional to the receivedjamming power.

1For a complete discussion on this see [38, Section V].

D. Metric

Since the quantization noise is uniformly distributed[39, Section 5], the derivation of the capacity of the chan-nel between transmitter and receiver, and the channelbetween transmitter and eavesdropper, is not straightfor-ward. Thus, we apply an upper-bound and a lower-boundof the capacity of a channel with independent additivenoise as described in [44] and [45]. Suppose that theresolution of the A/D of receiver D is δD. The capacityof the channel between the transmitter S and the receiverD conditioned on the fading coefficient can be lowerbounded as [38]:

CS,D(|hS,D|2

)≥log

PS |hS,D|2dαS,D

+ σ2J + σ2

D +δ2D12

σ2J + σ2

D +δ2D12

,=log

PS |hS,D|2dαS,D

+θ2PJ |hS,D|2

dαS,D+ σ2

D +δ2D12

θ2PJ |hS,D|2dαS,D

+ σ2D +

δ2D12

, (4)

and the capacity of the channel between the transmitterS and the eavesdropper E can be upper bounded as [38]:

CS,E(|hS,E |2

)≤ log

PS |hS,E |2dαS,E

+ σ2E + δ

′2E

12

σ2E + δ

′2E

2πe

. (5)

In order to guarantee proper signal reception at thelegitimate receiver, the capacity of the main channelshould be greater than a predetermined threshold. Letus define,

γD =

PS |hS,D|2dαS,D

+ θ2PJ |hS,D|2dαS,D

+ σ2D + δ2D

12

θ2PJ |hS,D|2dαS,D

+ σ2D + δ2D

12

.

Hence, the communication between source and destina-tion is reliable if,

γD ≥ γ∗D. (6)

We define the average outage probability between S andD as,

pout = P (γD < γ∗D) .

In order to guarantee secrecy, the capacity of the channelbetween the transmitter and eavesdropper should be lessthan a predetermined threshold. We define,

γE =

PS |hS,E |2dαS,E

+ σ2E + δ

′2E

12

σ2E + δ

′2E

2πe

. (7)

Hence, the communication between source and destina-tion is secure if,

γE ≤ γ∗E . (8)

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We define the average secrecy-outage probability (i.e.eavesdropping probability) as,

peav = P (γE ≥ γ∗E) .

From (6) and (8) we conclude that if reliability andsecrecy constraints are satisfied, the secrecy rate of atleast,

Rs = log(γ∗D)− log(γ∗E), (9)

can be achieved. However as described above, instead ofconsidering a constraint on the secrecy rate, we considerconstraints on the individual success probabilities of thereceiver and the eavesdropper. If we instead put theconstraint on the secrecy rate, for a single secrecy ratemany (γD, γE) would satisfy the constraint. But codesare designed to work on a specific (γD, γE) pair, andthere is no universal wiretap code which is effective forall the pairs (γD, γE) that satisfy (9) [46], [47]. Hence,we consider (6) and (8) as our reliability and secrecyconstraints, respectively.

III. SERJ: SECURE ENERGY-EFFICIENT ROUTING

USING JAMMING

Our goal is to find the optimal path with minimumenergy consumption that connects the source to thedestination:

Π∗ = arg minΠ∈PC (Π) ,

where P is the set of all routes from the source to thedestination, and C (Π) is the cost of establishing the pathΠ. In other words, C (Π) is the total power of the sourceand the relay nodes, which consists of the power totransmit the message

∑li∈Π PSi , and the jamming power∑

li∈Π PJi . Hence, our optimization objective is,

C (Π) = min∑li∈Π

PSi + PJi . (10)

Suppose Π = 〈l1, . . . , lH〉. By applying the coding tech-nique described in [48], securing each hop is sufficientto ensure end-to-end secrecy. Hence, we consider thefollowing secrecy constraints,

γEi,j < γ∗E , ∀li ∈ Π and ∀Ej ∈ E , (11)

which means that for all eavesdroppers in the networkEj ∈ E , the secrecy constraint must be satisfied. Trans-mission is reliable provided that an end-to-end averageoutage probability is guaranteed, i.e.

pSDOUT = 1−∏li∈Π

(1− piout

)≤ ε. (12)

where piout denotes the average outage probability of thelink li = 〈Si, Di〉. Also, the following constraints shouldbe satisfied,

PSi ≥ 0, and PJi ≥ 0. (13)

Suppose that the number of key bits per jammingsymbol that Si utilizes is denoted by Ki. From (1),

PJi =2l2(22Ki+1 − 3× 2Ki + 1

)3

PSi .

Define,

βi =2l2(22Ki+1 − 3× 2Ki + 1

)3

.

Hence, the optimization objective can be written as,

C (Π) = min∑li∈Π

PSi (1 + βi) . (14)

A. Analysis of Secrecy

Consider the secrecy constraint. Substituting δ′E from(3) into (8) and (11), we have,

γEi,j =

PS |hSi,Ej |2dαSi,Ej

(1 + 1

12×22bE−2K

)+ σ2

E

PS |hSi,Ej |22πedαSi,Ej 22bE−2K + σ2

E

. (15)

Since we do not want to make assumptions on the eaves-droppers’s noise characteristics or the distance of anyeavesdropper from a system node, we assume σ2

E = 0.Hence, our goal is to have,

γEi,j =

PSi |hSi,Ej |2dαSi,E

(1 + 1

12×22bE−2Ki

)PSi |hSi,Ej |2

2πedαSi,Ej 22bE−2Ki

=1 + 1

12×22bE−2Ki

12πe22bE−2Ki

< γ∗E . (16)

Note that (16) is a deterministic function and does notdepend on |hSi,Ej |2. Thus, if we choose Ki such that (16)is satisfied, none of the eavesdroppers in E can interceptthe message being transmitted over the link li. In orderto guarantee secrecy, the following lower bound for thenumber of key bits per jamming symbol Ki at each hopmust be satisfied:

Ki >1

2log2

(2πe22bE

γ∗E − πe/6

). (17)

This bound only depends on the resolution of the eaves-dropper’s A/D (which is assumed to be bounded by bE ,as discussed in Section II-B), and does not depend on theeavesdropper’s location or its CSI. Intuitively, when thenumber of key bits per jamming symbol is sufficientlylarge, the quantization noise becomes large enough to

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protect the message against the eavesdropper regardlessof its location or its CSI. From (17), the same numberof key bits per jamming symbol K can be used in allhops,

K =

⌈1

2log2

(2πe22bE

γ∗E − πe/6

)⌉. (18)

Because βi only depends on Ki, βi = β, i = 1, . . . ,H ,where β is defined in (1). Hence, (14) becomes,

C (Π) = min∑li∈Π

PSi (1 + βi)

= min (1 + β)∑li∈Π

PSi (19)

Since β is an increasing function of K, using theK of (18) is equivalent to minimizing β. Hence, ouroptimization objective turns into,

min∑li∈Π

PSi . (20)

B. Analysis of Reliability

For the reliability constraint in (12), the probability ofoutage at Di is,

piout = P( PSi |hSi,Di |2

dαSi,Di+ θ2 PJi |hSi,Di |2

dαSi,Di+ σ2

D + δ2D12

θ2 PJi |hSi,Di |2dαSi,Di

+ σ2D + δ2D

12

<γ∗D

)

= P( PSi |hSi,Di |2

dαSi,Di+ θ2 βPSi |hSi,Di |2

dαSi,Di+ σ2

D + δ2D12

θ2 βPSi |hSi,Di |2dαSi,Di

+ σ2D + δ2D

12

<γ∗D

)

= P(|hSi,Di |2 <

(γ∗D − 1)(σ2D + δ2D

12

)PSi(1− (γ∗D − 1)θ2β)/dαSi,Di

)

= 1− e−

(γ∗D−1)(σ2D+

δ2D12

)PSi

(1−(γ∗D−1)θ2β)/dαSi,Di , (21)

where the last equality follows because, for Raleighfading, |hSi,Di |2 is exponentially distributed. Substituting(21) into (12), the end-to-end outage probability con-straint is,

pSDOUT = 1−∏li∈Π

e−

(γ∗D−1)(σ2D+

δ2D12

)PSi

(1−(γ∗D−1)θ2β)/dαSi,Di

= 1− exp

(−∑li∈Π

(γ∗D − 1)(σ2D + δ2D

12

)PSi

(1− (γ∗D − 1)θ2β

)/dαSi,Di

)≤ ε.

Thus, the end-to-end reliability constraint turns into,∑li∈Π

dαSi,DiPSi

≤ η, (22)

where,

η =log(

11−ε) (

1− (γ∗D − 1) θ2β)

(γ∗D − 1)(σ2D + δ2D

12

) . (23)

C. Optimal Cost of a Given Path

Our goal is to find the optimal path, which requiresthe minimum transmission and jamming power to satisfyboth outage and reliability constraints. The optimal pathis not known in advance. Hence, first we find the optimaltransmit and jamming power allocation for a given pathΠ, and then we use it to design a routing algorithm thatfinds the optimal path. From (20)-(23), in order to findthe optimal transmit and jamming power allocation for agiven path, we should solve the following optimizationproblem,

minPSi≥0

∑li∈Π

PSi (24)

subject to: ∑li∈Π

dαSi,DiPSi

≤ η (25)

The left side of (25) is a decreasing function of PSi andour goal is to find the minimum PSi . Hence, we cansubstitute the inequality with an equality,∑

li∈Π

dαSi,DiPSi

= η (26)

This optimization problem can be solved using thetechnique of Lagrange multipliers. We must solve (24)and the following equations simultaneously,

∂

∂PSi

{∑li∈Π

PSi + λ

(∑li∈Π

dαSi,DiPSi

− η

)}= 0,

for i = 1, . . . ,H.

Taking derivatives we have,

1− λdαSi,DiP 2Si

= 0, i = 1, . . . ,H, (27)

and thus,

PSi =√λdαSi,Di (28)

Substituting PSi , i = 1, . . . ,H from (28) into (26), weobtain that,

λ =1

η2

(√dαSi,Di

)2(29)

7

Substituting λ from (29) into (28), the optimal transmitpower at each link is given by,

PSi =1

η

√dαSi,Di

∑lk∈Π

√dαSk,Dk (30)

Hence, the aggregate cost of transmitting the message is,∑li∈Π

PSi =1

η

(∑lk∈Π

√dαSk,Dk

)2

, (31)

and the cost of jamming is,∑li∈Π

PJi =β

η

(∑lk∈Π

√dαSk,Dk

)2

. (32)

The total cost of establishing the optimal path Π is,

C (Π) =1 + β

η

(∑lk∈Π

√dαSk,Dk

)2

(33)

D. Routing Algorithm

Considering the structure of the total cost of establish-ing the optimal path (33) yields the routing algorithm.Assign the link weight

√dαSi,Di to each potential link

li between nodes Si and Di of the network. Then, runa classical shortest-path algorithm such as Dijkstra’salgorithm to find the route Π with minimum total weight,which is

∑li∈Π

√dαSi,Di . Clearly this route also mini-

mizes the total cost in (33). From (30), each node alongΠ forwards the message to the next node using the total(transmit and jamming) power,

C (li) =1 + β

η

√dαSi,Di

∑lk∈Π

√dαSk,Dk . (34)

IV. NUMERICAL RESULTS

In this section we numerically compare the perfor-mance of our algorithm with that of the SMER algorithm[35] in different scenarios.

SMER Algorithm. In SMER, the system nodes em-ploy cooperative jamming to establish a secure path,and, if the eavesdroppers get very close to a transmitter,the secrecy is compromised. Hence, while the SERJalgorithm proposed here has no need or sense of a“guard region”, to employ SMER we must introducesuch into the scenario. Thus, for the sake of comparisonto SMER, assume a guard region with radius rmin > 0around each transmitter and assume that no eavesdroppercan enter the guard regions. Further, in SMER a set oflocations and the probability that an eavesdropper existsin each location must be known. In order to address thisrequirement of SMER, we divide a circle centered at the

!"

!"

!"

#"

$%&'

"

$%()

"

Fig. 1: Three sectors have an eavesdropper with probabil-ity one, and the rest of the sectors have an eavesdropperwith probability zero.

transmitter S and with radius rmax into many sectors.Each sector is a location where an eavesdropper mightexist. For instance, when three eavesdroppers are present,three sectors have an eavesdropper with probability one,and the rest of the sectors have an eavesdropper withprobability zero (Fig. 1). Unlike the SERJ algorithmproposed in this paper, the secrecy outage probabilityof SMER is non-zero. In the next section, we will seehow this non-zero eavesdropping probability affects thepower consumption of secret communication.

To get more insight into the problem, first we considersecure one-hop transmission from a transmitter S to areceiver D in the presence of eavesdroppers. Next, wewill consider multi-hop minimum energy routing in anetwork and in the presence of multiple eavesdroppers.In both cases, we assume that the system nodes and theeavesdroppers use 14-bit A/Ds, and we set θ = 10−6. Weset the source-destination outage probability π = 0.1,receiver noise power N0 = 1 (eavesdropper noise poweris zero), γB = 42 and γE = 34, which results in thesecrecy rate Rs = 0.2. We consider different propagationattenuation scenarios: α = 2 which is the path-lossexponent corresponding to free space, and α = 3 andα = 4 which are the path-loss exponents correspondingto a terrestrial environment.

A. One-Hop Communication

Consider a single hop in a wireless network, consistingof a transmitter S and a receiver D (Fig. 2). For SMER,suppose two jammers J1 and J2 help the the transmitterto convey its message to the receiver securely [35]. Thedistance between each jammer and the source is denoted

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Fig. 2: One-hop communication between source S anddestination D in the presence of eavesdroppers (Es). InSMER, two jammers J1 and J2 help to make the linksecure.

1 2 3 4 5 6 7 8 9 1022

24

26

28

30

32

34

36

38

40

Number of eavesdroppers

log

(P)

SERJ

SMER, α=4

SMER, α=3

SMER, α=2

Fig. 3: Power consumption of SERJ and SMER versusthe number of eavesdroppers for various values of path-loss exponent α.

by d. In the remainder of this section, we consider theeffect of various parameters of the network on the energyconsumption of our scheme and SMER.

Number of Eavesdroppers. Fig. 3 shows the trans-mission power versus the number of eavesdroppersaround the transmitter2. In this figure, peav = 10−5,rmin = 0.01, rmax = 2, and dSD = 1. As shown in

2In all figures in this section, P denotes the aggregate powerconsumed by the algorithm.

10−3

10−2

10−1

20

25

30

35

40

45

log(rmin

)

log(P

)

SERJ

SMER, α=4

SMER, α=3

SMER, α=2

Fig. 4: Power consumption of SERJ and SMER versusthe radious rmin of the guard region for various valuesof α and when only one eavesdropper is present. Aswe allow the eavesdropper to become closer to thetransmitter (i.e. as rmin gets smaller), the power neededto make the link secure using SMER becomes higher. Onthe other hand, with SERJ there is no need to assume aguard region around the transmitter.

Section III, the power required when employing SERJdoes not depend on the number of eavesdroppers. On theother hand, when the number of eavesdroppers increases,the power needed to establish a secure link using SMERincreases dramatically. Since the cost of communicationusing SERJ only depends on the distance between thetransmitter and the receiver which is normalized todSD = 1, the cost of using SERJ does not change withthe change of path-loss exponent in these plots.

Guard Region Radius. Whereas the proposed algo-rithm (SERJ) does not require a guard region, recall thatSMER cannot be utilized without such. Fig. 4 showsthe power versus rmin in the presence of nE = 5eavesdroppers, and for various values of the path-lossexponent α. We set dSD = 1, peav = 10−5 andrmax = 2. We observe that when rmin gets small, thepower needed to establish a secure link using SMERincreases dramatically, while the power needed to es-tablish a secure link using SERJ does not depend onthe allowable location of the eavesdropper. In fact asis shown in Section III, the power used by SERJ isindependent of the distance between the transmitter andthe eavesdroppers, and, even if the eavesdroppers getvery close to the transmitter, they cannot intercept themessage.

Uncertainty in the Location of Eavesdroppers.In Fig. 5, the power needed to transmit the messagesecurely versus rmax for various values of the path-lossexponent α is depicted. For SMER we set peav = 10−5

9

1 1.5 2 2.5 3 3.5 4 4.5 524

26

28

30

32

34

36

38

40

42

rmax

log(P

)

SERJ

SMER, α=4

SMER, α=3

SMER, α=2

Fig. 5: Power consumption of SERJ and SMER versusrmax for various values of α and when nE = 5eavesdroppers are present. The performance of SMERis closely dependent on the uncertainity in the locationsof the eavesdroppers, while the performance of SERJdoes not depend on the locations of the eavedroppers.

0 ... 1e−10 1e−9 1e−8 1e−7 1e−6 1e−5 1e−4 1e−3 1e−2 1e−115

20

25

30

35

40

45

50

Eavesdropping probability

log(P

)

SERJ

SMER, α=4

SMER, α=3

SMER, α=2

Fig. 6: Power consumption of SERJ and SMER versuseavesdropping probability for various values of α andwhen nE = 5 eavesdroppers are present. For smallsecrecy outage probabilities, the power consumption ofSMER is substantially higher than the power consump-tion of SERJ.

and rmin = 0.01. As rmax increases, the uncertaintyin the location of the eavesdroppers increases, and thusin SMER the jammers need to consume more power tocover a larger area. On the other hand, with SERJ, thetransmit power is independent of the locations of theeavesdroppers.

Eavesdropping Probability. As was shown in SectionIII, the eavesdropping probability of SERJ is zero. But,the eavesdropping probability of SMER is not zero.Fig. 6 shows the power needed to establish a securelink versus the eavesdropping probability when nE = 5

0.5 1 1.5 2 2.5 3 3.5 420

25

30

35

40

45

50

Distance between source and destination (dSD

)

log(P

)

SERJ, α=4

SERJ, α=3

SERJ, α=2

SMER, α=4

SMER, α=3

SMER, α=2

Fig. 7: Power consumption of SERJ and SMER versusthe distance between source and destination dSD forvarious values of α and when nE = 5 eavesdroppers arepresent. As the distance between the transmitter and thereceiver gets longer, the transmit power of both schemesincreases.

eavesdroppers are present, and when for SMER rmin =.01, and rmax = 2. It can be seen that the powerconsumption of SMER dramatically changes when thesecrecy outage probability changes. In particular, forsmall secrecy outage probabilities, the power consump-tion of SMER is substantially higher than the powerconsumption of SERJ.

Distance between Source and Destination. Fig. 7shows the transmission power versus the distance be-tween source and destination dSD for various values ofα. For SMER, we set peav = 10−5, rmin = .01, andrmax = 2dSD. As the distance between the transmitterand the receiver gets longer, the transmit power of bothschemes increases.

B. Multi-Hop Communication

We consider a wireless network that consists of n sys-tem nodes and nE eavesdroppers which are distributeduniformly at random on a 5 × 5 square. Our goal is tofind a secure path with minimum aggregate energy fromthe source to the destination, using SERJ and SMER.Before we proceed to the numerical results for a multi-hop network, we compare the complexity of SERJ andSMER algorithms in a network that consists of n systemnodes.

Running Time. In order to find the optimal path usingSERJ we need to apply the Dijkstra’s algorithm, whichis a polynomial algorithm with running time O(n2). Onthe other hand, SMER is a pseudo-polynomial algorithmof order O(n2B), where B is the maximum cost of any

10

5 10 15 20 25 30 35 40 45 5027.5

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29

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30

30.5

31

31.5

32

Number of eavesdroppers

log

(P)

SERJ

SMER

Fig. 8: Power consumption of SERJ and SMER versusthe number of eavesdroppers. As the number of eaves-droppers increases, the amount of power that SMERuses increases, while the amount of power that SERJuses does not depend on the number and location of theeavesdroppers.

path in the network. Note that, while the running time ofSMER is polynomial in B, the actual value of B growsexponentially with the size of the input (i.e., the numberof bits used to represent link costs). That is, if l bits areused to represent the link cost values then B will be oforder 2l.

For the remainder of this section, we assume that inSMER, for every node two friendly jammers exist thathelp the node to establish a secure link. We averagethe results over 10 random realizations of the network.In each realization, the system nodes are distributeduniformly at random, and the closest system node to thepoint (0, 0) is the source of the message and the closestsystem node to the point (5, 5) is the destination. Weconsider the path-loss exponent α = 3, since α = 2corresponds to non-terrestrial environments, and α = 4leads to very high link costs of SMER, which make therunning time of this algorithm excessively high. In thesequel, we investigate the effect of various parameterson the total energy consumption of SERJ and SMER,and compare their performance.

Number of Eavesdroppers. The average power Pversus the number of eavesdroppers for SERJ and SMERis shown in Fig. 8. There are n = 25 system nodes inaddition to the eavesdroppers. The path-loss exponent ofthe environment is α = 3. For SMER, we set peav =10−5, rmin = .03, and rmax = 2. It can be seen that forvery small numbers of eavesdroppers, the performanceof SMER is better than that of SERJ. However, as thenumber of eavesdroppers increases, the amount of power

5 10 15 20 25 30 35 40 45 5020

22

24

26

28

30

32

34

36

38

Number of system nodes

log(P

)

SMER

SERJ

Fig. 9: Power consumption of SERJ and SMER versusthe number of system nodes. For both algorithms theaverage power is not sensitive to the number of systemnodes.

that SMER uses increases and becomes more than thepower that SERJ consumes. As is shown in Section III,the amount of power that SERJ uses does not depend onthe number and location of the eavesdroppers.

Number of System Nodes. The effect of the numberof system nodes on the average aggregate power con-sumption is shown in Fig. 9. There are nE = 25 eaves-droppers, and the path-loss exponent of the environmentis α = 3. For SMER, we set peav = 10−5, rmin = .03,and rmax = 2.

It can be seen that the performance of SERJ is alwayssuperior to the performance of SMER. For both algo-rithms the average power is not sensitive to the numberof system nodes. The fluctuations in this figure are dueto the random generation of network configurations.

Uncertainty in the Location of the Eavesdroppers.In Fig. 10, the power needed to transmit the messagesecurely versus rmax is shown. There are n = 25system nodes and nE = 25 eavesdroppers, and thepath-loss exponent of the environment is α = 3. ForSMER we set peav = 10−5 and rmin = 0.03. WithSERJ, the transmit power is independent of the locationof the eavesdroppers. With SMER, as rmax increases,the uncertainty in the location of the eavesdroppersincreases, and thus the jammers need to consume morepower to cover a larger area. For the case that SMERis secure against any eavesdropper in the network (i.e.rmax = 5, if we do not consider the guard regionsaround the transmitters), the power spent by SMER issubstantially higher than the power spent by SERJ.

11

1 1.5 2 2.5 3 3.5 4 4.5 524

26

28

30

32

34

36

rmax

log

(P)

SERJ

SMER

Fig. 10: Power consumption of SERJ and SMER versusthe uncertainity in the location of the eavesdropper(i.e. rmax around each transmitter in the network). Thetransmit power using SERJ is independent of the locationof the eavesdroppers. But with SMER, as the uncertaintyin the location of the eavesdroppers increases the powerconsumption increases.

V. CONCLUSIONS

In this paper, we have considered secure energy-efficient routing in a quasi-static multi-path fading en-vironment in the presence of passive eavesdroppers.Since the eavesdroppers are passive, their locations andCSIs are not known to the legitimate nodes. Thus welooked for approaches that do not rely on the locationsand quality of the channels of the eavesdroppers. Wedeveloped an energy-efficient routing algorithm basedon the random jamming to exploit non-idealities of theeavesdropper’s receiver to provide secrecy. Our routingalgorithm is fast (finds the optimal path in polynomialtime), and does not depend on the number of eavesdrop-pers and their location and/or channel state information.

We have performed several simulations over single-hop and multi-hop networks with various network pa-rameters, and compared the performance of our proposedalgorithm with that of the SMER algorithm of [35],[36]. A major weakness of SMER is that it requiresthe definition of a guard region that restricts how closeeavesdroppers can come to system nodes. Even withsuch a guard region, which SERJ does not require, weobserved that when the uncertainty in the location ofthe eavesdroppers is high and in disadvantaged wirelessenvironments, the energy consumption of our algorithmis substantially less than that of the SMER algorithm.Gains of SERJ over SMER would be even more sub-stantial in environments with "smart" eavesdroppers; forexample, eavesdroppers that located themselves close to

system nodes or pointed directional antennas at systemnodes would significantly degrade the performance ofSMER, but there would be no impact on the performanceof SERJ. Hence, the proposed algorithm directly ad-dresses one of the key roadblocks to the implementationof information-theoretic security in wireless networks:robustness tot he operating environment.

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