Detection of Misbehavior in Cooperative SpectrumSensing

Erfan Soltanmohammadi, Mort Naraghi-PourDepartment of Electrical and Computer EngineeringLouisiana State University, Baton Rouge, LA 70803

{e-mail: esolta1, naraghi@lsu.edu}.

Abstract—The problem of cooperative spectrum sensing in thepresence of multiple classes of honest and misbehaving cognitiveradios (CRs) is investigated. The CRs transmit their binarydecisions regarding the state of the channel to a fusion center(FC) which must classify the CRs and determine whether thechannel is vacant of the primary user. We present a novelapproach based on the expectation maximization (EM) algorithmin order to detect the presence of the primary user, to classifythe cognitive radios, and to compute their detection and falsealarm probabilities. In contrast to reputation-based classi�ers(RBCs), our approach can classify the radios into more thantwo classes of honest and malicious CRs. Numerical results showsigni�cant improvements over RBC. In particular, with only afew decisions from the CRs, the proposed algorithm can quicklyand ef�ciently classify the CRs whereas RBC fails in many caseseven for networks with a large number of honest CRs.

Index Terms—Cooperative spectrum sensing, Malicious user,Expectation-Maximization, Hypothesis testing.

I. INTRODUCTION

In cooperative spectrum sensing (SS) multiple cognitiveradios (CR) attempt to detect the primary user (PU) signaland report their decisions to a fusion center (FC), which fusesthe received messages to detect the presence or absence of thePU [1], [2]. Cooperative SS provides signi�cant performanceimprovement over single-user techniques by taking advan-tage of spatial diversity and overcoming the hidden terminalproblem. It also requires lower complexity for the individualsecondary user (SU) terminals at the cost of a small increasein communication between the SU terminals and the FC.Unfortunately cooperative SS is vulnerable to misbehavior

by the individual radios which may send false information:due to the malfunctioning of their terminal; to gain unfairaccess to the channel; or to completely disrupt the spectrumsensing process [3], [4]. Several methods have been recentlyproposed to counter such malicious attacks. An outlier-baseddetection mechanism is proposed in [5] where it is assumedthat energy detection is used at the CRs who transmit their rawmeasurements (without quantization) to the FC. Reputation-based detection is another widely used approach [6]–[8]. Hereeach CR sends its binary decision to the FC which makesthe �nal decision using a voting scheme (see Section IV).At the same time the FC constructs a reputation metric foreach CR based on the similarity of CR’s decisions to the�nal decisions of the FC. The reputation metric is updatedover time and compared with a threshold in order to identifythe radios. In [9] a weighted sequential probability ratio test

(WSPRT) is proposed where the weights are derived from aradio’s reputation metric.In [10] the authors de�ne a reputation metric based on two

types of attackers: type-1, which report the channel to be busywhen it is detected to be free, and type-0 which report thechannel to be free when it is detected to be busy. This modeldoes not include malicious radios that attempt to confuse theFC under both hypotheses. In addition, it is assumed that thesubset of honest CRs is known to the FC.A drawback of the reputation-based methods, which results

in a loss of performance, is that the detection of the hypothesesis separated from the detection of the malicious CRs. Inaddition, as shown in our numerical results in Section IV,for the reputation metric to be reliable, the FC must assemblea large number of decisions from each CR, and the algorithmfails when the number of decisions is small. Furthermore,as explained in Section IV, in the absence of any priorinformation on the parameters of the network, the reputation-based method may fail even in cases where the honest nodesare in majority. Finally reputation-based methods can onlyclassify the radios into two classes.In this paper, we consider a network with several classes of

CRs where all the nodes in a class have the same detectionand false alarm probabilities. In particular, there may be morethan one class of misbehaving nodes due to the presence ofmalicious as well as malfunctioning radios. There may alsobe more than one class of honest CRs. This may arise whenSUs employ different SS techniques, or when, due to theirgeographic position with respect to the PU, they experiencedifferent path loss and fading from the PU, resulting indifferent detection and false alarm probabilities. No parametersof the network are assumed to be known except that there is aclass of honest CRs with more nodes than any other class. Wepresent a method for detecting the hypotheses and classifyingthe CRs based on the expectation maximization algorithm.We also compute the detection and false alarm probabilitiesof each class. Our results show that the proposed algorithmsigni�cantly outperforms the reputation-based method andeven in cases where the reputation-based method fails, theproposed method is able to properly classify the CRs anddetect the hypotheses.The rest of this paper is organized as follows. In Section II,

we describe the system model and our notations. The proposedalgorithmn is presented in Section III and its performanceis evaluated in Section IV with numerical examples. Finally

2013 IEEE Military Communications Conference

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DOI 10.1109/MILCOM.2013.163

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2013 IEEE Military Communications Conference

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DOI 10.1109/MILCOM.2013.163

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concluding remarks are drawn in Section V.

II. SYSTEM MODEL AND NOTATIONS

We consider a cognitive radio network of L independentradios monitoring a spectral band in order to detect thepresence (H1) or absence (H0) of a PU. There are K classesof CRs c1, c2, . . . cK , where c1 denotes the class of honestradios and the remaining classes represent honest, malicious orotherwise misbehaving radios. At time t radio l makes a binarydecision rlt regarding the presence of the PU where rlt = 0if H0 and rlt = 1, otherwise. De�ne H = [hit], i = 0, 1,t = 1, 2, · · · , T as the hypothesis matrix where column t

represents the state of the hypothesis at time t = 1, 2, · · · , T .At each time t, one of the element in column t is 1 and theother is 0. For example if h0t = 0, then h1t = 1, indicating thatwe have hypothesis H1. Similarly, if h0t = 1 then h1t = 0,indicating that at time t we have hypothesis H0.It is assumed that given H , the radios’ decisions {rlt, l =

1, 2, · · · , L, t = 1, 2, · · · , T} are independent. The probabil-ities of detection and false alarm denoted by p1k and p0k,respectively, for class ck are given by

pηk = Pr(rlt = 1|h1t = η, l ∈ ck), η = 0, 1 (1)

At time t, radio l transmits a single bit dlt ∈ {0, 1} to theFC. While for the honest radios we have dlt = rlt, radios inother classes may alter their decision before transmission tothe FC. For η = 0, 1 let ρη(k) � Pr(dlt = 1|rlt = η, l ∈ ck).Note that for the honest radios, ρ0(1) = 0 and ρ1(1) = 1 andfor a malicious radio in class, say ι, which �ips its decisionsbefore transmission, ρ0(ι) = 1 and ρ1(ι) = 0.

The probabilities of detection and false alarm “perceived”by the FC for a radio in class ck, and denoted by p1k and p0k,respectively, can be written as

pηk � Pr(dlt = 1|h1t = η, l ∈ ck) (2)ρ1(k)pη(k) + ρ0(k)(1− pη(k)), η = 0, 1

It is assumed that for the malicious nodes the values ρ0(k)and ρ1(k) do not change during the observation interval. Asnoted in [10], frequent changes of attack probabilities will notproduce predictable deleterious effects on the FC.The FC collects T transmissions to form a decision matrix

D = [dlt], l = 1, 2, . . . , L, t = 1, 2, . . . , T from which it triesto classify the radios and detect the hypotheses matrix H . Inthis process the FC will also determine the probabilities of de-tection and false alarm (p1k, p0k) for each class ck. Hereafterwe refer to the pair (p1k, p0k) as the operating point of theradios in class ck, and let P � [pik] i = 0, 1, k = 1, 2, . . . ,K.

Let zl,k = 1 if radio l ∈ ck, and zero, otherwise and de�nethe matrix Z = [zlk], l = 1, 2, . . . , L, k = 1, 2, . . . ,K as theradio identi�cation matrix. Also let πk denote the probabilitythat a node belongs to class ck, i.e., πk = Pr(zlk = 1) andlet Π = [πk], k = 1, 2, . . . ,K. In order to formulate ourdetection problem we allocate probabilities φit = Pr(hit = 1)for the hypothesis at time t, where φ0t + φ1t = 1, and letΦ = [φit], i = 0, 1, t = 1, 2, . . . , T . Finally the three-tupleΘ � {P,Π,Φ} is de�ned as the parameter set.

We would like to detect the matrices Z and H from thedecision matrix D. However, since the parameter set Θ isunknown, it must be �rst estimated from the received decisionmatrix D. A maximum likelihood estimate of Θ is given byΘ = argmaxΘ Pr(D|Θ), where

Pr(D|Θ) =∑H,Z

Pr(D,Z,H|Θ). (3)

Due to the complexity of the mixture model in (3), the estimateof Θ cannot be obtained in closed form. Therefore we employthe iterative EM algorithm to estimate Θ with H and Z aslatent variables [11]. For a discussion of the convergenceproperties of the EM algorithm we refer to [11], [12].A remark is in order here. Due to the presence of unknown

parameters such as the radio identi�cation matrix and theoperating points of each class, the hypothesis testing problemcannot be formulated with a Bayesian or Neyman-Pearsoncriterion. In particular the probability matrix Φ is used onlyas an artifact in our detection problem and is not assumed tobe the true prior of the hypotheses.

III. PARAMETER ESTIMATION, CLASSIFICATION, ANDHYPOTHESES TESTING

To estimate Θ, we �rst need to evaluate Pr(D,Z,H|Θ)which is given in (4) and from which the log-likelihoodfunction, denoted by L(Θ;D,Z,H), is obtained in (5). Aniteration of EM involves the following two steps [12].1) Expectation step: Expectation of the log-likelihood func-

tion, denoted by Q(Θ;Θold) in (5), is evaluated withrespect to the conditional distribution P (Z,H|D,Θold)of the latent variables (Z,H), where Θold is the previousestimate for Θ. This is shown in (6).

2) Maximization step: Q(Θ;Θold) is maximized with re-spect to Θ.

To perform the expectation step, let β(l, k) �E[zlk|D; Θold] and α(l, k, i, t) � E[zlkhit|D; Θold]. Then,

β(l, k) = Pr(zlk = 1|D; Θold) (7)

=πoldk

∏T

t=1 Pr(dlt|zlk = 1;Θold)∑K

j=1 πoldj

∏T

t=1 Pr (dlt|zlj = 1;Θold)

=πoldk

∏Tt=1

∑1i=0 φ

oldit

((poldik )

dlt

(1− poldik

)(1−dlt))

∑K

k′=1 πoldk′

∏T

t=1

∑1i=0 φ

oldit

((poldik′)

dlt

(1− poldik′

)(1−dlt)) .

Moreover,

α(l, k, i, t) = Pr(zlk = 1, hit = 1|D; Θold) (8)= Pr(hit = 1|zlk = 1, D; Θold)× Pr(zlk = 1|D; Θold)

where Pr(hit = 1|zlk = 1, D; Θold) is derived in (9) in whichdt = [d1t, d2t, · · · , dLt]

tr denotes the tth column of D, i.e., thedecision vector received from all the radios at time t. Thereforefrom (6)-(9), Q(Θ;Θold) can be obtained as in (10).To perform the maximization step of the EM algorithm we

need to maximize Q(Θ;Θold) with respect to the parameter

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Pr(D,Z,H|Θ) = Pr(D|Z,H; Θ)Pr(Z,H|Θ) =L∏

l=1

K∏k=1

[T∏

t=1

1∏i=0

π1

2T

k

(pdlt

ik (1− pik)(1−dlt) φ

1

L

it

)hit

]zlk(4)

L(Θ;D,Z,H) � logPr(D,Z,H|Θ)L∑

l=1

K∑k=1

zlk

T∑t=1

1∑i=0

{log πk

2T+ hit

[dlt log pik + (1− dlt) log (1− pik) +

1

Llog φit

]}(5)

Q(Θ;Θold) � E(Z,H)|D;Θold [L(Θ;D,Z,H)] =

L∑l=1

K∑k=1

T∑t=1

1∑i=0

{E(Z,H)|D;Θold [zlk]

1

2Tlog πk

+E(Z,H)|D;Θold [zlkhit]

[dlt log pik + (1− dlt) log (1− pik) +

1

Llog φit

]}(6)

Pr(hit = 1|zlk = 1, D; Θold) = Pr(hit = 1|zlk = 1, dt; Θold)

=Pr(dt|hit = 1, zlk = 1;Θ

old)Pr(hit = 1|zlk = 1;Θold)∑1j=0 Pr(dt|hjt = 1, zlk = 1;Θold)Pr(hjt = 1|zlk = 1;Θold)

=φoldit

((poldik )

dlt

(1− poldik

)(1−dlt))∏

l′ �=l

[∑K

k′=1

((poldik′)

dl′t

(1− poldik′

)(1−dl′t

))πoldk′

]∑1

j=0

{φoldjt

((poldjk )

dlt

(1− poldjk

)(1−dlt))∏

l′ �=l

[∑Kk′=1

((poldjk′)

dl′t

(1− poldjk′

)(1−dl′t

))πoldk′

]}(9)

Q(Θ;Θold) =

L∑l=1

K∑k=1

β(l, k) log πk +1

L

L∑l=1

K∑k=1

T∑t=1

1∑i=0

α(l, k, i, t) log φit

+L∑

l=1

K∑k=1

T∑t=1

1∑i=0

α(l, k, i, t) [dlt log pik + (1− dlt) log (1− pik)] (10)

set Θ. Maximization with respect to the operating points isachieved from

∂Q

∂pik=

L∑l=1

T∑t=1

α(l, k, i, t)

(dlt

pik− (1− dlt)

(1− pik)

)= 0 (11)

which results in

pnewik =

∑L

l=1

∑T

t=1 α(l, k, i, t)dlt∑L

l=1

∑T

t=1 α(l, k, i, t)(12)

To maximize Q(Θ;Θold) with respect to πk we must alsosatisfy the constraint

∑Kk=1 πk = 1. We de�ne the Lagrangian

Q by

Q(Θ, λ; Θold) � Q(Θ;Θold) + λ

{K∑

k=1

πk − 1}

(13)

Differentiating with respect to πk results in

∂Q

∂πk

=

L∑l=1

β(l, k)1

πk

+ λ = 0 (14)

Now multiplying both sides by πk and summing over k givesλ = −L, from which

πnewk =

1

L

L∑l=1

β(l, k) (15)

We note that since Q(Θ;Θold) is a concave function ofthe πk’s and the constraint

∑K

k=1 πk = 1 is linear, theLagrange multiplier method achieves the optimal solution [13].Similarly, Q(Θ;Θold) can be maximized with respect to φit

to get

φnewit =

1

L

L∑l=1

K∑k=1

α(l, k, i, t) (16)

Again, the above method achieves the optimal solution for theφit’s.

A. Resolving the Ambiguity in Parameter Estimation

Let Θc � {P c,Π,Φc} be the counterpart of the parameterset Θ, where P c and Φc are given by pcik = p(1−i)k andφcit = φ(1−i)t, i = 0, 1. It can be veri�ed that Pr(D|Θ) =

Pr(D|Θc). Therefore, we always have two possible solutionsΘc and Θ for the parameter set. Note that this ambiguity isnot speci�c to our method and is inherent in any estimationprocedure for the parameter set. The EM algorithm convergesto one of these two possible solutions. This ambiguity canbe resolved if the class of honest radios has the highestpopulation among all the classes and its operating point isabove the chance line, i.e., p11 > p01. We note that forpractical networks, these assumption are not unrealistic.

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B. Node Identi�cation and Hypotheses Testing

Let Θ denote the parameter set estimated by the EMalgorithm. Then the MAP detection rule for (Z,H) is givenby

(Z, H) = argmaxZ,H

logPr(D,Z,H|Θ) (17)

Using (5) we get,

(Z, H) = argmaxZ,H

L∑l=1

K∑k=1

T∑t=1

1∑i=0

{zlk2Tlog πk (18)

+zlkhit

[dlt log pik + (1− dlt) log (1− pik) +

1

Llog φit

]}

Unfortunately the complexity of the optimization in (18) isprohibitive. Therefore we employ a suboptimal algorithm forthe estimation of Z and H as follows. Let

h0t =

{1, φ0t > φ1t0, otherwise. (19)

We note that the above is equivalent to setting H =argmaxH Pr(H|Θ). Next, let N �

∑Tt=1 h0t, M �∑T

t=1 h1t, nl �∑T

t=1 h0tdlt, and ml �∑T

t=1 h1tdlt. Also letdl � [dl1, dl2, . . . , dlT ] denote the vector of decisions receivedfrom node l. Then, the class of radio l is estimated as ck∗ (i.e.,zlk∗ = 1), where

k∗ = argmaxk

Pr(zlk = 1|dl, H; Θ) (20)

= argmaxk

Pr(dl|zlk = 1, H; Θ)

= argmaxk

pnl

0k(1− p0k)(N−nl)pml

1k (1− p1k)(M−ml)

IV. NUMERICAL RESULTS

In this section we compare the results obtained from theproposed algorithm (denoted EMC) with those from thereputation-based classi�er (RBC) [6]–[9] which can be usedto classify the radios into two groups of honest and Byzantine(malicious) radios. In RBC, the hypothesis is �rst estimated ateach time through a voting scheme1 as h1t = 1 if

∑L

l=1 dlt >

q and h1t = 0, otherwise for t = 1, 2, . . . , T , where q isa threshold. Then, the operating points are obtained frompik =

∑T

t=1hitdlt

∑T

t=1hit

. Finally radio classi�cation is performedby comparing the reputation Rl of radio l with a thresholdwhere

Rl �T∑

t=1

(1− dlt)h0t + dlth1tHonest≷

Byzantineλ (21)

The value of λ affects the probability of misclassifying anhonest radio as Byzantine and vice versa. In the followingresults we set λ = 0.5 so the probability of misclassifyingan honest node as Byzantine is the same as misclassifying aByzantine as honest.

1Voting or q-out-of-L rule is the only available rule for RBC when the FCdoes not have any prior information regarding the radios’ parameters [10].

TABLE ICLASS PARAMETERS OF EACH OPERATING POINT.

Set ck p0k p1k πk

OP1 c1 0.1 0.9 0.6c2 0.9 0.4 0.4

OP2 c1 0.2 0.7 0.7c2 0.9 0.2 0.3

OP3

c1 0.2 0.9 0.5c2 0.8 0.3 0.3c3 0.05 0.05 0.2

OP4

c1 0.2 0.8 0.4c2 0.9 0.2 0.15c3 0.05 0.05 0.2c4 0.95 0.95 0.25

We evaluate the performance of the classi�ers using dis-criminability, ΔZ , which is a measure of the misclassi�cationrate and is given by

ΔZ �1

2L

L∑l=1

K∑k=1

|zl,k − zl,k| (22)

and the hypothesis discriminability given by

ΔH �1

2T

1∑i=0

T∑t=1

|hit − hit| (23)

Finally, to measure the reliability of the classi�ers, we de�nethe estimation error for the radios’ operating points as

ΔOP �1√2

K∑k=1

πk

√(p0k − p0k)2 + (p1k − p1k)2 (24)

These measures are normalized so that they are in the interval[0, 1] where the smaller values indicate better estimates.In the simulations we use the four sets of operating points

shown in Table I. In the �rst set denoted OP1, there aretwo classes of honest (c1) and Byzantine radios (c2), whereByzantines try to confuse the FC under both hypotheses. Inthe second set, OP2, there are also two classes of honest andByzantine radios but with different populations from OP1.Moreover, the honest radios are not as effective in detectingthe hypotheses. In the case of OP3 and OP4, c3 represents theclass of “almost-always-no” radios and c4 represents the classof “almost-always-yes” radios.The performance of the classi�ers with respect to the

number of decisions, T , is shown in Fig. 1 and Fig. 2 for theestimation error ΔOP and the misclassi�cation rate ΔZ forL = 20 radios. It can be seen that the accuracy of estimationand radio classi�cation improves with T and that the proposedmethod outperforms RBC. Note that RBC is not capable ofclassifying the radios into more than two classes; thus forRBC, ΔZ is not de�ned for OP3, and OP4. As shown in Figs.1 and 2, the performance of the classi�ers for OP1 is betterthan the other cases since the malicious radios are collectivelyweaker in the case of OP1. By this we mean that given thefraction of Byzantines and their operating points, the averagenumber of radios that provide false information to the FCunder each hypothesis is smaller in the case of OP1.To evaluate the performance of the classi�er with respect

to the fraction of honest radios in the network, we consider

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Fig. 1. Estimation error of the operating point vs. T for L = 20.

Fig. 2. Misclassi�cation rate vs. T for L = 20.

OP1 and OP2 and evaluate the estimation error and misclassi-�cation rate as a function of π1 for T = 18 and L = 20. Theresults are shown in Figs. 3 and 4. From Fig. 4 it can be seenthat for OP2, the performance of RBC is not acceptable whenπ1 ≤ 0.6. In general RBC fails to classify the malicious radioscorrectly when the class of malicious radios is collectivelystronger than the class of honest radios. In such cases eventhough the honest CRs are in majority, their performance interms of detection and false alarm probability is not very good(i.e., it is close to the chance line). On the other hand themalicious CRs have good detection and false alarm probabilityand try to completely mislead the FC. As a result the FCreceives more decisions in favor of the alternative hypothesisH1−η when Hη is the true hypothesis.

Performance of the classi�er with respect to the number ofradios in the network, L, is evaluated by the estimation error,misclassi�cation rate, and hypothesis discriminability and theresults are shown in Figs. 5-7. As expected, in all the cases

Fig. 3. Estimation error of the operating point vs. π1 for T = 18 andL = 20.

Fig. 4. Misclassi�cation rate vs. π1 for T = 18 and L = 20.

the performance improves with L. However, it can be seenfrom these �gures that for small number of decisions T , theperformance of RBC is not acceptable for OP2 for any valuesof L.

V. CONCLUSION

We studied the problem of cooperative spectrum sensingin the presence of misbehaving cognitive radios (CRs). Noprior information on the parameters of the network is assumedexcept that a class of honest radios is in majority. We developthe expectation maximization (EM) for the detection of thehypotheses and the classi�cation of the radios. In contrastto other recently proposed methods, our approach can clas-sify the radios into more than two classes. This applies incases where the honest CRs may employ different spectrumsensing techniques or encounter dissimilar channel and noiseconditions resulting in different detection and false alarm

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Fig. 5. Estimation error vs. L for T = 4, 8.

Fig. 6. Misclassi�cation rate vs. L for T = 4, 8.

probabilities. Another case is when the network includes morethan one type of misbehaving CRs such as both maliciousand malfunctioning radios. Our numerical results show signi�-cant improvements over the popular reputation-based classi�er(RBC). In particular, with only a few decisions from the radiosthe proposed algorithm can quickly and ef�ciently classify theCRs whereas the RBC method fails.

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