Covert Wireless Communication in Presence of a Multi-Antenna Adversary and Delay Constraints
11 Covert Wireless Communication in Presence ofa Multi-Antenna Adversary and Delay Constraints
Khurram Shahzad,
Student Member, IEEE , Xiangyun Zhou,
Senior Member, IEEE and Shihao Yan,
Member, IEEE
Abstract —Covert communication hides the transmission ofa message from a watchful adversary, while ensuring reliableinformation decoding at the receiver, providing enhanced securityin wireless communications. In this letter, covert communicationin the presence of a multi-antenna adversary and under delayconstraints is considered. Under the assumption of quasi-staticwireless fading channels, we analyze the effect of increasing thenumber of antennas employed at the adversary on the achievablethroughput of covert communication. It is shown that in contrastto a single-antenna adversary, a slight increase in the number ofadversary’s antennas drastically reduces the covert throughput,even for relaxed covertness requirements.
Index Terms —
Physical layer security, covert communication,multiple antennas.
I. I
NTRODUCTION
The broadcast nature of wireless transmission makes itprone to unauthorized access, raising serious concerns about itssecurity and privacy. With an ever-increasing dependence onwireless devices not only for communication but also activitiesrelated to health, finance and sharing private information, thereis a renewed interest in the security and privacy of wirelesstransmissions. Circumstances exist where instead of protectingthe information content of the transmission, it is imperativeto hide the transmission itself. In such situations, traditionalsecurity schemes employing cryptography and physical layersecurity [1] are deemed insufficient, as such schemes causesuspicion, drawing further probing from an adversary. Hid-ing communications in sensitive or hostile environments isof paramount importance to military and law enforcementagencies. On the other hand, detecting any malicious covertcommunications is also highly desired by law enforcement andcyber task forces since even the presence of such activitiesoffers sufficient incentive for them to take action [2]. In allsuch scenarios, covert communication approaches [3] offer ahighly viable solution, with applications not only of interestto military organizations but general public as well.Recent research efforts have explored this nascent approachto security under different communication scenarios establish-ing the fundamental limits in additive white Gaussian noise(AWGN) channels [4], under channel and noise uncertainty[5, 6], use of jamming and artificial noise [7, 8] and underrelay networks [9]. Optimality of Gaussian signalling in covert
Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]. Shahzad and X. Zhou are with the Research School of Electrical, Energyand Materials Engineering, Australian National University, Canberra, ACT2601, Australia (emails: { khurram.shahzad, xiangyun.zhou } @anu.edu.au).S. Yan is with the School of Engineering, Macquarie University, Sydney,NSW 2109, Australia (e-mail: [email protected]). communication has been analyzed in [10], while [11] offersa first study in considering a UAV as the transmitter inthe context of covert communications. Multi-antenna covertcommunications under AWGN channels has been consideredin [12], while a recent work in [13] has considered theirperformance in random wireless networks considering bothcentralized and distributed antenna systems at the transmitter.The aforementioned works consider covert communicationunder the assumption of an infinite number of channel uses.However, limited storage resources and requirements of quickupdates in modern communication systems require a finite,sometimes small, number of channel uses, and hence theresults in the infinite blocklength regime do not hold anymore.Under finite blocklength, covert communication has beenconsidered in the literature under AWGN and fading channels[14–16], where a single-antenna adversary was considered.In this work, we consider covert communication under fad-ing channels with a finite blocklength, in presence of a multi-antenna adversary. Equipping the adversary with multipleantennas makes him a stronger adversary in fading conditions,yielding the task of communicating covertly even harder.Although [13] has considered achieving covertness for a multi-antenna transmitter in the presence of multiple adversaries,the adversaries are non-colluding, whereas we consider anadversary utilizing a centralized detection approach. Further-more, the analysis in [13] is presented under the assumption ofan infinite blocklength as compared to our finite blocklengthassumption. We analyze the achievable throughput under strictdelay constraints and study the impact of adversary’s multipleantennas on covertness, when the covert communication pairis equipped with single antennas. In particular, we show thatin contrast to a single antenna case, the covert throughputquickly reduces to zero with a slight increase in number ofantennas at the adversary, even for highly relaxed covertnessrequirements. II. S YSTEM M ODEL
We consider a scenario where the adversary, Willie, uses M ≥ antennas for detection, whereas each of Alice andBob is equipped with a single antenna. A communication slotis a block of time in which transmission of a message fromAlice to Bob is complete. If Alice transmits in a slot, shesends a finite number, L M , of symbols to Bob, with a symbolindex of l , while Willie observes silently, looking to decidewhether Alice has transmitted or not. The maximum numberof symbols in a slot is denoted by L max , and hence, L M ≤ L max . We assume that Alice’s transmitted signal samples areindependent, with a distribution given by CN (0 , P a ) while the a r X i v : . [ c s . I T ] O c t distribution of AWGN at Bob’s receiver is given by CN (0 , .The additive noise samples at the different antennas at Willieare also considered to be independent, with a distribution of CN (0 , . Furthermore, it is assumed that Willie’s receivedsignals and his noise are independent.The wireless links from Alice to Bob and Alice to Willieare subject to quasi-static Rayleigh fading channels, whichconstitute a suitable model for covert communications scenariounder NLOS communications, and is commonly adopted inthe literature [7, 16]. Resultantly, corresponding to a largecoherence time, the channel coefficients remain constant ina slot and change independently from one slot to the next.The vector of channel coefficients from Alice to Willie’s M antennas is denoted by h aw ∈ C M × , and for each entry h aw of h aw , the mean of | h aw | is denoted by /λ . We followthe common assumption that a secret of sufficient length isshared between Alice and Bob [4], which is unknown toWillie. Employing random coding arguments, Alice generatescodewords by independently drawing symbols from a zero-mean complex Gaussian distribution with a variance of P a .Here, each codebook is known to Alice and Bob and is usedonly once. When Alice transmits in a slot, she selects thecodeword corresponding to her message and transmits theresulting sequence.Let Y = [ y , y , . . . , y L M ] ∈ C M × L M be the matrixcontaining Willie’s observed signals at M antennas. Williefaces a binary hypothesis test regarding Alice’s actions andwe denote Willie’s hypotheses of Alice transmitting or notby H and H , respectively. The observation model at Willieregarding Alice’s transmission state can be expressed as (cid:40) H : y l ∼ CN ( , I M ) ,H : y l ∼ CN ( , P a h aw h Haw + I M ) , (1)where I M is an M × M identity matrix. We assume thatin a given slot, P a is fixed and known to Willie. Under theassumption of an equal probability of Alice transmitting or notin a slot, achieving covertness requires P F A + P MD ≥ − (cid:15) forsome arbitrarily small (cid:15) [4, 14]. Here, P F A and P MD denotethe Probability of False Alarm and Probability of MissedDetection, respectively.As per [17], the decoding error probability at Bob is notnegligible for finite blocklengths. For a given decoding errorprobability, δ , the channel coding rate (in bits per channeluse) for a given channel realization and for duration of L M symbols is given as [18] R ≈ log (cid:0) P a | h ab | (cid:1) − (cid:115) L M (cid:18) − P a | h ab | ) (cid:19) Q − ( δ )ln 2 , (2)where h ab is the fading channel coefficient from Alice toBob and Q − ( · ) is the inverse Gaussian Q-function. Sincethe transmission rate achieved here for a finite blocklength isless than the Shannon capacity, the decoding error probability, δ , is considered to be less than . . We consider the amountof information (in bits), given by L M R (1 − δ ) , as our perfor-mance metric while P F A + P MD ≥ − (cid:15) is the covertness constraint. Alice being part of a wider network, periodicallybroadcasts pilot signals enabling other users to estimate theirchannels. Though this enables Bob to know his instantaneouschannel from Alice, facilitating his message decoding, it alsogives Willie the channel information from Alice. For ease ofexposition, we assume that Alice is also aware of her channelto Bob. In comparison to [13], we note that in case of multipletransmit antennas at Alice, the assumption of CSI availabilityat all the antennas is even harder to justify since trainingby covert receivers may expose their existence and violatecovertness requirements. We also note that while under infiniteblocklength, having perfect channel state information (CSI)at the transmitter and receiver results in no decoding errorsat the receiver, this is not the case under finite blocklengthscenario, where decoding errors still occur even in the presenceof perfect CSI at both the transmitter and the receiver.III. D ETECTION AT W ILLIE
Since Willie is aware of his channels from Alice, hertransmit power and his own receiver’s noise variance, heis able to design an optimal detector, which represents theworst case scenario from the covert communication designperspective. In the following, we present Willie’s optimaldetector and the corresponding detection error probabilities.
Lemma 1.
The optimal detector at Willie has the decisionrule given as (cid:107) h Haw Y (cid:107) H ≷ H θ ∗ , (3) where θ ∗ = L M (cid:18) P a + (cid:107) h aw (cid:107) (cid:19) ln (cid:0) P a (cid:107) h aw (cid:107) + 1 (cid:1) (4) is the optimal decision threshold.Proof. Since Willie has complete statistical knowledge of hisobservations, hence resorting to the Neyman Pearson criterion[19], the optimal test for Willie to minimize his detection errorprobability is the likelihood ratio test. Under H , the pdf ofthe observation matrix at Willie, Y , is given by f ( Y | H ) = L (cid:89) l =1 π M (cid:112) | K | exp (cid:2) − y Hl K − y l (cid:3) , (5)where K (cid:44) I M is the covariance matrix of Willie’s observa-tions under H . We have f ( Y | H ) = L (cid:89) l =1 π M exp (cid:2) − y Hl y l (cid:3) = 1 π ML exp (cid:2) − tr ( YY H ) (cid:3) , (6)where tr ( · ) denotes the trace of a matrix. By taking thelogarithm of the pdf under H , we have L ( Y ) = − M L ln( π ) − tr ( YY H ) . (7) Similarly, the pdf of Y under H is written as f ( Y | H , h , P a ) = L (cid:89) l =1 π M (cid:112) | K | exp (cid:2) − y Hl K − y l (cid:3) = 1 π ML (cid:112) | K | L exp (cid:2) − tr (cid:0) K − YY H (cid:1)(cid:3) (8)where K (cid:44) P a h aw h Haw + I M is the covariance matrix ofWillie’s observations under H . Here, we have | K | = P a (cid:107) h aw (cid:107) + 1 , (9)and K − = I M − h aw h Haw P a + (cid:107) h aw (cid:107) , (10)where K − is obtained using Woodbury Matrix Identity formatrix inversion [20].Putting in the expressions of | K | and K − in (8) and takingthe logarithm of the pdf under H , L ( Y ) = − M L ln( π ) − L (cid:0) P a (cid:107) h aw (cid:107) + 1 (cid:1) − tr (cid:0) YY H (cid:1) + (cid:107) h Haw Y (cid:107) (cid:16) P a + (cid:107) h aw (cid:107) (cid:17) . (11)The log likelihood ratio (LLR) can be thus written asLLR = L ( Y ) − L ( Y )= (cid:107) h Haw Y (cid:107) (cid:16) P a + (cid:107) h aw (cid:107) (cid:17) − L (cid:0) P a (cid:107) h aw (cid:107) + 1 (cid:1) . (12)Comparing the LLR to a threshold results in the followingoptimal decision rule (cid:107) h Haw Y (cid:107) (cid:16) P a + (cid:107) h aw (cid:107) (cid:17) − L (cid:0) P a (cid:107) h aw (cid:107) + 1 (cid:1) H ≷ H ϕ, (13)where ϕ = 0 for the LLR under the assumption of equalprobability of whether Alice transmits or not. The optimaldecision rule can then be obtained by a rearrangement. (cid:4) We note from (3) that the optimal detector at Willie is amaximum ratio combiner [21], which assigns weightage tothe observations at different antennas as per the correspondingchannel gains. Thus the antennas with better channel gainfrom Alice have higher contribution in the decision statistic in(3). We next present the detection error probabilities at Willieunder the optimal detector.
Lemma 2.
The detection error probabilities at Willie, i.e., P F A and P MD are given as P F A = 1 − γ (cid:16) L M , θ ∗ (cid:107) h aw (cid:107) (cid:17) Γ ( L M ) , (14) and P MD = γ (cid:16) L M , θ ∗ (cid:107) h aw (cid:107) ( (cid:107) h aw (cid:107) P a +1) (cid:17) Γ ( L M ) , (15) respectively, where γ ( a, b ) is the lower incomplete Gammafunction, Γ( x ) is the complete Gamma function, and θ ∗ is theoptimal threshold of Willie’s detector given in (4). Proof. Under hypothesis H , y l has a distribution given by CN ( , I M ) . Conditioned on the known channel coefficientsfrom Alice, the vector h Haw Y has a complex Gaussian distri-bution given by CN ( , (cid:107) h aw (cid:107) I L M ) . Thus, (cid:107) h Haw Y (cid:107) ∼ (cid:107) h aw (cid:107) χ L M , (16)where χ L M denotes a chi-squared random variable (RV) with L M degrees of freedom. Hence, P F A = P (cid:2) (cid:107) h Haw Y (cid:107) > θ ∗ | H (cid:3) . (17)Under hypothesis H , y l ∼ CN ( , P a h aw h Haw + I M ) , andconditioned on the known channels, the distribution of h Haw Y is given by CN (cid:0) , (cid:107) h aw (cid:107) ( (cid:107) h aw (cid:107) P a + 1) I L M (cid:1) . As a result,we have (cid:107) h Haw Y (cid:107) ∼ ( (cid:107) h aw (cid:107) ( (cid:107) h aw (cid:107) P a + 1)) χ L M , (18)giving P MD = P (cid:2) (cid:107) h Haw Y (cid:107) ≤ θ ∗ | H (cid:3) . (19)Calculating the probabilities in (17) and (19) gives the desiredresult, hence completing the proof. (cid:4) We note here that the analysis of the optimal detector atWillie under finite blocklength in fading scenarios is quitedifferent as compared to the infinite blocklength case. Ashighlighted in [13], for infinite observations at the adversary,uncertainties of transmitted signals and receiver noise vanishand the analysis is simplified, whereas this does not hold underthe finite blocklength scenario.IV. A
LICE ’ S A PPROACH TO A CHIEVE C OVERTNESS
Due to the involvement of incomplete Gamma function, thedetection performance at Willie does not lend itself well forfurther analysis. To proceed, we lower bound Willie’s detectionperformance using Pinsker’s inequality [19], giving P F A + P MD ≥ − (cid:114) D (cid:16) P L M || P L M (cid:17) , (20)where P L M and P L M denote the probability density functions(pdfs) of Willie’s observation vectors under hypothesis H and H for L M independent channel uses, respectively, and D ( P L M || P L M ) is the Kullback-Leibler (KL) Divergence from P L M to P L M . As per [14], D ( P L M || P L M ) ≤ (cid:15) must beensured to guarantee P F A + P MD ≥ − (cid:15) . Since Alice isunaware of her channels to Willie, she looks to minimize theexpected value of D ( P L M || P L M ) over all possible realizationsof her channels to Willie. The optimization problem at Aliceis thus stated as:P1 maximize L M ,P a L M R (1 − δ )subject to E (cid:107) h aw (cid:107) (cid:104) D (cid:16) P L M || P L M (cid:17)(cid:105) ≤ (cid:15) ,L M ≤ L max . (21)where E [ · ] denotes the statistical expectation, and is taken overall the channels from Alice to the multiple antennas at Willie.The expression for the KL-divergence for M ≥ is given as D (cid:16) P L M || P L M (cid:17) = L M D ( P || P )= L M (cid:18) ln (cid:0) (cid:107) h aw (cid:107) P a + 1 (cid:1) − (cid:107) h aw (cid:107) P a (cid:107) h aw (cid:107) P a + 1 (cid:19) , (22) g ( P a ) = λ M ( M − (cid:34) λ M ( P a ) meijer-G (cid:26) , , [0 , , M ] , [] , λP a (cid:27) − P (2 − M ) a Γ( M + 1) e λPa γ ( − M, λP a ) (cid:35) (28)meijer-G ( a, b, c, d, z ) = meijer-G ([ a , . . . , a n ] , [ a n +1 , . . . , a p ] , [ b , . . . , b m ] , [ b m +1 , . . . , b q ] , z )= G m,np,q ( a , . . . , a p , b , . . . , b q | z )= 12 πi (cid:90) (cid:16)(cid:81) mj =1 Γ( b j − s ) (cid:17) (cid:16)(cid:81) nj =1 Γ(1 − a j + s ) (cid:17)(cid:16)(cid:81) qj = m +1 Γ(1 − b j + s ) (cid:17) (cid:16)(cid:81) pj = n +1 Γ( a j − s ) (cid:17) z s ds (29)where, (cid:107) h aw (cid:107) is the sum of M independent exponential RVs,and for M > , constitutes an Erlang RV, with pdf given by f (cid:107) h aw (cid:107) ( h ; M, λ ) = E M ( h, λ )= λ M h M − e − λh ( M − h, λ ≥ . (23)In the following, we present the optimal choice of P a and L M for Alice to achieve a desired level of covertness. Proposition 1.
The optimal number of Alice’s channel uses, L ∗ M , in terms of Alice’s transmit power, maximizing thethroughput to Bob while satisfying a given covertness require-ment, (cid:15) , is given by L ∗ M = min (cid:18) L max , (cid:15) g ( P a ) (cid:19) , (24) while the optimal transmit power at Alice is the solution to maximize P a L ∗ M R (1 − δ ) , (25) where g ( P a ) is as given in (28).Proof. The expectation in (21) for
M > is calculated as E (cid:107) h aw (cid:107) (cid:104) D (cid:16) P L M || P L M (cid:17)(cid:105) = L M g ( P a ) , (26)where the function g ( P A ) and meijer-G ( a, b, c, d, z ) within g ( P a ) are defined as in (28) and (29), respectively [22]. Dueto the covertness constraint, P1 is now restated asP1.1 maximize L M ,P a L M R (1 − δ )subject to L M ≤ min (cid:18) L max , (cid:15) g ( P a ) (cid:19) . (27)Since R is an increasing function of P a and L M , the covert-ness requirement puts an upper limit on the number of channeluses by Alice in a block for a given P a . The upper limit for L M is thus determined by the tighter bound between the covertnessconstraint and L max . Accommodating the optimal L M resultsin the optimization problem in (25) which is of one dimensionand can be solved by methods of efficient numerical search,hence concluding the proof. (cid:4) We note from (24) that the optimal choice of blocklength, L ∗ M , is a decreasing function of the transmit power used byAlice and is upper-bounded by the delay constraint determinedby L max . This fact will be more evident and explained in thenumerical results section. . . . . . · − P a O p t i m a l N o . o f A li ce ’ s T r a n s m i t S y m b o l s , L ∗ M L max = 200 , (cid:15) = 0 . L max = 500 , (cid:15) = 0 . L max = 200 , (cid:15) = 0 . L max = 500 , (cid:15) = 0 . Fig. 1. Number of Alice’s optimal transmit symbols, L ∗ M , versus the transmitpower, P a . Corollary 1.
The optimal number of Alice’s channel uses,for M = 1 , in terms of Alice’s transmit power, maximizingthe throughput to Bob while satisfying a given covertnessrequirement, is given by L ∗ = min (cid:18) L max , (cid:15) f ( P a ) (cid:19) , (30) while the optimal transmit power at Alice can be foundsimilarly as in Proposition . Here, f ( P a ) is given as f ( P a ) = − (cid:20) (cid:18) λP a (cid:19) e λPa Ei (cid:18) − λP a (cid:19)(cid:21) , (31) and Ei ( · ) is the Exponential Integral function.Proof. The proof follows by simply putting M = 1 in theresults given in Proposition . (cid:4) V. R
ESULTS A ND D ISCUSSIONS
In this section, we present numerical results examining theeffect of reliability and covertness on the optimal parametersfor Alice and the achievable covert throughput. Unless oth-erwise stated, the parameters are set as follows: δ = 0 . , λ = 1 , and | h ab | = 1 . It should be noted here that dueto the assumption of Bob and Willie’s receiver noise being CN (0 , , the transmit power at Alice has units relative to theconsidered additive noise.Fig. 1 shows the optimal number of Alice’s transmit sym-bols against a range of her transmit power, P a , for differentvalues of L max and (cid:15) . First, the effect of limiting Alice’s M C o v e rt T h r o u g hpu t , L M R ( − δ )( b i t s ) (cid:15) = 0 .
1, Optimal L M (cid:15) = 0 .
1, Fixed L M (cid:15) = 0 .
2, Optimal L M (cid:15) = 0 .
2, Fixed L M (cid:15) = 0 .
3, Optimal L M (cid:15) = 0 .
3, Fixed L M Fig. 2. Achievable covert throughput from Alice to Bob, L M R (1 − δ ) , versusthe number of antennas at Willie, M , for varying covertness requirements. maximum transmit symbols is evident for lower values of P a ,as decreasing P a allows for Alice to use a higher blocklengthunder a certain covertness constraint. Furthermore, for a givenvalue of (cid:15) , an increase in P a requires using lower number oftransmit symbols for a given covertness requirement which isin agreement with the intuition that if Alice chooses to transmitat a higher power, the transmission time should be reduced.This also represents a trade-off between achieving covertnessand maximizing the throughput since the covert throughput isan increasing function of both P a and L M , while achievingcovertness requires a decrease in both.In Fig. 2, we show the achievable throughput againsta multi-antenna Willie for varying covertness requirementsunder the optimal choice of L M and P a . We also show theachievable throughput for the case where L M is fixed whilethe transmit power is optimized to satisfy a required covertcriteria. It should be noted here that in calculating thesethroughput results, we consider a lower bound on L M , whichis due to the use of approximated expression of R as given in(2), and this bound ensures that the calculated rate is alwaysnon-negative. We note that (cid:15) = 0 . represents a relativelyrelaxed covertness requirement.As we can see, the difference between utilizing the optimaland fixed value of L M is evident in terms of the differencein throughput from Alice to Bob. Furthermore, as the numberof antennas at Willie grows beyond a single antenna, thereis a very sharp decrease in the achievable covert throughputin both cases and depending on the covertness requirement,it reaches zero very quickly. This shows the effectiveness ofWillie in detecting any covert transmissions using more thanone antenna. As we see, even for (cid:15) = 0 . , the throughputdecreases fairly quickly to zero and hence covertness can notbe achieved beyond M = 16 .VI. C ONCLUSION
In this work, we have considered the performance of covertcommunication in the presence of a multi-antenna adversarywhile under strict delay constraints. We have analyzed theeffect of covertness requirement and the number of antennas at Willie on Alice’s achievable covert throughput. It has beenshown that the improved detection capability of Willie dras-tically reduces the covert throughput. This letter presented aninitial work on covert communication under delay constraintsand in the presence of a multi-antenna adversary. Futurework will focus on devising improved covertness schemes forachieving non-zero throughput despite the higher number ofantennas at Willie, and under more complex fading modelsencompassing both LOS and NLOS scenarios.R
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