Covert Communication with Finite Blocklength in AWGN Channels

0.32 MB·

by  Shihao Yan

Covert Communication with Finite Blocklength in
AWGN Channels
Shihao Yan†, Biao He‡, Yirui Cong†, and Xiangyun Zhou†
†Research School of Engineering, The Australian National University, Canberra, ACT, Australia
‡Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong
Emails: shihao.yan, yirui.cong, xiangyun.zhou @anu.edu.au,[email protected]
{ }
7 Abstract—Covertcommunicationistoachieveareliabletrans- respect to the square root of n was characterized for a broad
1 mission from a transmitter to a receiver while guaranteeing an class of discrete memoryless channels (DMCs) and AWGN
0 arbitrarily small probability of this transmission being detected channels in [6]. We note that this square root law requires
2 by a warden. In this work, we study the covert communication
a pre-shared secret to be established between Alice and Bob
inAWGNchannelswithfiniteblocklength,inwhichthenumber
n of channel uses is finite. Specifically, we analytically prove that prior to Alice’s transmission. This pre-sharedsecret is proved
a theentireblock(all available channel uses) shouldbeutilizedto to be unnecessary for the square root law when the channel
J
maximize the effective throughput of the transmission subject quality from Alice to Bob is higher than that from Alice to
1 to a predetermined covert requirement. This is a nontrivial
Willie,forbinarysymmetricchannel(BSC)[7],DMC[8],and
3 result because more channel uses results in more observations
at thewarden for detectingthetransmission. Wealso determine AWGN channel [8].

] the maximum allowable transmit power per channel use, which In the square root law we have (√n)/n 0 as n ,
T O → →∞
is shown to decrease as the blocklength increases. Despite the which states that the rate is asymptotically zero (i.e., the
.I decrease inthemaximum allowable transmit power perchannel average number of bits that can be covertly and reliably
s use, themaximum allowable total power over theentireblockis
transmitted per channel use asymptotically approaches zero).
c
proved to increase with the blocklength, which leads to the fact
[ However,insomescenariosa positiveratehasbeenprovedto
that the effective throughput increases with the blocklength.
be achievable (e.g., [7,9–13]). For example, it is proved that
1
v I. INTRODUCTION a positive rate can be obtained when Willie has uncertainty
1 Infuturewirelessnetworks,thedemandforwirelessdatais aboutthe receivernoise variancein AWGN channels[11,13],
9 growingat such a rate that requires1000xtoday’s capacity in when Willie does not exactly know the receiver noise model
8 thenextfivetotenyears.Againstthisbackground,crucialcon- in BSC channels [7], or when Willie lacks knowledge of his
8
cerns on the security and privacy of wireless communications channel characteristics in AWGN and block fading channels
0
. are emergingsince a largeamountof confidentialinformation [12,13]. In addition to the noise or channel uncertainty, as
1
(e.g., email/bank account information and password, credit provedin[10]apositiveratecanalsobeachievedwhenWillie
0
card details) is transferredoverwireless networks.In addition has uncertainty on the time instant of the communication.
7
1 to the secrecy and integrity of the transmitted information, In the literature as seen in the aforementioned works, only
: in some scenarios a user may wish to transmit messages [11] mentioned the impact of finite samples (i.e., finite n) on
v
over wireless networks without being detected. This is due to the detection performance at Willie. It is numerically shown
i
X the fact that (for example) the exposure of this transmission thatwithnoiseuncertaintyatWillietheremayexistanoptimal
r may disclose the user’s location information, which probably number of samples that maximizes the communication rate
a
violates the privacy of the user. Therefore, covert communi- subject to ξ 1 ǫ, where ξ is the sum of false positive and
≥ −
cation is attracting an increasing amount of research interests miss detection rates at Willie and 0 < ǫ 1 is an arbitrarily
≤
recently (e.g., [1–3]). In covert communication, a transmitter small number. Besides the detection performance at Willie,
(Alice) intends to communicate with a legitimate receiver finite n also hassignificantimpactonthe maximalachievable
(Bob) without being detected by a warden (Willie), who is rate R of the channel from Alice to Bob (i.e., the maximal
observing this communication. achievable rate decreases as n decreases for a fixed decoding
In fact, covert communication was addressed by spread error probability δ) [14], which has not been considered
spectrum techniques in the early 20th century and a review in the literature of covert communication (including [11]).
on spread spectrum techniques can be found in [4]. However, Therefore, the impact of finite n on covert communication
the performance limit of covert communication has not been has not been well examined. This leaves a significant gap in
fully examined in the literature and recently attracts much ourunderstandingof theperformancelimitof practicalcovert
research attention. Considering additive white Gaussian noise communication, since in practice the length of a codeword is
(AWGN) channels, a square root law has been derived in [5], always finite. For example, to achieve transmission efficiency
which states that Alice can transmit no more than (√n) (e.g., short delay) we may require the codeword to be short
O
bitsin n channelusescovertlyand reliablyto Bob.Following (e.g., in the order of 100 channeluses) for vehicle-to-vehicle
[5], the scaling constant of the amount of information with communication or real-time video processing [15].
is equipped with a single antenna. We assume the channel
from Alice to Bob and the channel from Alice to Willie are
only subject to AWGN. In the covert communication, Alice
transmits n complex-valued symbols x[i] (i = 1,2,...,n) in
each codeword to Bob, while Willie is passively collecting
n observations on Alice’s transmission to detect her presence
(i.e.,whether Alice is transmitting).In this work,we consider
that the length of a codeword is constrained by a maximum
blocklength denoted by N. Thus, we have n N as a
≤
constraint on n. We denote the AWGN at Bob and Willie
as r [i] and r [i], respectively, where r [i] (0,σ2),
r [i]b (0w,σ2), σ2 and σ2 are thebnois∼e vCarNiancesbat
w ∼ CN w b w
Fig.1. Illustrationofthesystemmodelofinterestforcovertcommunication. Bob and Willie, respectively. In addition, we assume that
x[i], r [i], and r [i] are mutually independent. We denote
b w
the transmit power of Alice as P (i.e., E x[i]2 = P).
{| | }
Furthermore,weassumethatAliceadoptsGaussiansignaling,
A. Our Contributions
i.e., x[i] (0,P).
ConsideringAWGNchannels,westudytheimpactoffinite ∼CN
n on both the maximal achievable rate at Bob and detection B. Channel Coding Rate for Finite Blocklength
performance at Willie in covert communication. To this end, The received signal at Bob for each signal symbol is given
noting that the decoding error probability δ is not negligible by
when n is finite, we first propose to adopt the effective
y [i]=x[i]+r [i]. (1)
throughput η (i.e., η = nR(1 δ)) subject to ξ 1 ǫ, b b
− ≥ −
as the performance metric to evaluate covert communication.
As pointed out by [14], the decoding error probabilityat Bob
As can be seen from the definition of η, it explicitly captures isnotnegligiblewhennisfinite.Assuch,foragivendecoding
thetradeoffamongn,R,andδ foragivencovertrequirement. errorprobabilityδ thechannelcodingrateofthechannelfrom
We consider a maximum blocklength of N channel uses, Alice to Bob can be approximated by [14,16]
in which the covert information needs to be transmitted.
Hence, the actual number of channel uses n is constrained R log (1+γ ) γb(γb+2)Q−1(δ) + log2(n), (2)
by n ≤ N. Although a larger n offers more observations to ≈ 2 b −sn(γb+1)2 ln(2) 2n
Willie for detecting the transmission, we analytically prove
where γ = P/σ2 is the signal-to-noise ratio (SNR) at Bob,
that the optimal value of n that maximizes η subject to the b b
andQ−1()istheinverseQ-function.Equivalently,foragiven
given covert requirement is N (i.e., the entire block with
·
channel coding rate R, the decoding error probability at Bob
all available channel uses). We also determine the maximum
allowable transmit power per channel use (denoted by P∗) is given by
that achieves the maximum η. Our examination shows that √n(1+γ ) ln(1+γ )+ 1ln(n) Rln2
P∗ decreases as N increases, which is due to the fact that δ =Q b b 2 − .
increasing N forces Alice to allocate less power for each (cid:0) γb(γb+2) (cid:1)!
(3)
channel use to meet the covert requirement. Nevertheless, we p
show that the maximum allowable total transmit power (i.e., C. Binary Hypothesis Testing at Willie
NP∗) increases as N increases, which leads to the fact that In order to detect Alice’s presence, Willie is to distinguish
the effective throughput of the communication from Alice to
the following two hypotheses
BobincreasesasN increases.Theresultsinthispaper,forthe
first time, provide important insights on the design of covert 0 : yw[i]=rw[i]
H (4)
communication with a finite blocklength. ( 1 : yw[i]=x[i]+rw[i],
H
Notations: Scalar variables are denoted by italic symbols.
where denotes the null hypothesis where Alice is not
Vectorsandmatricesaredenotedbylower-caseandupper-case H0
transmitting, denotes the alternative hypothesis where
boldfacesymbols,respectively.Given a vector x, x[i] denotes H1
thei-thelementofx. TheexpectationisdenotedbyE and Alice is transmitting, and yw[i] is the received signal at
(0,σ2) denotes the circularly-symmetric complex{n·o}rmal Willie. Following the assumptions detailed in Section II-A,
CdiNstribution with zerIoI.mSeaYnSTanEdMvMarOiaDnEceLσ2. waσsw2ef)h,(ayrvewes[pit]eh|cHeti0vli)ekl=eyl.ihCIonNodt(h0ef,uσcnw2ocv)tieoarnnscdoofmf(myywwu[n[iii]]c|Hautni1od)ne=,rtHhCeN0 ua(l0nt,idmPHa+t1e
goal of Willie is to minimize the total error rate, which is
A. Channel Model
given by
The system model of interest for covert communication is
ξ =P +P , (5)
illustrated in Fig. 1, where each of Alice, Bob, and Willie F M
where PF , Pr( 1 0) is the false positive rate, PM , D. Covert Requirement
D |H
Pr( 0 1) is the miss detection rate, 1 and 0 are the Covertcommunicationrequiresξ 1 ǫforsomearbitrar-
binaDry|Hdecisions that infer whether AlicDe is presDent or not, ily small ǫ. As per (12), we can en≥sure− (P0 P1) 2ǫ2 in
respectively. We assume that Willie knows both P and σw2 order to guarantee ξ 1 ǫ. We note thDat (kP0 P≤1) 2ǫ2
exactly, and thus the optimal test that minimizes ξ is the is a more strict con≥strai−nt relative to ξ D 1 k ǫ a≤s per
likelihood ratio test with λ = 1 as the threshold1, which is (12). From a conservative point of view a≥nd to−avoid the
given by complex expressions for P and P , in this work we adopt
F M
PP10 ,,Qnini==11ff((yyww[[ii]]||HH10)) DD≥<10 1. (6) tDpiro(onPv.0idAkePls1go)o,o≤tdhec2ovǫv2aeluratesneothsfse.ǫTrieshquuessi,rpeienmcietahnliltsyfwovroerrckyowvsmeeratollcnoliynmcmoorundnseiirdcteaor-
After performing somQe algebraic manipulations, (6) can be ǫ (0,0.5] because ǫ > 0.5 means that Willie is allowed to
∈
reformulated as achieve more than 50% success detection rate.
n D1 III. COVERT COMMUNICATION WITH A FINITENUMBER
1
T , n |yw[i]|2 ≥< Γ, (7) OFCHANNEL USES
Xi=1 D0 In this section, we first adopt the effective throughput to
where T is the average power of each received symbol at evaluate the performanceof covert communicationin AWGN
Willie and Γ is the threshold for T, which is given by channels with finite blocklength. Then, we determine the
optimal n and P that maximize this effective throughput
(P +σ2)σ2 P +σ2
Γ= w w ln w . (8) subject to the covert requirement.
P σ2
(cid:18) w (cid:19) A. Effective Throughput
As per (6) and (7), we note that the radiometer is indeed the
The square root law states that Alice can transmit no more
optimal detector when Willie knows the likelihood functions
than (√n) bits in n channel uses covertly and reliably to
exactly (i.e., there are no nuisance parameters embedded in O
Bob. Such scaling-law results are obtained when n .
the likelihoodfunctions).Following(7)and notingthatT isa → ∞
As such, these square-law results cannot be applied in the
chi-squared random variable with 2n degrees of freedom, the
covert communication with finite n. In this work, we focus
false positive rate and miss detection rate are given by [9,11]
on the amount of information that can be transmitted reliably
γ n, nΓ from Alice to Bob for a given positive ǫ. Noting that the
PF =Pr(T >Γ|H0)=1− (cid:16)Γ(nσ)w2 (cid:17), (9) decodingerrorprobabilityofachannelwithfiniteblocklength
is not negligible, we adopt the effective throughput from
γ n, nΓ Alice to Bob as the main performance metric for the covert
PM =Pr(T <Γ|H1)= (cid:16) Γ(Pn+)σw2 (cid:17), (10) communication with finite blocklength, while utilizing the
covertrequirementas the constraint. The effective throughput
where Γ(n) = (n 1)! is the gamma function and γ(, ) is from Alice to Bob is defined as [19,20]
− · ·
the incomplete gamma function given by
η =nR(1 δ). (14)
x −
γ(n,x)= e−ttn−1dt. (11) We note that η gives the average number of information bits
Z0 that can be transmitted from Alice to Bob reliably (excluding
With the radiometer as the optimal detector, following information bits suffering from decoding errors) by utilizing
Pinsker’s inequality, we have a lower bound on ξ, which is a codeword with finite length n.
given by [5,17,18]
B. Optimal Number of Channel Uses and Transmit Power
ξ 1 1 (P0 P1), (12) The ultimate goal of our design in covert communication
≥ − 2D k is to achieve the maximum effective throughput while guar-
r
where (P0 P1) is the Kullback-Leibler (KL) divergence anteeingthecovertrequirement.To this end,we first consider
from PDto Pk, which can be expressed as a fixed channel coding rate R and focus on the design of
0 1
the number of channel uses and the transmit power P, since
P +σ2 P
D(P0kP1)=n ln σ2 w − P +σ2 . (13) tfhroemdeAsilgicneotfonBoabndanPd athffeecdtestebcottihonthpeereffofremctaivnecethartouWgihllpiue.t
(cid:20) (cid:18) w (cid:19) w(cid:21)
Assuch, fora givenR the optimizationproblemin the covert
1Wenotethatλ=1isduetotheunknownorequalaprioriprobabilities, communication of interest can be written as
i.e.,P0andP1areunknownorequal,whereP0istheaprioriprobabilitythat
H0 istrue,P1 istheaprioriprobability thatH1 istrue,andP0+P1=1. argmaxη, (15)
If both P0 and P1 are known, the total error rate is reformulated as ξ = n,P
tPh0ePlFike+lihPoo1dPMratiaontdestthewiothptiλm=al Pte1st/Pth0a.tWmeinaimlsoizensottehitsharteftohremauslsautemdpξtioins s.t. (P0 P1) 2ǫ2, (16)
D k ≤
ofequalaprioriprobabilitiesiscommonlyadoptedintheliteratureofcovert
n N. (17)
communication (e.g.,[5,10,11]). ≤
Theorem 1: The optimal valuesof n and P that maximize 101
the effective throughput η subject to (P0 P1) 2ǫ2 and ξ≥1−ǫ
n N are derived as D k ≤ D(P0kP1)≤2ǫ2
≤
n∗ =N, (18)
P∗ 100
P∗ =(σ2 +P∗) ln +1 2ǫ2N , (19)
w σ2 −
(cid:20) (cid:18) w (cid:19) (cid:21) ∗
P
where P∗ is the solution to the fixed-point equation (19). N
N =2000
Proof: The detailed proof is provided in Appendix.
Based on Theorem 1, we see that it is best for Alice to 10-1
transmitoverallavailablechannelusesforcovertcommunica-
N =500
tion,providedthatthetransmitpowerisoptimizedtomaintain
the same level of covertness despite that Willie has more
N =100
observations when n is larger. The same level of covertness
is achieved by reducing the transmit power when n becomes 10-120-3 10-2 10-1
larger, which can be seen from (19) that P∗ decreases with ǫ
N. It is interesting to observe that both n∗ and P∗ are not
Fig.2. MaximumallowabletotaltransmitpowerNP∗versusǫfordifferent
functions of R. This demonstrates that the obtained n∗ and values ofN,whereσb2=σw2 =1.
P∗ are globally optimal, regardless the value of the channel
codingrateR.Assuch,theoptimalvalueofRthatmaximizes
(a) (c)
the effective throughputsubject to the covertrequirementcan 10 -17
be obtained through ξ≥1−ǫ ξ≥1−ǫ
8 D(P0kP1)≤2ǫ2 -18 D(P0kP1)≤2ǫ2
R∗ =ar0g≤mRinNR[1−δ(P∗,N,R)], (20) ∗NP6 ∗P(dB)--2109
where δ(P∗,N,R) is obtained by substituting P = P∗ and 4 -21
n∗ =N into(3).WenotethatR∗canbealsoobtainedthrough
2 -22
searchingtheoptimalvalueofδ thatmaximizesη forn∗ =N 200 400 600 800 1000 200 400 600 800 1000
N N
and P = P∗. We define δ∗ = δ(P∗,N,R∗) and denote the (b) ×10-3 (d)
4 5
maximum effective throughput as η∗, which is achieved by ξ≥1−ǫ ξ≥1−ǫ
substituting P∗, n∗, R∗, and δ∗ into (14). 3 D(P0kP1)≤2ǫ2 4 D(P0kP1)≤2ǫ2
IV. NUMERICAL RESULTS η(bit)2 /N(bit)23
η
Inthissection,weprovidenumericalresultsontheeffective 1
1
throughput subject to ξ 1 ǫ to verify our analysis on the
covertcommunicationw≥ith −(P0 P1) 2ǫ2astheconstraint. 0200 400 600 800 1000 0200 400 600 800 1000
D k ≤ N N
In Fig. 2 we plot the maximum allowable total transmit
power NP∗ over the entire block versus ǫ. In this figure and Fig.3. NP∗,η,P∗,andη/N versusN,whereσb2=σw2 =1,δ=0.01,
thefollowingfigures,thecurvesforξ 1 ǫareachievedby andǫ=0.1.
≥ −
numerically evaluating the false positive and detection rates
as per (9) and (10), respectively. In this figure, we observe
that the NP∗ with ξ 1 ǫ as the constraint is higher in Fig. 3 (a) and Fig. 3 (b), respectively. Although NP∗
than that with (P0 P≥1) −2ǫ2 as the constraint. This is increases as shown in Fig. 3 (a), it is interesting to observe
D k ≤
due to the fact that the equality in (12) cannot be achieved thatthemaximumallowabletransmitpowerP∗monotonically
in the considered system model, and hence (P0 P1) 2ǫ2 decreases as N increases in Fig. 3 (c). This can be explained
D k ≤
is a more strict constraint than ξ 1 ǫ. We also observe by(19)inourTheorem1.Intuitively,thisisduetothefactthat
≥ −
thatNP∗ increases(hencethe effectivethroughputincreases) asthenumberofobservationsatWillieincreases,Alicehasto
as N increases, which can be explained by our Theorem 1. reducehertransmitpowerinorderto meetthesame detection
Finally, we observe that NP∗ decreases (hence the effective performance at Willie. In Fig. 3 (d), we observe that the
throughputdecreases) as ǫ decreases, which demonstrates the effectivethroughputperchanneluse(i.e.,η/N)monotonically
tradeoff between the covert requirement and the achievable increases as N increases. This is due to the fact that the
effective throughput (e.g., a more strict covert requirement decrease in δ (i.e., the decoding error probability given in
leads to a smaller effective throughput). (3)) caused by increasing N is more than the increase in δ
In Fig. 3, we plot NP∗, η, P∗, and η/N versus N in caused by the reduction of P∗ as shown in Fig. 3 (c). These
different sub-figures, respectively. As expected, we first ob- aforementionedobservationsbased onFig. 3 demonstratethat
serve that NP∗ and η monotonically increase as N increases increasing N not only helps Alice to allocate less transmit
10 ×10-3
12
σ2=−2dB
9 b
σb2=0dB ǫ=0.08
8 10 σb2=2dB
7
8
η(bit) 56 ∗η/N 6
4
4
3
N=1000
N=500
2
N=100 2 ǫ=0.02
1
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 100 200 300 400 500 600 700 800 900 1000
δ N
Fig. 4. Effective throughput η versus the decoding error probability δ for Fig.5. Maximumeffectivethroughputperchanneluseη∗/N versusN for
different values ofN,whereσb2=σw2 =1andǫ=0.1. different values ofǫandσb2,whereσw2 =1.
powerto eachchannelusein orderto maintainthesame level to each channel use, but also decreases the decoding error
of covertness, but also reduces the decoding error probability probability of the communication from Alice to Bob.
in the communication from Alice to Bob, which turns out to
ACKNOWLEDGMENTS
improvetheeffectivethroughputofthecovertcommunication.
ThisworkwassupportedbytheAustralianResearchCoun-
In Fig. 4, we plot the effective throughput η subject to
cil’s Discovery Projects (DP150103905).
ξ 1 ǫ versus the decoding error probability δ. We first
≥ −
observe that the optimal value of δ that maximizes η indeed APPENDIX
exists, based on which we can determine the optimal R. We
WepresentourproofofTheorem1inthefollowing6steps.
also observe that the optimal value of δ decreases as N
increases. As shown in Fig. 3 (c), the maximum allowable Step 1: We note that η and D(P0kP1) are both mono-
tonically increasing functions of P and n. As such, we can
transmit power P∗ decreases as N increases. As per (2), the
concludethat the equalityin the constraint(16)is alwaysmet
observation, that both δ∗ and P∗ decreases as N increases,
indicatesthattheoptimalchannelcodingrateR∗ decreasesas in order to maximize η. Thus, we have D(P0kP1)=2ǫ2 and
following (13) we have
N increases. We also plot the maximum effective throughput
perchanneluse(i.e.,η∗/N)versusN inFig.5.Inthisfigure, 2ǫ2
n= , (21)
we first observe that as N increases η∗/N increases, which f(γ )
w
is consistent with our observation found in Fig. 3 (d). We
where
also observe that as ǫ increases slightly (e.g., from 0.02 to
0.08)η∗/N significantlyincreases.Thisdemonstratesthatthe f(γw), D(P0kP1) =ln(γw+1) γw , (22)
achievable effective throughput is very sensitive to the the n − γw+1
covert requirement. and γ =P/σ2 is the SNR at Willie.
w w
Step2:We notef(0)=0 andwe derivethe firstderivative
V. CONCLUSION
of f(γ ) with respect to γ as
w w
Thisworkinvestigatedthecovertcommunicationwithfinite
∂f(γ ) γ
blocklength (i.e., a finite number of channel uses n N) w = w 0, (23)
≤ ∂γ (γ +1)2 ≥
overAWGNchannels.Weprovedthattheeffectivethroughput w w
of covert communication is maximized when all available which leadsto the factthat f(γ ) is a monotonicallyincreas-
w
channel uses are utilized, i.e., n∗ = N. To guarantee the ing function of γw. With the constraint (P0 P1) = 2ǫ2, n
D k
same level of covertness, the maximum allowable transmit is a monotonically decreasing function of f(γ ) as per (21),
w
power per channel use decreases as N increases, while the whichresults in thatn is a monotonicallydecreasingfunction
maximumallowabletotaltransmitpoweroverallchanneluses of γ (thus of P).
w
increases as N increases. In contrast, we found that both the Step3:Insteadofdirectlyprovingn∗ =N formaximizing
effective throughput and the effective throughput per channel the effective throughput, we next prove that n∗ = N maxi-
use increase as N increases. This is due to the fact that mizes nγ (i.e., maximizes nP) under the constraint (21) in
w
increasing N not only reduces the transmit power allocated the remaining steps. This is due to the fact that nP is the
total transmit power for the n channel uses and the effective (0.4835)2 as γn=2 >1.16, which leads to nγ >2.32 when
w w
throughputincreasesasthetotaltransmitpowerincreases[14]. n = 2. As such, we have nγ < 2.3145 when n = 1 and
w
Step 4: We next prove that either n = 1 or n = N max- nγ > 2.32 when n = 2, which results in nγ for n = 2 is
w w
imizes nγ . To this end, in the following we first show that larger than nγ for n=1. We recall that nγ monotonically
w w w
nγ initially decreases and then increases with n. Following increases with n when n n†. Therefore, for 0.4835 ǫ
w
≥ ≤ ≤
(21) and (22), we have 0.5 the optimal value of n that maximizes nγ is N.
w
2ǫ2 Step 6: So far, we haveprovedn∗ =N. Then,substituting
nγw = , (24) n∗ =N into (21), we obtain the fixed-pointequation in (19).
g(γ )
w
where g(γ ) is given by REFERENCES
w
[1] P. H. Che, S. Kadhe, M. Bakshi, C. Chan, S. Jaggi, and A. Sprintson,
ln(1+γ ) 1
g(γ )= w . (25) “Reliable,deniableandhidablecommunication:Aquicksurvey,”inProc.
w
γw − 1+γw IEEEInf.TheoryWorkshop,Nov.2014,pp.227–231.
[2] B.A.Bash,D.Goeckel, D.Towsley,andS.Guha, “Hidinginformation
We thenderivethefirstderivativeofg(γ )withrespecttoγ
w w in noise: Fundamental limits of covert wireless communication,” IEEE
as Commun.Mag.,vol.53,no.12,pp.26–31,Dec.2015.
[3] B. He, S. Yan, X. Zhou, and V. Lau, “On covert communication
∂g(γ ) h(γ )
w = w , (26) with noise uncertainty,” IEEECommun. Lett., accepted to appear, DOI:
∂γ γ2(1+γ )2 10.1109/LCOMM.2016.2647716, Dec.2016.
w w w
[4] M. K. Simon, Jim K. Omura, Robert A. Scholtz, and Barry K. Levitt,
where SpreadSpectrum Communications Handbook, McGraw-Hill, 1994.
[5] B.Bash,D.Goeckel,andD.Towsley,“Limitsofreliablecommunication
h(γw)=2γw2 +γw−(1+γw)2ln(1+γw). (27) withlowprobabilityofdetectiononAWGNchannels,”IEEEJ.Sel.Areas
Commun.,vol.31,no.9,pp.1921–1930, Sep.2013.
We note that there are only two solutions to h(γw) = 0 for [6] L. Wang, G. Wornell, and L. Zheng, “Fundamental limits of communi-
γ 0, including γ = 0 and γ = γ†.2 We also note that cation with low probability of detection,” IEEETrans. Inf. Theory, vol.
w ≥ w w w 62,no.6,pp.3493–3503,Jun.2016.
as γ we have h(γ ) . Then, we can conclude
w → ∞ w → −∞ [7] P.H.Che, M.Bakshi, and S.Jaggi, “Reliable deniable communication:
tAhsatshuc(γhw, )no≥tin0gfγor2(γ1w+<γγw†)2and0h(aγnwd)f≤oll0owfoinrgγw(26≥),γww†e. 2H0i1d3in,gppm.e2s9sa4g5e–s29in49n.oise, in Proc. IEEEInt’l. Symp. Info. Theory, Jul.
have ∂g(γ )/∂γ w 0 forwγ ≥< γ† and ∂g(γ )/∂γ 0 [8] M.R.Bloch,“Covertcommunicationovernoisychannels:Aresolvability
w w ≥ w w w w ≤ perspective,”IEEETrans.Inf.Theory,vol.62,no.5,pp.2334–2354,May
for γw ≥ γw†. This indicates that g(γw) initially increases 2016.
and then decreases with γ . As per (24), we know that nγ [9] S.LeeandR.Baxley,“Achievingpositiveratewithundetectablecommu-
w w
nicationoverAWGNandRayleighchannels,”inProc.IEEEICC,2014,
monotonically decreases with g(γ ), which leads to the fact
w Jun.2014,pp.780–785.
that nγw first decreases and then increases as γw increases [10] B. A. Bash, D. Goeckel, and D. Towsley, LPD communication when
(i.e., nγ has one minimum value but no maximum value). the warden does not know when, in Proc. IEEEInt. Symp. Inf. Theory,
w
Jul.2014,pp.606–610.
We recall that n is a monotonically decreasing function of
[11] S.Lee,R.J.Baxley,M.A.Weitnauer,andB.Walkenhorst,“Achieving
γw under the constraint (21), which is proved following (23). undetectable communication,” IEEE J. Sel. Topics Signal Process., vol.
Therefore, we conclude that nγ first decreases and then 9,no.7,pp.1195–1205, Oct.2015.
w
[12] T. Sobers, B. Bash, D. Goeckel, S. Guha, and D. Towsley, “Covert
increasesasn increases, andthusthe maximumvalueof nγ
w communicationwiththehelpofanuninformedjammerachievespositive
is achieved either at n=1 or n=N. rate,” in Proc. Asilomar Conf. on Signals, Systems, and Comput., Nov.
Step 5: We next prove that n=N (not n=1) maximizes 2015,pp.625–629.
nγ .Substitutingγ† into(21),wehaven† =2ǫ2/f(γ†).For [13] D.Goeckel,B.Bash,S.Guha,andD.Towsley,Covertcommunications
w w w when the warden does not know the background noise power, IEEE
0 < ǫ < 0.4835, we have n† < 1 due to f(γw†) > 0.4675. Commun.Lett.,vol.20,no.2,pp.236–239, Feb.2016.
When n† < 1, nγ increases with n due to n 1. As such, [14] Y.Polyanskiy,H.Poor,andS.Verdu,“Channelcodingrateinthefinite
w
≥ blocklength regime,” IEEETrans.Inf. Theory, vol.56,no.5,pp.2307–
for 0 < ǫ < 0.4835 the optimal value of n that maximizes
2359,May2010.
nγw is N (i.e., n∗ = N). For 0.4835 ǫ 0.5, we have [15] B. Makki, T.Svensson, and M.Zorzi, “Finite block-length analysis of
≤ ≤
n† < 2 again due to f(γ†) > 0.4675. We next confirm that spectrumsharingnetworksusingrateadaptation,”IEEETrans.Commun.,
w
even for n† < 2 we still have n∗ = N. To this end, we only vol.63,no.8,pp.2823–2835, Aug.2015.
[16] G. Ozcan andM. C. Gursoy, “Throughput of cognitive radio systems
have to confirm nγw for n=2 is larger than that for n =1. withfiniteblocklengthcodes,”IEEEJ.Sel.AreasCommun.,vol.31,no.
When n = 1, following (21) we have f(γ ) = 2ǫ2. The 11,pp.2541–2554, Nov.2013.
w
maximumvalue of γ that guaranteesf(γ )=2ǫ2 (i.e., n= [17] E.LehmannandJ.Romano,TestingStatisticalHypotheses,3rded.New
w w York:Springer, 2005.
1) is obtained when ǫ = 0.5 since f(γw) is a monotonically [18] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd
increasing function of γ as proved by (23). We obtain this ed.JohnWiley&Sons,Hoboken, NJ,2002.
w
maximum value by solving f(γ ) = 0.5 as γn=1 < 2.3145, [19] X.Zhou,M.R.McKay,B.Maham,andA.Hjørungnes,“Rethinkingthe
w w secrecy outage formulation: A secure transmission design perspective,”
which leads to nγw < 2.3145 when n = 1. When n = 2, IEEECommun.Lett.,vol.15,no.3,pp.302–304,Mar.2011.
following(21)wehavef(γ )=ǫ2.Theminimumvalueofγ [20] S.Yan,N.Yang,G.Geraci,R.Malaney,andJ.Yuan,“Optimization of
w w
thatguaranteesf(γ )=ǫ2 (i.e.,n=2)is obtainedwhenǫ= coderatesinSISOMEwiretapchannels,”IEEETrans.WirelessCommun.,
w vol.14,no.11,pp.6377–6388, Nov.2015.
0.4835. We obtain this minimum value by solving f(γ ) =
w
2Weobtainγw† ≈2.1626bynumerically solvingh(γw)=0.