achieving covertness and secrecy: a new paradigm for

1

Achieving Covertness and Secrecy: A NewParadigm for Secure Wireless Communication

Huihui Wu, Yuanyu Zhang, Member, IEEE, Yulong Shen, Member, IEEE and Xiaohong Jiang, Senior Member,IEEE

Abstract—This paper explores a new secure wireless commu-nication paradigm where the physical layer security technologyis applied to counteract both the detection and eavesdroppingattacks, such that the critical covertness and secrecy propertiesof the communication are jointly guaranteed. We first providetheoretical modeling for covertness outage probability (COP),secrecy outage probability (SOP) and transmission probability(TP) to depict the covertness, secrecy and transmission per-formances of the paradigm. To understand the fundamentalsecurity performance under the new paradigm, we then define anew metric - covert secrecy rate (CSR), which characterizes themaximum transmission rate subject to the constraints of COP,SOP and TP. We further conduct detailed theoretical analysisto identify the CSR under various scenarios determined by thedetector-eavesdropper relationships and the secure transmissionschemes adopted by transmitters. Finally, numerical results areprovided to illustrate the achievable performances under the newsecure communication paradigm.

Index Terms—Wireless communication, covertness, secrecy,physical layer security.

I. INTRODUCTION

THE fundamental research of wireless communicationsecurity is of great importance for the development

of secure network communication, information security andcommunication privacy [1], [2]. It is notable that in modernsecure wireless communication applications, covertness andsecrecy serve as two typical properties. Covertness concernswith the protection of wireless communication from detectionattacks that attempt to detect the existence of the commu-nication [3], [4], while secrecy deals with the protection ofwireless communication from eavesdropping attacks [5], [6]which manage to intercept the information conveyed by thecommunication. With the wide application of secure wirelesscommunication, how to ensure the covertness and secrecyof such communication has become an increasingly urgentdemand.

Thanks to the rapid progress of information and commu-nication technologies, physical layer security (PLS) techniqueis now regarded as a highly promising approach to counteract

H. Wu is with the School of Computer Science and Technology, XidianUniversity, Xi’an, Shaanxi, China, and also with the School of SystemsInformation Science, Future University Hakodate, Hakodate, Hokkaido, Japan(Emails: [email protected]).

Y. Zhang is with the School of Computer Science and Technology, XidianUniversity, Xi’an, Shaanxi, China, and also with the Graduate School ofScience and Technology, Nara Institute of Science and Technology, 8916-5Takayama, Ikoma, Nara, 630-0192, Japan (Email: [email protected]).

Y. Shen is with the School of Computer Science and Technology, XidianUniversity, Xi’an, Shaanxi, China (Email: [email protected]).

X. Jiang is with the School of Systems Information Science, FutureUniversity Hakodate, Hakodate, Hokkaido, Japan (Emails: [email protected]).

the detection and eavesdropping attacks and thus to ensure thecovertness and secrecy properties of wireless communications.The basic principle behind the PLS technology is to exploitthe inherent physical layer randomness of wireless channels(e.g., noise and fading) to implement the secure and covertcommunications [7]. For example, transmitters can intention-ally inject artificial noise (AN) into their channels to hide theirsignals from detectors or to add uncertainty to the informationintercepted by eavesdroppers. The PLS technology realizes se-cure wireless communications from the information-theoreticperspective and thus provides stronger form of covertness andsecrecy guarantees than traditional security technologies likethe cryptography and spread spectrum [8]–[10]. Actually, thePLS technology serves as an effective supplement for thetraditional security technologies to significantly improve thecovertness and secrecy of wireless communications [4], [11].

By now, extensive research efforts have been devoted tostudy of covertness or secrecy guarantee for wireless com-munication based on the PLS technology. In [12]–[17], theAN technique or cooperative jamming technique was adoptedfor covert wireless communication in the typical three-nodescenario with a transmitter, a receiver and a malicious detector.In these works, the AN may be initiated by the transmitter[12], [13], by the (full-duplex) receiver [14], [15], or by someexternal helper nodes [16], [17] to avoid the communicationsignal from being detected by the detector. The works in[13], [18]–[20] show that the covert wireless communicationcan be implemented by exploiting the detector’s uncertaintyabout its channel state information, like the instantaneouschannel coefficient [18], statistical channel coefficient [13]or background noise [19], [20]. Such uncertainty makes itdifficult for the detector to determine the received signalpower or the background noise power, and thus unable todistinguish between the scenarios with or without wirelesscommunication by examining the power difference in thesescenarios. Some recent works also explored the possibility ofensuring covertness based on other PLS technologies, suchas multi-antenna technique [21], [22], coding scheme [23],[24], relay selection [25], [26] and resource (i.e., channel use)allocation [27].

The PLS technology has also been widely adopted forachieving secrecy in various wireless communication sce-narios, such as ad-hoc networks [28], [29], device-to-device(D2D) communications [30], [31], cellular networks [32], [33]and the Internet of Things (IoT) [34], [35]. These worksmainly exploited the application of AN technique to createa relatively better channel to the receiver than that to theeavesdropper with the aim of achieving a positive secrecy

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rate. In [36], [37], the beamforming technique was exploredfor secure wireless communication in multi-antenna scenarios,where the transmit power of signals was concentrated towardthe direction of intended receiver such that a much bettersignal quality at the receiver can be created than that atthe eavesdropper. The work in [38] further combined thebeamforming and AN techniques to achieve a significantsignal advantage at the receiver, while the works in [39], [40]considered the multi-user scenarios and applied relay selectiontechnique to create a transmitter-receiver channel advantageover the transmitter-eavesdropper channel. Some other worksin [41]–[43] also studied the secure wireless communicationbased on the technique of resource allocation (e.g., powerallocation, time slot allocation, energy allocation).

The above works help us understand the great potentialsof the PLS technology in ensuring the covertness or secrecyof wireless communication. It is notable that these worksmainly focus on the traditional paradigms of secure wirelesscommunication where only one type of attack may exist,be it detection or eavesdropping, and concern with eitherthe covertness guarantee or secrecy guarantee for wirelesscommunications. In practice, however, both detection or eaves-dropping attacks may coexist, especially in some criticalcommunication scenarios consisting of multiple groups withcommon or conflicting interests, like military communicationsand coastal surveillance. Therefore, in this paper we aremotivated to explore a new secure wireless communicationparadigm where the PLS technology is applied to counteractboth the detection and eavesdropping attacks. To the best ofour knowledge, this is the first paper that studies the jointguarantee for the critical covertness and secrecy propertiesof wireless communications at the physical layer. The maincontributions of this paper are summarized as follows.

• A new secure wireless communication paradigm: In thisparadigm, the PLS technology is applied to counteractboth the detection and eavesdropping attacks and thus tojointly guarantee the covertness and secrecy propertiesof wireless communications. To demonstrate the newparadigm, we consider four representative communicationscenarios of the paradigm, which are categorized by thedetector-eavesdropper relationships (i.e., independenceand friend) and the secure transmission schemes adoptedby the transmitters (i.e., a power control (PC)-basedscheme and an AN-based scheme). In the friend rela-tionship case, the detector group and eavesdropper groupshare their signals received from the target transmittersin the hope of enhancing the attack performance ofboth sides, while in the independence relationship case,the two groups independently conduct their own attackwithout such signal sharing.

• Theoretical modeling for the new paradigm: To depictthe covertness, secrecy and transmission performances ofthe new paradigm, for each concerned communicationscenario we provide the corresponding theoretical mod-eling of covertness outage probability (COP) (i.e., theprobability that detectors detect the transmitted signals),the secrecy outage probability (SOP) (i.e., the probability

that eavesdroppers recover the conveyed information) andthe transmission probability (TP) (i.e., the probability ofconducting transmissions), respectively.

• A novel security metric characterizing the covertness,secrecy and transmission performances: This paperdefines a novel security metric-covert secrecy rate (CSR),which characterizes the maximum transmission rate sub-ject to the constraints of COP, SOP and TP, and thus canserve as the fundamental security criterion for this newcommunication paradigm. We further conduct detailedtheoretical analysis to identify the CSR for each of thefour communication scenarios. Finally, extensive numer-ical results are provided to illustrate the CSR perfor-mances under the new secure communication paradigm.

The rest of this paper is organized as follows. SectionII presents an example system for the new paradigm andthe definition of CSR. Theoretical analyses for the CSRperformance under the four scenarios are given in Section IIIand Section IV, respectively. Section V provides numericalresults to illustrate the CSR performances and Section VIconcludes this paper.

II. NEW PARADIGM AND SECURITY METRIC

To demonstrate the new secure wireless communicationparadigm, we consider a system (as illustrated in Fig. 1) wherea transmitter Alice sends messages to a receiver Bob in thepresence of a detector Willie and an eavesdropper Eve. Willieattempts to detect the existence of the signals transmitted fromAlice, while Eve targets the messages contained in the signals.Alice and Bob operate in the half-duplex mode, while Willieand Eve can operate in the full-duplex mode. All nodes areassumed to be equipped with a single omnidirectional antenna.For notation simplicity, we use a, b, e and w to represent Alice,Bob, Eve and Willie, respectively, throughout this paper.

Time is divided into successive slots with the same durationthat is long enough for Alice to transmit multiple symbols. Tocharacterize the channels, we adopt the quasi-static Rayleighfading channel model, where the channel coefficients remainconstant in one slot and change independently from one slot toanother at random. We use hij to denote the coefficient of thechannel from i to j, where i ∈ {a, b, e, w} and j ∈ {a, b, e, w}.As assumed in [12], the corresponding channel gain |hij |2 fol-lows the exponential distribution with unit mean. We assumethat Alice and Bob know the instantaneous and statisticalchannel coefficient hab but only the statistical coefficients ofother channels including those to Eve and Willie. We alsoassume that Eve knows the instantaneous channel coefficienthae, while Willie knows only the statistical channel coefficientof haw and hew. These assumptions are widely used inprevious research related to PLS and covert communication.

A. Secure Transmission Schemes

Alice employs two transmission schemes based on powercontrol (PC) and artificial noise (AN), respectively. In thePC-based scheme, Alice controls her transmit power Pa inorder to hide the message signals into the background noise toachieve covertness and secrecy. In the AN-based scheme, Alice

3

intentionally injects AN into the message signals to confuseWillie and Eve so as to reduce their attack effects. Differentfrom the PC-based scheme, in the AN-based scheme, Aliceuses a constant transmit power (also denoted by Pa) and splitsthe power between message and noise transmissions. We useρ ∈ (0, 1] to denote the fraction of transmit power used for themessage transmission. In addition to the strategies of transmitpower, Alice also adopts the Wyner encoding scheme [44] toresist the eavesdropping of Eve. To transmit a message, Alicechooses a target secrecy rate Rs for this message and anotherrate Rt for the whole transmitted symbol. The differenceRt −Rs represents the rate sacrificed to confuse Eve.

The goal of Alice is to ensure a positive and constantsecrecy rate Rs. Thus, Alice will send messages to Bob onlywhen the instantaneous capacity Cb of the Alice-Bob channelcan support the secrecy rate Rs (i.e., Cb ≥ Rs). In thissituation, Alice will set Rt arbitrarily close to Cb to causeas much confusion to Eve as possible, while ensuring reliablemessage transmission to Bob. Thus, the probability of Alicetransmitting messages in a certain time slot can be defined as

ptx = P(Cb ≥ Rs). (1)

Note that the transmission probability (TP) ptx can be inter-preted as a metric to measure the transmission performance.

B. Attacking ModelIn practice, Willie and Eve can belong to different organi-

zations with unrelated or common goals, resulting in variousrelationships between them. In this paper, we consider tworepresentative relationships, i.e., independence and friend. Asshown in Fig. 1, in the independence relationship, Eve andWillie care only about their own attack without helping orhindering the other. In the friend relationship, Willie and Evewill share their signals received from Alice to help improvethe attack power of the other.

To detect the existence of signals transmitted from Alice ineach slot, Willie adopts the commonly-used likelihood ratiotest [16], in which he first determines a threshold θ and thenmeasures the average power P̄w of the symbols received fromAlice in this slot. If P̄w ≥ θ, Willie accepts a hypothesis H1

that Alice transmitted messages to Bob in this slot. If P̄w ≤ θ,Willie accepts a hypothesis H0 that Alice did not transmitmessages. Formally, the likelihood ratio test can be given by

P̄wH1

≷H0

θ. (2)

In general, the likelihood test introduces two types of detectionerrors. One is called false alarm, which means that Williereports a detected transmission whilst the transmission doesnot exist in fact. The other is called missed detection, whichmeans that Willie reports no detected transmission whilstthe transmission exists indeed. We use pFA and pMD todenote the probabilities of false alarm and missed detection,respectively. If neither false alarm nor missed detection occurs,the transmission from Alice to Bob is said to suffer fromcovertness outage. Thus, the covertness outage probability(COP) is given by

pco = 1− (pFA + pMD). (3)

(a) Independence relationship

hab

haw hae

Alice

Willie Eve

Bob

hab

haw hae

Alice

Willie

Share signals

(b) Friend relationship

Eve

Bob

Fig. 1. Two relationships between Willie and Eve.

The smaller the COP is, the higher the covertness of thetransmission is. Note that 1 − pco can be interpreted as thedetection error probability of Willie.

Compared with the detection of Willie, the eavesdroppingattack of Eve is relatively simpler. To intercept the transmittedmessages, Eve tries to decode the signals received from Alice.If Eve is able to recover the messages (i.e., the instantaneoussecrecy capacity Cs [45] of the Alice-Bob channel falls belowthe target secrecy rate Rs), the transmission from Alice to Bobis said to suffer from secrecy outage. Note that secrecy outageoccurs only when Alice actually transmits a message (i.e.,Cb ≥ Rs). Thus, we can define the secrecy outage probability(SOP) as the following conditional probability:

pso = P(Cs < Rs | Cb ≥ Rs). (4)

Similarly, the smaller the SOP is, the stronger the secrecy ofthe transmission is.

C. Covert Secrecy RateTo understand the fundamental security performance under

the new paradigm, we propose a novel metric, called covertsecrecy rate (CSR), by jointly considering the covertness,secrecy and transmission performances. The CSR is definedas the maximum transmission rate under which the constraintsof COP, SOP and TP can be ensured. To obtain the CSR, weformulate two optimization problems for the PC-based andAN-based transmission schemes, respectively, which are givenby

P1 (PC-based): Rcs = maxPa,Rs

Rsptx(Pa, Rs), (5a)

s.t. pco(Pa) ≤ εc, (5b)pso(Rs) ≤ εs, (5c)ptx(Pa, Rs) ≥ 1− εt, (5d)

4

and

P2 (AN-based): Rcs = maxρ∈[0,1],Rs

Rsptx(ρ,Rs), (6a)

s.t. pco(ρ) ≤ εc, (6b)pso(ρ,Rs) ≤ εs, (6c)ptx(ρ,Rs) ≥ 1− εt, (6d)

where Rcs denotes the CSR, εc, εs and εt denote the con-straints of COP, SOP and TP. Note that Problem P1 optimizesthe transmission rate over the transmit power Pa and thesecrecy rate Rs, while Problem P2 conducts the optimizationover the power allocation parameter ρ and the secrecy rate Rs.

III. CSR ANALYSIS: INDEPENDENCE RELATIONSHIP CASE

In this section, we investigate the CSR performance underthe independence relationship case, for which we focus on thePC-based and AN-based transmission schemes in SubsectionsIII-A and III-B, respectively.

A. PC-Based Transmission Scheme

As mentioned in Section II-A, Alice decides to transmit ina certain time slot only when the instantaneous capacity Cb ofAlice-Bob channel can support the secrecy rate Rs. To do this,Alice measures the instantaneous channel coefficient |hab|2and determines the Alice-Bob channel capacity Cb based onthe well-known Shannon Capacity formula [45], i.e.,

Cb = log

(1 +

Pa|hab|2

σ2b

), (7)

where log is to the base of 2. Since |hab|2 is exponentiallydistributed, the transmission probability ptx of Alice underthe PC-based transmission scheme is

pIPtx (Pa,Rs)=P(Cb ≥ Rs)=exp

(− (2Rs−1)σ2

b

Pa

). (8)

When Alice chooses to transmit, she sends n symbols toBob, represented by a complex vector x, where each symbolx[i] (i = 1, 2, · · · , n) is subject to the unit power constraint,i.e., E[|x[i]|2] = 1. Thus, the signal vectors received at Bob,Willie and Eve are given by

yκ =√Pahaκx + nκ, (9)

where the subscript κ ∈ {b, w, e} stands for Bob, Willie orEve, a represents Alice, and nκ denotes the noise at κ with thei-th element nκ[i] being the complex additive Gaussian noisewith zero mean and variance σ2

κ, i.e., nκ[i] ∼ CN (0, σ2κ).

According to the detection scheme in Subsection II-B,Willie makes a decision on the existence of transmitted signalsbased on the average power P̄w of the received symbols yw.In this case, P̄w is given by

P̄w =

∑ni=1|yw[i]|2

n= limn→∞

(Pa|haw|2 + σ2w)χ2

2n/n

= Pa|haw|2 + σ2w, (10)

where χ22n is a chi-squared random variable with 2n degrees

of freedom. By the Strong Law of Large Numbers [46], χ22n

n

converges in probability to 1 as n tends to infinity. If P̄w ≤ θ,Willie accepts the hypothesis H0 that Alice did not transmitmessages, leading to a missed detection. Thus, the probabilityof missed detection pMD is given by

pMD = P(Pa|haw|2 + σ2

w ≤ θ)

=

{1− exp

(− θ−σ

2w

Pa

), θ > σ2

w,

0, θ ≤ σ2w.

(11)

The eavesdropping result of Eve depends on the instanta-neous secrecy capacity Cs of the Alice-Bob channel, which isthe difference between the channel capacity of the Alice-Bobchannel and that of the Alice-Eve channel [45]. Thus, Cs isformulated as

Cs = log

(1 +

Pa|hab|2

σ2b

)− log

(1 +

Pa|hae|2

σ2e

). (12)

Note that |hab|2 and |hae|2 are random variables here. Basedon the definition of the SOP in Subsection II-B, the SOP underthe PC-based scheme can be given by

pIPso(Rs)=P (Rs < Cb < Ce +Rs)

P (Cb > Rs)= 1− P (Cs > Rs)

P (Cb > Rs)

=1−e(2Rs−1)σ2b

Pa P(Pa|hab|2

σ2b

− 2RsPa|hae|2

σ2e

>2Rs−1

)=

2Rsσ2b

2Rsσ2b + σ2

e

. (13)

When Alice does not transmit, security performance is not aconcern and thus we only focus on the covertness performance.In this case, Willie receives only noise, i.e., yw = nw andthus the the average power P̄w of the received symbols yw isP̄w = σ2

w. If P̄w ≥ θ, Willie accepts the hypothesis H1 thatAlice transmitted messages, leading to a false alarm. Thus, theprobability of false alarm pFA is given by

pFA = P(σ2w ≥ θ

)=

{0, θ > σ2

w,

1, θ ≤ σ2w.

(14)

Combining the pMD in (11) and the pFA in (14), we obtainthe COP under the PC-based scheme as

pIPco (Pa, θ) =

{exp

(− θ−σ

2w

Pa

), θ > σ2

w,

0, θ ≤ σ2w.

(15)

Note that the COP is identical for Alice and Willie, sincethey have the same knowledge about |haw|2, i.e., the statistical|haw|2. To maximize the COP pIPco , Willie will choose theoptimal detection threshold θ, denoted by θ∗IP. We can seefrom (15) that pIPco is a decreasing function of θ and is largerthan or equal to 0 for θ > σ2

w. Thus, the optimal θ∗IP exists in(σ2w,∞) and is thus given by θ∗IP = υ + σ2

w, where υ > 0 isan arbitrarily small value.

Under the condition that Willie chooses the optimal detec-tion threshold θ∗IP, Alice solves the optimization problem in(5) to obtain the CSR. The main result is summarized in thefollowing theorem.

Theorem 1. Under the scenario where Willie and Eve are inthe independence relationship and Alice adopts the PC-based

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RIPcs =

1ln 2W0

(− υσ2b ln εc

)exp

− 1

W0

(− υ

σ2b

ln εc

) − σ2b ln εcυ

, R∗s,IP = R0s,IP ≤ min

{RSOPs,IP, R

TPs,IP

},

log(

σ2eεs

(1−εs)σ2b

)exp

((σ2eεs−(1−εs)σ

2b) ln εc

(1−εs)υ

), R∗s,IP = RSOP

s,IP ≤ min{R0s,IP, R

TPs,IP

},

(1− εt) log(

1 + υ ln(1−εt)σ2b ln εc

), R∗s,IP = RTP

s,IP ≤ min{R0s,IP, R

SOPs,IP

},

(16)

secure transmission scheme, the CSR of the system can begiven by (16), where

RSOPs,IP = log

(σ2eεs

(1− εs)σ2b

), (17)

RTPs,IP = log

(1−

P ∗a,IP ln(1− εt)σ2b

), (18)

R0s,IP =

1

ln 2W0

(P ∗a,IPσ2b

), (19)

W0(·) is the principal branch of Lambert’s W function, andP ∗a,IP = − υ

ln εcis the optimal transmit power.

Proof. As can be seen from (5a), the optimal transmit powerPa and optimal target secrecy rate Rs are required to solvethe optimization problem P1. We first derive the optimal Pa.It is easy to see from (8) and (15) that both pIPtx and pIPcomonotonically increase as Pa increases. Thus, the covertnessconstraint in (5b) results in an upper bound on Pa, which is

Pmaxa,IP = − υ

ln εc, (20)

and the TP constraint in (5d) leads to a lower bound on Pa,which is

Pmina,IP = − (2Rs − 1)σ2

b

ln(1− εt). (21)

Note that the inequality Pmina,IP ≤ Pmax

a,IP must hold, which givesthe following condition on Rs:

Rs ≤ log

(1 +

υ ln(1− εt)σ2b ln εc

). (22)

Since the objective function in (5a) is an increasing functionof Pa, the optimal Pa is the upper bound, i.e., P ∗a,IP = Pmax

a,IP .Next, we derive the optimal Rs by analyzing the feasible

region of Rs and the monotonicity of the objective functionwith respect to Rs. We can see that as Rs increases, pIPtxin (8) monotonically decreases while pIPso in (13) monotoni-cally increases. Thus, based on the constraints (5c) and (5d),the regions of Rs for ensuring secrecy and transmissionperformances are [0, RSOP

s,IP] and [0, RTPs,IP] with RSOP

s,IP andRTPs,IP given by (17) and (18), respectively. Note that RTP

s,IP

is obtained at Pa = P ∗a,IP = − υln εc

and thus the region[0, RTP

s,IP] is equivalent to (22). Hence, the feasible region ofRs is [0,min{RSOP

s,IP, RTPs,IP}]. Taking the first derivative of the

objective function in (5a) in terms of Rs gives

∂Rcs∂Rs

=

(1−Rs2

Rsσ2b ln 2

Pa

)exp

(− (2Rs−1)σ2

b

Pa

). (23)

Solving ∂Rcs∂Rs

= 0, we can obtain the stationary point R0s,IP

in (19). We can see that the objective function is increasingover [0, R0

s,IP) and decreasing over [R0s,IP,∞). This implies

that if R0s,IP falls inside the feasible region of Rs, i.e.,

R0s,IP ≤ min{RSOP

s,IP, RTPs,IP}, the optimal Rs is R∗s,IP = R0

s,IP.Otherwise, the optimal Rs is R∗s,IP = min{RSOP

s,IP, RTPs,IP}.

Finally, substituting the optimal Pa and Rs into the objectivefunction in (5a) completes the proof.

B. AN-Based Transmission SchemeSuppose Alice transmits, in addition to the message sym-

bols, she will also inject AN, represented by a complex vectorz, where each symbol z[i] (i = 1, 2, · · · , n) is subject to theunit power constraint, i.e., E[|z[i]|2] = 1. Alice will use afraction ρ of her transmit power Pa for message transmissionand the remaining power for AN radiation. Thus, the signalvectors received at Bob will be given by

yb =√ρPahabx +

√(1− ρ)Pahabz + nb. (24)

Based on (24), Alice measures the instantaneous Alice-Bobchannel capacity Cb as

Cb = log

(1 +

ρPa|hab|2

(1− ρ)Pa|hab|2 + σ2b

), (25)

and decides to transmit when Cb ≥ Rs. Thus, the transmissionprobability under the AN-based scheme can be given by

pIAtx (ρ,Rs) = P (Cb ≥ Rs)

= P(

ρPa|hab|2


≥ 2Rs − 1

)= exp

(− (2Rs − 1)σ2

b

ρPa − (2Rs − 1)(1− ρ)Pa

). (26)

Next, we analyze the secrecy and covertness performanceswhen Alice transmits messages. In this situation, the signalvectors received at Willie and Eve have the same form of thatreceived at Bob, which are given by

yκ =√ρPahaκx +

√(1− ρ)Pahaκz + nκ, (27)

where the subscript κ ∈ {w, e} stands for Willie or Eve. From(27), we can see that the average power P̄w of the receivedsymbols yκ at Willie is the same as that given in (10). Thus,the probability of missed detection pMD under the AN-basedscheme can also be given by (11).

According to (27), the secrecy capacity Cs under the AN-based scheme can be formulated as

Cs=log

(1+

ρPa|hab|2

(1−ρ)Pa|hab|2+σ2b

)−log

(1+

ρPa|hae|2

(1−ρ)Pa|hae|2+σ2e

).

(28)

6

Thus, following the definition of SOP in (4), we derive theSOP under the AN-based scheme as

pIAso (ρ,Rs)=1− exp

((2Rs − 1)σ2

b

ρPa − (2Rs − 1)(1− ρ)Pa

)(29)

×P(

ρPa|hab|2


− 2RsρPa|hae|2

(1−ρ)Pa|hae|2+σ2e

>2Rs−1

)=1−exp

((2Rs − 1)σ2

b

ρPa−(2Rs−1)(1−ρ)Pa− (2Rs+ρ−1)σ2

b

(1−2Rs)(1−ρ)Pa

)∫ φ

0

exp

(2Rs+ρ−1)(1−(1−ρ)2Rs)σ2bσ

2e

(1−2Rs )(1−ρ) −(2Rs−1)σ2bσ

2e

(1−2Rs)(1−ρ)P 2a y+(1−(1−ρ)2Rs)Paσ2

e

−y

dy,

where φ =(1−2Rs (1−ρ))σ2

e

(2Rs−1)(1−ρ)Pa .Finally, we analyze the covertness performance when Alice

does not transmit messages. In this situation, Alice still gen-erates AN to confuse Willie, which is different from the PC-based scheme. Thus, the signal vector yw received by Willieconsists of both the AN z and background noise, i.e.,

yw =√

(1− ρ)Pahawz + nw. (30)

In this case, the average power of the received symbols ofWillie is P̄w = (1−ρ)Pa|haw|2+σ2

w, and thus the probabilityof false alarm is given by

pFA = P((1− ρ)Pa|haw|2 + σ2

w ≥ θ)

=

{exp

(− (θ−σ2

w)(1−ρ)Pa

), θ > σ2

w,

1, θ ≤ σ2w.

(31)

Combining the pFA in (31) and the pMD in (11), we obtainthe COP pIAco under the AN-based scheme as

pIAco (ρ, θ)=

{exp(− (θ−σ2

w)Pa

)−exp

(− (θ−σ2

w)(1−ρ)Pa

), θ>σ2

w,

0, θ≤σ2w.

(32)

We can see from (32) that the optimal detection thresholdθ∗IA for Willie exists when θ > σ2

w and can be obtained bysolving ∂pIAco

∂θ = 0. Thus, θ∗IA is given by

θ∗IA = σ2w +

(ρ− 1)Paρ

ln(1− ρ). (33)

By solving the optimization problem in (6) with θ = θ∗IA, wecan obtain the CSR, which is given in the following theorem.

Theorem 2. Under the scenario where Willie and Eve are inthe independence relationship and Alice adopts the AN-basedsecure transmission scheme, the CSR of the system is

RIAcs=R∗s,IA(ρ∗IA)exp

(− (2R

∗s,IA(ρ∗IA) − 1)σ2

b

ρ∗IAPa−(2R∗s,IA(ρ

∗IA)−1)(1−ρ∗IA)Pa

),

(34)where ρ∗IA is the optimal power allocation parameter andR∗s,IA is the optimal secrecy rate. Here, ρ∗IA can be obtainedby solving pIAco (ρ, θ∗IA) = εc with θ∗IA given by (33). R∗s,IA isgiven by

R∗s,IA(ρ∗IA)=

R0s,IA(ρ∗IA), R∗s,IA=R0

s,IA≤min{RSOPs,IA, R

TPs,IA

},

RSOPs,IA(ρ∗IA), R∗s,IA=RSOP

s,IA≤min{R0s,IA, R

TPs,IA

},

RTPs,IA(ρ∗IA), R∗s,IA=RTP

s,IA≤min{R0s,IA, R

SOPs,IA

},

(35)

where the stationary point R0s,IA can be obtained by solving

∂Rcs∂Rs

= 0, RSOPs,IA is the solution of pIAso (Rs) = εs and RTP

s,IA isgiven by

RTPs,IA(ρ∗IA) = log

(Pa ln(1− εt)− σ2

b

(1− ρ∗IA)Pa ln(1− εt)− σ2b

). (36)

Proof. The proof follows the same idea as the one for Theorem1. The only difference is to derive the optimal power allocationparameter ρ instead of optimal transmit power Pa. Here, wefocus on the derivation of the optimal ρ and omit the analysisof the optimal Rs. We can see that the objective function in(6a) is an increasing function of ρ, implying that the upperbound on ρ is needed. Substituting θ = θ∗IA into (32) yields

pIAco = ρ(1− ρ)1−ρρ . (37)

Taking the first derivative of (37) in terms of ρ, we have

∂pIAco∂ρ

=− ln(1− ρ)

ρ(1− ρ)

1−ρρ > 0, (38)

which shows that pIAco is an increasing function of ρ. We cansee from (26) and (29) that pIAtx is also an increasing functionof ρ, while ρSOP

IA is a decreasing function. Thus, only thecovertness constraint (6b) gives an upper bound ρmax

IA on ρ,while the TP and SOP constraints in (6d)) and (6c) give twolower bounds ρTP

IA and ρSOPIA respectively. Hence, the optimal ρ

is ρ∗IA = ρmaxIA . Note that ρmax

IA ≥ max{ρTPIA, ρ

SOPIA

}must hold,

which imposes a constraint (or region) on Rs. However, thisregion is equivalent to the one obtained from the TP and SOPconstraints in (6d)) and (6c), and thus can be neglected in theanalysis of optimal Rs.

IV. CSR ANALYSIS: FRIEND RELATIONSHIP CASE

The CSR performance of the friend relationship case isinvestigated in this section, for which the CSR analyses for thePC-based and AN-based transmission schemes are provided inSubsections IV-A and IV-B, respectively. To depict the friendrelationship, we interpret Willie and Eve as two antennas of asuper attacker. This model is widely used to characterize thecollusion among eavesdroppers [47].

A. PC-Based Transmission Scheme

Alice follows the same decision process as introduced inSection III-A to decide whether to transmit messages ornot. Note that the instantaneous Alice-Bob channel capacityCb in this case is identical to that in (7), which meansthat the transmission probability is also the same. Thus, thetransmission probability pFPtx in the friend relationship scenariounder the PC-based scheme is given by (8).

Next, we analyze the covertness and secrecy performanceswhen Alice transmits messages. When Alice chooses to trans-mit a signal vector x, Willie and Eve receive the same signalvectors yw and ye as that given in (9). Since Willie and Eveshare their received signals in this case, the signal vectorsreceived at Willie and Eve contain the one from the otherside. Thus, based on the signal vector yκ in (9), the averagepower of the received symbols at Willie can be given byP̄w =

∑κ∈{w,e} |yκ|2 = Pa|haw|2 + Pa|hae|2 + σ2

e + σ2w.

7

Note that |haw|2 and |hae|2 are random variables for Willie.Thus, the probability of missed detection pMD is given by

pMD = P(Pa|haw|2 + Pa|hae|2 + σ2

e + σ2w ≤ θ

)(39)

=

{1− Pa+θ−σ2

e−σ2w

Paexp(− θ−σ

2e−σ

2w

Pa

), θ > σ2

e+σ2w,

0, θ ≤ σ2e+σ2

w.

According to [34], the signal sharing results in an im-proved Signal-to-Noise Ratio (SNR) for Eve, which isPa|hae|2+Pa|haw|2

σ2e+σ

2w

. Thus, the secrecy capacity Cs is

Cs=log

(1+

Pa|hab|2

σ2b

)−log

(1+

Pa|hae|2 + Pa|haw|2

σ2e + σ2

w

).

(40)Since |hab|2, |hae|2 and |haw|2 are independent, the SOP underthe PC-based scheme is given by

pFPso (Rs) = 1− exp

((2Rs − 1)σ2

b

Pa

)× P

(Pa|hab|2

σ2b

−2RsPa|haw|2+Pa|hae|2

σ2w+σ2

e

>2Rs−1

)=

2Rsσ2b (2Rsσ2

b + 2σ2w + 2σ2

e)

(2Rsσ2b + σ2

w + σ2e)2

. (41)

Finally, we focus on the covertness performance when Alicesuspends her transmission. Since the decision of suspendingtransmission is unknown to Willie and Eve, they still sharetheir signals, which contain only background noises. Thus,the received signal at Willie is given by yw = ne + nw andthe average received power is P̄w = σ2

e + σ2w. Hence, the

probability of false alarm pFA can be given by

pFA = P(σ2e + σ2

w ≥ θ)

=

{0, θ > σ2

e + σ2w,

1, θ ≤ σ2e + σ2

w.(42)

Combining the pFA in (42) and the pMD in (39), we obtainthe COP as

pFPco (Pa,θ)=

{Pa+θ−σ2

e−σ2w

Paexp(− θ−σ

2e−σ

2w

Pa

), θ > σ2

e + σ2w,

0, θ ≤ σ2e + σ2

w.(43)

Taking the derivative of the pFPco in (43) gives

∂pFPco∂θ

= −θ − σ2e − σ2

w

P 2a

exp

(−θ − σ

2e − σ2

w

Pa

). (44)

This shows that pFPco is a decreasing function of θ when θ >σ2e + σ2

w. Thus, the optimal detection threshold is

θ∗FP = υ + σ2e + σ2

w, (45)

where υ > 0 is an arbitrarily small value.Given the θ∗FP, the ptx in (8), the SOP in (41) and the COP

in (43), the problem in (5) can now be solved to obtain theCSR. The result is given in the following theorem.

Theorem 3. Under the scenario where Willie and Eve are inthe friend relationship and Alice adopts the PC-based securetransmission scheme, the CSR of the system is given in (46),Here,

RSOPs,FP = log

((1−

√1− εs)(σ2

w + σ2e)

σ2b

√1− εs

), (47)

RTPs,FP and R0

s,FP are the same as those given in (18) and (19),respectively, with the optimal transmit power P ∗a,FP given by

P ∗a,FP = − υ

1 + W−1(− εce ). (48)

W0(·) and W−1(·) are the principal branch and the non-principle branch of Lambert’s W function, respectively, and eis Euler’s number.

Proof. The proof is similar to that of Theorem 1 and thusomitted here.

B. AN-Based Transmission Scheme

We first derive the transmission probability to characterizethe transmission performance of the transmission. SupposeAlice transmits under the AN-based scheme, Bob will receivethe same signal as that given in (27), yielding the sameinstantaneous Alice-Bob channel capacity Cb as that given in(25). This means that the transmission probability pFAtx underthe AN-based scheme in the friend relationship scenario isidentical to that in the independence scenario, which is givenin (26).

We proceed to analyze the miss detection probability andSOP when Alice transmits messages. When Alice transmitsa signal vector x, the signal vectors at Willie and Eve arethe same as that given in (27). After receiving the sharedsignals from Eve, the average power P̄w of the receivedsymbols at Willie is given by P̄w =

∑κ∈{w,e} |yκ|2 =

Pa|hae|2 + Pa|haw|2 + σ2e + σ2

w, which is identical to (10),i.e., the average power in the independence case. Thus, theprobability of missed detection pMD can be given by (39).

After Eve receives the signals from Willie, the Signal-to-Noise-plus-Interference Ratio (SINR) is

ρPa|hae|2+ρPa|haw|2

(1−ρ)Pa|hae|2+(1−ρ)Pa|haw|2+σ2e+σ2

w

. (49)

Thus, the secrecy capacity Cs under the AN-based scheme is

Cs=log

(1 +

ρPa|hab|2


)(50)

− log

(1+

ρPa|hae|2+ρPa|haw|2


w

),

According to the definition in (4), the SOP is given by (51).When Alice does not transmit messages, we consider only

the covertness of the transmission by analyzing the probabilityof false alarm. In this case, Alice still sends AN to confuseWillie. Thus, based on (30), the signal vector yw contains boththe signals (i.e., AN and background noise) shared by Eve, ANand background noise. In this case, the average power of thereceived symbols at Willie is P̄w = (1 − ρ)Pa|haw|2 + (1 −ρ)Pa|hae|2 + σ2

e + σ2w. Thus, the probability of false alarm

pFA is given by

pFA=P((1−ρ)Pa|haw|2+(1−ρ)Pa|hae|2+σ2

e+σ2w≥θ

)(52)

=

{(1+

θ−σ2e−σ

2w

(1−ρ)Pa

)exp(− θ−σ

2e−σ

2w

(1−ρ)Pa

), θ > σ2

e + σ2w,

1, θ ≤ σ2e + σ2

w.

8

RFPcs =

1ln 2W0

(− υ

(1+W−1(− εce ))σ2b

)exp

− 1

W0

(− υ

(1+W−1(−εce

))σ2b

)− (1+W−1(− εce ))σ2b

υ

, R∗s,FP =R0s,FP≤min

{RSOPs,FP, R

TPs,FP

},

log(

(1−√1−εs)(σ2

w+σ2e)

σ2b

√1−εs

)exp

(((1−

√1−εs)(σ2

w+σ2e)−√1−εsσ2

b)(1+W−1(− εce ))υ√1−εs

), R∗s,FP =RSOP

s,FP≤min{R0s,FP, R

TPs,FP

},

(1− εt) log

(1 + υ ln(1−εt)

σ2b(1+W−1(− εce ))

), R∗s,FP =RTP

s,FP≤min{R0s,FP, R

SOPs,FP

},

(46)

pFAso (ρ,Rs)=1− exp

((2Rs − 1)σ2

b

ρPa−(2Rs−1)(1−ρ)Pa

)P(

ρPa|hab|2


− 2Rs(ρPa|hae|2+ρPa|haw|2)


w

>2Rs−1

)

=1− exp

((2Rs − 1)σ2

b

ρPa − (2Rs − 1)(1− ρ)Pa− (2Rs + ρ− 1)σ2

b

(1− 2Rs)(1− ρ)Pa

)×∫ (1−2Rs (1−ρ))(σ2w+σ2e)

(2Rs−1)(1−ρ)Pa

0

∫ (1−2Rs (1−ρ))(σ2w+σ2e)

(2Rs−1)(1−ρ)Pa−z

0

× exp

−y − (2Rs − 1)σ2b (σ2

w + σ2e)− (2Rs+ρ−1)(1−(1−ρ)2Rs )σ2

b (σ2w+σ2

e)(1−2Rs )(1−ρ)

(1− 2Rs)(1− ρ)P 2a (y + z) + (1− (1− ρ)2Rs)Pa(σ2

w + σ2e)− z

dy dz, (51)

Combining the pFA in (52) and the pMD in (39), the COPcan be given by

pFAco (ρ,θ)=

(

1+θ−σ2

e−σ2w

Pa

)exp(− θ−σ

2e−σ

2w

Pa

)−(1+

θ−σ2e−σ

2w

(1−ρ)Pa

)exp(− θ−σ

2e−σ

2w

(1−ρ)Pa

), θ>σ2

e+σ2w,

0, θ≤σ2e+σ2

w.(53)

We can see from (53) that the optimal detection thresholdθ∗FA can be obtained by solving ∂pFAco

∂θ = 0, which is

θ∗FA = σ2e + σ2

w +2(ρ− 1)Pa

ρln(1− ρ). (54)

Given the θ∗FA in (54), we solve the optimization problem in(6) to obtain the CSR, which is given in the following theorem.

Theorem 4. Under the scenario where Willie and Eve are inthe friend relationship and Alice adopts the AN-based securetransmission scheme, the CSR of the system is

RFAcs =R∗s,FA(ρ∗FA) exp

(− (2R

∗s,FA(ρ∗FA) − 1)σ2

b

ρ∗FAPa−(2R∗s,FA(ρ

∗FA)−1)(1−ρ∗FA)Pa

).

(55)Here, the optimal power allocation parameter ρ∗FA solvespFAco (ρ, θ∗FA) = εc with θ∗FA given by (54). The optimal secrecyrate R∗s,FA is given in (35), where R0

s,FA can be obtained bysolving ∂Rcs

∂Rs= 0, RSOP

s,FA is the solution of pFAso (Rs) = εs andRTPs,FA is given in (36).

Proof. The proof is similar to that of Theorem 2 and thusomitted here.

V. NUMERICAL RESULTS

In this section, we provide extensive numerical results toillustrate the CSR performances of the four representative

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

COP constraint c

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8C

SR

(bi

ts p

er c

hann

el u

se)

s = 0.01,

t = 0.1

s = 0.02,

t = 0.1

s = 0.03,

t = 0.5

0.001 0.02 0.04 0.06 0.08 0.10.0135

0.014

0.001 0.02 0.04 0.06 0.08 0.10.02

0.04

Rs,IP* = R

s,IPSOP

Rs,IP* = R

s,IP0

Rs,IP* = R

s,IPSOP

Rs,IP* = R

s,IPTP

(a) Independence relationship scenario.

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

COP constraint c

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

CS

R (

bits

per

cha

nnel

use

)

s = 0.01,

t = 0.1

s = 0.012,

t = 0.1

s = 0.03,

t = 0.53

0.001 0.02 0.04 0.06 0.08 0.10.01

0.02

0.03

0.001 0.02 0.04 0.06 0.08 0.1

0.0102

0.0104

Rs,FP* = R

s,FP0

Rs,FP* = R

s,FPSOP

Rs,FP* = R

s,FPSOP

Rs,FP* = R

s,FPTP

(b) Friend relationship scenario.

Fig. 2. CSR Rcs vs. COP constraint εc (PC-based transmission scheme).

scenarios under the new secure communication paradigm. Wealso show the impacts of various system parameters (e.g., COP

9

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

COP constraint c

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

CS

R (

bits

per

cha

nnel

use

)s = 0.01,

t = 0.01

s = 0.01,

t = 0.1

s = 0.022,

t = 0.5

0.001 0.02 0.04 0.06 0.08 0.10

0.005

0.01

Rs,IA* = R

s,IASOP

Rs,IA* = R

s,IASOP

Rs,IA* = R

s,IA0

Pa = -20 (dB)

Rs,IA* = R

s,IATP

Rs,IA* = R

s,IATP


0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

COP constraint c

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

CS

R (

bits

per

cha

nnel

use

)

s = 0.01,

t = 0.01

s = 0.01,

t = 0.1

s = 0.022,

t = 0.5

0.001 0.02 0.04 0.06 0.08 0.10

0.005

0.01

0.015

Rs,FA* = R

s,FASOP

Rs,FA* = R

s,FASOP

Pa = -20 (dB)

Rs,FA* = R

s,FA0

Rs,FA* = R

s,FATP

Rs,FA* = R

s,FATP


Fig. 3. CSR Rcs vs. COP constraint εc (AN-based transmission scheme).

constraint εc, SOP constraint εs, TP constraint εt and transmitpower Pa) on the CSR performance. Unless otherwise stated,we set the parameter υ to υ = 0.01 and the noise powers atBob, Willie and Eve to σ2

b = −20 dB and σ2w = σ2

e = 0 dB.To explore the impact of the COP constraint εc on the

CSR performance, we show in Fig. 2 Rcs vs. εc in theindependence relationship case under the PC-based and AN-based transmission schemes, respectively. The results for thefriend relationship case under both transmission schemes arepresented in Fig. 3. We set the transmit power of Alice toPa = −20 dB in Fig. 3. In each subfigure of Fig. 2 andFig. 3, we also plot the CSR curves under different settingsof SOP constraint εs and TP constraint εt. We can see fromFig. 2 and Fig. 3 that the CSRs achieved under different SOPand TP constraints always increase as εc increases. This isbecause a looser COP constraint results in a larger optimaltransmit power in the PC-based scheme (resp. a larger optimalpower allocation parameter in the AN-based scheme) and thusa larger CSR.

We can also observe from Fig. 2 and Fig. 3 that the shapeof the CSR curve varies as the values of the SOP constraintεs and TP constraint εt change. For example, the CSR curveunder the setting of εs = 0.03 and εt = 0.5 (dashed line)in Fig. 2 exhibits an exponential growth and that under thesetting of εs = 0.02 and εt = 0.1 (dotted line) grows in a

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SOP constraint s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CS

R (

bits

per

cha

nnel

use

)

c = 0.01,

t = 0.01

c = 0.05,

t = 0.05

c = 0.01,

t = 0.49

0

0.05

0.1

0.15

0.2

0.25 Rs,IP* = R

s,IP0

Rs,IP* = R

s,IPSOP

b2 = -30 (dB)

0.001 0.0011 0.0012 0.0013 0.0014 0.0015

Rs,IP* = R

s,IPSOP

Rs,IP* = R

s,IPTP


0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SOP constraint s

0

0.1

0.2

0.3

0.4

0.5

0.6

CS

R (

bits

per

cha

nnel

use

)

c = 0.01,

t = 0.01

c = 0.1,

t = 0.1

c = 0.01,

t = 0.51

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Rs,FP* = R

s,FP0R

s,FP* = R

s,FPSOP

b2 = -30 (dB)

0.001 0.0011 0.0012 0.0013 0.0014 0.0015

Rs,FP* = R

s,FPSOP

Rs,FP* = R

s,FPTP


Fig. 4. CSR Rcs vs. SOP constraint εs (PC-based transmission scheme).

piecewise fashion. This is because different values of εs, εtand the COP constraint εc result in different RSOP

s,IP, RTPs,IP and

R0s,IP in (17-19) (resp. RSOP

s,FP, RTPs,FP, R0

s,FP in (47,18,19),RSOPs,IA, RTP

s,IA, R0s,IA in (35) and RSOP

s,FA, RTPs,FA, R0

s,FA in (35)),which further lead to different optimal target secrecy rates (aslabeled in Fig. 2 and Fig. 3) and thus different CSR curves.

Next, we investigate the impact of the SOP constraint εs onthe CSR performance, for which we show Rcs vs. εs in theindependence and friend relationship cases under the PC-basedtransmission scheme in Fig. 4 and those under the AN-basedtransmission scheme in Fig. 5. We set the noise power at Bobto σ2

b = −30 dB in Fig. 4 and that to σ2b = −31 dB in Fig.

5. We set the transmit power of Alice to Pa = −20 dB inFig. 5. For both figures, we consider three different settingsof COP constraint εc and TP constraint εt, respectively. Wecan see from Fig. 4 and Fig. 5 that, when both εc and εt arerelatively small (e.g., εc = 0.01 and εt = 0.01 in Fig. 4(a)),the CSR stays unchanged as the SOP constraint εs increases,which implies that the SOP constraint εs has no impacts on theCSR performance. This is because, in this situation, the CSR isachieved at only the optimal target secrecy rate R∗s,IP = RTP

s,IP

(as labeled in Fig. 4(a)), which is independent of εs as can beseen from (18). On the other hand, when either εc or εt is large,the CSR first increases sharply and then remains constant asthe SOP constraint εs increases. This is because the optimal

10

1 1.1 1.2 1.3 1.4 1.5

10-3

0

0.05

0.1

0.15

0.2

0.25

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SOP constraint s

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

CS

R (

bits

per

cha

nnel

use

)

c = 0.01,

t = 0.01

c = 0.01,

t = 0.22

c = 0.05,

t = 0.04

0

0.02

0.04

0.06

Rs,IA* = R

s,IA0

b2 = -31 (dB), P

a = -20 (dB)

Rs,IA* = R

s,IASOP

0.001 0.0015 0.002 0.0025

Rs,IA* = R

s,IATP

Rs,IA* = R

s,IATPR

s,IA* = R

s,IASOP


0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SOP constraint s

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

CS

R (

bits

per

cha

nnel

use

)

c = 0.01,

t = 0.01

c = 0.01,

t = 0.49

c = 0.05,

t = 0.05

0

0.02

0.04

0.06

Rs,FA* = R

s,FA0

b2 = -31 (dB), P

a = -20 (dB)

0.001 0.0015 0.002 0.0025

Rs,FA* = R

s,FASOP

Rs,FA* = R

s,FATP

Rs,FA* = R

s,FATPR

s,FA* = R

s,FASOP


Fig. 5. CSR Rcs vs. SOP constraint εs (AN-based transmission scheme).

target secrecy rate is R∗s,IP = RSoPs,IP for small εs, which

increases as εs increases, and then changes to R∗s,IP = R0s,IP

or R∗s,IP = RTPs,IP for large εs, which is independent of εs.

Such phenomenon indicates that, when either εc or εt is large,the CSR is sensitive to the change of the SOP constraint εsin an extremely small region, e.g., from 0 to about 0.00115 inFig. 4(a). Similar phenomena can be observed from Fig. 4(b),Fig. 5(a) and Fig. 5(b).

We now show the impact of the TP constraint εt on the CSRperformance in Fig. 6 and Fig. 7, where we plot Rcs vs. εt forthe two relationship cases under the PC-based and AN-basedtransmission schemes, respectively. Three different settings ofCOP constraint εc and SOP constraint εs are adopted for eachsubfigure in Fig. 6 and Fig. 7. We set the transmit powerof Alice to Pa = −20 dB in Fig. 7. We can see from Fig.6(a) that, if the COP constraint εc is much larger than theSOP constraint εs, the CSR stays constant as the constraint εtincreases, i.e., the CSR is independent of εt. Otherwise, theCSR first increases and then stays constant as εt increases.This is because, for the former case, the CSR is achieved atonly the optimal target secrecy rate R∗s,IP = RSOP

s,IP (as labeledin Fig. 6(a)), which is independent of εt as can be seen from(17). For the latter case, the optimal target secrecy rate isR∗s,IP = RTP

s,IP for small εt, which increases as εt increases,and then changes to R∗s,IP = R0

s,IP or R∗s,IP = RSOPs,IP for

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP constraint t

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

CS

R (

bits

per

cha

nnel

use

)

c = 0.95,

s = 0.01

c = 0.05,

s = 0.05

c = 0.1,

s = 0.013

Rs,IP* = R

s,IPSOP

Rs,IP* = R

s,IPTP

Rs,IP* = R

s,IPSOP

Rs,IP* = R

s,IP0


0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP constraint t

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

CS

R (

bits

per

cha

nnel

use

)

c = 0.992,

s = 0.01

c = 0.05,

s = 0.05

c = 0.1,

s = 0.013

Rs,FP* = R

s,FPSOP

Rs,FP* = R

s,FPTP

Rs,FP* = R

s,FPSOP

Rs,FP* = R

s,FP0


Fig. 6. CSR Rcs vs. TP constraint εt (PC-based transmission scheme).

large εt, which is independent of εt. We can observe similarphenomena from Fig. 6(b), Fig. 7(a) and Fig. 7(b).

We proceed to compare the CSR performance achievedin the independence relationship scenario and that achievedin the friend relationship scenario, for which we show Rcsvs. εc for both relationship scenarios under the PC-basedtransmission scheme in Fig. 8(a) and those under the AN-based transmission scheme in Fig. 8(b), respectively. We setthe SOP constraint and TP constraint to εs = εt = 0.1 inboth figures. In addition, we set the parameter υ to υ = 0.01and 0.001 in Fig. 8(a) and the transmit power of Alice Pato Pa = −5 dB and −20 dB in Fig. 8(b). We can observefrom both subfigures that the CSRs in the independencerelationship case are always larger than those in the friendrelationship case under all the parameter settings and bothtransmission schemes. This is intuitive since Willie and Evecan improve their attacking abilities by sharing their signals.The above observations indicate that being friends is the betterchoice than being independent for the eavesdropper group anddetector group.

Finally, we compare the PC-based transmission schemeand the AN-based transmission scheme in terms of the CSRperformance. To do so, we show Rcs vs. εc in Fig. 9 (resp.Rcs vs. εs in Fig. 10 and Rcs vs. εt in Fig. 11) underboth transmission schemes in the independence and friend

11

0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP constraint t

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

CS

R (

bits

per

cha

nnel

use

)c = 0.07,

s = 0.01

c = 0.05,

s = 0.05

c = 0.95,

s = 0.01

0.001 0.02 0.04 0.06 0.08 0.10

0.005

0.01

0.015

Rs,IA* = R

s,IASOP

Rs,IA* = R

s,IA0 R

s,IA* = R

s,IASOP

Pa = -20 (dB) R

s,IA* = R

s,IATP

Rs,IA* = R

s,IATP

Rs,IA* = R

s,IATP


0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

TP constraint t

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

CS

R (

bits

per

cha

nnel

use

)

c = 0.07,

s = 0.01

c = 0.05,

s = 0.05

c = 0.8,

s = 0.01

0.001 0.02 0.04 0.06 0.08 0.10

0.005

0.01

Rs,FA* = R

s,FASOP

Rs,FA* = R

s,FA0 R

s,FA* = R

s,FASOP

Pa = -20 (dB) R

s,FA* = R

s,FATP

Rs,FA* = R

s,FATP

Rs,FA* = R

s,FATP


Fig. 7. CSR Rcs vs. TP constraint εt (AN-based transmission scheme).

relationship scenarios, respectively. We set εs = εt = 0.1 inFig. 9, εc = εt = 0.1 in Fig. 10 and εc = εs = 0.1 in Fig.11. For each figure, we consider two different settings of thetransmit power of Alice Pa for the AN-based scheme. We canobserve from Fig. 9 that, in both relationship scenarios, the PC-based scheme achieves better CSR performance than the AN-based scheme, when a small transmit power (e.g., Pa = −20dB) is adopted in the AN-based scheme. However, when thetransmit power of AN-based scheme is relatively larger (e.g.,Pa = −15 dB), the PC-based scheme achieves better CSRperformance than the AN-based scheme under stringent COPconstraints (e.g., less than about 0.055 in Fig. 9(a)), while theAN-based scheme achieves better CSR performance than thePC-based scheme under less strict COP constraints.

Similar results can be obtained from Fig. 10, which showsthat the PC-based scheme outperforms the AN-based schemeif either the transmit power of the AN-based scheme or theSOP constraint is small. Otherwise, the AN-based scheme out-performs the PC-based scheme. However, the results obtainedfrom Fig. 11 are different. We can see from Fig. 11 that theAN-based scheme outperforms the PC-based scheme whenadopting a large transmit power (i.e., Pa = −15 dB), while itachieves worse CSR performance than the PC-based schemewhen adopting a small transmit power (i.e., Pa = −20 dB).Based on the above observations from Fig. 9, Fig. 10 and 11,

0 0.02 0.04 0.06 0.08 0.1

COP constraint c

0.01

0.02

0.03

0.04

0.05

0.06

CS

R (

bits

per

cha

nnel

use

)

Independence scenarioFriend scenario

= 0.01

= 0.001

s =

t = 0.1

(a) PC-based transmission scheme.

0 0.02 0.04 0.06 0.08 0.1

COP constraint c

0

0.05

0.1

0.15

0.2

0.25

0.3

CS

R (

bits

per

cha

nnel

use

)

Independence scenarioFriend scenario

Pa = -20 (dB)

Pa = -5 (dB)

s =

t = 0.1

(b) AN-based transmission scheme.

Fig. 8. Comparisons of the CSR performances in two relationship cases.

we can conclude that when the transmit power is not a bigconcern, transmitters may prefer the AN-based transmissionscheme to achieve better CSR performance, especially forless strict covertness, secrecy and transmission performanceconstraints. On the other hand, when the transmit power isconstrained (e.g., in IoT and sensor networks), the PC-basedscheme is more preferable for transmitters.

VI. CONCLUSION

This paper explores a new secure wireless communicationparadigm, where the physical layer security technology isapplied to ensure both the covertness and secrecy of the com-munication. We define a novel metric of covert secrecy rate(CSR) to depict the security performance of the new paradigm,and also provide solid theoretical analysis on CSR under twotransmission schemes (i.e., artificial noise (AN)-based one andpower control (PC)-based one) and two detector-eavesdropperrelationships (i.e., independence and friend). The results inthis paper indicate that in general the CSR performance canbe improved when the constraints on covertness, secrecy andtransmission performance become less strict. In particular,the PC-based transmission scheme outperforms the AN-basedtransmission scheme in terms of the CSR performance whenstrict constraints are applied to the covertness, secrecy andtransmission performance. On the other hand, when these

12

0 0.02 0.04 0.06 0.08 0.1COP constraint

c

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

CS

R (

bits

per

cha

nnel

use

)PC-based P

a = P

a,IP*

AN-based Pa = -20 dB


s =

t = 0.1


0 0.02 0.04 0.06 0.08 0.1

COP constraint c

0

0.01

0.02

0.03

0.04

0.05

0.06

CS

R (

bits

per

cha

nnel

use

)

PC-based Pa = P

a,FP*



s =

t = 0.1


Fig. 9. Comparisons of the CSR performances in the PC-based and AN-basedtransmission schemes (Rcs vs. εc).

constraints become less strict, the AN-based scheme mayachieve better CSR performance than the PC-based one byproperly adjusting the message transmit power. We expect thatthis work can shed light on the future studies of new securewireless communication paradigms.

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0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017

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0 0.02 0.04 0.06 0.08 0.1

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