1 Covert Communication Gains from Adversarys Uncertainty of Phase Angles

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Covert Communication Gains from Adversary’s

Uncertainty of Phase Angles

Sen Qiao, Daming Cao, Qiaosheng Zhang, Yinfei Xu, Member, IEEE, and Guangjie Liu

Abstract—This work investigates the phase gain of intelligent

reﬂecting surface (IRS) covert communication over complex-

valued additive white Gaussian noise (AWGN) channels. The

transmitter Alice intends to transmit covert messages to the

legitimate receiver Bob via reﬂecting the broadcast signals from

a radio frequency (RF) source, while rendering the adversary

Willie’s detector arbitrarily close to ineffective. Our analyses

show that, compared to the covert capacity for classical AWGN

channels, we can achieve a covertness gain of value 2 by

leveraging Willie’s uncertainty of phase angles. This covertness

gain is achieved when the number of possible phase angle pairs

N= 2. More interestingly, our results show that the covertness

gain will not further increase with Nas long as N≥2, even if

it approaches inﬁnity.

Index Terms—Covert communication, Intelligent reﬂecting

surface, Phase shift keying, Phase deﬂection, Covertness gain

from Phase.

I. INTRODUCTION

IN certain complicated antagonistic realms (such as mil-

itary communications), even a little exposed intention

of communication may lead to signiﬁcant strategic failures.

Consequently, the military has developed diverse techniques

(e.g. the spread spectrum technique [1]–[4]) to ensure the

covertness of communication, i.e., to hide the very presence

of communication from watchful adversaries. From the the-

oretical perspective, the information-theoretic limit of covert

communication was ﬁrst investigated by [5], which discovered

asquare root law (SRL) for additive white Gaussian noise

(AWGN) channels. This seminal theorem has subsequently

been extended to various channel models, including binary

symmetric channels [6], discrete memoryless channels [7]–[9]

and multiuser channels [10]–[14], etc.

In the covert communication scenario, the transmitter Alice

occasionally wishes to transmit a message to the legitimate re-

ceiver Bob over a noisy channel, while simultaneously ensures

that the adversary Willie is not able to detect the transmission

(if exists). The SRL states that to ensure both covertness and

This work was supported by the National Key R&D Program of China

(Grants No. 2021QY0700), the National Natural Science Foundation of China

(Grants No. U21B2003, 62072250), the Startup Foundation for Introducing

Talent of NUIST (Grants No. 2023r014) and Zhi Shan Young Scholar Program

of Southeast University.

Sen Qiao, Daming Cao and Guangjie Liu are with the School of Electrical

and Information Engineering, Nanjing University of Information Science

and Technology, Nanjing, 210044, China (e-mail: sensariel@nuist.edu.cn;

dmcao@nuist.edu.cn; gjieliu@gmail.com).

Qiaosheng Zhang is with Shanghai Artiﬁcial Intelligence Laboratory,

Shanghai, 200032, China (e-mail: zhangqiaosheng@pjlab.org.cn).

Yinfei Xu is with the School of Information Science and Engineering,

Southeast University, Nanjing 210096, China (e-mail: yinfeixu@seu.edu.cn).

Corresponding author: Daming Cao.

reliability, only O√nbits can be transmitted over nchannel

uses. Note that the transmission rate approaches zero as n

grows to inﬁnity.

Prior works have put forth diverse strategies to improve

the performance of covert communication, including relaying

networks [15], [16], multiple interference networks [17], [18],

unmanned aerial vehicle (UAV) networks [19], [20], multi-user

networks [21], etc. In particular, Lu et al. [22] noticed that the

intelligent reﬂecting surface (IRS) (a.k.a. the reconﬁgurable

intelligent surface (RIS)) has the capability of enhancing the

received signal at the receiver side while simultaneously dete-

riorating the signal at the warden side. In their setting, Alice

transmits messages covertly by reﬂecting her signal to Bob, or

reﬂecting additional noise to Willie via IRS devices. Following

their pioneering work, recent works [23]–[25] further show

that for IRS covert communication, Willie’s uncertainty about

noise can be appropriately leveraged to enhance the covert

performance. Besides, the optimization of transmission power

and reﬂection beamforming in IRS networks have also been

investigated in [26]–[28] and [23], [29], [30], respectively.

Moreover, the covert communication in UAV mounted IRS

(UIRS) communication systems has also been investigated in

[31], [32].

Different from the methods in the aforementioned works

[15]–[33], utilizing other resources, such as the spectrum and

time resource, has also been proven to be effective approaches

to improve the performance of covert communication. In [34],

Wang et al. investigated the problem of covert communication

over Multiple-Input Multiple-Output (MIMO) AWGN chan-

nels, where all users are equipped with multiple antennas.

Furthermore, the authors in [35] considered utilizing the time

resource to enhance the performance of covert communication.

In their setting, Alice and Bob are allowed to secretly choose

one single time slot (out of T(n)slots) to communicate, and

[35] showed that they can transmit Omin{pnlog T(n), n}

bits reliably and covertly when Willie does not have the

knowledge of the chosen slot.

In addition to spectrum and time, phase is another resource

that can be utilized, however many communication scenarios

studied in literature locate on real-valued AWGN channels

[5], [9], [35], [36]. Further, to the best of our knowledge,

no work has realized the beneﬁts of utilizing phase in covert

communication. Since the IRS can transmit information by

varying the amplitude and/or phase of signals [37], it is

interesting to investigate whether one can transmit more covert

information by utilizing the phase resource via IRS. We

note that most existing works on IRS covert communication

arXiv:2210.05084v3 [cs.IT] 6 May 2023

H0H1y

(a) Not deﬂect symbols

H1y

(b) Deﬂect each symbol

H1y

Fig. 1. Hypothesis testing with and without phase

focus on the optimization of transmission power and reﬂection

beamforming, without taking the effect of phases into con-

sideration. For example, the pioneers in [38] considered IRS

covert communication over complex-valued AWGN channels

with Binary Phase Shift Keying (BPSK) codebook, but no

phase changes has been employed. Here we show how Alice

can leverage Willie’s ignorance of the exact phase angle to

improve the covertness.

In our scenario, Alice communicates to Bob with an IRS

device and a shared secret of sufﬁcient length, and Willie may

not know the exact phase angles that Alice and Bob select

in advance. Willie observes over a complex-valued Gaussian

channel and performs a binary hypothesis test to detect the

communication. When Alice transmits with a BPSK codebook,

like the setting in [35], [36], [38], Willie can perform a binary

hypothesis test as shown in Fig. 1(a). For ease of notation, we

express the phase angle with values between zero and 2π. The

whole transmission consists of nsymbols. All symbols are sent

with two supplementary initial angles; for concreteness we set

the angles to be π/4and 5π/4. The choice of different initial

angles does not change Willie’s ability of detection due to

the symmetry of Gaussian noise. However, by utilizing IRS,

Alice can also reﬂect each symbol with other phase angles

except for the two initial angles, e.g., 3π/4and 7π/4(this is

equivalent to deﬂect the initial angles by π/2), as shown in

Fig. 1(b). Thus, Willie does not know whether the initial angles

(π/4,5π/4) or the deﬂected angles (3π/4,7π/4) are used by

Alice. Since this increases the uncertainty of Willie, Alice can

achieve an improvement over a naive application of the SRL.

Perhaps surprisingly, we show that it is not necessary for Alice

to reﬂect each symbol with a different angle. Instead, Alice

can achieve an improvement by confusing Willie whether all

the nsymbols are deﬂected by a same phase angle (e.g., π/2)

or not, as shown in Fig. 1(c). We refer the readers to Sec.

II-B and Sec. II-C for more details about the two deﬂection

methods.

Deﬂect, or not deﬂect? Just one-bit information can provide

a substantial covertness gain. The improvement stems from the

fact that Willie does not know the exact phase and has to detect

all possible phase angles. In this work, we investigate the effect

of phases in covert communication for the ﬁrst time. We name

the covertness improvement achieved from phase resources as

phase gain. Our results highlight the phase gain by comparing

covert performance between codebooks with a single phase

pair and codebooks with multiple phase pairs. We provide

detailed achievability proofs of three different codebooks. In

particular, we would like to point out that the calculation of

the KL divergence of one particular codebook (with multiple

phase pairs) requires non-trivial analytical techniques, since

its induced output distribution cannot be transformed to n

single-letter distributions by the chain rule. To circumvent this

difﬁculty, we take a non-trivial approach—using the Taylor

series expansion lim

x→0log (1 + x) = x−1

2x2+O(x3)to

approximate the KL divergence so that it can be calculated

by summing up the approximations of three terms. Moreover,

we generalize our results to inﬁnite phase angles and prove

that further increasing the number of phase angles does not

lead to a larger phase gain.

The rest of the paper is organized as follows. In Section

II, we introduce the system model and codebooks for IRS

covert communication over complex-valued AWGN channels.

Section III provides the main results of this work. In Section

IV, we present the detailed proofs of our results. Section V

presents numerical results that validate our theoretical results.

Section VI concludes this work and proposes several directions

that are fertile avenues for future research.

II. PREREQUISITES

A. System Model

We consider a complex-valued discrete-time AWGN chan-

nel model, as shown in Fig. 2. The RF source continu-

ously broadcasts ncomplex-valued random variable SR=

{SR,i}n

i=1. The IRS transmitter (Alice) intends to transmit

ncomplex-valued symbols c={ci}n

i=1 to a receiver (Bob)

by utilizing the broadcast signal from the RF source, while

the detector (Willie) seeks to detect the existence of the

transmission. Then, the i-th symbol that Bob and Willie

received can be expressed as

SB,i =hRBSR,i +hRAhABSR,ici+ZB,i,(1)

SW,i =hRW SR,i +hRAhAW SR,ici+ZW,i,(2)

where ZB,i and ZW,i are independent and identically dis-

tributed (i.i.d.) Gaussian noise with variance 2σ2, i.e.,

ZB,i, ZW,i ∼ CN(0,2σ2). The channel coefﬁcients from the

RF source to Alice, Bob and Willie are denoted by hRA,

hRB and hRW , respectively. The channel coefﬁcients from

Alice to Bob and to Willie are denoted by hAB and hAW ,

respectively. Without loss of generality, we assume the RF

source broadcasts nsymbols with zero phase angle, and all

channel coefﬁcients are known to everyone.

Alice(IRS)

RF source

Bob

Willie H0:Qn

H1:Qn

SRc

Fig. 2. System model for IRS covert communication

B. Codebook Construction

Alice transmits covert information to Bob by varying the

reﬂection coefﬁcient of IRS, including the amplitude and

phase coefﬁcient. Speciﬁcally, Alice transmits a uniformly-

distributed message W∈[[1, M]] to Bob by encoding it into

a codeword cn=c1, c2, ..., cnof blocklength n. For each

symbol ciof the codeword, we use its amplitude kcikand

phase θci(instead of its x-component ci,x and y-component

ci,y) to describe it. Clearly, we have

kcik2=kci,xk2+kci,yk2,(3)

tan(θci) = ci,y/ci,x.(4)

In the following section, we use ciand (kcik, θci)inter-

changeably when there is no confusion. In this work, we

consider three codebooks, and the constructions of the three

codebooks are provided as follows:

1) The BPSK codebook: In this codebook, each symbol

cihas the same amplitude βand two possible phases θand

θ+π. That is, ciequals either (β, θ)or (β, θ +π). All

symbols in this work are constructed in pairs, such as (β, θ)

and (β, θ +π). For ease of notation, we re-denote the symbol

(β, θ +π)as (−β, θ), and we also call these two angles, e.g.,

θand θ+π, as one angle pair. We sample the codeword

cnindependently and randomly according to the distribution

B(cn) = Qn

i=1 PB(ci)with PB(ci= (β, θ)) = PB(ci=

(−β, θ)) = 1/2.

2) The 2N-PSK codebook: In this codebook, each symbol

cihas the same amplitude βand 2Npossible phases tπ

t= 1,2,...,2N. Equivalently, each symbol can be described

as ci= (β, tπ

N)or ci= (−β, tπ

N),t= 1,2, . . . , N. Now,

we sample the codeword cnindependently and randomly

according to Pn

2N(cn) = Qn

i=1 P2N(ci)with P2N(ci=

(β, tπ

N)) = P2N(ci= (−β, tπ

N)) = 1

2N, t = 1,2, ..., N .

3) The N-BPSK codebook: This codebook is transformed

from the BPSK codebook. The phase angle of each codeword

is additionally added a uniformly random distributed phase

angle. In other words, Alice transmits each codeword in BPSK

with an additional phase angle b

θ, which is selected uniformly

at random from {π

N,2π

N, ..., (N−1)π

N, π}. This additional phase

angle is shared conﬁdentially with Bob in advance.1Compared

to the 2N-PSK codebook, each symbol in a codeword from

N-BPSK only has two possible phase angles instead of 2N

possible phase angles. Speciﬁcally, in the 2N-PSK codebook,

the phases of different symbols can belong to different angle

pairs, e.g., c1= (β, π

N)and c2= (β, 2π

N). However, in the N-

BPSK codebook, the phases of different symbols must belong

to one angle pair, e.g., if c1= (β, π

N), we have c2= (β, π

or c2= (−β, π

N).

It is assumed that the codebook is revealed to Willie,

including the value of amplitude gain β, and the set of all

possible angles, i.e., {θ, π

N,2π

N, ..., (N−1)π

N, π}.

C. Hypothesis test

We make a practical assumption that Willie can recover the

broadcast information SRand subtract it from his observations

to enhance the detection performance. Hence, the i-th symbol

received by Willie can be equivalently rewritten as

SW,i =hRAhAW SR,ici+ZW,i.(5)

Considering a quasi-static ﬂat fading channel with coefﬁ-

cients hRA and hAW , we deﬁne the expected amplitude of

complex-value hRAhAW SRas A, i.e,

E(|hRAhAW SR|2) = A2,(6)

and we deﬁne the expected phase of hRAhAW SRas θ0. To

determine whether Alice is communicating, Willie performs a

binary hypothesis test [39] based on nsuccessive observations

SW={SW,i}n

i=1. Speciﬁcally, let the null hypothesis (H0)

denote that no communication is taking place, where each

sample SW,i =ZW,i is an i.i.d. complex-Gaussian random

variable distributed according to CN(0,2σ2). The alternative

hypothesis (H1) denotes that communication is taking place

and each sample SW,i =hRAhAW SR,ici+ZW,i. Willie aims

to distinguish these two hypotheses:

H0:SW,i =ZW,i,(7)

H1:SW,i =hRAhAW SR,ici+ZW,i.(8)

Let Qn

0(resp. ¯

Q(n)

1) denote the probability distribution of

Willie’s nobservations when H0(resp. H1) is true. The proba-

bility of false alarm (i.e., rejecting H0when it is true) is denote

by PF A, and the probability of missed detection (i.e., accepting

H0when it is false) is denoted by PMD. We assume that

the distribution Qn

0and ¯

Q(n)

1are known to Willie, and Willie

can perform an optimal statistical hypothesis test that satisﬁes

PF A +PM D = 1 −V(¯

Q(n)

1kQn

0)[36]. By using the deﬁnition

of KL divergence and Pinsker’s inequality, we can obtain that

the optimal test satisﬁes PF A +PMD ≥1−qD(¯

Q(n)

1kQn

0). It

is possible for Willie to perform a blind test when the sum of

error probabilities equal one, i.e., PF A +PMD = 1. And the

objective of covert communication is to guarantee that Willie’s

statistical test is not much better than blind test. Therefore, we

1In many covert communication scenarios, Alice shares a key with Bob

before the communication, typically of size O(√n)bits. So it is reasonable

to assume that Alice and Bob share an additional key of size O(log N)

perfectly and conﬁdentially to reach a consensus on the phase angle.

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1CovertCommunicationGainsfromAdversary'sUncertaintyofPhaseAnglesSenQiao,DamingCao,QiaoshengZhang,YinfeiXu,Member,IEEE,andGuangjieLiuAbstractThisworkinvestigatesthephasegainofintelligentreectingsurface(IRS)covertcommunicationovercomplex-valuedadditivewhiteGaussiannoise(AWGN)channels.ThetransmitterA...

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