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IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020 99
Enhancing Physical Layer Security in Internet of
Things via Feedback: A General Framework
Bin Dai , Associate Member, IEEE, Zheng Ma , Yuan Luo , Xuxun Liu , Member, IEEE,
Zhuojun Zhuang, and Ming Xiao , Senior Member, IEEE
Abstract—In this article, a general framework for enhancing
the physical layer security (PLS) in the Internet of Things (IoT)
systems via channel feedback is established. To be specific, first,
we study the compound wiretap channel (WTC) with feedback,
which can be viewed as an ideal model for enhancing the PLS in
the downlink transmission of IoT systems via feedback. A novel
feedback strategy is proposed and a corresponding lower bound
on the secrecy capacity is constructed for this ideal model. Next,
we generalize the ideal model (i.e., the compound WTC with
feedback) by considering channel states and feedback delay, and
this generalized model is called the finite state compound WTC
with delayed feedback. The lower bounds on the secrecy capac-
ities of this generalized model with or without delayed channel
output feedback are provided, and they are constructed accord-
ing to variations of the previously proposed feedback scheme
for the ideal model. Finally, from a Gaussian fading example,
we show that the delayed channel output feedback enhances the
achievable secrecy rate of the finite state compound WTC with
only delayed state feedback, which implies that feedback helps
Manuscript received June 17, 2019; revised September 6, 2019; accepted
September 19, 2019. Date of publication October 3, 2019; date of current
version January 10, 2020. The work of B. Dai was supported in part by the
National Natural Science Foundation of China under Grant 61671391, in part
by the China Scholarship Council under Grant 201807005013, and in part
by the 111 Project under Grant 111-2-14. The work of Z. Ma was supported
in part by the National Natural Science Foundation of China under Grant
U1734209, in part by the Key International Cooperation Project of Sichuan
Province under Grant 2017HH0002, in part by the EU Marie Skłodowska-
Curie Individual Fellowship under Grant 796426, and in part by the NSFC
China-Swedish Project under Grant 6161101297. The work of Y. Luo was
supported by the National Natural Science Foundation of China under Grant
61871264. The work of M. Xiao was supported in part by the Swedish
Strategic Research Foundation Project “High-Reliable Low-Latency Industrial
Wireless Communications,” in part by the EU Marie Skłodowska-Curie
Actions Project titled “High-Reliability Low-Latency Communications With
Network Coding,” and in part by the ERA-NET Smart Energy Systems SG+
2017 Program, “SMART-MLA” under Project 89029 (and SWEA 42811-2).
(Corresponding author: Bin Dai.)
B. Dai is with the School of Information Science and Technology,
Southwest Jiaotong University, Chengdu 611756, China (e-mail:
daibin@home.swjtu.edu.cn).
Z. Ma is with the School of Electrical Engineering and Computer Science,
Royal Institute of Technology (KTH), SE-10044 Stockholm, Sweden (e-mail:
zma@home.swjtu.edu.cn).
Y. Luo is with the Computer Science and Engineering Department,
Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:
luoyuan@cs.sjtu.edu.cn).
X. Liu is with the School of Electronic and Information Engineering,
South China University of Technology, Guangzhou 510641, China (e-mail:
liuxuxun@scut.edu.cn).
Z. Zhuang is with the Innovation Academy for Microsatellites,
Chinese Academy of Sciences, Shanghai 201203, China
(e-mail: zhuangzhuojun@163.com).
M. Xiao is with the School of Electrical Engineering and the ACCESS
Linnaeus Center, Royal Institute of Technology (KTH), SE-10044 Stockholm,
Sweden (e-mail: mingx@kth.se).
Digital Object Identifier 10.1109/JIOT.2019.2945503
to enhance the PLS in the downlink transmission of the IoT
systems.
Index Terms—Compound channel, feedback, secrecy capacity,
wiretap channel (WTC).
I. INT RODU CTI ON
INTERNET of Things (IoT) is taking the center stage of
the upcoming 5G as the devices are expected to be a major
component of 5G network. Due to the broadcasting nature of
wireless communication, signals in the IoT systems are more
vulnerable to eavesdropping, and hence the secure commu-
nication over the IoT systems is one of the most pressing
problems needed to be solved. The study of the secure trans-
mission over communication systems started from Wyner [1]
in his groundbreaking work on the wiretap channel (WTC),
where a transmitter broadcasts its message Wover Nchan-
nel uses to a legitimate receiver and an eavesdropper via a
degraded broadcast channel (BC), and the perfect secrecy is
guaranteed if the information leakage rate (1/N)I(W;ZN),
where ZNdenotes the received signal at the eavesdropper, van-
ishes as the codeword length Ntends to infinity.1The secrecy
capacity, defined as the channel capacity with perfect secrecy
constraint, was established in [1]. Subsequently, Csiszár and
Körner [3] generalized the model studied in [1] by consid-
ering a general (not degraded) BC and the transmission of a
common message which can be decoded by both legitimate
receiver and eavesdropper. The follow-up studies of [1]–[3]
include the Gaussian WTC [4], the BC with two secret mes-
sages [5], [6], and one transmitter broadcasts a secret message
to multiple legitimate receivers and one eavesdropper (wiretap
BC) [7], [8].
Here note that Wyner [1] further pointed out that the secrecy
capacity is positive if the legitimate receiver’s channel is less
noisy than the eavesdropper’s. Then it is natural to ask the
following two questions.
1) How to achieve positive secrecy capacity when the
eavesdropper’s channel is less noisy than the legitimate
receiver’s?
1Here the perfect secrecy defined in [1] is in fact weak perfect secrecy.
Another definition of the perfect secrecy is strong perfect secrecy [2], which
is defined as the information leakage I(W;ZN)at the eavesdropper vanishes
as Ntends to infinity.
2327-4662 c
2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
100 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
2) When the legitimate receiver’s channel is less noisy than
the eavesdropper’s, how to further enhance the secrecy
capacity?
The answer to both questions is artificial noise (AN) [9]–[16]
and channel feedback (CF). However, we should notice that
since the IoT devices (e.g., sensors and actuators) have sig-
nificant energy constraint [17], [18], AN may not be suitable
for IoT systems, and hence CF is of particular interest for
enhancing the physical layer security (PLS) in IoT systems.
The study of the effect of CF on the PLS of communica-
tion channels started from [19], where the pioneering work [1]
has been revisited by considering the situation that the legit-
imate receiver’s received channel output is sent back to the
transmitter through an additional noiseless feedback channel
which is not known by the eavesdropper. Since the legitimate
receiver’s channel output is perfectly known by the transmitter
and completely not known by the eavesdropper, we can gen-
erate the secret key from it, and this key helps to protect the
transmitted message. Combining the above idea of generating
the secret key from the CF with the random binning scheme
for the WTC [1], Ahlswede and Cai [19] proposed a coding
scheme which splits the transmitted message into two parts,
where one submessage is encoded in the same way as that
of the WTC [1], and the other is encrypted by the secret key.
Compared with the secrecy capacity of the WTC [1], it is easy
to see that encrypting part of the message by the key leads to
the channel output feedback enhancing the secrecy capacity
of the WTC.
Note that in [19], the feedback channel is only used to
send the legitimate receiver’s channel output. What happens
when the feedback channel can transmit anything as the legal
receiver wishes? Ardestanizadeh et al. [20] studied this case
and pointed out that for the legal receiver, the best way is to
transmit randomly generated sequences (served as secret keys)
over the feedback channel. Assumeing that the transmission
rate of the feedback channel is up to Rf, using a coding scheme
in the same way as that in [19], Ardestanizadeh et al. [20]
showed that sending the pure secret key is better than send-
ing legal receiver’s channel output if Rfis larger than the
key rate in [19], and vice versa. The works of [19] and [20]
indicate that there is no difference between sending the pure
secret key and sending legitimate receiver’s channel output,
and the main purpose of the feedback is to allow the legitimate
receiver and the transmitter to share the secret key. In recent
years, the above secret-key-based feedback coding scheme
has been widely used in communication systems with feed-
back channel. To be specific, for the communication channels
with legal receiver’s channel output feedback, Dai et al. [23],
[24] studied PLS of the channels with memory or memoryless
states and legitimate receiver’s channel output feedback, and
proposed variations of the secret-key-based feedback coding
scheme in [19]. For the communication systems with feedback
channels directly transmitting pure secret keys, Schaefer et al.
[21] extended the work of [20] to a broadcast situation, where
two legitimate receivers of the BC independently send their
secret keys to the transmitter via two noiseless feedback chan-
nels, and these keys help to increase the achievable secrecy
rate region of the broadcast WTC [7]. Cohen and Cohen [22]
Fig. 1. PLS in the downlink transmission of the IoT systems.
introduced memoryless channel state into the work of [20],
and showed that the transmitted message can be protected by
two keys, where one is from the feedback channel, and the
other is generated by the channel state. Very recently, Dai and
Luo [25] showed that for the general WTC with CF, a bet-
ter choice of the transmitter is to produce not only the secret
key but also the auxiliary message from CF, where the auxil-
iary message is used to improve the legal receiver’s decoding
performance. Dai and Luo [25] proved that for the WTC with
channel output feedback, this new feedback scheme performs
better than the widely used secret-key-based feedback scheme.
Moreover, Li et al. [26] and Gundüz et al. [27] showed that
the classical Schalkwijk–Kailath (SK) [28] feedback scheme
for the Gaussian channel achieves the secrecy capacity of the
Gaussian WTC with channel output feedback, and it equals
the capacity of the same channel model without the secrecy
constraint. However, we should notice that the results of [26]
and [27] only work in the Gaussian case.
In IoT systems, the uplink transmission is from sensors to
controllers, and the downlink transmission is from controllers
to actuators. A classical scenario for the PLS in the downlink
transmission of the IoT systems is depicted in Fig. 1, where
a controller tries to send a secret message to several actuators
in the presence of an eavesdropper. An ideal model charac-
terizing this classical scenario is called the compound WTC,
where the channels for all actuators and the eavesdropper are
independent of one another. Achievable secrecy rates (lower
bounds on the secrecy capacities) of various compound WTCs
were provided in [29]–[31], and it is shown that if the eaves-
dropper’s channel is less noisy than all actuators’ (legitimate
receivers’) channels, the achievable secrecy rate equals zero.
As we mentioned before, AN is not suitable for the scenario
in Fig. 1. Moreover, we should notice that the already exist-
ing feedback schemes in [19]–[25] cannot be applied to the
communication scenario in Fig. 1 due to the reason that each
actuator does not know others’ CF, and hence the feedback of
all channels cannot be used to generate a common secret key
shared between the transmitter and all the actuators.
In this article, we try to answer the following two funda-
mental questions.
1) How to increase the achievable secrecy rate of the
compound WTC model by using CF?
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 101
Fig. 2. Compound WTC with feedback.
2) Is there a more practical model for the scenario shown
in Fig. 1? If so, can we apply the feedback scheme of
1) to this more practical model?
This article provides the comprehensive answers to the afore-
mentioned questions. Our main contributions are summarized
as follows.
1) We study the compound WTC with feedback (see
Fig. 2). An achievable secrecy rate, which is constructed
according to a novel feedback strategy, is provided for
the model of Fig. 2.
2) A more practical model for the scenario in Fig. 1 is
provided (see Fig. 3), and an achievable secrecy rate for
this new model is obtained according to a modified feed-
back scheme of the model of Fig. 2. From a Gaussian
fading example, we show that the achievable secrecy
rate of the new model is larger than that of the same
model without channel output feedback, which implies
that the proposed feedback scheme enhances the PLS in
the downlink transmission of the IoT systems.
In the remainder of this article, random variables (RVs),
values, and alphabet are denoted by uppercase letters, lower-
case letters, and calligraphic letters, respectively. The random
vector and its value are written in a similar way. For exam-
ple, suppose that X1is an RV, and x1is a real value in the
alphabet X1. Similarly, suppose that XN
1,1is a random vector
(X1,1,...,X1,N), and xN
1,1=(x1,1,...,x1,N)is a vector value
in XN
1(the Nth Cartesian power of X1). Moreover, for sim-
plicity, the probability Pr{X=x}is denoted by P(x), and in
the remainder of this article, the base of the log function is 2.
The outline of this article is organized as follows. Section II
investigates the compound WTC with feedback (see Fig. 2),
and provides bounds on the secrecy capacity of this ideal
model. Section III studies a generalized version of the model
of Fig. 2, called the finite state compound WTC with delayed
feedback (see Fig. 3), and also provides bounds on the secrecy
capacity of this generalized model. In Section IV, the capac-
ity results on the model of Fig. 3 are further illustrated by
a Gaussian fading example. The final conclusion is given in
Section V.
Fig. 3. Finite state compound WTC with delayed feedback.
II. COMPOUN D WIRETAP CHANNEL WITH FEEDBACK
The discrete memoryless compound WTC with feedback
is shown in Fig. 2, where a transmitter wishes to broadcast
his/her secret message to Llegitimate receivers and an eaves-
dropper attempts to eavesdrop this secret message via a WTC.
The overall channel transition probability of the model of
Fig. 2 is given by
PyN
1,1,yN
2,1,...,yN
L,1,zN|xN=PzN|xNL
j=1
PyN
j,1|xN
=
N
i=1
P(zi|xi)
L
j=1
Pyj,i|xi
(1)
where xi∈X,yj,i∈Yj, and zi∈Z. Here note that zN(xN) is
an abbreviation of zN
1(xN
1), and a similar convention is applied
to ZN
1(XN
1).
The transmitted message Wis uniformly distributed over
W= {1,2, . . . , |W|}. Since all legitimate receivers send their
received channel outputs back to the transmitter via feed-
back channels, the ith (i∈ {1,2,...,N}) channel input Xi
is given by
Xi=fiW,Yi−1
1,1,Yi−1
2,1,...,Yi−1
L,1(2)
where fiis a stochastic encoding function, and Yi−1
j,1(j∈
{1,2,...,L}) is the jth legitimate receiver’s channel output
feedback at time i.
For the jth (j∈ {1,2,...,L}) legitimate receiver, after
receiving YN
j,1, he/she produces an estimation ˆ
W(j)=ψj(YN
j,1)
(ψjis the jth legitimate receiver’s decoding function), and the
average decoding error probability equals
Pe,j=1
|W|
i∈W
PrψjyN
j,1= i|isent.(3)
The secrecy level of the transmitted message Wat the
eavesdropper is formulated as
=1
NHW|ZN.(4)
102 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
Fig. 4. Feedback scheme for the compound WTC.
Given a non negative number R, if for any > 0, there exist
encoder and decoders such that
log |W|
N≥R−, ≥R−
Pe,j≤for all j∈ {1,2,...,L}.(5)
Ris achievable under weak perfect secrecy constraint. The
secrecy capacity Cf
sis composed of all such achievable R
defined in (5), and bounds on Cf
sare given in the remainder
of this section.
Theorem 1 (Lower Bound on Cf
s): Cf
s≥Rf
s, where
Rf
s=max min
jminIVj;Yj,Uj−IVj;Z+,IVj;Yj.(6)
[a]+=afor a≥0, [a]+=0 for a<0, and the joint
probability is defined as
P(z,y1, . . . , yL,x,v1, . . . , vL,u1, . . . , uL)
=P(z|x)P(u1, . . . , uL|v1, . . . , vL,y1, . . . , yL)
×P(y1, . . . , yL|x)P(x|v1, . . . , vL)P(v1, . . . , vL)
=P(z|x)P(x|v1, . . . , vL)P(v1,. . . , vL)×
L
j=1
Puj|vj,yjPyj|x.
(7)
Proof Sketch: The lower bound Rf
sis constructed accord-
ing to a block Markov coding scheme, and the encoding-
decoding procedure of each block is briefly explained in Fig. 4.
From Fig. 4, we see that in each block, after receiving the
CF of receiver j(j∈ {1,2,...,L}), the transmitter encodes
the transmitted message of current block and the feedback of
receiver jas a codeword vN
j,1, and the channel input of current
block is generated via an auxiliary discrete memoryless chan-
nel with inputs vN
1,1,...,vN
L,1, and output xN. For receiver j,
after receiving the channel outputs of all blocks, he/she uses
the backward and jointly typical decoding scheme to decode
the transmitted vN
j,1for all blocks. If vN
j,1is decoded without
error, the messages of all blocks can be obtained by receiver j.
The details about the encoding-decoding scheme of Theorem 1
are in Appendix A.
Remark 1:
1) In [29, Th. 1], it has been shown that a lower bound
Rson the secrecy capacity Csof the compound WTC is
given by
Cs≥Rs=max min
jIU;Yj−I(U;Z)+(8)
where the joint distribution is denoted by
P(z,y1, . . . , yL,x,u)=P(z|x)P(y1,...,yL|x)P(x|u)P(u)
=P(z|x)P(x|u)P(u)
L
j=1
Pyj|x.(9)
Fig. 5. Binary symmetric case of the compound WTC with noiseless
feedback.
In general, we do not know whether Rf
sis larger than
Rsor not. In Section IV, from a Gaussian fading exam-
ple, we show that Rf
sis larger than Rs, which indicates
that CF may increase the achievable secrecy rate of the
compound WTC.
2) For the compound WTC with noiseless feedback, it
is natural to ask: When not directly using the noise-
less feedback channels for secure communication, is the
total secrecy rate of the original compound WTC and
the noiseless feedback channels larger than Rf
sgiven
in Theorem 1? To answer this question, a binary sym-
metric case of the model of Fig. 2 is investigated (see
Fig. 5), and we show that for this special case, directly
using noiseless feedback channels for secure communi-
cation may not always be the best choice. In Fig. 5,
one transmitter wishes to send a message Wto two
legitimate receivers via two BSCs with crossover proba-
bilities p1and p2, respectively, and an eavesdropper tries
to eavesdrop Wvia another BSC with crossover proba-
bility q. In addition, the legitimate receivers send their
received signals back to the transmitter via two noise-
less feedback channels. From Theorem 1, we see that
an achievable secrecy rate Rf
sfor the model of Fig. 5 is
given by
Rf
s=max min
j=1,2minIVj;Yj,Uj−IVj;Z+,IVj;Yj.
(10)
Then defining P(V1=0)=α,P(V1=1)=1−α,
P(V2=0)=β, and P(V2=1)=1−β,V1is inde-
pendent of V2, and letting X=V1+V2,U1=V1+Y1,
U2=V2+Y2, we have
Rf
s=max
α,β min
minH(α)−H(α β q)
+H(β q)+,
H(α β p1)−H(β p1)},
minH(β)−H(α β q)
+H(α q)+,
sH(α β p2)−H(α p2)}
(11)
where H(a)= −alog(a)−(1−a)log(1−a)and
ab=a(1−b)+(1−a)b. Next, if the noiseless feedback
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 103
Fig. 6. Binary symmetric compound WTC model with noiseless feedback
channels for direct transmission.
channels of Fig. 5 are used for direct transmission, the
model of Fig. 5 should be revised as the model in Fig. 6.
In Fig. 6, two messages W1and W2are transmitted,
where W1is transmitted through the binary symmet-
ric compound WTC in the model of Fig. 5 without
feedback, and W2is transmitted through the noiseless
feedback channels and due to the broadcast nature of
wireless communication, W2can also be eavesdropped
by the eavesdropper via a binary symmetric WTC with
crossover probability q. From [29, Th. 1], an achievable
secrecy rate R∗
1of W1is given by
R∗
1=max
P(x)min
j=1,2IX;Yj−I(X;Z)+
(a)
=minH(q)−H(p1)+,H(q)−H(p2)+(12)
where (a) is from defining P(X=0)=α,P(X=1)=
1−α, and using the fact that the maximum is achieved
when α=(1/2). Analogously, an achievable secrecy
rate R∗
2of W2is given by
R∗
2=max
P(x)
min
j=1,2IX;Y
j−IX;Z+
(b)
=H(q)(13)
where (b) follows from substituting p1=p2=0
into (12). Hence the total secrecy rate R∗=R∗
1+R∗
2
is given by
R∗=minH(q)−H(p1)+,H(q)−H(p2)++H(q).
(14)
Fig. 7 plots the achievable secrecy rate Rf
sof the model
of Fig. 5 and the total secrecy rate R∗of the model
Fig. 7. Comparison of the secrecy rates Rf
sand R∗for p1=0.0001, p2=0.5,
and several values of q.
Fig. 8. Comparison of the secrecy rates Rf
sand R∗for p1=0.0001, p2=0.3,
and several values of q.
of Fig. 6 for p1=0.0001, p2=0.5, and several
values of q. From this figure, we see that the feed-
back scheme proposed in Theorem 1 performs no better
than directly using the feedback channels for secure
transmission when one legitimate receiver’s channel is
completely noisy (i.e., p2=0.5). Fig. 8 plots Rf
sand R∗
for p1=0.0001, p2=0.3, and several values of q. From
this figure, we see that the feedback scheme proposed
in Theorem 1 performs better than directly using the
feedback channel for secure transmission when qis very
small. From the above figures, we see that directly using
the feedback channels for secure transmission may not
always be the best choice, and sometimes the feedback
scheme of Theorem 1 may perform better.
Theorem 2 (Upper Bound on Cf
s): Cf
s≤Cf−out
s, where
Cf−out
s=min
jmax
P(x)IX;Yj(15)
and the joint probability is defined as
P(z,y1, . . . , yL,x)=P(z|x)
L
i=1
P(yi|x).(16)
Proof: This outer bound can be directly obtained by using
the fact that the secrecy capacity cannot exceed the capacity of
each channel and feedback does not increase the capacity of
a discrete memoryless channel. Hence the secrecy capacity Cf
s
is upper bounded by the minimum of each channel’s capacity
[here note that the capacity of channel jis maxP(x)I(X;Yj)],
and the proof is completed.
104 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
Here note that the compound WTC with feedback investi-
gated in this section is only an ideal model for the PLS in
the downlink transmission of the IoT system. In the next sec-
tion, we will study a more practical model, which we call the
finite state compound WTC with delayed feedback. The lower
bound on the secrecy capacity of this more practical model is
constructed according to a variation of the feedback strategy
in Theorem 1 (see the remainder of this article).
III. FINITE STATE COMP OUND WIRETAP CHANNEL WITH
DELAYED FEEDBACK
The practical IoT systems often consist of time-varying and
fading channels, and the states of these channels are often
obtained by the transmitter via receivers’s delayed feedback.
In [33], the time-varying fading channel in the presence of
one transmitter, one legitimate receiver, one eavesdropper and
delayed CF is modeled as the finite state Markov WTC (FSM-
WTC) with delayed feedback. In this section, we extend the
FSM-WTC with delayed feedback to a more general case, i.e.,
the finite state compound WTC with delayed feedback (see
Fig. 3). In Fig. 3, the channel consists of multiple legitimate
receivers and one eavesdropper, and each legitimate receiver
sends his/her received signal back to the transmitter via a cor-
responding feedback channel with different delayed feedback
time. In the remainder of this section, we first give formal
definition of the model of Fig. 3, and then we show bounds
on the secrecy capacity of this new model.
Model Formulation:
1) The overall channel transition probability of the model
of Fig. 3 is given by
PyN
1,1,yN
2,1,...,yN
L,1,zN|xN,sN
1,1, . . . , sN
L,1,sN
e,1
=PzN|xN,sN
e,1L
j=1
PyN
j,1|xN,sN
j,1
=
N
i=1
Pzi|xi,se,iL
j=1
Pyj,i|xi,sj,i
(17)
where xi∈X,yj,i∈Yj,zi∈Z,se,i∈Se, and sj,i∈Sj.
2) The finite state processes {Se,i}and {Sj,i}(j∈
{1,2,...,L}) are supposed to be stationary irreducible
aperiodic ergodic Markov chains. The state processes are
independent of one another, and they are independent
of the transmitted message. Moreover, the state process
{Sj,i}is independent of the channel input and outputs
given the previous states, i.e.,
Psj,i|xi,yi
1,1, . . . , yi
L,1,si−dj
j,1=Psj,i|sj,i−dj(18)
where 1 ≤dj≤i−1. The state process {Se,i}is indepen-
dent of the channel input and the eavesdropper’s channel
output given the previous states, i.e.,
Pse,i|xi,zi,si−1
e,1=Pse,i|se,i−1.(19)
Define the one-step transition probability matrix of the
state process {Sj,i}as Kj. Denote the steady state prob-
abilities of {Sj,i}and {Se,i}by πjand πe, respectively.
Note that
PrSj,i=sm,Sj,i−dj=sl=πj(sl)Kdj
j(sl,sm)(20)
where sm,sl∈Sj,smis the mth element of Sj,slis the
lth element of Sj, and Kdj
j(sl,sm)is the (l,m)th element
of the dj-step transition probability matrix Kdj
jof the
Markov process.
3) The transmitted message Wis uniformly distributed over
W= {1,2,...,|W|}, and it is independent of the state
processes {Se,i}and {Sj,i}(j∈ {1,2,...,L}). receiver
j∈ {1,2,...,L}sends his/her received signal back to the
transmitter via a feedback channel after a delay time dj.
Without loss of generality (W.L.O.G.), assume that 1 ≤
d1≤d2≤ · · · ≤ dL≤N. For the case that all legitimate
receivers only send their received channel states back to
the transmitter via feedback channels with delay times
d1,...,dL, the ith (i∈ {1,2,...,N}) channel input Xi
is given by
Xi=
fi(W),1≤i≤d1
fiW,Si−d1
1,1,d1≤i≤d2
. . . . . .
fiW,Si−d1
1,1,...,Si−dL−1
L−1,1,dL−1≤i≤dL
fiW,Si−d1
1,1,...,Si−dL
L,1,dL≤i≤N.
(21)
For the case that all legitimate receivers send their
received channel outputs and channel states back to
the transmitter via feedback channels with delay times
d1,...,dL, the ith (i∈ {1,2,...,N}) channel input Xi
is given by
Xi=
fi(W), 1≤i≤d1
fiW,Si−d1
1,1,Yi−d1
1,1,d1≤i≤d2
··· ···
fiW,Si−d1
1,1,Yi−d1
1,1
. . . , Si−dL
L,1,Yi−dL
L,1,dL≤i≤N.
(22)
Here note that fiin (21) and (22) is a stochastic encoding
function.
4) For receiver j(j∈ {1,2,...,L}), after receiving YN
j,1and
SN
j,1, he/she produces an estimation ˆ
W(j)=ψj(YN
j,1,SN
j,1),
and his/her average decoding error probability is defined
as
Pe,j=1
|W|
i∈W
Prψj(yN
j,1,sN
j,1)= i|isent.(23)
The secrecy level of the transmitted message Wat the
eavesdropper is formulated as
=1
NHW|ZN,SN
e,1.(24)
The definition of a non-negative number Rachieving
weak perfect secrecy is the same as that in (5).
The secrecy capacity of the model of Fig. 3 with delayed
channel output feedback is denoted by Cf−dy
s, and without
delayed channel output feedback is denoted by Cf−d
s. Bounds
on Cf−dy
sand Cf−d
sare given in the following theorems.
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 105
Fig. 9. Feedback scheme for the finite state compound WTC with delayed
states and channel output feedback.
Theorem 3 (Lower Bound on Cf−dy
s): Cf−dy
s≥Rf−dy
s, where
Rf−dy
s=max min
jminIVj;Yj,Uj|Sj,˜
Sj−IVj;Z|Se+
IVj;Yj|Sj,˜
Sj(25)
the auxiliary RV ˜
Sjrepresents Sj,i−dL,Sjrepresents Sj,i, and
the joint probability is defined as
P(z,y1, . . . , yL,x,v1, . . . , vL,u1, . . . , uL,s1, . . . , sL,˜s1, . . . , ˜sL,se)
=P(z|x,se)P(x|v1, . . . , vL)P(se)
×
L
j=1Pyj|x,sjPuj|vj,yj,˜sjPvj|˜sj×P˜sjKdL
j˜sj,sj.
(26)
Proof Sketch: The lower bound Rf−dy
sis constructed
by combining the coding scheme of Theorem 1 with the
multiplexing encoding-decoding scheme for the FSM-WTC
with delayed feedback [33], and the encoding–decoding pro-
cedure is briefly explained in Fig. 9. From Fig. 9, we see
that in block i(i∈ {1,2, . . . , N}), the transmitted message
Wifor receiver j(j∈ {1,2,...,L}) is further divided into kj
submessages (Wi=(Wi,1,Wi,2, . . . , Wi,kj)), where kjis the
size of the alphabet Sj, i.e., kj= |Sj|. Moreover, in block
i, after receiving the delayed feedback channel output and
state of receiver j, the transmitter encodes each submessage
Wi,l(l∈ {1,2,...,kj}) and the delayed feedback as a sub-
codeword vNl
j,i,1(similar to the coding scheme in the proof of
Theorem 1, see Appendix A), where Nlis the subcodeword
length for Wi,l, and kj
l=1Nl=N. Hence the total codeword
vN
j,i,1for Wiis the multiplexing of all subcodewords vNl
j,i,1for
l∈ {1,2, . . . , kj}, i.e., vN
j,i,1=(vN1
j,i,1,vN2
j,i,1, . . . , vNkj
j,i,1). Similar
to the coding scheme of Theorem 1, the channel input of block
iis generated via an auxiliary discrete memoryless channel
with inputs vN
1,i,1,...,vN
L,i,1, and output xN.
In the decoding procedure, for receiver j, after receiving the
channel outputs and states of all blocks, he/she uses the back-
ward, de-multiplexing and jointly typical decoding scheme to
decode the transmitted vn
j,1=(vN
j,1,1,vN
j,2,1,...,vN
j,n,1)of all
blocks. If vn
j,1is decoded without error, the messages for all
blocks can be obtained by receiver j. The details about the
encoding-decoding scheme of Theorem 3 are in Appendix B.
Theorem 4 (Lower Bound on Cf−d
s): Cf−d
s≥Rf−d
s, where
Rf−d
s=max min
jIVj;Yj|Sj,˜
Sj−IVj;Z|Se+(27)
the auxiliary RV ˜
Sjrepresents Sj,i−dL,Sjrepresents Sj,i, and
the joint probability is defined as
P(z,y1,...,yL,x,v1,...,vL,s1,...,sL,˜s1,...,˜sL,se)
=P(z|x,se)P(x|v1,...,vL)P(se)
×
L
j=1Pyj|x,sjPvj|˜sjP˜sjKdL
j˜sj,sj.(28)
Proof: First, recall that in the proof of Theorems 1 and 3,
the auxiliary RV U1, . . . , ULare generated by the channel
output feedback and they are used to improve the legiti-
mate receivers’ decoding performance. Then, note that in
Theorem 4, there is no channel output feedback, which indi-
cates that U1,...,ULare useless. Finally, substituting U1=
U2= · · · = UL=const into Rf−dy
s, and along the lines of the
proof of Theorem 3, the lower bound Rf−d
sis obtained. The
proof of Theorem 4 is completed.
The following Theorem 5 provides an upper bound for both
Cf−dy
sand Cf−d
s.
Theorem 5 (Upper Bound on Both Cf−dy
sand Cf−d
s):
Cf−dy
s≤Cf−out
sand Cf−d
s≤Cf−out
s, where
Cf−out
s=min
jmax
Px|˜s∗
jIX;Yj|Sj,˜
S∗
j(29)
the auxiliary RV ˜
S∗
jrepresents Sj,i−dj,Sjrepresents Sj,i
Pyj,x,sj,˜s∗
j=Pyj|x,sjPx|˜s∗
jP˜s∗
jKdj
j˜s∗
j,sj(30)
for all j∈ {1,2,...,L}.
Proof: This outer bound can be directly obtained by
using the fact that the secrecy capacities Cf−dy
sand Cf−d
s
cannot exceed the capacity of each channel without secrecy
constraint. To be specific, first, note that in the model of
Fig. 3, the channel j(j∈ {1,2,...,L}) with delayed feedback
and without eavesdropper has already been investigated by
Viswanathan [34]. It has been shown in [34] that the capacity
Cf−dof each channel with only delayed state feedback equals
the capacity Cf−dy of the same channel with delayed both state
and channel output feedback, and they are given by
Cf−d=Cf−dy =max
Px|˜s∗
jIX;Yj|Sj,˜
S∗
j.(31)
Then, using the fact that Cf−dy
sand Cf−d
scannot exceed (31)
for all j∈ {1,2,...,L}, the upper bound Cf−out
sis obtained.
The proof is completed.
106 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
IV. GAUSSIAN FADIN G EX AMP LE OF THE FINITE STATE
COMPO UND WIRETAP CHAN NEL WITH DELAYED
FEEDBACK
In this section, we compute the capacity bounds in
Section III via a Gaussian fading example, and we would like
to know how the delayed feedback time affects the capacity
bounds. The remainder of this section is organized as follows.
In Section IV-A, we show bounds on the secrecy capacities
of the Gaussian fading case of the model of Fig. 3 with or
without delayed channel output feedback. In Section IV-B, the
bounds in Section IV-A are further explained via numerical
results.
A. Gaussian Fading Case of the Model of Fig. 3
For the Gaussian fading case of the model of Fig. 3, at time
i(1 ≤i≤N), the channel inputs and outputs are given by
Yj,i=hjsj,iXi+Nsj,i,Zi=gse,iXi+Nse,i(32)
where j∈ {1,2,...,L}, and sj,i,hj(sj,i),se,i,g(se,i),Nsj,i, and
Nse,iare defined as follows.
1) sj,iis the ith time state of the channel for receiver j, and
hj(sj,i)is the fading coefficient of the channel from the
transmitter to receiver jand it depends on the ith time
state sj,i.
2) g(se,i)is the fading coefficient of the channel from the
transmitter to the eavesdropper and it depends on the ith
time state se,i.
3) Nsj,i∼N(0, σ 2
sj,i)is the noise of the channel from the
transmitter to receiver jand it is Gaussian distributed
with zero mean and variance σ2
sj,iwhich depends on the
ith time state sj,i.
4) Nse,i∼N(0, σ 2
se,i)is the noise of the channel from the
transmitter to the eavesdropper and it is Gaussian dis-
tributed with zero mean and variance σ2
se,iwhich depends
on the ith time state se,i.
Let Pbe the transmitter’s power constraint satisfying
EX2≤P.(33)
At the ith time, receiver j∈ {1,2,...,L}obtains Sj,iand the
channel output Yj,i, and then he/she transmits Sj,i(or Sj,iand
Yj,i) back to the transmitter via a feedback channel after a delay
time dj(W.L.O.G., assume that 1 ≤d1≤d2≤ · · · ≤ dL≤N).
The following Corollary 1 shows the lower bound Rf−dy∗
son
the secrecy capacity Cf−dy∗
sof the Gaussian fading case of
the model of Fig. 3 with delayed states and channel output
feedback. Corollary 2 shows the lower bound Rf−d∗
son the
secrecy capacity Cf−d∗
sof the Gaussian fading case of the
model of Fig. 3 with only delayed state feedback. Corollary 3
shows an upper bound for both Cf−dy∗
sand Cf−d∗
s.
Corollary 1: A lower bound Rf−dy∗
son Cf−dy∗
sis given by
Rf−dy∗
s=max
P(˜s1),...,P(˜sL),α1,...,αL:
˜s1π1(˜s1)P(˜s1)≤P
···
˜sLπL(˜sL)P(˜sL)≤P
α1+···+αL=1, α1,...,αL≥0
min
jmin
×
˜sj
sj
πj˜sjKdL
j˜sj,sj1
2log2πeαjP˜sj
−
se
πe(se)1
2log g2(se)P+σ2
se
g2(se)P1−αj+σ2
se+
˜sj
sj
πj˜sjKdL
j˜sj,sj
×1
2log h2
jsjP˜sj+σ2
sj
h2
jsjP˜sj1−αj+σ2
sj(34)
where [x]+=xfor x≥0, [x]+=0 for x<0, and P(˜sj)
(j∈ {1,2,...,L}) is the transmitter’s power allocated to the
state ˜sj.
Proof: First, for j∈ {1,2,...,L}, define
X=
L
j=1
Vj(35)
where Vj∼N(0,Pj),Pj≤αjP,αj≥0, and
L
j=1αj=1. Here note that V1,...,VLare independent
of one another. From the definition (35), it is easy
to check that the power constraint (33) holds. Next,
define
EX2|˜sj=P˜sj(36)
where P(˜sj)is the transmitter’s power allocated to the state ˜sj,
and it satisfies
˜sj
πj˜sjP˜sj=
˜sj
πj˜sjEX2|˜sj=EX2≤P.(37)
Further define
EV2
j|˜sj=αjP˜sj.(38)
From (37) and (38), it is easy to check that
Pj=EV2
j=
˜sj
πj˜sjEV2
j|˜sj
=
˜sj
πj˜sjαjP˜sj=αj
˜sj
πj˜sjP˜sj
≤αjP.(39)
Finally, note that for j∈ {1,2,...,L},Ujis generated
from the feedback Yjand the transmitted codeword Vj,
define
Uj=Vj+Yj.(40)
Now substituting the above definitions (32), (35), (36), (38),
and (40) into Theorem 3, Rf−dy∗
sis obtained. The proof of
Corollary 1 is completed.
Corollary 2: A lower bound Rf−d∗
son Cf−d∗
sis given by
Rf−d∗
s=max
P(˜s1),...,P(˜sL),α1,...,αL:
˜s1π1(˜s1)P(˜s1)≤P
···
˜sLπL(˜sL)P(˜sL)≤P
α1+···+αL=1, α1,...,αL≥0
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 107
×min
j
˜sj
sj
πj˜sjKdL
j˜sj,sj
×1
2log h2
jsjP˜sj+σ2
sj
h2
jsjP˜sj1−αj+σ2
sj
−
se
πe(se)1
2log g2(se)P+σ2
se
g2(se)P1−αj+σ2
se+
(41)
where [x]+=xfor x≥0, [x]+=0 for x<0, and P(˜sj)
(j∈ {1,2,...,L}) is the transmitter’s power allocated to the
state ˜sj.
Proof: Substituting the definitions (32), (35), (36),
and (38) into Theorem 4, Rf−dy∗
sis obtained. The proof of
Corollary 2 is completed.
The following Corollary 3 provides an upper bound for both
Cf−dy∗
sand Cf−d∗
s.
Corollary 3: An upper bound Cf−out∗
son both Cf−dy∗
sand
Cf−d∗
sis given by
Cf−out∗
s=min
jmax
P(˜sj):
˜sjπj(˜sj)P(˜sj)≤P
˜sj
sj
πj˜sjKdj
j˜sj,sj
×1
2log h2
jsjP˜sj+σ2
sj
σ2
sj
.(42)
Proof: Substituting the definitions (32) and (36) into
Theorem 5, Cf−out∗
sis obtained. The proof of Corollary 3 is
completed.
B. Numerical Results
In this section, we investigate a two-state example, i.e., for
j∈ {1,2,...,L},Sjconsists of two elements Gj(good state)
and Bj(bad state), and Sealso consists of two elements Ge
and Be. The state process of {Sj}is given by
PGj|Gj=1−bj,PBj|Gj=bj
PBj|Bj=1−gj,PGj|Bj=gj(43)
and the steady probabilities of Gjand Bjare given by
πjGj=gj
gj+bj, πjBj=bj
gj+bj.(44)
In addition, the state process of {Se}is given by
P(Ge|Ge)=1−be,P(Be|Ge)=be
P(Be|Be)=1−ge,P(Ge|Be)=ge(45)
and the steady probabilities of Geand Beare given by
πe(Ge)=ge
ge+be, πe(Be)=be
ge+be.(46)
For the noise Nsjof the channel from the transmitter to receiver
j, its variance σ2
sjin state Gjis σ2
Gj, and in state Bjis σ2
Bj.
Similarly, the noise Nseof the channel from the transmitter to
the eavesdropper, its variance σ2
sein state Geis σ2
Ge, and in
state Beis σ2
Be.
Fig. 10. Comparison of the bounds in Theorems 1–3 for σ2
G1=0.1, σ2
B1=1,
σ2
G2=0.3, σ2
B2=0.66, σ2
G3=1, σ2
B3=2, σ2
Ge=2000, σ2
Be=6000,
g1=0.05, b1=0.05, g2=0.1, b2=0.08, g3=0.2, b3=0.08, ge=0.3,
be=0.5, h1(G1)=1, h1(B1)=0.5, h2(G2)=0.9, h2(B2)=0.6, h3(G3)=
0.8, h3(B3)=0.4, g(Ge)=0.8, g(Be)=0.7, d1=1, d2=2, d3=3, and
several values of P.
Fig. 11. Comparison of the bounds in Theorems 1–3 for the same values of
the parameters in Fig. 10 except that σ2
Ge=1 and σ2
Be=2.5.
For L=3, which indicates that there are three legitimate
receivers in the model of Fig. 3, Fig. 10 plots the lower
and upper bounds on the secrecy capacities of the Gaussian
fading case of the model of Fig. 3 with delayed state feed-
back, and with or without delayed channel output feedback
for σ2
G1=0.1, σ2
B1=1, σ2
G2=0.3, σ2
B2=0.66, σ2
G3=1,
σ2
B3=2, σ2
Ge=2000, σ2
Be=6000, g1=0.05, b1=0.05,
g2=0.1, b2=0.08, g3=0.2, b3=0.08, ge=0.3, be=0.5,
h1(G1)=1, h1(B1)=0.5, h2(G2)=0.9, h2(B2)=0.6,
h3(G3)=0.8, h3(B3)=0.4, g(Ge)=0.8, g(Be)=0.7,
d1=1, d2=2, d3=3, and several values of P. As
depicted in this figure, if the eavesdropper’s channel noise
variance (σ2
Ge=2000, σ2
Be=6000) is large, the lower bound
Rf−dy∗
smeets the upper bound Cf−out∗
s, which indicates that
the secrecy capacity of the Gaussian fading case of the model
of Fig. 3 with delayed both states and channel output feedback
is determined. Moreover, we see that channel output feedback
helps to enhance the achievable secrecy rate of the Gaussian
fading case of the model of Fig. 3 with only delayed state
feedback.
Fig. 11 plots the bounds for the same values of the param-
eters given in Fig. 10 except that σ2
Ge=1 and σ2
Be=2.5.
As depicted in this figure, if the eavesdropper’s channel noise
variance (σ2
Ge=1, σ2
Be=2.5) is decreasing, the gap between
the lower and upper bounds on Cf−dy∗
sis increasing. In addi-
tion, we see that channel output feedback still helps to enhance
the achievable secrecy rate Rf−d∗
sof the Gaussian fading case
of the model of Fig. 3 with only delayed state feedback.
108 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
Fig. 12. Comparison of the bounds in Theorems 1–3 for the same values of
the parameters in Fig. 10 except that σ2
Ge=0.03 and σ2
Be=0.05.
Fig. 13. Comparison of the lower bounds in Theorems 1 and 2 for P=20,
σ2
G1=0.1, σ2
B1=1, σ2
G2=0.3, σ2
B2=0.66, σ2
G3=1, σ2
B3=2, σ2
Ge=1,
σ2
Be=2.5, g1=0.05, b1=0.05, g2=0.1, b2=0.08, g3=0.2, b3=0.08,
ge=0.3, be=0.5, h1(G1)=1, h1(B1)=0.5, h2(G2)=0.9, h2(B2)=0.6,
h3(G3)=0.8, h3(B3)=0.4, g(Ge)=0.8, g(Be)=0.7, and several values
of d(d1=d2=d3=d).
Fig. 12 plots the bounds for the same values of the param-
eters given in Fig. 10 except that σ2
Ge=0.03 and σ2
Be=0.05.
As depicted in this figure, if the eavesdropper’s channel noise
variance (σ2
Ge=0.03, σ2
Be=0.05) is sufficiently small, the
achievable secrecy rate Rf−d∗
sof the Gaussian fading case of
the model of Fig. 3 with only delayed state feedback equals
0, which implies that the perfect secrecy cannot be guaranteed
for this case. Using the channel output feedback, the positive
achievable secrecy rate Rf−dy∗
sis derived, and hence the PLS
of the Gaussian fading case of the model of Fig. 3 with only
delayed state feedback is enhanced. In addition, we should
notice that there still exists a huge gap between the lower and
upper bounds on Cf−dy∗
s.
To investigate how the delayed feedback time dj(j∈
{1,2,...,L}) affects the secrecy rates of the model of Fig. 3,
Fig. 13 plots the lower bounds in Theorems 1 and 2 for the
case that L=3 (three legitimate receivers) and d1=d2=
d3=d, which implies that the delayed feedback times of
the three legitimate receivers are the same and equal d. As
depicted in this figure, the achievable secrecy rates of the
Gaussian fading case of the model of Fig. 3 with or with-
out delayed channel output feedback are decreasing while the
delay time dis increasing. However, we should notice that both
Rf−dy∗
sand Rf−d∗
sare approaching their infinite asymptotes
while dis sufficiently large.
Fig. 14. Comparison of the lower bounds in Theorems 1 and 2 for P=20,
σ2
G1=0.1, σ2
B1=1, σ2
G2=0.3, σ2
B2=0.66, σ2
G3=1, σ2
B3=2, σ2
Ge=1,
σ2
Be=2.5, g1=0.05, b1=0.05, g2=0.1, b2=0.08, g3=0.2, b3=0.08,
ge=0.3, be=0.5, h1(G1)=1, h1(B1)=0.5, h2(G2)=0.9, h2(B2)=0.6,
h3(G3)=0.8, h3(B3)=0.4, g(Ge)=0.8, g(Be)=0.7, and several values
of d(d1=0, d2=d,d3=2d).
Fig. 14 plots the lower bounds in Theorems 1 and 2 for
the case that L=3 (three legitimate receivers) and d1=0,
d2=d,d3=2d, which implies that the delayed feedback
times of the three legitimate receivers are different. Similar to
Fig. 13, Rf−dy∗
sand Rf−d∗
sare monotonic decreasing functions
of d, and both of them approach their infinite asymptotes when
dis sufficiently large.
V. CO NCL USION
This article established a general framework for enhanc-
ing the PLS in the downlink transmission of IoT systems via
feedback. Two models, including the compound WTC with
feedback and the finite state compound WTC with delayed
feedback, were studied, and bounds on the secrecy capaci-
ties of the two models were given. From a Gaussian fading
example, we see that the delayed channel output feedback
enhances the lower bound on the secrecy capacity of the finite
state compound WTC with only delayed state feedback, and
the corresponding feedback strategy may achieve the secrecy
capacity if the eavesdropper’s channel noise variance is suf-
ficiently large. Moreover, numerical results indicate that the
secrecy rates are decreasing while the feedback delay time is
increasing, and the secrecy rates are approaching their infinite
asymptotes while the feedback delay time is sufficiently large.
However, we should notice that all the capacity results given
in this article only work well under the perfect weak secrecy
condition, and how to design the corresponding encoding-
decoding schemes under the strong perfect secrecy condition
is of further interest to us.
APPEN DIX A
PROO F OF THE OREM 1
The messages are conveyed to the receivers via nblocks.
The blocklength of block i∈ {1,2,...,n−1}is N, and for
block n(the last block), its blocklength is γN, where γis a
positive real number and will be determined later. In block i
(i∈ {1,2,...,n−1}), the random sequences XN,ZN,YN
1,1,
YN
2,1,...,YN
L,1,UN
1,1,UN
2,1, . . . , UN
L,1,VN
1,1, and VN
2,1,...,VN
L,1,
are denoted by ¯
Xi,¯
Zi,¯
Y1,i,¯
Y2,i, . . . , ¯
YL,i,¯
U1,i,¯
U2,i,..., ¯
UL,i,
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 109
Fig. 15. Some important notations in the proof of Theorem 1.
¯
V1,i, and ¯
V2,i,..., ¯
VL,i, respectively. Similarly, in block n, the
random sequences XγN,ZγN,YγN
1,1,YγN
2,1, . . . , YγN
L,1,VγN
1,1, and
VγN
2,1,...,VγN
L,1, are denoted by ¯
Xn,¯
Zn,¯
Y1,n,¯
Y2,n,..., ¯
YL,n,
¯
V1,n, and ¯
V2,n,..., ¯
VL,n, respectively. In addition, the value of
the random vector is written in lower case letter.
Code-Book Construction:
1) The message Wis sent to all legitimate receivers via
nblocks, i.e., the message Wis composed of ncom-
ponents (W=(W1,...,Wn)), and each component Wi
(i∈ {1,2, . . . , n}) is the message for block i. Here Wi
takes values in the set {1, . . . , 2NR}.
2) The dummy message W, which is used to con-
fuse the eavesdropper, is composed of ncomponents
(W=(W
1, . . . , W
n)), and the component W
i(i∈
{1,2,...,n}) is transmitted in block i. Here note that
W
iis randomly chosen from the set {1, . . . , 2NR}, i.e.,
Pr{W
i=l} = 2−NR, where l∈ {1,...,2NR}.
3) The auxiliary message W∗
j, where j∈ {1,2,...,L}, is
used to improve receiver j’s decoding performance. Here
note that W∗
jis composed of n−1 components (W∗
j=
(W∗
j,1,...,W∗
j,n−1)), where W∗
j,i(i∈ {1,2, . . . , n−1})
takes values in the set {1,...,2NR∗
j}.
4) In block i(1 ≤i≤n), randomly generate 2N(R+R+R∗)
independent identically distributed (i.i.d.) sequences ¯vj,i
(j∈ {1,2,...,L}) according to the probability P(vj),
and label them as ¯vj,i(wi,w
i,w∗
j,i−1), where wi∈
{1,2,...,2NR},w
i∈ {1,2, . . . , 2NR}and w∗
j,i−1∈
{1,2,...,2NR∗
j}.
5) In block i(1 ≤i≤n−1), for each possible
value of ¯vj,i(wi,w
i,w∗
j,i−1)and ¯yj,i, randomly generate
2N˜
Rji.i.d. codewords ¯uj,iaccording to the probability
P(uj|vj,yj). Then label these ¯uj,ias ¯uj,i(w∗
j,i,w∗∗
j,i), where
w∗
j,i∈ {1,2,...,2NR∗
j}and w∗∗
j,i∈ {1,2,...,2N(˜
Rj−R∗
j)}.
6) In block i(1 ≤i≤n), for given ¯v1,i,...,¯vL,i,
the channel input ¯xiis i.i.d. generated according to
P(x|v1,...,vL).
For convenience, Fig. 15 provides some important notations
in the proof of Theorem 1.
Encoding Procedure:
1) At block 1, the transmitter selects ¯vj,1(w1,w
1,1)(j∈
{1,2,...,L}). Here notice that w
1is randomly chosen
from the set {1,2,...,2NR}.
2) At block i(i∈ {2,3,...,n−1}), once the transmitter
receives ¯yj,i−1(j∈ {1,2,...,L}), he seeks a ¯uj,i−1such
Fig. 16. Encoding procedure for j∈ {1,2, . . . , L}.
that the triplet (¯uj,i−1,¯vj,i−1,¯yj,i−1)is jointly typical. If
there exist multiple ¯uj,i−1, randomly choose one, and
if no such ¯uj,i−1exists, an error occurs. Based on the
covering lemma [32], if
˜
Rj≥IUj;Vj,Yj(47)
this encoding error vanishes as Ntends to infinity. Once
the transmitter decodes such a ¯uj,i−1(w∗
j,i−1,w∗∗
j,i−1), he
chooses ¯vj,i(wi,w
i,w∗
j,i−1)for transmission.
3) At block n, once the transmitter receives the feedback
¯yj,n−1(j∈ {1,2, . . . , L}), he seeks a ¯uj,n−1such that
the triplet (¯uj,n−1,¯vj,n−1,¯yj,n−1)is jointly typical, and
the corresponding encoding error vanishes as Ntends
to infinity if (47) is guaranteed. Once the transmit-
ter decodes such a ¯uj,n−1(w∗
j,n−1,w∗∗
j,n−1), he chooses
¯vj,n(1,1,w∗
j,n−1)for transmission.
Decoding Procedure: receiver j’s (j∈ {1,2, . . . , L}) decod-
ing scheme begins from the last block. At block n, first, note
that due to the reason that the blocklength is γN, the actual
rate of W∗
j,n−1is given by
HW∗
j,n−1
γN=NR∗
j
γN=R∗
j
γ.(48)
Next, receiver jselects a unique ¯vj,nsuch that (¯vj,n,¯yj,n)are
joint typical. If multiple or no such ¯vj,nexists, an error occurs.
From the packing lemma [32], this error vanishes if
R∗
j
γ≤IVj;Yj.(49)
Since I(Vj;Yj)of (49) is finite, for any given R∗
j, we can
choose a sufficiently large γsuch that (49) is guaranteed.
After decoding ¯vj,n, receiver jpicks out w∗
j,n−1from it. Then
he/she tries to choose a unique ¯uj,n−1such that given w∗
j,n−1,
¯uj,n−1and ¯yj,n−1are jointly typical. If multiple or no such
¯uj,n−1exists, an error occurs. From the packing lemma [32],
this error vanishes if
˜
Rj−R∗
j≤IUj;Yj.(50)
Once such unique ¯uj,n−1is obtained, receiver jseeks a unique
¯vj,n−1such that (¯vj,n−1,¯yj,n−1,¯uj,n−1)are jointly typical. From
110 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
Fig. 17. Decoding procedure for receiver j(j∈ {1,2, . . . , L}).
the packing lemma [32], this error vanishes if
R+R+R∗
j≤IVj;Uj,Yj.(51)
After decoding ¯vj,n−1, receiver jpicks out wn−1, and w∗
j,n−2
from it. Repeating the above decoding procedure, the mes-
sages of all blocks are decoded by receiver j. The decod-
ing procedure is completed. Figs. 16 and 17 illustrate the
encoding–decoding procedure for receiver j(j∈ {1,2,...,L}).
Equivocation Analysis: The eavesdropper’s equivocation ,
defined as =(1/[(n−1)N+γN])H(W|¯
Z1,..., ¯
Zn)(the
overall length of all blocks is (n−1)N+γN), follows that:
=1
(n−1)N+γNHW|¯
Z1, . . . , ¯
Zn
(a)
=1
(n−1)N+γN
n−1
i=1
HWi|W1, . . . , Wi−1,¯
Z1, . . . , ¯
Zn
(b)
=1
(n−1)N+γN
n−1
i=1
HWi|¯
Zi
=1
(n−1)N+γN
n−1
i=1HWi,¯
Zi−H¯
Zi
=1
(n−1)N+γN
n−1
i=1HWi,¯
Zi,¯
Vj,i
−H¯
Vj,i|Wi,¯
Zi−H¯
Zi
(c)
=1
(n−1)N+γN
n−1
i=1H¯
Zi|¯
Vj,i+H¯
Vj,i
−H¯
Vj,i|Wi,¯
Zi−H¯
Zi
=1
(n−1)N+γN
n−1
i=1H¯
Vj,i−I¯
Vj,i;¯
Zi−H¯
Vj,i|Wi,¯
Zi
=1
(n−1)N+γNn−1
i=1
H¯
Vj,i−
n−1
i=1
I¯
Vj,i;¯
Zi
−
n−1
i=1
H¯
Vj,i|Wi,¯
Zi
(d)
≥1
(n−1)N+γNn−1
i=1
H¯
Vj,i−(n−1)NIVj;Z+1
−
n−1
i=1
H¯
Vj,i|Wi,¯
Zi
=1
(n−1)N+γNH¯
Vj,1+
n−1
i=2
H¯
Vj,i
−(n−1)NIVj;Z+1
−H¯
Vj,1|W1,¯
Z1−
n−1
i=2
H¯
Vj,i|Wi,¯
Zi
(e)
≥1
(n−1)N+γNNR+R−2
+(n−2)NR+R+R∗
j−3
−(n−1)NIVj;Z+1
−H¯
Vj,1|W1,¯
Z1−
n−1
i=2
H¯
Vj,i|Wi,¯
Zi
(f)
≥1
(n−1)N+γNNR+R−2
+(n−2)NR+R+R∗
j−3
−(n−1)NIVj;Z+1
−N4−(n−2)N5
=R+R−2
(n−1)+γ+
(n−2)R+R+R∗
j−3
(n−1)+γ
−(n−1)IVj;Z+1
(n−1)+γ−4+(n−2)5
(n−1)+γ(52)
where (a) follows from the fact that W=(W1, . . . , Wn)
and Wnis constant, (b) follows from the fact that given ¯
Zi,
the message Wiof block iis independent of other blocks’
(¯
Z1,..., ¯
Zi−1,¯
Zi+1,..., ¯
Zn)and previous blocks’ messages
(W1,...,Wi−1), i.e., the Markov chain Wi→¯
Zi→
(W1,...,Wi−1,¯
Z1,..., ¯
Zi−1,¯
Zi+1,..., ¯
Zn)holds, (c) follows
from the fact that H(Wi|¯
Vj,i)=0, (d) follows from a similar
argument in [5, Lemma 3], i.e., I(¯
Vj,i;¯
Zi)≤N(I(Vj;Z)+1),
where 1→0 as N→ ∞, (e) follows from the construction
of ¯
Vj,iand a similar argument in [3, eqs. (16) and (23)], i.e.,
H(¯
Vj,1)≥N(R+R−2),H(¯
Vj,i)≥N(R+R+R∗
j−3), where
2, 3→0 as N→ ∞, (f) follows from that given w1,¯z1,
the eavesdropper attempts to find a unique ¯vj,1jointly typical
with his/her received ¯z1, and from the packing lemma [32],
this decoding error vanishes if
R≤IVj;Z(53)
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 111
then applying Fano’s lemma, H(¯
Vj,1|W1,¯
Z1)≤N4is
obtained, where 4→0 while N→ ∞, and analogously,
for i∈ {2,...,n−1}given wi,¯zi, the eavesdropper attempts
to find a unique ¯vj,ijointly typical with his/her received ¯zi, and
from the packing lemma [32], this decoding error vanishes if
R+R∗
j≤IVj;Z(54)
then applying Fano’s lemma, H(¯
Vj,i|Wi,¯
Zi)≤N5is obtained,
where 5→0 while N→ ∞. Here note that (53) is included
in (54), and thus we only need to use (54) to derive the final
region.
The bound (52) implies that if
R+R∗
j≥IVj;Z.(55)
≥R−is satisfied by choosing sufficiently large nand N.
Now it remains to use the above condi-
tions (47), (50), (51), (54), and (55) to derive the lower
bound in Theorem 1, as follows.
First, note that from (47) and (50), we have
R∗
j≥IUj;Vj,Yj−IUj;Yj
=IUj;Vj|Yj.(56)
Next, substituting (56) into (51), we get
R≤R+R≤IVj;Uj,Yj−IUj;Vj|Yj
=IVj;Yj.(57)
Then, note that from (54) and (55), we can conclude that
R+R∗
j=IVj;Z.(58)
Now substituting (58) into (51), we have
R≤IVj;Uj,Yj−IVj;Z.(59)
From the above (57) and (59), we have
R≤minIVj;Uj,Yj−IVj;Z,IVj;Yj.(60)
Next, note that if I(Vj;Uj,Yj)≤I(Vj;Z), from (51), we have
R+R+R∗
j≤IVj;Uj,Yj≤IVj;Z.(61)
Combining (61) with (58), and observing that R≥0, we can
conclude that R=0 if I(Vj;Uj,Yj)≤I(Vj;Z). Hence (60)
should be rewritten as
R≤minIVj;Uj,Yj−IVj;Z+,IVj;Yj.(62)
Note that (62) should be satisfied for all j∈ {1,2, . . . , L},
hence we have
R≤min
jminIVj;Uj,Yj−IVj;Z+,IVj;Yj.(63)
Finally, note that the effective transmission rate is
H(W)
(n−1)N+γN=n−1
i=1H(Wi)
(n−1)N+γN
=(n−1)NR
(n−1)N+γN=n−1
n−1+γR(64)
which indicates that the effective transmission rate approaches
Ras the number of blocks n→ ∞, then maximizing the bound
in (63), Theorem 1 is proved, and the proof is completed.
APPEN DIX B
PROO F OF THE OREM 3
The encoding-decoding scheme of Theorem 3 combines
that of Theorem 1 with the multiplexing encoding-decoding
scheme for the finite state Markov channel with delayed
feedback [34]. The detail about the coding scheme is given
below.
Definition: Similar to the definitions in the proof of
Theorem 1, the messages are transmitted via nblocks. Since
0≤d1≤d2≤ · · · ≤ dL≤N, define the block-
length of block i∈ {1,2,...,n−dL}is N, and for block
i∈ {n−dL+1, . . . , n}, its blocklength is N=γN. For
j∈ {1,2, . . . , L}, define the alphabet Sjas Sj= {1,2,...,kj}
and the steady probability πj(l) > 0 for any l∈Sj. Moreover,
in block i∈ {1,2,...,n−dL}, define N˜sj(˜sj∈ {1,2,...,kj}) as
N˜sj=Nπj˜sj.(65)
Similarly, in block i∈ {n−dL+1, . . . , n}, define N
˜sj(˜sj∈
{1,2,...,kj}) as
N
˜sj=γN˜sj=γNπj˜sj.(66)
The random sequence XNof block i∈ {1,2,...,n−dL}
and XγNof block i∈ {n−dL+1,...,n}are denoted
by ¯
Xi, and similar convention is applied to other random
sequences. In addition, for block i∈ {1,2, . . . , n−dL}, the
random sequence XN˜sjis denoted by ¯
XN˜sj
i, and for block
i∈ {n−dL+1,...,n}, the random sequence XN
˜sjis denoted by
¯
XN
˜sj
i. Similar convention is applied to other random sequences,
and the values of the random sequences are written in lower
case letter.
The message Wis composed of ncomponents (W=
(W1,...,Wn)), and the component Wi(i∈ {1,2, . . . , n})
is the message for block i. Here Witakes values in
the set {1,...,2NR}. For j∈ {1,2, . . . , L}and given ˜sj
(˜sj∈ {1,2,...,kj}), further divide Wiinto kjsubmessages,
i.e., Wi=(Wi,1,...,Wi,kj), where Wi,˜sjtakes values in
{1,2,...,2N˜sjR(˜sj)}. Here note that
kj
˜sj=1
πj˜sjR˜sj=R.(67)
The dummy message Walso consists of ncomponents
(W=(W
1, . . . , W
n)), and W
i(i∈ {1,2,...,n}) is for
block i. Here note that W
iis uniformly drawn from the
set {1,...,2NR}, i.e., Pr{W
i=l} = 2−NR, where l∈
{1,...,2NR}. Similarly, for j∈ {1,2,...,L}and given
˜sj(˜sj∈ {1,2,...,kj}), further divide W
iinto kjsubmes-
sages, i.e., W
i=(W
i,1,...,W
i,kj)and W
i,˜sjtakes values in
{1,2,...,2N˜sjR(˜sj)}. Here note that
kj
˜sj=1
πj˜sjR˜sj=R.(68)
112 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
Fig. 18. Some important notations in the proof of Theorem 3.
The auxiliary message W∗
j(j∈ {1,2, . . . , L}) is com-
posed of ncomponents (W∗
j=(W∗
j,1, . . . , W∗
j,n)), where W∗
j,i
(i∈ {1,2,...,n}) takes values in {1,...,2NR∗
j}. Similarly,
given ˜sj(˜sj∈ {1,2,...,kj}), further divide W∗
j,iinto kjsubmes-
sages, i.e., W∗
j,i=(W∗
j,i,1, . . . , W∗
j,i,kj)and W∗
j,i,˜sjtakes values
in {1,2,...,2N˜sjR∗
j(˜sj)}. Here note that
kj
˜sj=1
πj˜sjR∗
j˜sj=R∗
j.(69)
For convenience, Fig. 18 provides some important notations
in the proof of Theorem 3.
Code-Book Construction:
1) In block i∈ {1,2,...,n−dL}, for j∈ {1,2,...,L}
and given ˜sj∈ {1,2,...,kj}, randomly produce
2N˜sj(R(˜sj)+R(˜sj)+R∗(˜sj)) i.i.d. sequences ¯vN˜sj
j,iaccord-
ing to the probability P(vj|˜sj), and label them as
¯vN˜sj
j,i(wi,˜sj,w
i,˜sj,w∗
j,i−dL,˜sj), where wi,˜sj∈ {1,2,...,
2N˜sjR(˜sj)},w
i,˜sj∈ {1,2,...,2N˜sjR(˜sj)}and w∗
j,i−dL,˜sj∈
{1,2,...,2N˜sjR∗
j(˜sj)}. In block i∈ {n−dL+1, . . . , n},
given ˜sj, randomly produce 2N˜sj(R(˜sj)+R(˜sj)+R∗(˜sj)) i.i.d.
sequences ¯vN
˜sj
j,iaccording to the probability P(vj|˜sj), and
label them as ¯vN
˜sj
j,i(wi,˜sj,w
i,˜sj,w∗
j,i−dL,˜sj), where wi,˜sj∈
{1,2,...,2N˜sjR(˜sj)},w
i,˜sj∈ {1,2,...,2N˜sjR(˜sj)}and
w∗
j,i−dL,˜sj∈ {1,2,...,2N˜sjR∗
j(˜sj)}.
2) In block i∈ {1,2, . . . , n−dL}, for j∈ {1,2, . . . , L},
˜sj∈ {1,2, . . . , kj}, and each possible value of
¯vN˜sj
j,i(wi,˜sj,w
i,˜sj,w∗
j,i−dL,˜sj)and ¯yN˜sj
j,i, randomly gener-
ate 2N˜sj˜
Rj(˜sj)i.i.d. codewords ¯uN˜sj
j,iaccording to the
probability P(uj|vj,yj,˜sj). Then label these ¯uN˜sj
j,ias
¯uN˜sj
j,i(w∗
j,i,˜sj,w∗∗
j,i,˜sj), where w∗
j,i,˜sj∈ {1,2,...,2N˜sjR∗
j(˜sj)}
and w∗∗
j,i,˜sj∈ {1,2,...,2N˜sj(˜
Rj(˜sj)−R∗
j(˜sj)) }. Here note that
kj
˜sj=1
πj˜sj˜
Rj˜sj=˜
Rj.(70)
3) In block i∈ {1,2,...,n−dL},¯vj,i(j∈ {1,2, . . . , L})
is generated by multiplexing different subsequences ¯vN˜sj
j,i
for all ˜sj∈ {1,2, . . . , kj}, i.e., ¯vj,i=(¯vN1
j,i,...,¯vNkj
j,i).
In block i∈ {n−dL+1,...,n},¯vj,iis generated by
multiplexing different subsequences ¯vN
˜sj
j,ifor all ˜sj∈
{1,2,...,kj}, i.e., ¯vj,i=(¯vN
1
j,i,...,¯vN
kj
j,i). Then for given
¯v1,i,...,¯vL,i, the channel input ¯xiis i.i.d. generated
according to P(x|v1,...,vL).
Encoding Procedure:
1) At block i∈ {1,...,2dL}, for each ˜sj∈ {1,2,...,kj},
the transmitter selects ¯vN˜sj
j,i(1,1,1)for transmission.
2) At block i∈ {2dL+1, . . . , n−dL}, for each ˜sj∈
{1,2,...,kj}and j∈ {1,2,...,L}, the delayed feed-
back state sequences ¯sN˜sj
j,i−2dL,¯sN˜sj
j,i−dL, the delayed feed-
back channel output ¯yj,i−dLand the previous trans-
mitted sequence ¯vN˜sj
j,i−dLhave already been known
by the transmitter. Here note that ¯sN˜sj
j,i−2dLis the
delayed feedback state de-multiplexing the delayed
feedback channel output ¯yj,i−dLinto ¯yN1
j,i−dL,...,¯yNkj
j,i−dL.
Given ˜sj, the transmitter seeks a ¯uN˜sj
j,i−dLsuch that
(¯uN˜sj
j,i−dL,¯vN˜sj
j,i−dL,¯yN˜sj
j,i−dL,¯sN˜sj
j,i−dL)are jointly typical. If
there exist multiple ¯uN˜sj
j,i−dL, randomly pick out one; if
no such ¯uN˜sj
j,i−dLexists, an error occurs. Based on the
covering Lemma [32], if
˜
Rj˜sj≥IUj;Vj,Yj|Sj,˜
Sj= ˜sj(71)
this encoding error vanishes as Ntends to
infinity. Once the transmitter decodes such
a¯uN˜sj
j,i−dL(w∗
j,i−dL,˜sj,w∗∗
j,i−dL,˜sj), he chooses
¯vN˜sj
j,i(wi,˜sj,w
i,˜sj,w∗
j,i−dL,˜sj)for transmission.
3) At block i∈ {n−dL+1, . . . , n}, using the previous
encoding scheme for i∈ {2dL+1,...,n−dL}, the trans-
mitter decodes ¯uN˜sj
j,i−dL(w∗
j,i−dL,˜sj,w∗∗
j,i−dL,˜sj)and chooses
¯vN
˜sj
j,i(wi,˜sj=1,w
i,˜sj=1,w∗
j,i−dL,˜sj)for transmission.
Decoding Procedure: Once receiver j∈ {1,2,...,L}
obtains all nblocks ¯yj,1,...,¯yj,nand ¯sj,1,...,¯sj,n, he/she
demultiplexes them into subsequences according to ˜sj∈
{1,2,...,kj}. receiver jdoes backward decoding, i.e., the
decoding procedure starts from block i∈ {n−dL+1,...,n}.
Given ˜sj∈ {1,2, . . . , kj}, receiver jselects a unique ¯vN
˜sj
j,isuch
that (¯vN
˜sj
j,i,¯yN
˜sj
j,i,¯sN
˜sj
j,i)are joint typical. If multiple or no such
¯vN
˜sj
j,iexists, an error occurs. From the packing lemma [32], this
DAI et al.: ENHANCING PLS IN IoT VIA FEEDBACK: GENERAL FRAMEWORK 113
error vanishes if
HW∗
j,i−dL,˜sj
N
˜sj
=
HW∗
j,i−dL,˜sj
γN˜sj
=N˜sjR∗
j˜sj
γN˜sj
=R∗
j˜sj
γ≤IVj;Yj|Sj,˜
Sj= ˜sj.(72)
Since I(Vj;Yj|Sj,˜
Sj= ˜sj)of (72) is finite, for any given
R∗
j(˜sj), we can choose a sufficiently large γsuch that (72)
is guaranteed.
After decoding ¯vN
˜sj
j,ifor block i∈ {n−dL+1, . . . , n}, receiver
jpicks out w∗
j,i−dL,˜sjfrom it. Then given ˜sj, he/she tries to
choose a unique ¯uN˜sj
j,i−dLsuch that given w∗
j,i−dL,˜sj,¯uN˜sj
j,i−dL,¯sN˜sj
j,i−dL
and ¯yN˜sj
j,i−dLare jointly typical. If multiple or no such ¯uN˜sj
j,i−dL
exists, an error occurs. From the packing lemma [32], this
error vanishes if
˜
Rj˜sj−R∗
j˜sj≤IUj;Yj|Sj,˜
Sj= ˜sj.(73)
Once such unique ¯uN˜sj
j,i−dLis decoded, given ˜sj, receiver j
seeks a unique ¯vN˜sj
j,i−dLsuch that (¯vN˜sj
j,i−dL,¯sN˜sj
j,i−dL,¯yN˜sj
j,i−dL,¯uN˜sj
j,i−dL)
are jointly typical. From the packing lemma [32], this error
vanishes if
R˜sj+R˜sj+R∗
j˜sj≤IVj;Uj,Yj|Sj,˜
Sj= ˜sj.(74)
After decoding ¯vN˜sj
j,i−dL, receiver jpicks out wi−dL,˜sjand
w∗
j,i−dL,˜sjfrom it. Repeating the above decoding procedure, the
messages of all blocks are decoded by receiver j. The decoding
procedure is completed.
Equivocation Analysis: The eavesdropper’s equiv-
ocation , defined as =(1/[(n−dL)N+
dLγN])H(W|¯
Z1, . . . , ¯
Zn,¯
Se,1,...,¯
Se,n)(the overall length of
all blocks is (n−dL)N+dLγN), follows that:
=1
(n−dL)N+dLγNHW|¯
Z1,..., ¯
Zn,¯
Se,1,...,¯
Se,n
(a)
=1
(n−dL)N+dLγN
n−dL
i=2dL+1
HWi|W2dL+1, . . .
Wi−1,¯
Z1, . . . , ¯
Zn,¯
Se,1,...,¯
Se,n
(b)
=1
(n−dL)N+dLγN
n−dL
i=2dL+1
HWi|¯
Zi,¯
Se,i(75)
where (a) follows from the fact that Wiis constant for block
i∈ {1,...,2dL}and i∈ {n−dL+1,...,n}, and (b) follows
from the fact that given ¯
Ziand ¯
Se,i, the message Wiof block
iis independent of other blocks’ (¯
Z1,..., ¯
Zi−1,¯
Zi+1,..., ¯
Zn),
(¯
Se,1,..., ¯
Se,i−1,¯
Se,i+1,...,¯
Se,n)and previous blocks’ mes-
sages (W2dL+1,...,Wi−1), i.e., the Markov chain Wi→
(¯
Zi,¯
Se,i)→(W2dL+1, . . . , Wi−1,¯
Z1, . . . , ¯
Zi−1,¯
Zi+1, . . . ,
¯
Zn,¯
Se,1, ..., ¯
Se,i−1,¯
Se,i+1,...,¯
Se,n)holds.
The term H(Wi|¯
Zi,¯
Se,i)in (75) is further bounded by
HWi|¯
Zi,¯
Se,i=HWi,¯
Zi,¯
Se,i−H¯
Zi,¯
Se,i
=HWi,¯
Zi,¯
Se,i,¯
Vj,i−H¯
Vj,i|Wi,¯
Zi,¯
Se,i
−H¯
Zi,¯
Se,i
(c)
=H¯
Zi|¯
Se,i,¯
Vj,i+H¯
Se,i,¯
Vj,i
−H¯
Vj,i|Wi,¯
Zi,¯
Se,i−H¯
Zi,¯
Se,i
(d)
=H¯
Zi|¯
Se,i,¯
Vj,i+H¯
Se,i+H¯
Vj,i
−H¯
Vj,i|Wi,¯
Zi,¯
Se,i−H¯
Zi,¯
Se,i
=H¯
Vj,i−I¯
Vj,i;¯
Zi|¯
Se,i−H¯
Vj,i|Wi,¯
Zi,¯
Se,i
(e)
≥NR+R+R∗
j−1−I¯
Vj,i;¯
Zi|¯
Se,i
−H¯
Vj,i|Wi,¯
Zi,¯
Se,i
(f)
≥NR+R+R∗
j−1−NIVj;Z|Se+2
−H¯
Vj,i|Wi,¯
Zi,¯
Se,i
(g)
≥NR+R+R∗
j−1−NIVj;Z|Se+2
−N3(76)
where (c) follows from the fact that H(Wi|¯
Vj,i)=0, (d)
follows from the fact that ¯
Se,iis independent of ¯
Vj,i, (e) fol-
lows from the construction of ¯
Vj,iand a similar argument
in [3, eqs. (16) and (23)], i.e., H(¯
Vj,i)≥N(R+R+R∗
j−1),
where 1→0 as N→ ∞, (f) follows from a similar argument
in [5, Lemma 3], i.e., I(¯
Vj,i;¯
Zi|¯
Se,i)≤N(I(Vj;Z|Se)+2),
where 2→0 as N→ ∞, and (g) follows from that given wi,
¯zi, the eavesdropper attempts to find a unique ¯vj,ijointly typi-
cal with his/her received ¯zi, and from the packing lemma [32],
this error vanishes if
R+R∗
j≤IVj;Z|Se(77)
then applying Fano’s lemma, H(¯
Vj,i|Wi,¯
Zi,¯
Se,i)≤N3is
obtained, where 3→0 while N→ ∞.
Substituting (76) into (75), we have
=n−3dL
n−dL+dLγR+R+R∗
j−IVj;Z|Se−1−2−3.
(78)
The bound (78) implies that if
R+R∗
j≥IVj;Z|Se.(79)
≥R−is satisfied by choosing sufficiently large nand N.
Now it remains to use the above condi-
tions (71), (73), (74), (77), and (79) to derive the lower
bound in Theorem 3, as follows.
First, note that from (70) and (71), we have
˜
Rj≥IUj;Vj,Yj|Sj,˜
Sj.(80)
Analogously, from (69), (70), and (73), we have
˜
Rj−R∗
j≤IUj;Yj|Sj,˜
Sj.(81)
From (67), (68), (69), and (74), we have
R+R+R∗
j≤IVj;Uj,Yj|Sj,˜
Sj.(82)
114 IEEE INTERNET OF THINGS JOURNAL, VOL. 7, NO. 1, JANUARY 2020
Next, from (80) and (81), we get
R∗
j≥IUj;Vj,Yj|Sj,˜
Sj−IUj;Yj|Sj,˜
Sj
=IUj;Vj|Yj,Sj,˜
Sj.(83)
Next, substituting (83) into (82), we get
R≤R+R
≤IVj;Uj,Yj|Sj,˜
Sj−IUj;Vj|Yj,Sj,˜
Sj
=IVj;Yj|Sj,˜
Sj.(84)
Then, note that from (77) and (79), we can conclude that
R+R∗
j=IVj;Z|Se.(85)
Now substituting (85) into (82), we have
R≤IVj;Uj,Yj|Sj,˜
Sj−IVj;Z|Se.(86)
From the above (84) and (86), we have
R≤minIVj;Uj,Yj|Sj,˜
Sj−IVj;Z|Se,IVj;Yj|Sj,˜
Sj.
(87)
Next, note that if I(Vj;Uj,Yj|Sj,˜
Sj)≤I(Vj;Z|Se), from (82),
we have
R+R+R∗
j≤IVj;Uj,Yj|Sj,˜
Sj≤IVj;Z|Se.(88)
Combining (88) with (85), and observing that R≥0, we
can conclude that R=0 if I(Vj;Uj,Yj|Sj,˜
Sj)≤I(Vj;Z|Se).
Hence (87) should be rewritten as
R≤min[IVj;Uj,Yj|Sj,˜
Sj−IVj;Z|Se]+,IVj;Yj|Sj,˜
Sj.
(89)
Note that (89) should be satisfied for all j∈ {1,2, . . . , L},
hence we have
R≤min
jmin[IVj;Uj,Yj|Sj,˜
Sj−IVj;Z|Se]+,IVj;Yj|Sj,˜
Sj.
(90)
Finally, note that the effective transmission rate is
H(W)
(n−dL)N+dLγN=n−dL
i=2dL+1H(Wi)
(n−dL)N+dLγN
=(n−3dL)NR
(n−dL)N+dLγN=n−3dL
n−dL+dLγR
(91)
which indicates that the effective transmission rate approaches
Ras the number of blocks n→ ∞, then maximizing the bound
in (90), Theorem 3 is proved, and the proof is completed.
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Bin Dai (A’15) received the B.Sc. degree in
communications and information systems from the
University of Electronic Science and Technology
of China, Chengdu, China, in 2004, and the M.Sc.
and Ph.D. degrees in computer science and technol-
ogy from Shanghai Jiao Tong University, Shanghai,
China, in 2007 and 2012, respectively.
In 2011 and 2012, he was a Visiting Scholar
with the Institute for Experimental Mathematics,
Duisburg–Essen University, Essen, Germany. He is
currently an Associate Professor with the Southwest
Jiaotong University, Chengdu, China. His current research interests include
information-theoretic security, and network information theory and coding.
Zheng Ma received the B.Sc. and Ph.D. degrees
in communications and information systems from
Southwest Jiaotong University, Chengdu, China, in
2000 and 2006, respectively.
He was a Visiting Scholar with the University of
Leeds, Leeds, U.K., in 2003. In 2003 and 2005,
he was a Visiting Scholar with the Hong Kong
University of Science and Technology, Hong Kong.
From 2008 to 2009, he was a Visiting Research
Fellow with the Department of Communication
Systems, Lancaster University, Lancaster, U.K. He is
currently a Professor with Southwest Jiaotong University. His current research
interests include information theory and coding, communication systems,
signal design and processing, field-programmable gate array/digital signal pro-
cessor implementation, and professional mobile radio.
Prof. Ma has been the Vice-Chairman of the IT Chapter of the IEEE
Chengdu Section since 2009.
Yuan Luo received the B.Sc. degree in (mathemat-
ics) information science, and the M.S. and Ph.D.
degrees in probability and mathematical statistics
from Nankai University, Tianjin, China, in 1993,
1996, and 1999, respectively.
He held a postdoctoral position at the Institute
of Systems Science, Chinese Academy of Sciences,
Shanghai, China, from July 1999 to April 2001,
and the Institute for Experimental Mathematics,
University of Duisburg–Essen, Essen, Germany,
from May 2001 to April 2003. He has been a Full
Professor with Shanghai Jiao Tong University, Shanghai, China, since 2006.
His current research interests include information theory (especially, Shannon
channel capacity and network coding), coding theory, and computer security
(especially, virtual machine security).
Xuxun Liu (M’14) received the Ph.D. degree in
communication and information system from Wuhan
University, Wuhan, China, in 2007.
He is currently an Associate Professor with the
School of Electronic and Information Engineering,
South China University of Technology, Guangzhou,
China. He has authored or coauthored over 30 scien-
tific papers in international journals and conference
proceedings. His current research interests include
wireless sensor networks, wireless communications,
computational intelligence, and mobile computing.
Zhuojun Zhuang received the B.Eng. degree from
the School of Software Engineering, Shanghai Jiao
Tong University, Shanghai, China, in 2006, and the
M.Eng. and Ph.D. degrees from the Department of
Computer Science and Engineering, Shanghai Jiao
Tong University, in 2009 and 2012, respectively.
His current research interests include cod-
ing theory, information theory, and network
communication.
Ming Xiao (S’02–M’07–SM’12) received the bach-
elor’s and master’s degrees in engineering from the
University of Electronic Science and Technology of
China, Chengdu, China, in 1997 and 2002, respec-
tively, and the Ph.D. degree from the Chalmers
University of Technology, Gothenburg, Sweden, in
2007.
From 1997 to 1999, he was a Network and
Software Engineer with China Telecom, Beijing,
China. From 2000 to 2002, he also held a posi-
tion with Sichuan Communications Administration,
Chengdu, China. Since 2007, he has been with Communication Theory
Department, School of Electrical Engineering, Royal Institute of Technology
(KTH), Stockholm, Sweden, where he is currently an Associate Professor.
Dr. Xiao was a recipient of the Best Paper Award from the International
Conference on Wireless Communications and Signal Processing in 2010 and
the IEEE International Conference on Computer Communication Networks
in 2011, the Chinese Government Award for Outstanding Self-Financed
Students Studying Abroad in 2007, the Hans Werthen Grant from the Royal
Swedish Academy of Engineering Science (IVA) in 2006, and the Ericsson
Research Funding from Ericsson in 2010. Since 2012, he has been an
Associate Editor of the IEEE TRANS ACT IO NS O N COM MU NI CAT IONS, IEEE
COMMUNICATI ON S LETTERS, and IEEE WI RELE SS COMMU NICATI ON S
LETT ER S, and has been a Senior Editor of IEE E COMMU NI CATI ON S
LETT ER S since 2015.
