Physical layer security from inter-session interference in large wireless networks

Abstract

Physical layer secrecy in wireless networks in the presence of eavesdroppers of unknown location is considered. In contrast to prior schemes, which have expended energy in the form of cooperative jamming to enable secrecy, we develop schemes where multiple transmitters send their signals in a cooperative fashion to confuse the eavesdroppers. Hence, power is not expended on “artificial noise”; rather, the signal of a given transmitter is protected by the aggregate interference produced by the other transmitters. We introduce a two-hop strategy for the case of equal path-loss between all pairs of nodes, and then consider its embedding within a multi-hop approach for the general case of an extended network. In each case, we derive an achievable number of eavesdroppers that can be present in the region while secure communication between all sources and intended destinations is ensured.

Full text

Physical Layer Security from Inter-Session
Interference in Large Wireless Networks
Azadeh Sheikholeslami
∗
, Dennis Goeckel
∗
, Hossein Pishro-Nik
∗
and Don Towsley
†
University of Massachusetts, Amherst, MA
∗
{sheikholesla,goeckel,pishro}@ecs.umass.edu,
†
towsley@cs.umass.edu
Abstract—Physical layer secrecy in wireless networks in the
presence of eavesdroppers of unknown location is considered.
Many schemes have been proposed to deal with eavesdroppers
based on the introduction of cooperative jamming. However, this
required energy expenditure that may not be tolerated. In this
paper, we develop schemes that multiple transmitters send their
signals in a cooperative fashion to confuse the eavesdroppers.
In contrast to previous approaches, power is not expended on
“artificial noise”; rather, the signal of a given transmitter is
protected by the aggregate interference produced by the other
transmitters. We introduce a two-hop strategy for the case of
equal path-loss between all pairs of nodes, and a multi-hop
strategy for the general case of an extended network. In each
case, we derive an achievable number of eavesdroppers that can
be present in the system while secure communication between
all sources and intended destinations is ensured.
I. INTRODUCTION
Because of the broadcast nature of wireless networks, any
node in the coverage range of a source can overhear any
message that it transmits. Consequently, one of the most
important and difficult considerations in wireless networks
is secrecy. The traditional approach to secrecy is encryption
of the plain message by means of special functions that are
assumed to be computationally infeasible for the adversary to
decrypt [1]. However, because of improvements in processors
and methods of breaking such encryption systems, there are
concerns that these assumptions no longer suffice. Especially
in sensitive applications requiring everlasting secrecy, users
might prefer a higher level of secrecy. Here we consider
methods of network design that inhibit the reception of the
transmitted signal at an eavesdropper. This might be used in
conjunction with traditional cryptographic approaches, as part
of a defense in depth approach, or it might enable information
theoretic secrecy, as described next.
In 1949, Shannon introduced the notion of perfect secrecy
[2]: if the eavesdropper’s uncertainty (entropy rate) about the
plain message after seeing the transmitted signal is equal to the
eavesdropper’s uncertainty about the message before seeing
the transmitted signal, then “perfect secrecy” is said to be
achieved. Based on this, Wyner introduced the wiretap channel
and showed that, in a degraded wiretap channel, adding
randomness to the codebook and using channel uncertainty
can protect the secure message from being intercepted by the
eavesdropper [3]. Later, the idea of the wiretap channel was
extended to more general cases [4], [5].
In all cases, the key to a positive secrecy rate is to have a bet-
ter channel from the transmitter to the intended receiver than to
the eavesdropper. However, in many cases the eavesdropper’s
channel is better than the legitimate channel; for example, the
eavesdropper might have a much better receiver than that of the
intended recipient or the eavesdropper might be much closer
to the transmitter than the intended receiver. Furthermore, in
many situations the location of the eavesdroppers are unknown
to the legitimate nodes.
To combat these problems, one must design algorithms
to produce the required advantage for the intended recipient
over the eavesdroppers. Negi and Goel introduced the idea of
adding artificial noise to the system by means of a multiple
antenna transmitter or a single antenna transmitter with some
helper nodes [6], [7]. The artificial noise is placed in the null
space of the channel from the transmitter to the intended re-
cipient and consequently does not affect the intended recipient
while at the same time degrading the eavesdropper’s channel
with high probability. Subsequently, cooperative jamming (by
producing artificial noise) [8]–[10] and using helper nodes [11]
to improve physical layer secrecy in small networks has been
extensively investigated.
However, in many network scenarios there are simultaneous
ongoing data flows between source-destination pairs. The
effect of their interference on the secrecy of wiretap channels
and methods to increase the secrecy rate region for small
networks consisting of at most two senders, two receivers and
one external eavesdropper have also been studied in recent
years [12]–[15].
The primary focus of this paper is to propose cooperative
strategies that use the mixed signal from the simultaneous
transmissions of multiple transmitters to hide each signal from
passive but intelligent and powerful eavesdroppers.
Although considerable research has been devoted to physi-
cal layer secrecy in scenarios with a few nodes, e.g. one trans-
mitter, one receiver, one eavesdropper and few helper nodes
that relay the message or generate artificial noise to help obtain
secrecy, rather less attention has been paid to security in large
multi-hop wireless networks with a large number of legitimate
nodes and eavesdroppers. In this case, the asymptotic results
for large networks are often studied as estimates for the utility
of approaches in finite size wireless networks. Consequently,
secrecy in large networks has recently been considered in the
work of [16] by introducing the secrecy-graph, which has been
extended in [17]–[19]. Furthermore, connectivity [20], [21],
coverage [22], scalability [23]–[26] and cooperative jamming
[27] in large wireless networks have been studied.

2
In this paper, we consider a cooperative strategy to improve
physical layer secrecy in large wireless networks with a large
number of sources, destinations, relay nodes and eavesdrop-
pers. In contrast to other cooperative scenarios where helper
nodes generate artificial noise, here each node transmits its
own (independent) message to the corresponding destination
in the presence of passive eavesdroppers. The signal of each
transmitter is concealed from the eavesdroppers by interfer-
ence produced from the aggregate signal of other transmitters.
As in [27], the legitimate nodes do not perform any interfer-
ence alignment or interference cancellation.
The legitimate nodes communicate with each other in such
a way that intended receivers receive the signal with signal-
to-interference-plus-noise-ratio (SINR) higher than a threshold
required to decode the message with arbitrarily low probability
of error, while at the same time keeping the SINR at the
eavesdroppers lower than another arbitrary threshold. This
approach is beneficial from different perspectives. From a
practical point of view, because the thresholds are separate, the
system designer has the freedom to choose threshold values
based on available equipment and security requirements. The
required SINR threshold for the eavesdropper can be chosen so
small that even a smart eavesdropper equipped with a suitable
modern decoder would not be able to decode the signal. From
an information theoretic point of view, the required SINR
thresholds at the legitimate and eavesdropper nodes can be
chosen such that the legitimate channel always has a required
advantage over the eavesdropper’s channel with probability
approaching one. This guarantees that we always can achieve
a desired secrecy rate from each source to its corresponding
destination.
In the literature of physical layer secrecy in large wireless
networks, usually geometric approaches based on the rela-
tive location of legitimate and eavesdropper nodes has been
considered and then the effect of multi-path fading on the
received signal has been ignored (e.g. [16]–[20]); however,
in our method, the presence of fading is very important for
the scheme to work properly. While in most other schemes
for enhancing physical layer secrecy, the location of the
eavesdroppers and/or channel state information (CSI) of the
source to eavesdropper channels are assumed to be known
by the legitimate nodes, we assume that the locations of
eavesdroppers are not known and legitimate nodes are not
aware of this CSI.
We propose two protocols to enhance physical layer secrecy
in the network described above and study their asymptotic
performance. First, we consider the case of equal path-loss
between all pairs of nodes. This scheme is applicable to the
situation when eavesdroppers cannot be closer to transmitters
than a specific distance or in networks where each transmitter
is able to deactivate the eavesdroppers within a neutralization
region around itself [19]. We consider a two-hop strategy and
propose a protocol to select a messaging relay for each source-
destination pair in such a way that the selected relay has
good links to both that source and that destination. In each
time period, a number of sources simultaneously transmit their
message to corresponding destinations using their selected
relay. The interference from the signals of other transmitters
hides the message of each transmitter from the eavesdroppers.
Next, we consider the general case, when all nodes are
placed uniformly and randomly in a square of area n.We
consider a multi-hop construction within a Gupta-Kumar
framework [28] by partitioning the whole area into small
square cells. By using the proposed protocol, nodes of each
cell choose suitable relays from the next cell in such a way
that each selected relay has a good link to its previous node.
Again the message of each transmitter is concealed from
eavesdroppers by the interference from the communication
of other nodes. We find an achievable number of allowable
eavesdroppers that the network can tolerate as a function of the
number of system nodes, while a required level of reliability
and secrecy is guaranteed.
The rest of the paper organized as follows. Section 2
describes the model and problem formulation. In Sections 3
and 4, the protocols for the case of equal path-loss between all
pairs of nodes and the general case are described and analyzed,
respectively. Discussion, ideas for future work and conclusions
are provided in Sections 5 and 6.
II. M
ODEL AND PROBLEM FORMULATION
A. Model
We consider an extended wireless network consisting of n
legitimate nodes placed uniformly at random on a 2-D plane
of area [0,
√
n]
2
. Each node is a source or destination of one
stream and source-destination pairs are randomly assigned.
In addition m eavesdroppers, E
1
, ..., E
m
are placed uni-
formly at random on this surface (Figure 1). The locations
of eavesdroppers and the CSIs of channels from legitimate
nodes to eavesdropper nodes are assumed to be unknown to
the legitimate nodes.
Denote the k
th
symbol transmitted by node A by x
(A)
k
.
All nodes transmit with the same power E
S
. The k
th
signal
received by node B is denoted by y
(B)
k
, and the distance
between nodes A and B is denoted by d
A,B
. We also denote
the noise at receiver B by n
(B)
k
and the multi-path fading from
a transmitter A to a receiver B by h
A,B
. Based on this model,
the k
th
signal received at node B from node A when all nodes
in a group of nodes, S
1
, transmit their signals is:
y
(B)
k
=
h
A,B
d
α
/2
A,B

E
S
x
(A)
k
+

A
i
∈S
1
,A
i
=A
h
A
i
,B
d
α
/2
A
i
,B

E
S
x
(A
i
)
k
+ n
(B)
k
where α>2 is the path loss exponent. The noise at each
receiver is assumed to be i.i.d complex Gaussian with power
N
0
. The multi-path fading h
A,B
is assumed to be frequency
non-selective Rayleigh, which remains constant during the
transmission of each packet. Then, |h
A,B
|
2
is exponentially
distributed with mean E[|h
A,B
|
2
] and, without loss of gener-
ality, we assume that E[|h
A,B
|
2
]=1. We also exploit channel

3
Fig. 1. Multiple sources, multiple destinations and relay nodes (circles) in
presence of a number of eavesdroppers (crosses)
reciprocity, i.e. h
A,B
= h
B,A
. The SINR from node A to node
B for any two nodes in the network is denoted by C
A,B
:
C
A,B
=
E
S
.|h
A,B
|
2
d
−α
A,B

i∈S
1
,i=j
E
s
|h
A
i
,B
|
2
d
−α
A
i
,B
+
N
0
/2
B. Problem Formulation
Our goal is to propose protocols to enhance physical layer
secrecy in the wireless networks described above and find the
achievable number, m(n), of eavesdroppers in the system as
a function of the number of nodes, that can be tolerated while
guaranteeing reliable and secure communication between all
source-destination pairs.
By reliability we mean that each packet is delivered from a
positive fraction of sources to corresponding destinations with
high probability as n →∞, i.e. a positive fraction of legitimate
nodes (relays or destinations) receive packets with signal-to-
interference-plus-noise-ratio (SINR) greater than a predefined
threshold γ, which is required for successful decoding (with
arbitrarily small probability of error) at a legitimate node. Let
P
(S→D)
OUT
denote the probability of the event {C
S,D
<γ}.For
any source S that transmits a message, reliable communication
with destination D is ensured if P
(S→D)
OUT
→ 0.
By secrecy we mean that with high probability, no eaves-
dropper can achieve a target SINR γ
E
from any of the sources
or relays as n →∞, i.e. all eavesdroppers are in outage
w.h.p. for large n; where, P
(E)
OUT
is the probability of the event

{C
S
j
,E
1
<γ
E
}∩... ∩{C
S
j
,E
m
<γ
E
},j=1, ..., n

.
III. EQUAL PATH LOSS BETWEEN ALL PAIRS OF NODES
In this section we consider the case that there is equal path
loss between all pairs of nodes. This assumption also applies
when we know that the eavesdroppers cannot come closer to
each transmitter (source or relay) than a specified distance or
each transmitter can deactivate the eavesdroppers within an
area around it. Without loss of generality, we assume that the
distance between all pairs of nodes is unity (i.e. d
A,B
=1,
for all A = B).
Among n system nodes, during each time period,
2logn
/t
of them are designated as sources and destinations, and the
other system nodes are the assisting nodes. Here t is a
constant to be determined later. We first propose a protocol
to convey messages from a positive fraction of designated
sources to corresponding destinations reliably and securely.
Then, we analyze the system and find the maximum number of
eavesdroppers that can be tolerated while secrecy is maintained
for all transmitted messages.
A. Protocol
The following protocol employed by sets of
log n
/t source-
destination (S-D) pairs to set up secure links between each
source and its corresponding destination.
1) Source-destination pair selection: Each of the assist-
ing nodes (relays) chooses one of the active log n/t
source-destination pairs uniformly at random and waits
to receive a pilot from that S-D pair. This results in
n
j
,j=1, ..., log n/t assisting nodes waiting for a pilot
from the j
th
S-D pair. We term the group of nodes
waiting to receive pilots from the j
th
S-D pair, the j
th
“waiting group” and denote it by A
j
,j=1, ..., log n/t.
2) Channel measurement between sources and relays: Each
source S
j
broadcasts a pilot signal. Each node in A
j
measures its channel to S
j
: h
S
j
,R
i
, ∀i, j such that
R
i
∈ A
j
and j =1, ..., log n/t. Each eavesdropper
also measures the link between itself and its randomly
selected source. Their measurements are assumed to be
exact.
3) Channel measurement between destinations and relays:
Each destination D
j
broadcasts a pilot signal. Each node
in A
j
measures its channel to D
j
: h
R
i
,D
j
∀i, j such
that R
i
∈ A
j
and j =1, ..., log n/t. Each eavesdropper
also measures the link between itself and its randomly
selected destination. Their measurements are assumed to
be exact.
4) Relay selection: In each waiting group of
nodes, A
j
j =1, ..., log n/t, the relays with
min

|h
S
j
,R
i
|
2
, |h
R
i
,D
j
|
2

>
1
2
log (
nt
log n
) are the
designated relays. Let M
j
denote the number of
designated relays and B − j denote the group of
designated relays for j
th
S-D pair. For each S-D
pair, the relay selection is successful if exactly one
designated relay exists, i.e. M
j
=1.
Each relay in the group B
j
sends a pilot to S
j
. Hence,
the relay selection for S
j
is successful if it receives
exactly one pilot. We indicate the group of sources with
successful relay selection by S
1
and the messaging relay
for the j
th
S-D pair by R
j
.
5) Message transmission from S
j
to R
j
: In this step, each
source in the group S
1
(sources with a successful relay
selection) transmits its message to the corresponding

4
messaging relay. The signal that relay R
j
receives is:
y
(R
j
)
k
= h
S
j
,R
j

E
S
x
(S
j
)
k
+

S
i
∈S
1
,i=j
h
S
i
,R
j

E
S
x
(S
i
)
k
+ n
(R
j
)
k
and eavesdropper E
l
,l=1,...,m− 1, receives:
y
(E
l
)
k
= h
S
j
,E
l

E
S
x
(S
j
)
k
+

S
i
∈S
1
,i=j
h
S
i
,E
l

E
S
x
(S
i
)
k
+ n
(E
l
)
k
6) Message transmission from relay R
j
to destinations D
j
:
In this step, each messaging relay transmits its message
to the intended destination in a manner similar to the
previous step.
B. Analysis
We first analyze the source-destination pair selection step.
The size of each waiting group is described in the following
lemma.
Lemma 3.1. With high probability, the number of relays in
each and every waiting group satisfies {
tn
2logn
<n
j
<
3tn
2logn
}.
Proof: Since log n/t S-D pairs exist, then the proba-
bility that a given relay belongs to a given waiting group
is
t
log n
; thus, the number of nodes in each waiting group
n
j
∼ Binomial(n,
t
log n
). Using a Chernoff bound for binomial
random variables:
P (n
j
> (1 + δ)
tn
log n
) <e
−
δ
2
tn
2logn
and
P (n
j
< (1 −δ)
tn
log n
) <e
−
δ
2
tn
4logn
then, setting δ =
1
2
,
P

(n
j
>
3tn
2logn
) ∪ (n
j
<
tn
2logn
)

<e
−
tn
8logn
Now we want to bound the number of nodes in each of the
waiting groups uniformly. Using a union bound yields:
P

log n
/t

j=1

(n
j
>
3tn
2logn
) ∪ (n
j
<
tn
2logn
)

<
log n
t
e
−
tn
8logn
→ 0.
Now consider the relay selection step. In the following
lemma, it is shown that the relay selection step for a positive
fraction of source-destination pairs is successful. The approach
which is used to prove this lemma is similar to the approach
used in [25].
Lemma 3.2. Let N
1
denote the total number of source-
destination pairs with M
j
=1. Then,
Pr(M
j
=1)→
1
e
> 0 as n →∞
and
log n
2et
<N
1
<
3logn
2et
with probability approaching 1.
Proof: Suppose that in a given waiting group with n
j
nodes, p is the probability that the minimum of fading of S
j
→
R
j
and R
j
→ D
j
links for a given S-D pair and given relay
R
j
is greater than
1
2
log(
nt
log n
), i.e.
p = Pr

min

|h
S
j
,R
i
|
2
, |h
R
i
,D
j
|
2

>
1
2
log (
nt
log n
)

The left side of the inequality is the minimum of two exponen-
tial random variables with mean 1. Thus, it is an exponential
random variable with mean
1
2
. Consequently,
p = e
−2
log(
nt
log n
)
2
=
log n
nt
From the independence of the fading, the probability that
exactly one relay R
j
∈ A
j
has this property is:
Pr(M
j
=1)=(
nt
log n
) × p(1 −p)
(
nt
log n
)
=(
nt
log n
) ×
1
(
nt
log n
)
(1 −
1
(
nt
log n
)
)
(
nt
log n
)
(
nt
log n
) →∞as n →∞; hence,
Pr(M
j
=1)→
1
e
as n →∞
Then the number of S-D pairs that have a single relay N
1
∼
Binomial(
log n
t
,
1
e
); by using the Chernoff bound for binomial
random variables:
P (N
1
< (1 −δ)
log n
te
) <e
−
δ
2
log n
2e
and,
P (N
1
> (1 + δ)
log n
te
) <e
−
δ
2
log n
4e
Choosing δ =1/2 and using union bound:
P (
log n
2et
<N
1
<
3logn
2et
) > 1 − 2e
−
log n
8e
Thus in the limit, the number of nodes that find a single relay
satisfies
log n
2et
<N
1
<
3logn
2et
with probability approaching 1.
Now consider the following lemma, which will be very
useful in our analysis [29]:
Lemma 3.3. Let Y
1
,...,Y
n
be a sequence of n i.i.d. ex-
ponential random variables, (more generally, each having an
exponential tail
¯
F (y) ∼ Ke
−ay
where K, a > 0). Let
M
n
=max(Y
1
,...,Y
n
). Then, lim
n→∞
M
n
log n
=
1
a
a.s.
The following theorem characterizes the maximum number
of eavesdroppers such that with high probability a positive
fraction of the S-D pairs can communicate and all communi-
cations are secure.
Theorem 3.1. Consider the case of equal path-loss between all
pairs of nodes. By applying the proposed protocol, the m(n)
eavesdroppers can be tolerated guaranteeing P
(S
j
→D
j
)
OUT
→ 0,

5
for each source S
j
which transmits a message, and P
(E)
OUT
→ 1
as n →∞is:
m(n)=o

n
log (1+γ
E
)
2et
log n

.
Proof: Because the relay-destination transmission is sim-
ilar to the source-relay transmission, we analyze only the
source-relay transmission. Also, by applying a coding tech-
nique used in [24], securing each hop is sufficient to ensure
source-destination secrecy.
First consider the probability of outage between a source
and its messaging relay, P
(S
j
→R
j
)
OUT
. Consider the SINR at
messaging relay R
j
during the transmissions from sources to
relays:
C
S
j
,R
j
=
E
S
.|h
S
j
,R
j
|
2

i∈S
1
,i=j
E
s
|h
S
i
,R
j
|
2
+
N
0
/2
From Lemma 3.1 with a =2and Lemma 3.3, |h
S
j
,R
j
|
2
>
log n
j
2
√
2
>
log n
4
w.h.p. as n →∞. In the denominator, S
1
is the
subset of sources that transmit their message; from Lemma
3.2 , |S
1
| = N
1
<
3logn
2et
and thus

i∈S
1
,i=j
|h
S
i
,R
j
|
2
<
3logn
et
as n →∞by the weak law of large numbers. The
interference term is the dominant term in the denominator of
C
S
j
,R
j
; hence,
C
S
j
,R
j
>
log n
/4
6logn
/et
>γ
with high probability if we choose t such that t>
24γ
e
, and
then P
(S
j
→R
j
)
OUT
→ 0.
Now consider the eavesdroppers. For each source S
j
,given
that the group S
1
of sources are transmitting and using a union
bound:
P
(S
j
→E)
OUT
=1− P
⎛
⎝
m(n)

i=1
{C
S
j
,E
i
≥ γ
E
}
⎞
⎠
≥ 1 −
m(n)

i=1
P (C
S
j
,E
i
≥ γ
E
)
Then, using the same approach as in [27], for each eavesdrop-
per:
P (C
S
j
,E
i
≥ γ
E
)
≤ P

E
S
|h
S
j
,E
i
|
2

S
k
∈S
r
,k=j
E
S
|h
S
k
,E
i
|
2
+
n
0
/2
>γ
E

<P

E
S
|h
S
j
,E
i
|
2

S
k
∈S
r
,k=j
E
S
|h
S
k
,E
i
|
2
>γ
E

= E
{|h
S
k
,E
i
|
2
,S
k
∈S
1
}

P

|h
S
j
,E
i
|
2
>γ
E

S
k
∈S
1
,k=j
|h
S
k
,E
i
|
2


=

S
k
∈S
1
,k=j
E
{|h
S
k
,E
i
|
2
}

e
−γ
E
|h
S
k
,E
i
|
2

=

1
1+γ
E

|S
1
|
Hence, by the law of total probability and using Lemma 3.2,
P (C
S
j
,E
i
≥ γ
E
) ≤

1
1+γ
E

log n
2et
w.h.p. as n →∞; and,
P
(E)
OUT
≥ 1 −
m(n)

i=1

1
1+γ
E

log n
2et
→ 1
as n →∞for any m(n)=o((1 + γ
E
)
log n
2et
). Using the union
bound again, we obtain P
(E)
OUT
, the probability that none of
the eavesdroppers can achieve the target SINR from any of
the transmitting sources. In this case:
m(n)=o

(1 + γ
E
)
log n
2et
log n

= o

n
log (1+γ
E
)
2et
log n

.
IV. GENERAL CASE
Now consider the case that the path-loss between pairs of
nodes is based on their relative locations. An extended network
is considered where n nodes placed in 2-D plane uniformly at
random on a square of side
√
n. Also m passive eavesdroppers
of unknown channels and locations which are placed uniformly
at random exist. Our goal is to find maximum achievable
number of eavesdroppers that can be tolerated while we still
have reliability and secrecy.
We tessellate the square [0,
√
n]
2
into
n
N
square cells, where
N = k log n and each cell is of side
√
N =
√
k log n (Figure
2). Each source sends its message to a final destination in a
multi-hop fashion. Each packet travels cell by cell horizontally
until its x dimension equals the final destination’s x dimension
and then travels vertically until it reaches its final destination.
Using this model, each packets take roughly

n
log n
hops;
thus, the occurrence of a final destination in each hop is very
infrequent. In other words, most of the traffic in the network
is the “through traffic” and just a few of it is local traffic.
In this model, the cells which are at least Δ
√
N apart (Δ
is a constant) can transmit simultaneously with no destructive
interference (see Appendix). Thus, we do not consider the
out-cell interference in our analysis and only focus on the
interference from the nodes inside each cell.
As mentioned earlier, using a coding technique used in [24],
securing each hop is sufficient to ensure source-destination
secrecy. Hence, we focus on the security of the through traffic
between two adjacent cells, C and C

, and later we will extend
it to the whole network.
Consider two adjacent cells C and C

. From [30], the
number of nodes in each cell
1
/2N<N
C
,N
C

<
3
/2N w.h.p.
providing that k>8. The cell C is divided into
N
c(N)
square
sub-cells, each of area c(N)=k
1
log N (Figure 3). The time
slot which is assigned to cell C is divided into time periods.
During each time period, the nodes in one of its sub-cells
choose nodes in cell C

and send their messages to them
based on the following protocol securely. If one of packets

6
Fig. 2. The whole area is tessellated into squares of side
√
N. The square
which is centered at cell C is region A.
in an active sub-cell is reaching its final destination in C

, the
other nodes in this sub-cell stop transmitting their messages.
A protocol similar to [27] is employed to deliver the packet to
the final destination safe and secure by using some of nodes
as chatterers and then the protocol below resumes. Note that
the use of the protocol in [27] is so infrequent that it does not
impact the power consumption and throughput of our network.
We denote the number of nodes in the i
th
sub-cell of C by
N
c
i
. The number of nodes in each sub-cell is characterized by
the following lemma. The same approach as in [30] is used
here.
Lemma 4.1. The number of nodes in each sub-cell satisfies
c(N)
4
<N
c
i
<
9c(N)
4
as N →∞w.h.p.
Proof: The probability that a node belongs to a sub-cell c
i
is
c(N)
N
. Given that the number of nodes in cell C, N
C
= N

,
N
c
i
∼ Binomial(N

,
c(N)
N
). Using a Chernoff bound:
P (N
c
i
< (1 −δ)
N

N
c(N)) <e
−
δ
2
c(N )N

2N
and,
P (N
c
i
> (1 + δ)
N

N
c(N)) <e
−
δ
2
c(N )N

4N
Where
1
/2 <
N

N
<
3
/2 with probability approaching 1. Using
the law of total probability and choosing δ =
1
/2,
P

(N
c
i
<
c(N)
4
) ∪ (N
c
i
>
9c(N)
4
)

< 2e
−
c(N )
16
Using a union bound to bound the number of nodes per sub-
cell uniformly:
P


N
/c(N )

i=1

(N
c
i
<
c(N)
4
) ∪ (N
c
i
>
9c(N)
4
)


<
N
c(N)
2e
−
c(N )
16
=
2N
1−k
1
/16
k
1
log N
Hence, providing that k
1
> 16,
c(N)
4
<N
c
i
<
9c(N)
4
, ∀i
w.h.p as N →∞.
The protocol for communication between two adjacent cells
is described in the following section.
A. Protocol
Consider two neighboring cells C and C

(Figure 3). When
nodes within a sub-cell are transmitting their messages, we
refer to these nodes as sources and the nodes in the neighbor
cell C

that receive the messages as relays. During each time
period,
1
/t fraction of the sources in the active sub-cell c
i
of
cell C choose relays from the region [
√
N
4
,
√
N] × [0,
√
N]
of C

, where t is a constant to be determined later. Then, the
sources transmit their messages securely to these relays.
The proposed protocol for conveying the messages securely
on each hop consists of:
1) Channel measurement between sources and relays:A
constant fraction
1
/t of sources within sub-cell c
i
broad-
cast their pilots to the relays in the next cell. Each
node in the region [
√
N
4
,
√
N] × [0,
√
N] of C

and
each eavesdropper measure the link between itself and
its randomly selected source. Their measurements are
assumed to be exact.
2) Relay selection: A relay that receives a pilot with fading
greater than log N, i.e. |h
S
j
,R
k
|
2
> log N such that S
j
∈
c
i
and R
k
∈ C

, is a designated relay. We denote the
group of designated relays for source S
j
by B
j
.
Each relay in each group B
j
sends a pilot to S
j
.For
S
j
, the relay selection step is successful if it receives
exactly one pilot, i.e. from only one relay. We denote
this relay as R
j
. Let S
1
denote the set of sources that
each has exactly one relay.
3) Message transmission from sources to relays: In this
step, each source in S
1
transmits its message to its
corresponding relay. Relay R
j
receives the following
signal:
y
(R
j
)
k
= h
S
j
,R
j

E
S
d
−α
S
j
,R
j
x
(S
j
)
k
+

S
i
∈S
1
,i=j
h
S
i
,R
j

E
S
d
−α
S
i
,R
j
x
(S
i
)
k
+ n
(R
j
)
k
and eavesdropper E
l
,l=1,...,m− 1, receives:
y
(E
l
)
k
= h
S
j
,E
l

E
S
d
−α
S
j
,E
l
x
(S
j
)
k
+

S
i
∈S
1
,i=j
h
S
i
,E
l

E
S
d
−α
S
i
,E
l
x
(S
i
)
k
+ n
(E
l
)
k

7
B. Analysis
Relay selection only depends on the multi-path fading of
the links between sources and relays. Thus from Lemma 3.2,
the probability that exactly one relay is selected is greater
than
1
e
> 0. Consequently, a positive fraction of nodes can
always convey messages. Denote the number of such nodes
by N
1
= |S
1
|. Furthermore, from Lemmas 3.2 and 4.1 we
have
k
1
log N
8et
<N
1
<
27k
1
log N
8et
.
Recall from the previous section that P
(S
j
→R
j
)
OUT
is the
probability that the SINR from a given source S
j
to its
corresponding relay R
j
is less than γ. For each active cell
C, P
(E)
OUT
is the probability of the event that the SINR at none
of the eavesdropper nodes inside a square region A, centered
at C and of side Δ
√
N (Figure 3) from any of the sources
exceeds the required SINR for eavesdroppers, γ
E
.
The following theorem characterizes the maximum number
of eavesdroppers that the network can tolerate.
Theorem 4.1. Consider the general case, legitimate and eaves-
dropper nodes placed uniformly and randomly at unknown
locations on the square region A. Based on the proposed
protocol, the maximum achievable number of eavesdroppers
that can be tolerated guaranteeing P
(S
j
→R
j
)
OUT
→ 0, for each
source S
j
which transmit its message, and P
(E)
OUT
→ 1 as
N →∞is:
m(N)=o

N
log (1+γ
E
2
−α
)
8et
log N

.
Proof: Consider the active sub-cell c
i
within C.As
mentioned earlier, we denote the group of active nodes in c
i
by
S
1
and the nodes in the S
1
by S
j
,j =1, ..., N
1
. We denote the
corresponding messaging relay for source S
j
in the neighbor
cell by R
j
. Consider the SINR at each messaging relay:
C
S
j
,R
j
=
E
S
.|h
S
j
,R
j
|
2
d
−α
S
j
,R
j

S
i
∈S
1
,i=j
E
S
.|h
S
i
,R
j
|
2
d
−α
S
i
,R
j
+
N
0
/2
From Lemma 3.3 with a =1, |h
S
j
,R
j
|
2
>
log(
3
/8N )
√
2
; thus,
|h
S
j
,R
j
|
2
>
log N
2
as N →∞.
Now consider the denominator. For each pair of nodes
S
i
∈S
1
and R
j
, d
S
i
,R
j
>
√
N
4
>> 2
√
k
1
log N, then we
can assume d
S
i
,R
j
≈ d
i,j
, where d
i,j
is the distance between
sub-cell c
i
and relay R
j
. By the weak law of large num-
bers,

S
i
∈S
1
|h
S
i
,R
j
|
2
< 2N
1
; using Lemmas 3.2 and 4.1,

S
i
∈S
1
|h
S
i
,R
j
|
2
<
9k
1
log N
2et
as N →∞. Furthermore, the
noise term in denominator is constant while the interference
grows as N gets large; thus,
C
S
j
,R
j
>
E
S
. log(
3
/8N )d
−α
i,j
2.

S
i
∈S
1
E
S
.|h
S
i
,R
j
|
2
d
−α
i,j
>
log N
/2
9k
1
log N
/et
>γ as N →∞
w.h.p. provided t>
18k
1
γ
e
. Consequently, P
(S
j
,R
j
)
OUT
→ 0 as
N →∞.
Fig. 3. Two adjacent cells C and C

. During each time period, the nodes
in one sub-cell of cell C choose nodes from area [
√
N
4
,
√
N] × [0,
√
N] of
C

and transmit their messages to them.
Now consider the eavesdroppers. For each source S
j
, using
the union bound for the eavesdroppers in the region A:
P
(S
j
→E)
OUT
=1− P
⎛
⎝
m(N)

l=1
{C
S
j
,E
l
≥ γ
E
}
⎞
⎠
≥ 1 −
m(N)

l=1
P (C
S
j
,E
l
≥ γ
E
)
Assume that sub-cell c
i
is active. Considering the eavesdropper
E
l
, based on its location, we have:
P (C
S
j
,E
l
≥ γ
E
)=P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
1
)P (E
l
∈ A
1
)
+ P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
2
)P (E
l
∈ A
2
)
+ P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
3
)P (E
l
∈ A
3
)
where A = A
1
∪A
2
∪A
3
. Furthermore, A
1
is the circle that is
circumscribed around the sub-cell c
i
, A
2
is the annulus with
the same center with inner radius

c(N)
2
and outer radius
3

c(N)
2
and A
3
is the area of A minus A
1
+ A
2
(Figure 3).
Then:
P (C
S
j
,E
l
≥ γ
E
)
< 1.
πc(N)
2Δ
2
N
+1.
4πc(N)
Δ
2
N
+ P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
3
)
Δ
2
N − 4.5πc(N)
Δ
2
N
Thus, we have to bound P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
3
):
P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
3
)
= P

E
S
.|h
S
j
,E
i
|
2
d
−α
S
j
,E
l

S
i
∈S
1
E
S
.|h
S
i
,E
l
|
2
d
−α
S
i
,E
l
+
N
0
/2
≥ γ
E
|E
l
∈ A
3

In the denominator of the SINR, removing the noise benefits
the eavesdropper. Also, the distance of the interfering sources
to the eavesdropper is less than the distance of the closest
source to the eavesdropper plus the diagonal of the sub-cell. In

8
addition, multi-path fading of different links are independent.
Given that the nodes in the group S
1
are transmitting:
P (C
S
j
,E
l
≥ γ
E
|E
l
∈ A
3
)
= P

E
S
.|h
S
j
,E
l
|
2
d
−α
S
j
,E
l

S
i
∈S
1
E
S
.|h
S
i
,E
l
|
2
d
−α
S
i
,E
l
+
N
0
/2
≥ γ
E
|E
l
∈ A
3

<P

E
S
.|h
S
j
,E
l
|
2
d
−α
S
j
,E
l

S
i
∈S
1
E
S
.|h
S
i
,E
l
|
2
d
−α
S
i
,E
l
≥ γ
E
|E
l
∈ A
3

<P

|h
S
j
,E
l
|
2
d
−α

S
i
∈S
1
|h
S
i
,E
l
|
2
(d +

2c(N))
−α
≥ γ
E
|d>

2c(N)

<E
{|h
S
i
,E
l
|
2
,S
i
∈S
1
}

P (|h
S
j
,E
l
|
2
≥ γ
E
.

S
i
∈S
1
|h
S
i
,E
l
|
2
2
−α
)

= E
{|h
S
i
,E
l
|
2
,S
i
∈S
1
}

e
−γ
E
.

S
i
∈S
1
|h
S
i
,E
l
|
2
2
−α

=

S
i
∈S
1
E
{|h
S
i
,E
l
|
2
}

e
−γ
E
2
−α
.|h
S
i
,E
l
|
2

=

1
1+γ
E
.2
−α

|S
1
|
From Lemmas 3.2 and 4.1, the number of transmitting nodes
N
1
= |S
1
| >
k
1
log N
8et
. Then,
P
(S
j
→E)
OUT
≥ 1 −
m(N)

i=1

4.5π log N
tΔ
2
N
+

1
1+γ
E
.2
−α

k
1
log N/8et

Using the union bound again, considering that at most
27 log N
8et
sources transmit at the same time (Lemma 4.1), the number of
achievable eavesdroppers in the region A that cannot intercept
the message from any of the sources, as N →∞is:
m(N)=o

(1 + γ
E
2
−α
)
k
1
log N
8et
log N

= o
⎛
⎝
N
k
1
log (1+γ
E
2
−α
)
8et
log N
⎞
⎠
.
V. D ISCUSSION
(a) Comparison to previous works: Considering the large
picture in general case scenario, and using union bound
to find the number of eavesdroppers that can be tol-
erated while ensuring that none of them can inter-
cept a message from any of the transmitters results in
m(n)=o(
log n
c
log log n
) number of eavesdroppers. Although
here m(n) seems slightly less than the number of
eavesdroppers in the similar network that employ some
helper nodes which generate artificial noise m(n)=
o(log n
c
1
) [25], these values are essentially the same.
Because here we consider protecting log log n flows
at each time period, but [25] considers protecting one
flow. Furthermore, this method is superior from different
angles. The previous method is not power efficient and
the energy used by the helper nodes is wasted, but in
our method we have no waste of energy and each node
transmits its own message. In addition, in this scheme
in each cell log log n number of transmitters use the
same bandwidth simultaneously without any destructive
interference, then, we can achieve a higher data rate than
the previous method.
(b) Motivating the finite case: The approach proposed in
this paper to physical layer secrecy is advantageous
from different perspectives and applying this method to
the finite case is also interesting. Suppose we have a
finite number of nodes in a given area and there are
some source-destination pairs among them. By careful
routing and scheduling flows in this network, the ag-
gregate of flows will hide each flow from the external
eavesdroppers while the interference does not destroy
the communication between sources and destinations.
Thus, a similar effect as in this paper can be achieved.
VI. C
ONCLUSION
In this paper, we use a cooperative transmission strategy and
some relay nodes to improve secrecy. We first consider a sce-
nario of equal path-loss between all pairs of nodes. A protocol
for two-hop communication between a large number of source-
destination pairs via one relay for each pair is proposed. It is
shown that in this scenario m(n)=o

n
log (1+γ
E
)
2et
log n

number
of eavesdroppers can be tolerated, where n is the number of
system nodes, α is path-loss exponent, γ
E
is the required SINR
threshold at each eavesdropper and t is a constant.
Next, for the general case, we choose a multi-hop strategy
in an extended network. We tessellate the network into small
square cells and propose a protocol for communication be-
tween two neighboring cells; and then in discussion we expand
the result to the whole network. In this case, it is shown that
m(N)=o

N
log (1+γ
E
2
−α
)
8et
log N

number of eavesdroppers inside
and around each cell can be tolerated, where N is roughly the
number of nodes in each cell and t is a constant different from
above.
Although the achievable number of eavesdroppers here is
comparable to previous work on a similar network [25], the
idea of using interference from aggregate signal of other
transmitters makes this work different. From an energy con-
sumption point of view, in the previous work, the energy
that chattering nodes employ to generate artificial noise can
even become more than the energy employed to transmit the
message. Here, all the energy is used to transmit messages
and there is no waste of energy. Furthermore, by using the
proposed protocol, the large number of nodes use the same
bandwidth simultaneously without any destructive interference
and then a higher data rate is achievable.

9
APPENDIX
We denote the out-cell interference from the transmission
of nodes in other active cells at relay R
j
by I
R
j
out
. Then,
I
R
j
out
<
∞

l=1

S∈S
l
E
s
.(Δ
√
Nl)
−α
|h
S,R
j
|
2
= E
S
.(Δ
√
N)
−α
.
∞

l=1
l
−α

S∈S
l
|h
S,R
j
|
2
In each active cell we know that N
1
<
27 log N
8et
nodes transmit-
ting. Also, the number of concentric transmitting cells at the
distance lΔ
√
N is 8l; then, the number of nodes transmitting
from distance lΔ
√
N simultaneously: |S
l
| <
27l log N
et
. Using
the law of large numbers

S∈S
l
|h
S,R
j
|
2
<
27
√
2l log N
et
.
Besides,

∞
l=1
l
−α+1
converges to some constant k for α>2;
hence,
I
R
j
out
<
27
√
2E
S
k
etΔ
α
log N
N
α/2
→ 0 as N →∞.
R
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