-
Divide-and-Conquer Based Cooperative Jamming:Addressing Multiple
Eavesdroppers in Close
ProximityZhihong Liu∗, Jiajia Liu∗§, Nei Kato†, Jianfeng Ma∗,
and Qiping Huang‡
∗School of Cyber Engineering, Xidian University, Xi’an,
China†Graduate School of Information Sciences, Tohoku University,
Sendai, Japan‡School of Telecommunication Engineering, Xidian
University, Xi’an, China
§Email: [email protected]
Abstract—This paper investigates divide-and-conquer
basedcooperative jamming for physical-layer security enhancement
inthe presence of multiple eavesdroppers. Different from
previousworks, we consider a scenario where the eavesdroppers can
belocated anywhere inside the communication region of the source,no
location information of the eavesdroppers is available and
noconstraint on the number of eavesdroppers is presupposed.
Thebasic idea is to transmit the message in multiple rounds and
ex-ploit the helpful interference from the source and the
destinationto jam the eavesdroppers in close proximity. Stochastic
geometrybased analytic results as well as Monte Carlo simulations
arepresented to illustrate the achievable secrecy performances.
I. INTRODUCTION
A. Background and Motivation
The inherent randomness associated with wireless channelscan be
leveraged to provide intrinsic security at the physical-layer level
[1], [2]. The traditional physical-layer secrecy per-formances are
hampered by channel conditions. In particular,if the channel
between the source and the destination is worsethan the channel
between the source and the eavesdropper, thesecrecy rate is
typically zero. Some recent works have beendevoted to overcoming
this limitation by utilizing cooperativenetworking, which has
received significant attention as anemerging network design
strategy for future wireless networks.In cooperative networking
[3]–[5], individual network nodescan cooperate to achieve desired
network performance in acoordinated way, and the cooperation can
mainly be classifiedinto two type methods, i.e., cooperative
relaying [6], [7] andcooperative jamming.
In cooperative jamming, also known as artificial noise [8],[9],
jammers transmit interfering signals at the same timewhen the
source transmits the message, with the purposeof thwarting the
eavesdroppers. Many cooperative jammingschemes have been
investigated for different scenarios, and areused to create
wireless virtual barriers [10], protect implantablemedical devices
[11], sensor networks [12], and other wirelessnetworks [13].
Existing works on cooperative jamming have focused pri-marily on
a simple model with single source-destination pairand single
eavesdropper [14], [15], or investigated the secure
(a) (b)
Fig. 1. The network model. × denotes an eavesdropper E, △
denotes ajammer J , and ◦ denotes a legitimate node. The solid
circle centered at Sdenotes the communication region, and the
smaller shaded disc is the jammingregion of a jammer where any
eavesdropper located inside can be thwarted.(a) Division of
regions. The area inside the communication region of S isequally
divided into regions I, II, III, IV; while the area outside is
regionV. (b) Illustration of the divide-and-conquer based jamming
strategy, wherethe coded blocks for the message are transmitted in
three rounds, and thejammers J1, J2, and J3 are adopted separately
in each round to thwart theeavesdroppers located within the
associated jamming region.
connection from the information-theoretic perspective [16].
Asdepicted in Fig.1(a), the source S wants to transmit
secretmessage to the destination D. If the eavesdroppers are
locatedin the region V which is outside the communication regionof
S, there exists a positive secrecy transmission capacitybetween S
and D, and the secure communication is possible.In the case that
the eavesdroppers are located in the regionsII and III, we can find
jammers in II or III to confoundthe eavesdroppers. However, if the
eavesdropper is locatedwithin close proximity of S and D, e.g.,
within regions I,IV, or the rectangular region (denoted by red
dotted line), it ischallenging to thwart the eavesdropper, let
alone the case ofhaving multiple eavesdroppers in such regions.
There has been several pioneering works addressing the sce-nario
of multiple eavesdroppers. In [17] a cooperative wirelessnetwork
with multiple eavesdroppers was considered wherethe authors
restricted their attention to the decode-and-forwardprotocol, and
cooperatively jamming multiple eavesdroppers isdeemed as a
difficult problem and left for future study. Goelet al. in [8]
considered multiple eavesdroppers of unknown
-
location, using artificial noise and multi-user diversity
toenhance the security, but their scheme can only tolerate
thenumber of eavesdroppers whose growth is sub-linear in thenumber
of system nodes. Dong et al. in [3] addressed securecommunications
of one source-destination pair with the helpof multiple cooperating
relays in the presence of one or moreeavesdroppers, but they have
to presuppose that the numberof relays is greater than the number
of eavesdroppers. Zhouet al. in [9] assumed a multi-antenna
transmitter for jamming.
Besides the limitation on the number of eavesdroppers,many works
presuppose that the eavesdroppers cannot bearbitrarily close to the
source. In [18], Çapar et al. supposedthat the probability that
any S-D pair does not have any nearbyeavesdroppers is close to 1.
In [19], the source is assumed toknow a priori whether there is any
eavesdropper within someneighborhood or not. In [20], Zhou et al.
supposed a secrecyguard zone in which the legitimate nodes are able
to detectthe existence of eavesdroppers in their vicinities.
Based on the above observations, we consider in this papera
scenario where a source communicates with a destination inthe
presence of multiple eavesdroppers which can be locatedanywhere
inside the communication region of the source, i.e.,the regions of
I, II, III and IV of Fig. 1(a). Furthermore,the eavesdroppers
operate passively and independently, theirlocation information is
unknown and there is no constrict onthe number of
eavesdroppers.
B. Approach and Contributions
As we focus on the scenario of having multiple eavesdrop-pers
located within close proximity of S and D, it is difficultto adopt
cooperative jamming to thwart all eavesdroppers in asingle
transmission. We address this issue based on a divide-and-conquer
method, which requires the source to encode themessage into
multiple coded blocks, and to transmit eachcoded block one by one.
The message is encoded in such a waythat it can be recovered if and
only if having received all codedblocks. Therefore, the message can
be securely transmitted aslong as any eavesdropper is guaranteed to
miss at least onecoded blocks. An illustrative example can be found
in Fig.1(b).
Our main contributions are summarized as follows:• We propose a
divide-and-conquer based cooperative jam-
ming strategy to improve the secrecy performances ofwireless
links in the presence of multiple eavesdroppers.The strategy does
not presuppose the location of eaves-droppers, even if they are
located quite close to the sourceor the destination. Furthermore,
this strategy can tolerateany number of independent eavesdroppers
provided thatthe density of legitimate nodes satisfies certain
condition.
• With the help of stochastic geometry theory, we providean
analytic framework for comprehensively investigatingthe secrecy
performance enhancement brought by thedivide-and-conquer based
jamming strategy, in terms ofthe secure communication probability,
as well as thesecrecy rate, etc.
• We prove that the optimal method is to choose onejammer at
each transmission, identify the critical con-
dition under which secure communication is possible,and reveal
the relationship between the limiting securecommunication
probability and the source-destinationdistance, the densities of
eavesdroppers and legitimatenodes, as further corroborated by
extensive Monte Carlosimulations.
II. SYSTEM MODELA. Network Model
We consider a wireless network consisting of multiplelegitimate
nodes and eavesdroppers. Let Φl = {xi}∞i=1 ⊂ R2be the set of
legitimate nodes, and Φe = {ei}∞i=1 ⊂ R2 bethe set of
eavesdroppers. Nodes in Φl and Φe are distributedaccording to
independent Poisson Point Processes (PPPs) [21]with intensities λ
and λE , respectively. Let a disc of radiusR centered at a node x
be B(x,R). The Euclidean distancebetween two nodes x and y is
denoted by dxy = ∥x−y∥. Eachnode has no idea of the information of
surrounding nodes.All nodes are assumed to be static, and the
eavesdroppers arepassively operating independently of each other.
That is, thereexists no collusion among eavesdroppers.
Each node, legitimate node or eavesdropper, is equippedwith a
single omnidirectional antenna. Wireless fading channelis modeled
by large-scale fading with path loss exponent α(α > 2) as in
[18], [22]. The network is interference limited,i.e., the thermal
noise is negligible compared to the aggregateinterference from
jammers [23].
B. Secure Communication
Without loss of generality, we focus on a single link anddenote
by S and D, respectively, the source node and thedestination node.
In order for the data transmission from S tobe securely received at
D in the presence of an eavesdropperE, the signal to interference
ratio (SIR) SIRSE at the eaves-dropper E should be smaller than
SIRSD at the destination D,i.e., SIRSE < SIRSD. In particular,
given SIRSD ≥ γl andSIRSE < γe, where γe < γl and γe can be
arbitrarily small,the secrecy capacity [2] of the communication
between S andD at each separate transmission can be determined
by
CSD(E) = [log2(1 + SIRSD)− log2(1 + SIRSE)]+.
That is, both S and D can achieve secure communication
withsecrecy rate RSD < CSD by agreeing on a code.
For the link pair S-D, as the eavesdroppers follow a PPPprocess
Φe, we say that the data transmission from S to D issecure if and
only if
SIRSD ≥ γl and SIRSE∗ < γe,∀E∗ ∈ Φe. (1)
C. Jamming
In the presence of multiple eavesdroppers of which thelocation
information is unknown, we randomly select severallegitimate nodes
to act jointly as jammers, emitting interferingsignals
simultaneously during the transmission from S to D,so as to satisfy
(1).
More specifically, suppose k jammers, J1, ..., Jk, areadopted to
jointly emitting jamming signals. If we denote by
-
PS the transmit power of node S and by PJi the transmitpower of
jammer Ji, 1 ≤ i ≤ k, (1) can be rewritten as thefollowing system
of inequalities
SIRSD =PSd
−αSD∑k
i=1 PJid−αJiD
≥ γl, (2)
SIRSE∗ =PSd
−αSE∗∑k
i=1 PJid−αJiE∗
< γe,∀E∗ ∈ Φe. (3)
Note that an eavesdropper E∗ located within the region Vof Fig.
1(a) typically have a distance dSE∗ > dSD, and the se-crecy
performance of this scenario has been intensively studiedin
literature [1]. In this paper, we focus on the eavesdropperslocated
within the disc B(S, dSD), i.e., the regions I, II, III,and IV.
Since the eavesdroppers within B(S, dSD) can be veryclose to the
nodes S and D, it is regarded as a challenging taskto achieve a
non-zero secrecy capacity for the communicationbetween S and D
[18]–[20].
D. Our Basic Methodology
Depending on the concrete number of legitimate nodesand
eavesdroppers, as well as their actual spatial distributionwithin
B(S, dSD), it is extremely difficult (if not impossible)to
simultaneously jam all eavesdroppers by using one set ofjammers
when the location of eavesdroppers is unknown.
To address this issue, we adopt a divide-and-conquer
basedjamming scheme. In particular, the source S first applies
acoding algorithm to encode its message into multiple, say t,coded
blocks, then transmits the coded blocks one by one tothe
destination D. The message is encoded in such a waythat the
original message can be recovered only after success-fully
collecting all t blocks. Therefore, with our scheme, aneavesdropper
will fail to decode the message as long as itmisses at least one
blocks. In other words, we can achievenon-zero secrecy capacity for
the communication between Sand D, as long as we are able to
achieve: (1) SIRSD ≥ γlin each separate transmission; (2) for each
eavesdropper Elocated within B(S, dSD), we have SIRSE < γe in at
leastone separate transmission.
We provide Fig. 1(b) as an illustrative example of ourbasic
methodology. As shown in Fig. 1(b), there are threeeavesdroppers
and seven legitimate nodes within the discB(S, dSD). If we encode
the original message into t ≥ 3coded blocks, and randomly select
one legitimate node asjammer when transmitting each code block. As
long as nodesJ1, J2, and J3 are selected and tuned to proper
transmit power,no matter their order of being selected, the message
can besecurely transmitted from S to D, and each eavesdroppercannot
recover the message on its own.
III. DIVIDE-AND-CONQUER BASED SMART JAMMING
A. Smart Jamming Strategy
Although it is difficult to thwart all the eavesdroppers
withinB(S, dSD) in a single transmission, we can randomly
selectlegitimate nodes to jam them in a divide-and-conquer way.
Togain an advantage against eavesdroppers, we generate multiple
coded blocks for a single message such that the message canbe
decoded if and only if all blocks are received, and noinformation
about the message can be gained if any block ismissing. When
transmitting each coded block, we randomlyselect several jammers to
jointly emit jamming signals. Forsimplicity, we suppose the
selected jammers adopt the sametransmit power at each separate
transmission.
The smart jamming strategy consists of three steps:1) Block
construction: let M be a b-bit message. The node
S first generates t−1 random b-bit blocks M1, ...,Mt−1and sets
Mt such that the message M satisfies M =M1 ⊕M2 ⊕ · · · ⊕Mt, where ⊕
denotes bit-wise XORoperation.
2) Power adjustment: before transmitting any block Mn,1 ≤ n ≤ t,
we first randomly choose k legitimatenodes in B(S, dSD) as jammers,
denoted by set Jn ={J1, . . . , Jk}. S then broadcasts a pilot
signal usingpower PS , and all the k jammers in Jn adjust
theirtransmit power PnJ until (4) is satisfied
PSPnJ
= γl
∑ki=1 d
−αJiD
d−αSD. (4)
3) Block transmission: when S is transmitting block Mnto D, the
jammers belonging to set Jn emit interferingsignals with transmit
power PnJ . Steps 2) and 3) arerepeated until all t coded blocks
are transmitted.
Note that the condition (4) only guarantees the SIRSDduring the
transmission of block Mn. For an eavesdropperE within B(S, dSD), in
order to prevent E from receivingblock Mn, the k jammers in Jn
should satisfy(
dSEdSD
)−α·∑k
i=1 d−αJiD∑k
i=1 d−αJiE
<γeγl
. (5)
Intuitively, given the number of separate transmissions t andthe
number of jammers k at each separate transmission, if
eacheavesdropper in the region of B(S, dSD) can be
successfullyjammed during at least one block transmission, the
messagetransmission of M is secure provided that S and D both
agreeon a code with secrecy rate RroundSD = log2(1+γl)− log2(1+γe)
at each round of block transmission.
Obviously, given S-D pair, λ, λE , γl, and γe, the
secrecyperformance of smart jamming strategy depends heavily onthe
control parameters (t, k), as to be explored in
ensuingsections.
B. Optimal Number of Jammers at Each Round of Transmis-sion
For any eavesdropper E uniformly distributed over B =B(S,R)
where R = dSD, if we denote by P roundc theprobability of E being
jammed during the transmission ofcoded block Mn, 1 ≤ n ≤ t, then we
have
P roundc = P[(
dSEdSD
)−α·∑k
i=1 d−αJiD∑k
i=1 d−αJiE
<γeγl
]
= P[dSE > R
(γlγe
) 1α(∑k
i=1 d−αJiD∑k
i=1 d−αJiE
) 1α].
-
0 5 10 15 200.1
0.15
0.2
0.25
0.3
0.35
k − Number of jammers
Pro
und
c
Upper bound
Simulation result: γe=0dB, γ
l=3dB
Simulation result: γe=0dB, γ
l=10dB
Fig. 2. The simulation results of P roundc , i.e., the
probability of aneavesdropper E being jammed during a single round
of transmission, whereR = 100 m, and α = 4. The red dashed lines
denote the upper bound forP roundc and are obtained from the right
hand side of (9).
Since dSE has the probability density function (PDF)fdSE (x)
=
2xR2 , x ∈ (0, R), P
roundc can be written as
P roundc = EE,Ji[1−
(γlγe
) 2α(∑k
i=1 d−αJiD∑k
i=1 d−αJiE
) 2α].
Using Jensen’s inequality, we can obtain an upper boundfor P
roundc as
P roundc < 1−(γlγe
) 2α[EE,Ji
(∑ki=1 d
−αJiD∑k
i=1 d−αJiE
)] 2α
< 1−(γlγe
) 2α[( EJi(∑ki=1 d−αJiD)
EJi,E(∑k
i=1 d−αJiE
)
)] 2α
. (6)
Using Campbell’s theorem,
EJi( k∑
i=1
d−αJiD
)= λ′
∫Bd−αJiDdJi, (7)
where λ′ = kπR2 is the density of Poisson point process afterthe
thinning procedure when we choose k jammers out of alllegitimate
nodes in B.
As the eavesdropper E at (x, y) is uniformly distributedover
B(S, dSD), its PDF is fE(x, y) = 1πR2 . Therefore,
EJi,E( k∑
i=1
d−αJiE
)= EE
[λ′
∫Bd−αJiEdJi
]=
∫B
1
πR2
[λ′
∫Bd−αJiEdJi
]dE. (8)
Substituting (7) and (8) into (6), the upper bound of P
roundccan be expressed as
P roundc < 1−(πR2
γlγe
) 2α[ ∫
B d−αJiD
dJi∫B[∫B d
−αJiE
dJi]dE
] 2α
. (9)
The right hand side of (9) reveals that the upper boundof P
roundc is statistically independent of k. Extensive MonteCarlo
simulations have been conducted and the results aresummarized in
Fig. 2. We randomly select k jammers andcalculate the total area
that the jamming region of the jammerscan cover, where the term of
jamming region is defined as
−100 −50 0 50 100−100
−50
0
50
100
(a) k = 1
−100 −50 0 50 100−100
−50
0
50
100
(b) k = 2
−100 −50 0 50 100−100
−50
0
50
100
(c) k = 3
−100 −50 0 50 100−100
−50
0
50
100
(d) k = 4
Fig. 3. Illustration of how the jamming region varies with the
number ofjammers k during each round of transmission. The S-D pair
are denoted by�, the jammers are denoted by △ and the jamming
region is denoted by theshaded area. We set R = 100 m, α = 4, γe =
0 dB, and γl = 3 dB.
a region where any eavesdropper E located inside will haveSIRSE
< γe.
One can observe from Fig. 2 that more jammers at eachround does
not increase the probability that any eavesdropperis jammed. Fig. 3
also confirms this result, in which the totaljamming region does
not increase obviously as k. Now weshow that the P roundc is
actually maximized at k = 1.
Lemma 1: If we choose one jammer to emit interfering sig-nal at
each round of transmission, i.e., k = 1, the probabilityP roundc of
smart jamming strategy will be maximized.
Proof: Suppose we select k jammers J1, ..., Jk at thei-th round
of transmission. In the case of one jammer pertransmission round, k
jammers J1, ..., Jk emit noise at kseparate transmissions and
have(
dSEdSD
)−α·d−αJiDd−αJiE
<γeγl
, for i = 1, ..., k.
Since(dSEdSD
)−α· d−αJiD <
γeγl
d−αJiE , for i = 1, ..., k.
and (dSEdSD
)−α·
k∑i=1
d−αJiD <γeγl
k∑i=1
d−αJiE .
Therefore we can get(dSEdSD
)−α·∑k
i=1 d−αJiD∑k
i=1 d−αJiE
<γeγl
,
which means that equation (5) is satisfied. Hence, the
totaljamming region with one jammer at each transmission is noless
than the total jamming region with k jammers at each
-
transmission. Thus the probability P roundc is maximized atk =
1.
C. Secure Communication Probability and Secrecy Rate
Max-imization
Hereafter, we focus on the optimal setting of k = 1. Notethat
the probability result derived above for P roundc is for asingle
eavesdropper within a single round of transmission. Inthis section,
we proceed to derive the secure communicationprobability Pc, i.e.,
the probability that the message M can besecurely transmitted from
S to D with our smart jammingstrategy. Then, we explore the secrecy
rate maximizationproblem under given constraint on Pc.
Theorem 1: With the smart jamming strategy where amessage M is
first encoded into t coded blocks and thentransmitted in t separate
rounds, the probability that themessage M can be securely
transmitted from S to D canbe determined as
E(Pc) = exp[−∫Be−
tπR2
SEiλE dEi
](10)
where Ei is an eavesdropper at (xi, yi) in B = B(S,R),
SEi = πa[(xi −R)2 + y2i ]
(1− a)2(11)
and a = x2i+y
2i
R2 · (γeγl)2/α.
Proof: Without loss of generality, consider a typicalnode at the
origin as S and D at location (R, 0). Given aneavesdropper Ei at
(xi, yi), we can define a masking regionAEi for Ei, from which any
jammer selected can satisfy thecondition (5) for Ei. By setting k =
1 in (5) and substitutingthe coordinates of S, D, and Ei, the
masking region AEi canbe obtained as(
x− xi − aR1− a
)2+
(y − yi
1− a
)2≤ a[(xi −R)
2 + y2i ]
(1− a)2,
(12)which means a disk region centered at (xi−aR1−a ,
yi1−a ) of radius√
a[(xi−R)2+y2i ](1−a)2 , and a =
x2i+y2i
R2 · (γeγl)2/α.
If we denote by SEi the area of masking region AEi , then(11)
can be obtained. Note that the message M is transmittedin t rounds
of transmission and in each round a jammer israndomly selected from
the legitimate nodes located in B.In this case, the thinning
procedure transforms the PPP oflegitimate nodes with density λ into
a PPP with density tπR2 .Therefore, the probability that there is
at least one jammer inthe region AEi can be expressed as
P{∃ Jammer in AEi | Ei} = 1− e− t
πR2SEi .
The secure communication probability of link S-D is
Pc =∏Ei∈B
[1− e−t
πR2SEi ].
0 5 100
100
200
300
400
γl − Threshold of SIR
SD
t − N
umbe
r of
Jam
mer
s
ε=0.1ε=0.2
(a) t vs. γl
0 5 100.004
0.006
0.008
0.01
0.012
0.014
0.016
RS
D −
Sec
recy
Rat
e
γl − Threshold of SIR
SD
ε=0.1ε=0.2
(b) RSD vs. γl
Fig. 4. Smart jamming strategy. (a) The number of jammers t
required toachieve E(Pc) > 1− ϵ vs. the threshold γl; (b) The
secrecy rate RSD thatcan be achieved vs. γl. Here R = 100 m, γe =
−3 dB, λE = 0.0002.
According to the PGFL of a homogeneous PPP [21], wehave
E(Pc) = E[ ∏Ei∈B
[1− e−t
πR2SEi ]
]= exp
[−∫Be−
tπR2
SEiλE dEi
].
Then we finish the proof for Theorem 1.Since the secrecy rate at
each round of block transmission is
RroundSD = log2(1+γl)− log2(1+γe), and the smart jammingstrategy
takes t rounds of transmission to deliver a messageM , we can
achieve a secrecy rate for the S-D link
RSD =1
t[log2(1 + γl)− log2(1 + γe)] (13)
where a bigger t results in a smaller RSD.Given 0 < ϵ < 1
and γe < γl, we present below a secrecy
rate maximization problem
Maximize RSD =1
t[log2(1 + γl)− log2(1 + γe)]
s.t. t < πR2λ,
E(Pc) = exp[−∫Be−
tπR2
SEiλE dEi
]> 1− ϵ.
Fig. 4 illustrates the tradeoff between t and RSD by tuningthe
control parameter γl. As shown in Fig. 4(a), given ϵ andγe, a
bigger γl requires a larger t; whereas in Fig. 4(b), RSDfirst
increases with γl then decreases, reaching its maximalvalue when γl
≈ 4.
Note that as shown in Fig. 4(a), the smart jamming strategyneeds
more than 100 jammers (i.e., t > 100) to satisfythe given
constraint on E(Pc). As illustrated in Fig. 5, theeavesdropper near
S or D has a very small masking region,and it is highly possible
that there is no legitimate nodelocated inside. According to our
smart jamming strategy whichrandomly selects legitimate nodes as
jammers, consequently,a bigger intensity λ (necessarily a large
number of jammers,i.e., t) is required to jam the eavesdroppers
located withinclose proximity of nodes S and D. To address this
issue, wepropose later an enhanced cooperative jamming strategy, as
tobe detailed in Section IV.
-
−100 −50 0 50 100
−100
−50
0
50
100
Distance [m]
Dis
tanc
e [m
]
Ei
AE
i
DS
Fig. 5. Illustration of masking regions (denoted by solid
circles) associatedto eavesdroppers (denoted by ×) at different
location. One can easily see thatthe eavesdropper near S or D has a
very small masking region.
D. Limiting Performance
In this section, we proceed to explore the limiting perfor-mance
that can be achieved by our divide-and-conquer basedsmart jamming
strategy. Note that, the maximum value of t,i.e., the maximum
rounds of block transmission, is actuallylimited by the number of
legitimate nodes located withinB = B(S,R). Therefore, the limiting
case is to select all thelegitimate nodes within B as jammers.
If we denote by P⃗c the secure communication probability
inlimiting case, then after following similar derivations of
(10),we have
E(P⃗c) = exp[−∫Be−λSEiλE dEi
]. (14)
where SEi is as defined in (11). From (14), one cansee that as
long as the parameters (λ, λE , γl, γe, R) satisfyλE
∫B e
−λSEi dEi → 0, we have E(P⃗c) → 1.Next we investigate the
relationship between E(P⃗c) and the
distance R between S and D. Given an eavesdropper Ei at(x, y),
the area of its masking region, i.e., the equation (11),can be
rewritten as
SEi = πγ(x2 + y2)
R2[(x−R)2 + y2][R2 − γ(x2 + y2)]2
, γ =
(γeγl
) 2α
.
If R is large enough, we have R2[(x−R)2+y2]
[R2−γ(x2+y2)]2 → 1, whichimplies that SEi ≈ πγ(x2 + y2). Then
equation (14) can beapproximated as follows,
E(P⃗c) ≈ exp[−∫BλEe
−λπγ(x2+y2) dEi
]= exp
[− 1γ· λEλ
(1− e−πγλR2
)
]. (15)
Furthermore, if R2 ≫ 1πγλ , e−πγλR2 → 0, then
E(P⃗c) ≈ exp(− 1γ
λEλ
)= exp
[−(γlγe
) 2α λE
λ
]. (16)
Fig. 6 shows how the limiting secure communication proba-bility
E(P⃗c) varies with dSD. As dSD becomes larger, E(P⃗c)first
decreases, then increases up and converges to a value
0 200 400 600 800 1000
0.4
0.5
0.6
0.7
0.8
0.9
1
dSD
(m) − Distance between S and D
Sec
ure
Con
nect
ion
Pro
babi
lity
λ=0.001, λE=0.0001, γ
e=0dB, γ
l=0dB
λ=0.001, λE=0.0005, γ
e=0dB, γ
l =3dB
λ=0.001, λE=0.0001, γ
e=0dB, γ
l=3dB
Fig. 6. The limiting secure communication probability E(P⃗c) vs.
the distancebetween S and D. The blue curves represent the Monte
Carlo simulationresults, and the red lines denote the limiting
values calculated from (16).
which approximates to the value determined by the equation(16).
Another observation is that, there is a gap betweenE(P⃗c) and the
limiting value determined by (16) which isinterrelated with λ, λE ,
γe, and γl. When γl = γe, thegap disappears implying a limiting
secure communicationprobability E(P⃗c) = exp (−λEλ ).
Theorem 2: Given λE and large R (R2 > 1πλγlγe
), suggestthat we want E(Pc) > 1−ϵ (0 < ϵ < 1), then λ
should satisfy
λ ≥ λE ·−1
ln(1− ϵ)·(γlγe
) 2α
(17)
Proof: (17) follows directly after substituting (16) intoE(Pc)
> 1− ϵ.
We explore above the secure communication in the view
ofeavesdroppers: if at least one jammer can be selected in
eacheavesdropper’s masking region, the communication is
secure.Intuitively, if the jamming regions of all selected jammers
cancover the whole region B(S, dSD), the communication fromS to D
is secure, no matter how many eavesdroppers locatedwithin B(S,
dSD). The question is what is the minimum λrequired to have the
joint jamming regions cover all eaves-droppers, regardless of the
eavesdropper density λe.
Theorem 3: If we denote by Pcover the probability that
alleavesdroppers in B = B(S,R) are covered by the jammingregion, in
order for E(Pcover) > 1− ϵ (0 < ϵ < 1), then
λ >πR2(− ln ϵ)∫
B SJi dJi(18)
where Ji is a jammer at (xi, yi) in B, and
SJi = R2β + r2α− bR sinβ, (19)
b =
√x2i+y
2i
1−c , r =√
c(x2i+y2i )
1−c , α = arccosb2+r2−R2
2br , β =
arccos b2+R2−r2
2bR , and c =(xi−R)2+y2i
R2 (γeγl)2/α.
Proof: We denote by AJi the jamming region of jammerJi. By
setting k = 1 in (5), the jamming region AJi can berepresented
as(
x− xi1− c
)2+
(y − yi
1− c
)2≤ c(x
2i + y
2i )
(1− c)2,
where c = (xi−R)2+y2i
R2 (γeγl)2/α.
-
0 0.2 0.4 0.6 0.8 1
x 10−4
0
0.2
0.4
0.6
0.8
1
Cov
er P
roba
bilit
y P
cove
r
λ − Intensity of legitimate nodes
R=100R=150R=200
ε
λ0
Fig. 7. Illustration of the lower bound of λ (i.e., λ0) required
to achieveE(Pcover) > 1− ϵ. We set γl = 3 dB, γe = 0 dB, ϵ =
0.2.
We denote by SJi the area of the region AJi ∩B(S, dSD),then SJi
can be derived as in (19). For an eavesdropper Edistributed
uniformly over B(S, dSD), the probability that Eis covered by the
jamming region of Ji is
P (i)cover = P{E is located in AJi} =SJiπR2
.
Hence, the probability that each eavesdropper E is coveredby at
least one jammer can be expressed as
Pcover = 1−∏Ji∈B
(1− P (i)cover),
and its expectation can be derived as
E(Pcover) = 1− E[∏Ji∈B
(1− P (i)cover)]
= 1− exp[−∫BP (i)coverλ dJi
]= 1− exp
[− λπR2
∫BSJi dJi
]. (20)
(18) follows directly after substituting E(Pcover) > 1− ϵ
into(20), and we finish the proof for Theorem 3.
Fig.7 shows the probability E(Pcover) as the function of λ.One
can observe that the Theorem 3 can accurately determinethe minimum
density λ required for the given ϵ.
IV. COOPERATIVE JAMMING STRATEGY ADDRESSINGNEARBY
EAVESDROPPERS
A. A Helper-Based Two-Stage Scheme
In this section, we propose a helper-based two-stage schemeto
exclusively jam eavesdroppers within close proximity of
thereceiver. Note that network coding was adopted to reduce
theimpact of nearby eavesdroppers in [24]. However, when
theeavesdropper is quite close to the source, the strategy in
[24]cannot achieve high secrecy rate.
Given the S-D link and a selected helper H , the schemeconsists
of the following two stages:
Stage 1: the source S transmits signal XS . At the sametime, the
destination D transmits jamming signal XD. Thesignal received by a
helper H in Stage 1 is YH which is amixture of XS and XD.
Stage 2: the helper H transmits signal XH = βYH ,where β is the
amplification factor in Amplify-and-Forwardtransmission.
Since node D has perfect knowledge of the signal XDtransmitted
in Stage 1, it can cancel the jamming signal XDfrom XH [3] while an
eavesdropper cannot achieve this due tothe unknown jamming signal
XD. In the above helper-basedtwo-stage scheme, an eavesdropper E
will wiretap two noisyversions of the data and selects the one with
higher signalquality to decode. In particular,
SIR(1)SE =PSd
−αSE
PDd−αDE
,
SIR(2)SE =(PSd
−αSH)β · d
−αHE
(PDd−αDH)β · d
−αHE
=PSd
−αSH
PDd−αDH
,
where β = PHPSd
−αSH+PDd
−αDH
, PD and PH are the transmit powerof the destination D and the
helper H , respectively.
Let N0 be the noise, and PS = PD. In order to defeat
theeavesdropper E, we should satisfy
SIRSD =(PSd
−αSH)βd
−αHD
N0=
PH/N0
dαHD +PDPS
dαSH
=PH/N0
dαHD + dαSH
≥ γl, (21)
and
SIRSE = max{SIR(1)SE ,SIR(2)SE}
= max
{dDEdSE
,dDHdSH
}α< γe. (22)
According to (22), the above helper-based two-stage schemecan
only efficiently jam the eavesdroppers within close prox-imity of
the receiver, i.e., D. The helper H should tune itstransmit power
according to (21). To thwart the eavesdroppersnear S, we can
execute the above scheme in a reverse way,i.e., letting S transmit
jamming signal.
B. Divide-and-Conquer Based Cooperative Jamming Strategy
Recall that the helper-based two-stage scheme presented
inSection IV-A, can only thwart the eavesdroppers near the
re-ceiver. As depicted in Fig. 8, although we are able to
introducejamming region BS (resp. BD) by executing the
helper-basedtwo-stage scheme from D to S (resp. from S to D),
thereis still an uncovered region BG = B(S, dSD)\(BS ∪ BD).In order
to efficiently thwart the eavesdroppers in the wholeregion B(S,
dSD), we present below a divide-and-conquerbased cooperative
jamming strategy.
1) Helper selection: we choose two helpers, Hsd for
thetransmission from S to D, and Hds for the transmissionfrom D to
S.
2) Auxiliary message transmission from D to S: Dchooses an
auxiliary message MD, sends it to S usingthe helper-based two-stage
scheme with the aid of Hds,and S acts as the jammer.
3) Auxiliary message transmission from S to D: Schooses an
auxiliary message MS , sends it to D using
-
−100 −50 0 50 100
−100
−50
0
50
100
Distance [m]
Dis
tanc
e [m
]
J1
J2
J3
J4
J5
Hsd
BS
DS
BD
BG
Hds
Fig. 8. Illustration of the jamming regions associated to
helpers and jammersin the divide-and-conquer based cooperative
jamming strategy, where △denotes a jammer, and ♢ denotes a helper.
The regions BS and BD representthe jamming regions in which any
eavesdropper can be thwarted respectivelyin jamming rounds from D
to S and from S to D, and the region BG denotesthe uncovered region
B(S, dSD)\(BS ∪BD).
the helper-based two-stage scheme with the aid of Hsd,and D acts
as the jammer.
4) Block construction: let M be the b-bit message to bedelivered
from S to D. S generates t − 1 random b-bit blocks M1, ...,Mt−1 and
then sets Mt such that Msatisfies M = (MD⊕MS)⊕M1⊕M2⊕ ...⊕Mt, where⊕
denotes bit-wise XOR operation.
5) Jammer selection and power adjustment: the sourceS selects t
legitimate nodes from B(S, dSD) as jammersJ1, ..., Jt. For each
jammer Ji, S broadcasts a pilotsignal, and the jammer Ji emits
noise and adjusts itspower PJi until the SIRSD at D is at least
γl.
6) Block transmission: in this step, blocks M1, ...,Mt
aredelivered from S to D in separate transmissions. WhenMi is
transmitting, the jammer Ji transmits noise toconfuse the potential
eavesdroppers nearby using thetransmit power PJi determined in
previous step.
After all blocks have been transmitted, D can reconstructthe
message M because D knows MD and has received MS ,M1, ...,Mt from
S. According to the Crypto Lemma [1], anyeavesdropper missing at
least one blocks/auxiliary messagescannot get any information about
M .
C. Performance Evaluation
Since the analytical results of cooperative jamming can
beobtained similarly as that of smart jamming strategy, we omitthe
derivations here and present numerical results to illustratethe
secrecy performances of the cooperative jamming strategy.
As the two helpers Hsd and Hds in Fig. 8 can defeateavesdroppers
in regions BD and BS , the only region thatcould be wiretapped is
the remaining region BG. Hence, theoptimal method is to select
nodes located between S and D asjammers. Fig. 9 shows how the
secrecy coverage ratio varieswith the number of selected jammers,
i.e., t, where the secrecycoverage ratio is calculated as the ratio
of the total jamming
0 5 10 15 200.75
0.8
0.85
0.9
0.95
1
t − Number of Jammers
Cov
erag
e R
atio
OptimalRandom
Fig. 9. Illustration of how the secrecy coverage ratio varies
with the numberof selected jammers (i.e., t), where R = 100 m, α =
4, γe = −3 dB, andγl = 10 dB. The secrecy coverage ratio is
calculated as the ratio of the totaljamming region to the area of
B(S, dSD).
0 5 100
5
10
15
20
25
30
γl − Threshold of SIR
SD
t − N
umbe
r of
Jam
mer
s
ε=0.1ε=0.2
(a) t vs. γl
0 5 100.05
0.1
0.15
0.2
0.25
0.3
γl − Threshold of SIR
SD
RS
D −
Sec
recy
Rat
e
ε=0.1ε=0.2
(b) RSD vs. γl
Fig. 10. Cooperative jamming strategy. (a) The number of jammers
t requiredto achieve E(Pc) > 1 − ϵ vs. the threshold γl; (b) The
secrecy rate RSDthat can be achieved vs. γl. Here R = 100 m, γe =
−3 dB, λE = 0.0002.
region to the area of B(S, dSD). One can easily observe fromFig.
8 that by setting t = 5, even for the random jammerselection, our
cooperative jamming strategy is able to achievea secrecy coverage
ratio of better than 0.9 Furthermore, thegap between optimal jammer
selection and random jammerselection vanishes quickly as t
increases beyond t = 10.Therefore, the cooperative jamming strategy
can be adoptedto efficiently thwart eavesdroppers within close
proximity.
Fig. 10 shows the number of jammers t required to achievethe
secure communication probability E(Pc) > 1−ϵ as well asthe
achievable secrecy rate, under exactly the same parametersettings
as that in Fig. 4. A comparison with Fig. 4, clearlyindicates that
the number of jammers t has been reduced byaround 10 times and the
secrecy rate has been enhanced byaround 20 times in Fig. 10.
Therefore, significant secrecyperformance improvements over the
smart jamming strategy,can be achieved by the cooperative jamming
strategy, in termsof not only the number of required jammers t but
also theachievable secrecy rate.
Finally, we provide Fig. 11 to further illustrate the
per-formance advantages of cooperative jamming strategy oversmart
jamming strategy, in terms of the limiting secure com-munication
probability vs. (λ, dSD). One can observe fromFig. 11(a) that, for
both the settings of λE there, the coop-erative jamming strategy
exhibits much better performancesthan the smart jamming strategy.
In particular, its limitingsecure communication probability
converges quickly to 1 after
-
0 0.002 0.004 0.006 0.008 0.010
0.2
0.4
0.6
0.8
1
Sec
ure
Con
nect
ion
Pro
babi
lity
λ − Intensity of legitimate nodes
λE=0.0001,Cooperative Jamming
λE=0.0002, Cooperative Jamming
λE=0.0001, Simulation
λE=0.0002, Simulation
λE=0.0001, Smart Jamming
λE=0.0002, Smart Jamming
(a) The limiting secure communication probabilityvs. λ, where R
= 100m, α = 4, γe = −10dB,and γl = 3dB.
0 200 400 600 800 1000
0.5
0.6
0.7
0.8
0.9
1
dSD
(m) − Distance between S and D
Sec
ure
Con
nect
ion
Pro
babi
lity
λE=0.0001, Cooperative Jamming
λE=0.0005, Cooperative Jamming
λE=0.0001, Smart Jamming
λE=0.0005, Smart Jamming
(b) The limiting secure communication probabilityvs. dSD , where
R = 100 m, λ = 0.001, α = 4,γe = −3 dB, and γl = 10 dB.
Fig. 11. Illustration of the performance advantages of
cooperative jammingstrategy over smart jamming strategy.
λ = 0.004. Furthermore, the Monte Carlo simulation resultsappear
to match nicely with the analytical ones, which furthercorroborates
our stochastic geometrical derivations. Similaradvantages can also
be observed from Fig. 11(b), wherethe secure communication
probability of cooperative jammingstrategy converges to 1 at around
dSD ≥ 300 m.
V. CONCLUSIONS
In this paper, we have investigated the physical-layer secu-rity
challenges presented by the eavesdroppers which are lo-cated within
close proximity of the source and the destination,and devised
divide-and-conquer based jamming strategies toaddress them.
Analysis and simulation results indicated thatthe strategies can
improve the secrecy performances undervarious network settings. As
one of the future works, we willput constraints on the node
transmit power.
VI. ACKNOWLEDGMENTS
We would like to thank the anonymous reviewers fortheir
insightful comments. This work was sponsored in partby Key Program
of NSFC-Guangdong Union Foundation ofChina (U1135002), Key Program
of the National NaturalScience Foundation of China (U1405255), and
the NationalNatural Science Foundations of China (61372073,
61373043,61173135, 61472367, 61432015, and 61571370).
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