Secure communications with cooperative jamming: Optimal power allocation and secrecy outage analysis

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by  Kanapathippillai Cumanan

1
Secure Communications with Cooperative Jamming:
Optimal Power Allocation and Secrecy Outage
Analysis
Kanapathippillai Cumanan, Member, IEEE, George C. Alexandropoulos, Senior Member, IEEE,
Zhiguo Ding, Senior Member, IEEE, and George K. Karagiannidis Fellow, IEEE
Abstract—This paper studies the secrecy rate maximization In order to circumvent the performance limitations in-
7 problem of a secure wireless communication system, in the troduced by the unfavourable wireless channel conditions,
presence of multiple eavesdroppers. The security of the com-
1 cooperative jamming has been proposed as an enabler of
munication link is enhanced through cooperative jamming, with
0 secrecy communication [5]–[14]. Under this approach, jam-
the help of multiple jammers. First, a feasibility condition is
2
derived to achieve a positive secrecy rate at the destination. ming signals are transmitted to improve the secrecy rate

n Then, we solve the original secrecy rate maximization problem, performance, by introducing interference at the eavesdrop-
a whichisnotconvexintermsofpowerallocationatthejammers. pers. In [15], different secrecy rate optimization problems
J To circumvent this non-convexity, the achievable secrecy rate
have been solved for relay network based on cooperative
2 is approximated for a given power allocation at the jammers
jamming, where the relays transmit noise to confound the
1 and the approximated problem is formulated into a geometric
programming one. Based on this approximation, an iterative eavesdroppers. However, these optimization problems have

] algorithm has been developed to obtain the optimal power been considered with a total relay power constraint. For
T allocationatthejammers.Next,weprovideabisectionapproach, the same network, a cooperative jamming scheme has been
I basedonone-dimensionalsearch,tovalidatetheoptimalityofthe
proposedin[16]withnointerferenceleakagetothelegitimate
.
s proposed algorithm. In addition, by assuming Rayleigh fading,
user. Furthermore,in [17], opportunisticcooperativejamming
c thesecrecyoutageprobability(SOP)oftheproposedcooperative
[ jamming scheme is analyzed. More specifically, a single-integral and relay chatting schemes have been developed, without the

form expression for SOP is derived for the most general case knowledge of eavesdropper channel state information (CSI),
1
as well as a closed-form expression for the special case of two and the performance of these schemes have been evaluated
v
cooperative jammers and one eavesdropper. Simulation results
3 throughthesecrecyoutageprobability(SOP)criterion.Onthe
havebeenprovidedtovalidatetheconvergenceandtheoptimality
8 other hand, in [18], an uncoordinated cooperative jamming
of the proposed algorithm as well as the theoretical derivations
3 of the presented SOP analysis. scheme with multi-antenna relays has been investigated by
3
nulling the interference leakage at the destination and the
0
corresponding SOP has been quantified with eavesdroppers’
.
1 I. INTRODUCTION statistical CSI. In [19], optimal cooperative jamming scheme
0 P
has been proposed with multiple relays in the presence of a
7 hysical (PHY) layer security has recently received con-
single eavesdropper,wherethe optimalrelaycoefficientshave
1
siderable attention as a significant candidate to enhance the
: been obtained through an one-dimensional search scheme.
v quality of secure communication in emerging and future
The SOP of a multi-user wireless communication system,
i
X wirelessnetworks,includingthefifthgeneration(5G)standard that consists of multiple users who transmit to a base station,
[1]. In this new paradigm, the propagation characteristics of
r while multiple eavesdroppers attempt to tap their transmis-
a wireless channels are exploited against passive eavesdroppers
sions, has been analyzed over Rayleigh fading channels in
and active attacks through PHY layer secret key generation
[20].In[21],aclosed-formexpressionofSOPwasderivedfor
andauthenticationschemes,whilecomplementingtheconven-
Rayleigh fading channels in a secrecy network with a multi-
tional cryptographic methods [2]. The fundamental concept
antennasourceandasingle-antennadestinationinthepresence
of information-theoretic security was first investigated in [3]
of a single-antenna eavesdropper. Finally, in [22], the SOP
and [4], where it was shown that secure communication is
performance of the multiple-input multiple-output (MIMO)
feasiblewhenthechannelqualityoflegitimatepartiesisbetter
wiretap channel, employing transmit antenna selection and
thanthatofthe eavesdropper.However,inpractice,thisis not
receive generalized selection combining, has been analyzed
alwayspossibleandso,theperformanceofPHYlayersecurity
over Nakagami-m fading channels.
is limited.
In this paper, we consider a PHY layer security network
with single-antenna nodes, where a source-destination pair
K. Cumanan is with the Department of Electronics, University of York,
York,YO105DD,UK.(Email:[email protected]). establishes secured communication, with the help of multiple
G. C. Alexandropoulos is with the Mathematical and Algorithmic Sci- jammers in the presence of multiple eavesdroppers. For this
ences Lab, France Research Center, Huawei Technologies Co. Ltd., 92100
network setup, we first present a feasibility condition to
Boulogne-Billancourt, France(Email:[email protected]).
Z.DingiswiththeSchoolofComputingandCommunications, Lancaster achieve a positive secrecy rate at the destination. Then, the
University Lancaster, LA14WA,UK.(Email:[email protected]). secrecy rate maximization problem is solved to determine the
G. K. Karagiannidis is with Khalifa University, Abu Dhabi, UAE and
optimal power allocation at the jammers, which is a non-
with Aristotle University of Thessaloniki, Thessaloniki, Greece. (Email:
[email protected]). convex problem in nature. In order to overcome the non-
2
h
D N cooperative jammers, J1,J2,...,JN, in the presence of
(cid:1) hEM hE1 gD(1) (cid:2) tMranesamvietssdercouprpeedrsi,nEfo1r,mEa2ti,o.n..t,oEdMes.tiTnahteiosnouDrc.eItSiswaissshuemsetod
that all network nodes are equipped with a single antenna.
The channel coefficient between S and D is denoted by
(cid:3)(cid:0) (cid:5)(cid:0)
h , whereas h represents the channel gain between S
D Em
g(N) and the mth eavesdropper Em, with m = 1,2,...,M. In
D
g(N) addition, the channel coefficient between the nth cooperative
E1
jammer J and D as well as E are denoted by g(n) and
n m D
(cid:3)(cid:4) gE(NM) (cid:5)(cid:6) gE(nm), respectively. The CSI between all nodes are assumed
Fig. 1: The considered secrecy network with one source, one to be perfectly available at S, D and Em ∀m. The source S
destination and multiple jammers, in the presence of multiple transmits signals to destination D whereas all jammers send
eavesdroppers. interference signals to confound the eavesdroppers.
The received signals at D and E can be mathematically
m
expressed respectively, as
convexity of the secrecy rate function, we approximate it
N
for a given power allocation at the jammers and formulate y = P h x + P g(i)x(i)+η (1)
the problem into a geometric programming one. Based on D s D s i D c D
i=1
this approximation, an iterative algorithm is developed, by p Xp
N
updatingabetterpowerallocationateachiteration.Tovalidate y = P h x + P g(i) x(i)+η (2)
Em s Em s i Em c Em
theoptimalityofthepresentedresults,weuseone-dimensional
i=1
p Xp
search based on bisection to determine the optimal power
wherex (E{|x |2}=1)andx(i) (E{|x(i)|2}=1)denotethe
allocation of the original secrecy rate maximization problem. s s c c
transmitted signal from S to D and the jamming signal from
Both the proposedand the one-dimensionalsearch algorithms
the ith jammer J , respectively. In addition, η (E{|η |2}=
yield identical results, which confirms the optimality of the i D D
σ2) and η (E{|η |2} = σ2 ) represent the noise at
proposed algorithm. Moreover, the SOP of the proposed D Em Em Em
node D and mth eavesdropper E , respectively. The power
scheme is analyzed over Rayleigh fading channels. A single- m
allocationat J and S are denotedby P andP , respectively.
integral form expression for the SOP is presented for the i i s
Assuming white Gaussian noise, the achievable secrecy rate
most general scenario, whereas a closed-form expression is
at D is defined as
derived for the special case of two cooperative jammers and
one eavesdropper. Finally, numerical and simulation results R =[log (1+γ )−log (1+γ )]+ (3)
s 2 D 2 Emax
have been provided to validate the theoretical derivations.
where γ = max{γ ,γ ,...,γ } and γ , γ are
The remainder of the paper is organized as follows. The Emax E1 E2 EM D Em
the signal-to-interference plus noise ratios (SINR) at D and
system model and the secrecy rate maximization problem
E , respectively, given by
formulationarepresentedinSectionII.Afeasibilitycondition m
to achieve positive secrecy rate is provided in Section III, P |h |2
γ = s D (4)
whereas Section IV presents an iterative approach for an D N P |g(i)|2+σ2
approximated secrecy rate maximization problem. In Sec- i=1 i D D
P |h |2
tion V, the optimality of the proposed scheme is validated γ = P s Em . (5)
through one-dimensionalsearch. The SOP analysis is derived Em Ni=1Pi|gE(im) |2+σE2m
in Section VI for Rayleigh fading channels, whereas Section For the secrecy network studied in this paper, we consider
P
VII provides numerical and simulation results to validate secrecy rate maximization with transmit power constraint. In
the performance of the proposed algorithm and the derived particular, we intend to maximize the achievable secrecy rate
theoretical SOP expressions. Finally, Section VIII concludes at the destination node D, with the available transmit power
this paper. at the source node and all N available jammers. The secrecy
Notations:Weuselower-caseboldfacelettersforvectors.(·)T rate maximization problem can be therefore formulated as
and |·| denote the transpose of a vector and absolute value
ofacomplexnumber,respectively.[x]+ representsmax{x,0} P1: max Rs
p(cid:23)0
whereas E{·}, Pr[·] and ∇(·) denote expectation, probability s.t. P ≤P¯,∀i (6)
i i
andgradientoperator,respectively.Thecumulativedistribution
function (CDF) and the probability density function (PDF) where P¯i is the maximum available transmit power at Ji and
of a random variable (RV) X are represented as FX(·) and p=[P1P2 ··· PN]T.
f (·), respectively. Ei(·) is the exponential integral [23, eq.
X
(8.211/1)]. III. FEASIBILITY CONDITIONS
FORPOSITIVE SECRECY RATE
II. SYSTEMMODEL The optimization problem P1, formulated in (6), is valid
We consider a secrecy network, as shown in Fig. 1, with or worth to solve only when it is possible to achieve a
onesource,S,whichcommunicateswithadestination,D and positive secrecy rate for a given set of channels and transmit
3
N
powers at D and J s. Through verifying these feasibility
i × P |g(i) |2+σ2 , ψ(k) (12)
conditions, the source can make a decision whether to solve i Em Em Em
!
i=1 k
thesecrecyratemaximizationtoobtainapositivesecrecyrate X X
and
at the destination. Hence, we first investigate the feasibility
conditions. From (3), the following conditions need to be N
Φ (p) , P |g(i) |2+σ2 +P |h |2
satisfied for m=1,2,...,M: Em i Em Em s Em
!
i=1
P |h |2 P |h |2 X
s D > s Em (7) N
Ni=1Pi|gD(i)|2+σD2 Ni=1Pi|gE(im) |2+σE2m × Pi|gD(i)|2+σD2 . (13)
!
i=1
By arrangingthe terms in (7), the followingequality needs to X
P P
hold ∀m: In (12), ψ(k) represents the individual term in the summa-
Em
N N tion,obtainedby expandingfunctionΨEm(p). Theconstraint
|h |2 P |g(i) |2+σ2 >|h |2 P |g(i)|2+σ2 in(11)isaquadraticfractionalnon-convexfunction.However,
D i Em Em Em i D D
i=1 ! i=1 ! theproblemin(11)canbeconvertedintoaseriesofgeometric
X X
which can be expressed as programmingproblems by exploiting the single condensation
method [26]. A fractional constraint with a posynomial nu-
pT |h |2g −|h |2g >|h |2σ2 −|h |2σ2 (8)
D Em Em D Em D D Em merator and a monomial denominator is convex. The idea of
where (cid:0) (cid:1) approximating the denominator posynomial with a monomial
was presented in [24] in order to convert the aforementioned
T
gEm = |gE(1m) |2 |gE(2m) |2 ··· |gE(Nm)|2 constraintto a convexone.We hereinafteradoptthis idea and
h Ti we approximate ΨEm(p) (i.e., denominator of the constraint
gD = |gD(1)|2 |gD(2)|2 ··· |gD(N)|2 (9) in (11)) to the best monomial, for a given set of p. The
h i following lemma is required:
The feasibility conditionsgiven by (8) can be formulatedinto
Lemma 1: For a posynomialg(x), the followinginequality
the following linear programming problem [24]:
holds:
min 1Tp
p(cid:23)0 g(x)= K w (x)≥gˆ(xˆ)= K wk(x) ak (14)
s.t. pT |h |2g −|h |2g >|h |2σ2 −|h |2σ2 , k a
D Em Em D Em D D Em k=1 k=1(cid:20) k (cid:21)
X Y
∀m. (10)
(cid:0) (cid:1) where a > 0 and K a = 1. Notation gˆ(xˆ) represents
k k=1 k
Theaboveconvexproblemcanbeeasilysolvedusingexisting the best approximation of g(xˆ) at xˆ with a = w (xˆ)/g(xˆ),
k k
convex optimization software [24], [25]. A positive secrecy and the inequality inP(14) holds with an equality at this point.
rate can be only achieved at the destination node, if the Proof: The proof is provided in Appendix A.
problemin (10) is feasible. In the followingsection, we solve
Based on Lemma 1, the denominator polynomial function
the secrecy rate maximization problem, with the assumption Ψ (p) in (11), can be approximated as Ψˆ (p)
Em Em
that a positive secrecy rate is achievable.
K ψ(k) αk(m)
Ψ (p)≈Ψˆ (p), Em (15)
Em Em (m)
"α #
IV. AN ITERATIVEAPPROACH FORTHE SOLUTIONOF THE kY=1 k
SECRECY RATEMAXIMIZATION PROBLEM where
ψ(k)
α(m) , Em ∀k. (16)
The secrecy rate maximization problem P1 given by (6) k Ψˆ (p)
Em
is non-convex due to the non-convex secrecy rate function
Usingtheapproximationgivenby(15),theproblemP2can
and therefore it is challenging to obtain the optimal solution.
be reformulated for a given set of power allocation p as
In this section, we develop an iterative algorithm for the
power allocation p at the jammer nodes, that is based on an P3: min τ
p(cid:23)0,τ≥0
approximation to the original problem P1. By reformulating
Φ (p)
(6)andintroducinganewslackvariableτ,theoriginalsecrecy s.t. Γˆ (p), Em ≤τ, ∀m,
maximization problem P1 can be written as Em ΨˆEm(p)
P ≤P¯,∀i. (17)
P2: min τ i i
p(cid:23)0,τ≥0 The above optimization problem P3, which is an approx-
Φ (p)
s.t. Γ (p), Em ≤τ,∀m imation of the original P1, can be now formulated into a
Em ΨEm(p) standard geometric programmingone. The iterative algorithm
Pi ≤P¯i,∀i. (11) AisdevelopedforP3,wherethepowerallocationpisupdated
at each iteration.
where
The solution of the proposed Algorithm A satisfies the
N
Ψ (p), P |g(i)|2+P |h |2+σ2 Karush-Kuhn-Tucker(KKT)conditions.Thiscanbevalidated
Em i D s D D
! by proving the following three conditions [27]:
i=1
X
4
Algorithm A: Secrecy Rate Maximization previousiteration.Hence,p˜isalwaysafeasiblesolutionofthe
next iteration, and the optimal power allocation p∗ obtained
Step 1: Initialization of power allocation vector p for a given p˜ will achieve a secrecy rate, which is greater
Step 2: Repeat than or equal to that of the previous iteration. This reveals
1) Calculate Ψ (p), ∀m using (12). that the achieved secrecy rate will monotonically increase at
Em
2) Calculate α(m), ∀k, m using (16). eachiteration,whichcanbealsoobservedfromthesimulation
k
3) Determine Ψˆ (p), ∀m by using (15). results, presented in Fig. 2. Since, the achievable secrecy rate
Em
is upper bounded for a given transmit power at the jammers,
4) Solve the standard geometric programming problem in
this algorithm will converge to a solution. Fortunately, the
(17).
proposed Algorithm A converges to the optimal solution,
Step 3: Until required accuracy is achieved or the maximum
which is validated through an one-dimensional search, based
number of iterations is reached.
on bisection and provided in the following section.
1) Γ (p) ≤ Γˆ (p), ∀ m,p, where Γ (p) = V. OPTIMALITY VALIDATION OFTHE SECRECY RATE
Em Em Em
ΦEm(p). MAXIMIZATION ALGORITHM
ΨEm(p)
2) Γ (p˜) =Γˆ (p˜), ∀ m, where p˜ denotes the power In this section, we present an one-dimensional search ap-
Em Em
allocation obtained from the previousiteration of Algo- proachtovalidatetheoptimalityoftheproposedalgorithmA.
rithm A. The concept behind this approach is to fix the received total
3) ∇Γ (p˜)=∇Γˆ (p˜),∀ m. interferencepoweratthedestinationnodeandfindtheoptimal
Em Em
The first condition holds due to the fact that Ψ (p) ≤ power allocation at the jammers [19], [28]. The secrecy rate
Em
Ψˆ (p),whichistruefromLemma1.Inaddition,thesecond maximizationproblemP1canbeformulatedintothefollowing
Em
condition is satisfied from the equality condition in Lemma max-min one:
1. The third condition can be validated through proving
P4:R∗ =maxmin (t ,t ,...,t )
∇ΨˆEm(p˜)=∇ΨEm(p˜) for all m: p ti 1 2 M
∇ΨˆEm(p˜)="∂Ψˆ∂EPm1(p˜)(cid:12)(cid:12)(cid:12)P˜1∂Ψˆ∂EPm2(p˜)(cid:12)(cid:12)(cid:12)P˜·2··∂Ψˆ∂EPmN∀(p˜m),(cid:12)(cid:12)(cid:12)P˜(N1#8,) s.t. log211++PPNi=Ni=11PPPPsis||igh||hE(gE1DD(m)mi)||2||222++σσE2D2m ≥ tm,∀m
(cid:12) (cid:12) (cid:12)  P ≤ P¯i,∀i (20)
∂Ψˆ ψ(k) αk(m) ρ(k) i
Em = Em k Em where R∗ is the optimal achieved secrecy rate. By fixing
∂P1 (cid:12)(cid:12)P1=P˜1 Yk "α(km)# "PP1ΨˆEm(p˜)# the total received interference (i.e., Ni=1Pi|gD(i)|2) at the
(cid:12)(cid:12) = ΨˆEm(p˜) Pkαk(m) P Ψˆkρ(Ek(m)p˜) cdaenstibneatfioornmtuolaatepdaratisc:ular value t0, thePfollowing subproblem
P1 Em
h ρ(k) i ∂Ψ P5: q∗ =max t
= k Em = Em (19) p,t
PP1 ∂P1 (cid:12)(cid:12)P1=P˜1 s.t. N P |g(i)|2 =t ,
where ρ(k) are the differentiated ψ(cid:12)(k)’s with respect to i D 0
Em (cid:12)Em i=1
P1. Similarly, the rest of the partial derivatives in (18) can X 1+ Ps|hD|2
be derived and it can be easily proved to be equal to the R (t )= t0+σD2 ≥t,∀m
partialderivativesofΨEm(p˜),withrespecttothecorrespond- Em 0 1+ Ps|hEm|2
ing power allocation. Hence, the power allocation obtained fm(t0)+σE2m
P ≤P¯i,∀i (21)
through Algorithm A satisfies the KKT conditions of the i
original optimization problem P1. However, it is difficult to wheref (t )= N P |g(1) |2.Nextweshowthattheprob-
m 0 i=1 i Em
analyticallyproveglobaloptimality.Inaddition,thegeometric lem in (21) is quasi-convex in terms of t , and therefore, the
0
programming in Algorithm A can be solved with polynomial optimal t0 can bePobtained through one-dimensionalsearch.
time complexity. In order to validate the convergence of the
proposed algorithm, simulation results will be provided in Lemma 2: REm(t0)isaquasi-concavefunctionintermsof
t .
Section VII for different sets of wireless channels. 0
Proof: This can be proved by finding the second deriva-
tive of R (t ) with respect to t and easily providedthat it
A. Convergence Analysis Em 0 0
is negative for any t >0 [28].
0
The approximated secrecy rate maximization problem P3
given by (17) is convex, and the optimal power allocation Inaddition,thepoint-wiseinfimumofasetofquasi-concave
p∗ can be obtained by solving (17) for a given set of power functions is quasi-concave [24]. Therefore, the problem P5
allocation p˜. At each iteration, the power allocation p˜ is givenby(21)isquasi-convexandtheoptimalpowerallocation
updated from the optimal solution p∗ determined through the at the jammers can be obtained through Algorithm B.
5
Algorithm B: One-Dimensional Search Based on Bisection be shown that the CDF of γ is given by
D
∞
Step 1: Initialize t(0min),t(0max) and ǫ FγD(x)= Fz(xw)fy w−σD2 dw
Step 2: Solve the problem in P5 given by (21) with t0 = ZσD2 (cid:0) (cid:1)
t(0min)+43t(0max). =1− N Anexp σD2 ∞exp − x + 1 w dw
SStteepp 43:: RSeetpet∗at=t. nX=1 (cid:18)Pn(cid:19)ZσD2 (cid:20) (cid:18)Ps Pn(cid:19) (cid:21)
1) t = t(0min)+t(0max). (=a)1−Psexp −σD2x N AnPn
2) S0olvethep2roblemP5givenby(21)andobtainthevalue (cid:18) Ps (cid:19)nX=1Pnx+Ps
(24)
of t
3) If t∗ >t where (a) follows after using [23, eq. (3.381/3)] and the
4) t(min) = t(0min)+t(0max) definition
0 2
5) else −1
N
6) t(0max) = t(0min)+2t(0max) An ,Pn 1− PPj  . (25)
7) end j=1,j6=n(cid:18) n(cid:19)
Y
Step 5: Repeat until t(max)−t(min) ≥ǫ. By differentiating (24), the PDF of γ is easily derived as
0 0 D
σ2x
f (x)=exp − D
γD P
(cid:18) s (cid:19)
(26)
N σ2 P P
× A P D + s n .
n=1 n n"Pnx+Ps (Pnx+Ps)2#
X
VI. SOP ANALYSIS OVERRAYLEIGH FADING CHANNELS A closed-formexpressionfor the CDF of γ can be easily
Emax
obtainedusing the marginalCDFs of γ ∀ i andthe factthat
Ei
these RVs are independent. In particular, the latter CDFs are
derived in closed form similar to the CDF of γ and each is
D
givenby (24) after substituting σ2 with σ2 . Hence, the CDF
D Ei
of γ can be expressed as
Emax
M σ2 x N A P
In this section, we analyze the SOP performance of the F (x)= 1−P exp − Ei n n .
proposed cooperative jamming scheme over Rayleigh fading γEmax i=1" s (cid:18) Ps (cid:19)n=1Pnx+Ps#
channels. In particular, for the system model presented in Y X (27)
Sec. II, we assume that h as well as g(n) ∀ n=1,2,...,N
D D
and γ ∀ i = 1,2,...,M are standard circularly-symmetric
Ei
complex Gaussian RVs.
By using the SOP definition of [29], the SOP of the
proposed cooperative jamming scheme can be obtained as
γ +1
P =Pr log D <R γ >γ By substituting (26) and (27) into (23), an analytical ex-
out 2 γ +1 D Emax (22)
(cid:20) Emax (cid:12) (cid:21) pression in the form of a single integral for the SOP of the
×Pr[γD >γEmax]+Pr(cid:12)(cid:12)[γD ≤γEmax] proposed PHY-layer security scheme can be obtained as
where R denotes the rate in bits per second (bps) per Hertz. σ2µν
With the utilization of the auxiliary positive real parameter Pout =1−µexp D Y (28)
P
µ , 2R and the negative real parameter ν , 2−R−1, (22) (cid:18) s (cid:19)
where integral Y is given by
can be rewritten, as shown in Appendix B, as
Pout =1−Pr(cid:20)γEmax < γµD +ν(cid:21) (23) Y =Z0∞(iM=1"1−Psexp(cid:18)−σPE2isx(cid:19)nN=1x+Anλn#)
∞ Y X
=1−µ F (x)f (µx−µν)dx. N A σ2 P
γEmax γD ×exp(−ξx) n D + s dx
In order to solvZe0 the integral in (23), we first derive n=1 µ "x−κn µ(x−κn)2#
X
(29)
a closed-form expression for the PDF of γ as follows.
D
Since z , Ps|hD|2 is an exponentially distributed RV and with ξ , Ps−1σD2µ as well as, for n = 1,2,...,N, κn ,
y , N P |g(n)|2 is a generalized chi-squared one, by P /(µP ) − ν and λ , P /P . By using the closed-
n=1 n Ei s n n s n
obtainingtheCDFofz andthePDFofy byeasilyintegrating form solution for Y included in Appendix C, a closed-form
P
[30, eq. (2.7)]and from [31, eq. (19)]for distinct P ’s, it can expression for the SOP of the proposed scheme for arbitrary
n
6
positive integer values of N and M is given by noise variances at the destination and the eavesdroppers are
assumed to be 0.1.
N
A
P =1−µexp(ξν) n σ2I (ξ,κ ,0) Toassesstheconvergenceoftheproposedsecrecyratemax-
out ( µ D 1,0 n imization algorithm, the available maximum transmit powers
n=1
+PsI (ξ,κ ,0) + X Pi(cid:2) i! at the source and relays have been set to, Ps = 2, P1 = 1,
µ 2,0 n s N k ! P2 = 1 and P3 = 3. Fig. 2 depicts the convergence of the
(cid:21) {αXi}Mi=1 k1+k2+X···+kN=i n=1 n achievablesecrecyratesforasetofdifferentfeasiblechannels.
N N A Q As it is evident from this figure, the proposed algorithm
× Aktt! µn σD2I1,{kn}Nn=1 ψi,κn,{λn}Nn=1 converges, while the achievable secrecy rates increase with
tY=1 nX=1 h (cid:0) (cid:1) the iteration number. In addition, it has been observed that
+PsI ψ ,κ ,{λ }N theproposedAlgorithmAconvergesto thesame secrecyrate,
µ 2,{kn}Nn=1 i n n n=1 withdifferentinitializationoftransmitpowersatthejammers.
(cid:21)(cid:27)
(cid:0) (cid:1) (30) However,we couldnotprovideanalyticalresultstoprovethis
convergence.As we discussed in the convergence analysis of
where symbol is used for short-
hand representation {αio}fMi=1 the multiple summation the algorithm, it can be observed that the achievable secrecy
M M−i+1 M−Pi+2 ··· M and the sum rate monotonically increases with the iteration number.
i=1 α1=1 α2=α1+1 αi=αi−1+1 Next, we compare the performance of the proposed algo-
is taken over all combinations of
Pk1+kP2+···+kN=Pi P rithm with the existing scheme in [19] and the best jammer
nonnegative integer indices k through k such that the sum
1 N
oPf all k is i. Moreover, I α ,α ,{α }N is selection scheme. The cooperative jamming scheme in [19]
n ℓ,{kn}Nn=1 1 2 3,n n=1 has been developed using both convex optimization approach
given by (C.8) for ℓ = 1,2 as well a(cid:16)s for kn being pos(cid:17)itive and one dimensional search scheme in the presence of a
integer and α , α , α ∈ R∗ ∀ n = 1,2,...,N. As an
1 2 3,n + single eavesdropperwhereasthe best jammeris selected from
example, for the special case of N =2 and M =1, the latter
available cooperative jammers in the best jammer selection
SOP expression simplifies to
scheme.Inordertoevaluatethiscomparison,thesamesecrecy
P =1−µexp(ξν) networkintheprevioussimulationisconsideredwitha single
out
2 eavesdropper and with the same noise variance 0.1 at all the
A P
× n σ2I (ξ,κ ,0)+ sI (ξ,κ ,0) nodes. Fig. 3 depicts the achieved secrecy rates for different
µ D 1,0 n µ 2,0 n
(n=1 (cid:20) (cid:21) available transmit power at the source and the cooperative
X
2 P A2 P jammers for different sets of channels, where it is assumed
− sµ n σD2I1,1(ψ,κn,λn)+ µsI2,1(ψ,κn,λn) that the maximum available transmit power at the source and
n=1 (cid:20) (cid:21) the cooperativejammersare the same. Asseen in Fig. 3, both
X
− PsA1A2σD2 [I (ψ,κ ,λ )+I (ψ,κ ,λ )] the proposed algorithm and the scheme in [19] achieve the
1,1 1 2 1,1 2 1
µ samesecrecyratesfordifferentsetsofchannelswiththesame
P2A A transmit power constraints and better secrecy rates than the
− s 1 2 [I (ψ,κ ,λ )+I (ψ,κ ,λ )]
µ2 2,1 1 2 2,1 2 1 bestjammerselectionscheme.Thisconfirmsthattheproposed
(cid:27)
(31) algorithmshowsthesame performanceastheoptimalscheme
in [19] and outperforms the best jammer selection scheme.
where ψ ,ξ+P−1σ2 ,
s E1 Next, we evaluate the optimality of the power allocation
I (ξ,κ ,0)=−exp(ξκ )Ei(−ξκ ), (32a) obtained through the proposed Algorithm A. In order to do
1,0 n n n
this, we simulate Algorithm B for the same set of channels
I (ξ,κ ,0)=κ−1+ξexp(ξκ )Ei(−ξκ ), (32b)
2,0 n n n n considered for Algorithm A. Table I presents the power
and allocationandthesecrecyratesobtainedthroughAlgorithmB
thatis basedonone-dimensionalsearchandontheAlgorithm
I (ψ,κ )−I (ψ,λ )
I1,1(ψ,κn,λn)= 1,0 n 1,0 n , (32c) A. As we can conclude from this table, the power allocation
λ −κ
n n
and achieved secrecy rates are identical for different sets
I (ψ,λ )−I (ψ,κ ) I (ψ,κ )
I (ψ,κ ,λ )= 1,0 n 1,0 n − 2,0 n . of channels in both algorithms. Note that there are small
2,1 n n (κn−λn)2 κn−λn differencesinthepowerallocationandachievedsecrecyrates,
(32d)
due to the accuracy or precision of software used. However,
these results providedin Table I confirm the optimality of the
proposed secrecy rate maximization Algorithm A.
VII. NUMERICAL RESULTSAND DISCUSSIONS By numerically evaluating (31), Fig. 4 depicts SOP per-
In order to validate the performance of the proposed al- formance as a function of rate R in bps per Hertz for
gorithms, we consider the secrecy network shown in Fig. 1, N = 2 cooperative jammers, M = 1 eavesdropper and
with a source-destination pair, three (N = 3) cooperative various power levels. It is shown in this figure that computer
jammers and two (M = 2) eavesdroppers. In the following simulationresultsforSOPmatchperfectlywiththeequivalent
simulations, all the channel coefficients involved are gener- numerical ones, for all considered parameters. As expected,
ated using zero-mean circularly symmetric independent and SOP degrades with increasing values for R. In addition, as
identicallydistributedcomplexGaussianRVs. Inaddition,the the transmit power of the source S increases and the transmit
7
Algorithm B Algorithm A
Achieved Achieved
Channels P1 P2 P3 Secrecy Rate P1 P2 P3 Secrecy Rate
1 1.00 0 0.50 1.62 1.00 0 0.50 1.62
2 1.00 0 0.17 2.98 1.00 0 0.16 2.98
3 0.47 0 0.35 1.68 0.46 0 0.34 1.68
4 1.00 0.43 0 2.72 1.00 0.42 0 2.73
5 0 0.28 0.31 1.09 0 0.28 0.31 1.09
TABLE I: The optimal power allocation at the jammers based on Algorithm A and Algorithm B, for different sets of
wireless channels.
powers at the two cooperative jammers J and J decrease,
1 2
6
SOP improves. The best SOP performance in this figure for
Channel 1
all consideredR valuesis achievedwith P =15 dB, P =0 Channel 2
s 1
5 Channel 3
dB and P =2 dB, and the lower value for SOP is 0.5.
2 Channel 4
The SOP performance as a function of source S’s transmit Channel 5
power P is illustrated in Fig. 5. The following transmission 4
s
e
scenarios have been considered: i) Scenario 1: R = 1, Rat
Pan1d=4;−an4ddBii)aSncdenPairio=2P: 1R+=(i −0.011), dPB1 w=ith1id=B 2an,3d Secrecy 3
P = P + (i − 1) dB with i = 2,3 and 4. For the 2
i 1
SOP results, the single-integralexpressiongiven by (28) after
substituting(29)andtheclosed-formexpressiongivenby(30) 1
for arbitrary values of N and M as well as the closed-form
expression given by (31) for M = 2 and N = 1 have been 0
1 2 3 4 5 6 7 8 9 10
numericallyevaluated.Asclearlyshown,computersimulation Iteration Number
results for SOP coincide with the numerical ones, for all
Fig. 2: The convergence of the proposed secrecy rate
considered parameters. Furthermore, it is evident that, for the
maximization Algorithm A, for different sets of wireless
same values of N and M, the SOP performance of Scenario
channels.
2 is always better than that of Scenario 1. In both scenarios,
the minimum SOP is accomplished with N = M = 1 and
the maximum with N = M = 4. Also, as expected, SOP
improveswithincreasingvaluesofPs forallconsideredcases. 6 Prop Ch1
In addition, it is shown in this figure that, as M increases Ext Ch1
Prop Ch2
while N is kept constant, SOP degrades significantly. This
5 Ext Ch2
performancedegradationcan beconfrontedfor somerangeof Prop Ch3
P valuesbyincreasingN.However,increasingN introduces ate Ext Ch3
asSOP performance penalty, that needs to be taken under crecy r4 PErxot pC Ch4h4
considerationwhendesigninga cooperatingjammingscheme. se Prop Ch5
ved 3 Ext Ch5
e
hi
c
A
VIII. CONCLUSIONS 2
In this paper, we studied the power allocation problem
of secrecy rate maximization with cooperative jammers, in
1
the presence of multiple eavesdroppers. For this problem, a

feasibility condition was first derived for power allocation in 2 4 6 8 10 12
Power
order to achieve positive secrecy rate. Then, the original non-
convex secrecy rate maximization problem was solved to ob- Fig. 3: The achieved secrecy rates of Algorithm A, the
taintheoptimalpowerallocationatthejammers.Theproposed scheme in [19] and best jammer selection scheme for five
optimal iterative approach was developed by approximating sets of different wireless channels with different maximum
the secrecy rate function and formulating the corresponding available transmit power. The dotted lines denote the best
problem into a geometric programming problem for a given jammer selection scheme.
set of power allocation at the jammers. In order to validate
the optimality of the developed algorithm, we also developed
an one-dimensional search algorithm based on bisection. In and convergence of the proposed algorithm as well as the
addition, the SOP analysis of the proposed cooperative jam- theoretical derivation of SOP analysis. These results confirm
ming approach was derived for Rayleigh fading channels. that the proposed algorithm yields the optimal power alloca-
Simulation results were provided to validate the optimality tion at the jammers, whereas the numerical simulation results
8
1 K
= g(xˆ)PKk=1wgk(xˆ(xˆ)) =g(xˆ)
k=1
0.9 Y
y where
Probabilit0.8 a¯k = wgk((xˆxˆ)) and K wgk((xˆxˆ)) =1. (A.2)
ge Xk=1
a
ut This completes the proof of Lemma 1.
O0.7
y
c
cre Ps=5 dB, P1=-1 dB, P2=-2 dB
Se0.6 PPs==51 5d BdB, P, P1==100 ddBB,, PP2==125 d dBB DERAIVPAPTEINODNIXOFB(23)
s 1 2
P=15 dB, P=8 dB, P=10 dB
s 1 2
Simulations Starting from (22) and using the definition of conditional
0.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 probability results in
Rate in bps per Hertz
γ
D
P =1+Pr +ν <γ <γ −Pr[γ <γ ]
Fig. 4: Pout, as a function of the rate, R, in bps per Hz, for out µ Emax D Emax D
N =2 cooperative jammers, M =1 eavesdroppers and ∞(cid:20) y (cid:21)
various power levels. =1− F +ν f (y)dy.
γEmax µ γD
Z0 (cid:18) (cid:19)
(B.1)
1
By using the change of variables x → y/µ+ν and the fact
0.95
that ν <0, yields (23).
0.9
y
abilit0.85 Scenario 1 APPENDIX C
b
Pro 0.8 CLOSED-FORMSOLUTION FOR(29)
e
g0.75
a To solve integral Y given by (29) that appears in the SOP
ut
O 0.7 expressiongivenby(28),wefirstmakeuseofthemultinomial
y
crec0.65 M=1 and N=1 expansion [32, eq. (23)] for the M-factor product, yielding
Se 0.6 Scenario 2 MMM===212 aaannnddd NNN===122 M 1−Psexp −σE2ix N An
0.55 MSim=4u laantido nNs=4 Yi=1" (cid:18) Ps (cid:19)nX=1x+λn#
0.5 i N i
-10 -5 0 5 10 15 20 25 30 x A
Transmit Power in dB =1+ Psiexp−Ps σE2αj x+nλn! .
Fig. 5: SOP performance, Pout, as a function of source’s {αXi}Mi=1 Xj=1 nX=1
transmit power, P , in dB for both Scenarios 1 and 2 as well   (C.1)
s
as various numbers of cooperative jammers and Then, in the latter expression, we utilize the multinomial
eavesdroppers. theoremtoexpandtheithpoweroftheN-termsumasfollows
i
N
A i!
n
=
dSeOmPoannsatrlaytseist.he correctness of theoretical derivations of the nX=1x+λn! k1+k2+X···+kN=i Nn=1kn! (C.2)
Q N Akt
× t .
(x+λt)kt
t=1
APPENDIX A Y
Substituting(C.2)into(C.1)andtheninto(29),integralY can
PROOF OF LEMMA1
be rewritten as
Function g(x) can be written as N A P
Y = n σ2I (ξ,κ ,0) s +I (ξ,κ ,0)
wheregth(ex)in=eqkXKu=a1liatky(cid:20)inw(kaA(kx.1))(cid:21)is≥okbYK=ta1i(cid:20)newdkaf(krxo)m(cid:21)atkh=egˆa(rxˆit)hm(Aet.i1c-) nX+={1αXiµ}Mi=(cid:20)1PDsik11,+0k2+X···n+kN=µii!QNnNt=2=,101kAnkt!t n (cid:21) (C.3)
geometricmeaninequality.Thisinequalityholdswithequality
when ak = wgk((xˆxˆ)) as follows: × N Aµn σD2I1,{kn}Nn=1 ψi,Qκn,{λn}Nn=1
K w (xˆ) a¯k nX=1 h (cid:0) (cid:1)
gˆ(xˆ) = k +PsI ψ ,κ ,{λ }N
kY=1(cid:20) a¯k (cid:21) µ 2,{kn}Nn=1 i n n n=1 (cid:21)
(cid:0) (cid:1)
9
where ψi , ξ + Ps−1 ij=1σE2αj. In addition, [6] tLra.nWsmanisgs,ioMn.wiEthlkoapshtilmana,lpJ.owHeuraanlglo,cNat.ioTnriann,unatnrudstTe.dDreulaoyngn,et“wSoerckusr,”e
Ifoℓ,r{kkn}nNn=b1e(cid:16)inαg1,αpo2s,i{tiαv3e,ni}nNnt=eg1e(cid:17)r fPaonrd ℓα1=, α12,,2α3a,sn w∈ellRa∗+s [7] lIFeE.cEtAiEoln-CQfoaomhrtmasneuicn,r.eCcLy.etZet.nh,hovanonglc,.e3am,nednnotH.i3.n,Acpolpno.up2we8rea9itr–iiv2,e9“2nO,epJtwpuonor.rtku2sn0,”i1s4tIi.EcErEelaTyrasnes-.
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