On the Secrecy Capacity of Cooperative Wiretap Channel
Meysam Mirzaee, Soroush Akhlaghi Shahed University, Tehran, Iran Emails:{me.mirzaee,akhlaghi}@shahed.ac.ir
¨
Abstract—Secure communications using physical layer charac- teristics has attracted many attentions in recent years. Secrecy capacity of communication systems can be increased by using multiple-antenna nodes. In this paper, we concentrate on the cooperative wiretap channel in which confidential messages are sent from a source to a legitimate receiver with the help of a relay in the presence of an eavesdropper. We study the secrecy capacity of Amplify and Forward (AF) relaying under peak power constraint at the relay node. To this end, we first evaluate an achievable secrecy rate for Gaussian input by solving a non- convex optimization problem. Then, we prove the converse by using the secrecy capacity of genie-aided channel as an upper bound of the secrecy capacity of interested channel. We show that this upper bound is achievable and equal to achievable secrecy rate of Gaussian input.
Index Terms—Secrecy capacity, achievable secrecy rate, phys- ical layer security, cooperative wiretap channel.
I. INTRODUCTION
Enhancing security has received considerable attentions in recent years due to the nature of wireless communication in which any illegitimate receiver can hear transmitted signal.
Information theoretic security was first proposed by Shan- non in his landmark paper [1] using cryptographic approaches.
However, this approach may not be feasible for some of wire- less technologies [2]. This motivated Wyner in his pioneering work in [3] to propose using physical layer characteristics to secure the wireless communication networks.
Wyner introduced the wiretap channel and showed that when the eavesdropper’s channel is a degraded version of legitimate receiver’s channel, source can send its confidential message to legitimate receiver while the eavesdropper can not learn anything about the message. Also, Wyner defined the secrecy capacity as the maximum achievable secrecy rate, the rate below which the message can not be decoded at the eavesdropper, and investigated it for discrete memoryless wiretap channels. Also, Broadcast channels which are not necessarily degraded are studied by Csisz´ar and Korner and secrecy capacity region of them are established in [4].
One of the important issues in secure transmission is the ef- fect of channel condition. For instance, when the destination’s channel strength is weaker than eavesdropper’s, no confidential message can be transmitted. Using multiple antennas can alleviate this issue [5]–[7].
Limitation in cost and size motivated researchers to propose cooperative communications to benefit advantages of multiple antenna systems by single antenna nodes [8]–[10]. Two major
strategies of Cooperative Communication are Amplify and Forward (AF) and Decode and Forward (DF).
The AF strategy has considerable lower complexity than DF. Also, in some applications, the relay nodes may have low security level, named as untrusted relays, and transmitted message should be confidential for them [11], [12]. In this case, the AF strategy is more applicable than DF since the relay can not decode the received signal and do not have access to its information bits.
Physical layer security issues in cooperative communication networks have attracted a great deal of attentions in recent literature and it is demonstrated that the achievable secrecy rate can be improved by relaying [13], [14].
In this paper, we focus on the derivation of secrecy capacity for a simple cooperative wiretap channel in which transmission of confidential messages from source to legitimate destination with the help of an AF untrusted relay is desirable such that the eavesdropper is kept as ignorant of the secret messages as possible. Referring to Fig. 1, the received signal at the destination is a degraded version of the relay’s and so the DF strategy is optimal. On the other hand, because of using untrusted relay, one may want to keep the relay unaware of incorporated transmission codebooks and thus the AF strategy should be employed at this scenario. In this regard, the secrecy capacity is fully characterized. To this end, we first explore the achievable secrecy rate of Gaussian input. Then, it is shown that any rate greater than this achievable rate can not be attained. Accordingly, the achievable rate of AF relaying is compared to that of DF to get an indication regarding the rate loss due to using untrusted relays.
II. SYSTEMMODEL
Consider a wireless communication system that consists of single antenna nodes S, R, D and E, denoting the source, relay, destination and passive eavesdropper, respectively (see Fig. 1).
Also, it is assumed that data transmission is taken over two hops with the help of a relay and there is no direct link from S to D and E. All channels are assumed as quasi-static fading channels and statistically independent. Moreover, we suppose that the channel gains from R to D and E are completely known at R, in addition to the source-to-relay channel gain.
This is a common assumption in some of related works such as [15].
According to the system model illustrated in Fig. 1, in the first hop of transmission, S sends the message W, which has
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2013 Iran Workshop on Communication and Information Theory (IWCIT)
uniform distribution in the alphabet W = {1,2, ...,2nR}, to the relay inntransmission intervals, whereRandnRindicate, respectively, the transmission rate of source and the entropy of message. An encoder fn : W → χns, where χns is the transmitted vector space, maps the messageW to a codeword xns ∈χns. In one time slot, each source symbolxs(t)has zero mean and unit power. The received signal at the relay can be computed as,
yr(t) =p
Pshrxs(t) +zr(t) for t= 1, . . . , n , (1) where hr is the channel fading coefficient from source to the relay, zr is zero-mean Additive White Gaussian Noise (AWGN) at R with unit power. Also,Psindicates the transmit power per symbol.
In the next hop, the relay sends a variation of its received signal to D as well as E, depending on the incorporated strategy. We investigate, respectively, AF and DF strategies in the following subsections.
A. Amplify and Forward
In AF relaying, the relay transmits a scaled version of its received signal, i.e., yr, to the destination as follows1,
xr=ωyr , (2)
where xr is the transmitted signal and ω is a scaling factor.
So, the received signals at the nodes D and E can be written, respectively, as,
yd=hdxr+zd=p
Pshdωhrxs+hdωzr+zd , (3) ye=hexr+ze=p
Psheωhrxs+heωzr+ze , (4) wherehd andheare channel fading coefficients from R to D and E, respectively. Also, zd∼ CN(0,1) andze∼ CN(0,1) are additive white Gaussian noises at the nodes D and E, respectively, where CN(0, σ2) means a zero-mean circularly symmetric complex Gaussian distribution with varianceσ2. B. Decode and Forward
In DF strategy, the relay attempts to decode the source mes- sage and re-encodes the estimated messageW to a codeword xnr ∈χnr by an encodergn :W →χnr. According to channel coding theorem, for large values ofn, the relay can decode the information signal with no error as long as the transmission rate is not greater than the capacity of source-relay channel, which is given by,
CS−R= log2 1 +Ps|hr|2
. (5)
The relay broadcasts a weighted version of re-encoded sym- bols, i.e., ωxr, to D and E. Thus, the received signals at the nodes D and E are given, respectively, by,
yd=hdωxr+zd , (6) ye=heωxr+ze . (7)
1For notational convenience, we ignore the index of symbols in the rest of paper.
h
rh
dh
eFig. 1. System Model
We assume that the relay is subject to a peak power constraint Pr, meaning E[|xr|2]≤Pr for AF strategy andE[|ωxr|2]≤ Prfor DF strategy, whereE[.]denotes expectation. As a result, the scaling factor at the relay should satisfy the following constraints,
|ω|2≤ 1+|hPr
r|2Ps for AF strategy ,
|ω|2≤ Pr for DF strategy , (8) whereE[|xr|2] = 1is assumed in DF strategy.
In the sequel, we aim to compute the secrecy capacity of this network.
III. SECRECY CAPACITY OF CHANNEL
In this section, we intend to address the secrecy capacity of considered channel for both AF and DF strategies in subsections III.A and III.B, respectively.
A. Amplify and Forward
We can consider the AF cooperative wiretap SISO channel as a degraded broadcast channel. Hence, the corresponding secrecy capacity can be computed as [16],
Cs(Pr) =
max
E{|xr|2}≤Pr
p(xs)∈ρ
1 2
hI(xs;yd)−I(xs;ye)i
+
, (9) where {x}+ is equivalent to max{0, x}, p(xs) is the Prob- ability Density Function (PDF) of xs andρ is the set of all possible PDFs with zero mean and unit variance. Also, the factor 12 is due to the two hop transmission.
Computing the secrecy capacity of interested channel using (9) may be computationally infeasible. Thus, we propose the following theorem which tends to address this issue by an indirect approach.
Theorem 1: The secrecy capacity of cooperative amplify and forward wiretap channel is given by,
Cs(Pr) =
0 α≤β
1 2log2
αβPr2+(αµ+β)Pr+µ αβPr2+(α+βµ)Pr+µ
α > βandPr≤q µ
αβ 1
2log2
2√
αβµ+αµ+β 2√
αβµ+α+βµ
α > βandPr>q µ
αβ , (10)
whereα=|hd|2, β=|he|2 andµ= 1 +Ps|hr|2.
Proof: Proof of this theorem is done in two steps. In the first step, proof of the achievability, we show that (10) is an achievable rate for Gaussian distribution using (9). In the next step, proof of the converse, we propose an upper bound and show that any transmission rate greater than (10) could not be achieved.
1) Proof of Achievability: For Gaussian input, the achiev- able secrecy rate can be evaluated as,
Rs(Pr) =
E{|xmaxr|2}≤Pr
1 2
hI(xs;yd)−I(xs;ye)i+
. (11) Thus, using (3) and noting xs∼ N(0,1), we have,
I(xs;yd) =log2
1 + Ps|hd|2|ω|2|hr|2 1 +|hd|2|ω|2
=log2
1 +αµ|ω|2 1 +α|ω|2
. (12)
where I(x;y) denotes the mutual information of random variablesxandy.
Similarly, noting (4), we arrive at the following, I(xs;ye) =log2
1 +βµ|ω|2 1 +β|ω|2
. (13)
Substituting (12) and (13) into (11), it turns out that the achievable secrecy rate becomes,
Rs(Pr) = (
max
|ω|2≤Prµ
1 2log2
αβµ|ω|4+ (αµ+β)|ω|2+ 1 αβµ|ω|4+ (α+βµ)|ω|2+ 1
)+
. (14) To find the optimal solution of (14), we should solve the following optimization problem,
max
|ω|2≤Prµ
αβµ|ω|4+ (αµ+β)|ω|2+ 1
αβµ|ω|4+ (α+βµ)|ω|2+ 1 , (15) which can be reformulated as,
maxx f(x) = αβµx2+ (αµ+β)x+ 1 αβµx2+ (α+βµ)x+ 1 ,
subject to 0≤x≤X (16)
wherex=|ω|2 andX = Pµr. The objective function of (16) is the ratio of two convex quadratic functions which is not convex in general [17]; hence, we can not employ the method of Lagrange Multipliers to get the optimal solution. To find the optimal value of x, i.e., x, we consider, separately, twoˆ possible cases of α≤β andα > β as the following.
Case α ≤ β: In this case, it is shown that the optimal solution of (16) isxˆ= 0. Noting the definition ofµ, indicating µ≥1, yields,
α(µ−1)≤β(µ−1), (17) or equivalently,
αµ+β≤α+βµ . (18)
Thus, for0< x≤X, the denominator off(x)is greater than the nominator which means f(x) < 1. On the other hand, sincef(0) = 1, the optimal value of xbecomes,
ˆ
x= 0 . (19)
Caseα > β: In this case, we show that the optimal value of (16) is given by,
ˆ x=
Pr
µ Pr≤q µ
αβ
√1
αβµ Pr>q
µ
αβ , (20)
wherexˆis derived through using the following theorem.
Theorem 2: Consider the following optimization problem, maxx∈Rnf(x) = xTQx+qTx+q0
xTPx+pTx+p0 , (21) where P and Q are n×n symmetric positive semi-definite matrices. To find the optimal solution, we define the following function,
F(x, λ) =xTQx+qTx+q0−λ(xTPx+pTx+p0), λ >0. (22) Also, we define the functions,
x(λ) =argmax
x∈RnF(x, λ)∀λ >0, (23) and
π(λ) = max
x∈RnF(x, λ) =F(x(λ), λ). (24) If there existsλ >ˆ 0 for which π(ˆλ) = 0, then ˆx ≡x(ˆλ) is the optimal solution of (21).
Proof: see [17].
According to the Theorem 2 and referring to (16), we define F(x, λ)as follows,
F(x, λ) =αβµ(1−λ)x2+
αµ+β−λ(α+βµ)
x+ 1−λ , (25) where we assume λ >0 and0≤x≤X.
Claim 1: The optimal value ofλ, i.e.,ˆλ, falls in the interval [1,αµ+βα+βµ).
Proof: see [18].
Based on claim 1, it is sufficient to merely investigate F(x, λ)for 1≤λ < αµ+βα+βµ.
Since 1−λ≤0, (25) is a concave function ofxand has two positive roots2. As a result,F(x, λ)can be represented as one of two curves depicted in Fig. 2, depending on the value ofλ. Assumingx˜maximizesF(x, λ), according to Fig. 2, we have
x(λ) =
(X X ≤x˜
˜
x X >x ,˜ (26)
2The number of positive roots of a polynomial with real coefficients ordered in terms of ascending power of the variable is either equal to the number of variations in sign of consecutive non-zero coefficients or less than this by a multiple of 2 [19].
x X x F(x,)
x
X x
F(x,)
(a) (b)
Fig. 2. Illustration of functionF(x, λ)for two possible cases
wherex, by using the first derivative of˜ F(x, λ)with respect tox, can be written as,
˜
x= λ(α+βµ)−(αµ+β)
2αβµ(1−λ) , (27)
Therefore, using (27) and claim 1, (26) can be expressed as, x(λ) =
(X 1≤λ≤2αβµX+αµ+β2αβµX+α+βµ
λ(α+βµ)−(αµ+β) 2αβµ(1−λ)
2αβµX+αµ+β
2αβµX+α+βµ < λ < αµ+βα+βµ . (28) Also, using (23) and (24), π(λ)can be obtained by,
π(λ) =F
x(λ), λ
=
(π1(λ) 1≤λ≤ 2αβµX+αµ+β2αβµX+α+βµ
π2(λ) 2αβµX+αµ+β2αβµX+α+βµ < λ < αµ+βα+βµ , (29) where
π1(λ) =
−αβµX2− α+βµ X−1
λ +αβµX2+ αµ+β
X+ 1 , (30) and
π2(λ) =
λ(α+βµ)−(αµ+β)2
4αβµ(λ−1) −λ+ 1. (31) Claim 2: λˆ can be written as,
λˆ=
λ1=αβµXαβµX22+(αµ+β)X+1+(α+βµ)X+1 X ≤√1 αβµ
λ2=2(α+βµ)(αµ+β)−8αβµ−√
∆
2(α−βµ)2 X > √1
αβµ , (32) where
∆ = (8αβµ−2(α+βµ)(αµ+β))2−4(α−βµ)2(αµ−β)2. (33) Proof: see [18].
Finally, substituting (32) into (28) yields, ˆ
x=
(X X ≤√1 αβµ λ2(α+βµ)−(αµ+β)
2αβµ(1−λ) X > √1
αβµ . (34) Also, by comparing (34) with (26), we conclude,
ˆ x=
(X X ≤√1 αβµ
√1
αβµ X > √αβµ1 , (35)
or equivalently, we have, ˆ x=
Pr
µ Pr≤q µ
αβ
√1
αβµ Pr>q µ
αβ . (36)
As a result, notingxˆ=|ωopt|2, it turns out that ifPr>q µ
αβ, all of available power at the relay isn’t used since relay sends a noisy version ofxsand additional transmit power may enhance the noise and decrease the secrecy rate.
Finally, using (19) and (36) and after some calculations, it can be observed that (10) is the achievable secrecy rate of the AF relaying for Gaussian input.
2) Proof of the Converse: For the converse part, we inves- tigate the capacity of genie-aided channel as an upper bound on the secrecy capacity of interested channel, using following lemma, to show that any rate greater than Rs(Pr), see (14), is not achievable.
Lemma 1 [5]: An upper bound on the secrecy capacity of cooperative wiretap channel is,
Cs(Pr)≤ max
p(xs)∈ρ
|ω|2≤Prµ
1
2I(xs;yd|ye). (37) We intend to show that Gaussian distribution maximizes I(xs;yd|ye). To this end, we have,
I(xs;yd|ye) =h(yd|ye)−h(yd|xs, ye). (38) The second term in the right hand side of (38) can be expressed, using (3) and (4), as,
h(yd|xs, ye) =h(p
Pshdωhrxs+hdωzr+zd|xs, ye)
=h(hdωzr+zd|xs, ye)
=h(hdωzr+zd|heωzr+ze). (39) It is observed that (39) does not depend on the distribution of xs. Thus, for maximization of (38), it is sufficient to choose p(xs) such that only the first term in the right hand side of (38), i.e.,h(yd|ye), is maximized. On the other hand, we have,
h(yd|ye)=a h(yd−αLMMSEye|ye)
≤b h(yd−αLMMSEye)
≤log2(πeλLMMSE), (40) where (a) comes from the fact that adding a known value to a random variable does not change the entropy and (b) holds since the entropy is not changed by conditioning. Moreover, αLMMSE and λLMMSE are, respectively, Linear Minimum Mean Square Error (LMMSE) estimator of yd by ye and variance of corresponding error, i.e.,E[|yd−αLMMSEye|2|ye]. The last inequality in (40) is due to the fact that the maximum differential entropy is attained by Gaussian distribution.
In the LMMSE estimation, ifydandyeare jointly Gaussian, the estimation error, i.e.,yd−αLMMSEye, is independent of every linear function ofye. This is due to the orthogonality principle [20]. Thus for Gaussian input, we haveh(yd−αLMMSEye|ye) = h(yd−αLMMSEye). Hence, the inequalities in (40) are held with
equality for Gaussian inputxsandI(xs;yd|ye)is maximized.
So, we can rewrite (37) as, Cs(Pr)≤ max
|ω|2≤Prµ
1
2I(xs;yd|ye)
= max
|ω|2≤Prµ
1
2log2(πeλLMMSE)
−1
2h(hdωzr+zd|heωzr+ze). (41) If zd andzehave following joint distribution,
zd
ze
∼ CN(0, Kφ), Kφ= 1 φ∗
φ 1
(42) then, λLMMSE is given by,
λLMMSE=1 + (α+β)µx− |φ|2−2ℜ{µxhdh∗eφ}
1 +βµx . (43)
The proof is provided in [18].
Moreover, the second term in (41) can be computed as, h(hdωzr+zd|heωzr+ze)
=h(hdωzr+zd, heωzr+ze)−h(heωzr+ze)
= log2πe 1 + (α+β)x− |φ|2−2ℜ{xhdh∗eφ}
1 +βx .
(44) The proof is given in [18].
Substituting (43) and (44) into (41) yields, Cs(Pr)≤
0max≤x≤X
1 2log2
( 1 +βx 1 +βµx
×1 + (α+β)µx− |φ|2−2ℜ{µxhdh∗eφ}
1 + (α+β)x− |φ|2−2ℜ{xhdh∗eφ}
) . (45) Kφ is covariance matrix and should be positive semi-definite.
Thus we need to have
|φ| ≤1 . (46)
Now, we compute upper bound of secrecy capacity for two casesα≤β andα > β, separately.
Caseα≤β: In this case, we choose, φ= h∗d
h∗e , |φ|2=α
β ≤1, (47)
thus (45), by using (47), is written as, Cs(Pr)≤
0max≤x≤X
1
2log2(1 +βx)(1−αβ + (β−α)µx) (1 +βµx)(1−αβ + (β−α)x)
= max
0≤x≤X
1
2log2βµ(β−α)x2+ (β−α)(µ+ 1)x+ 1−αβ βµ(β−α)x2+ (β−α)(µ+ 1)x+ 1−αβ
= 0 . (48)
This results in,
Cs(Pr) = 0. (49) Caseα > β: In this case,φ is set to,
φ= he hd
, |φ|2= β
α <1. (50) Substituting (50) into (45), we arrive at the following,
Cs(Pr)≤ max
0≤x≤X
1 2log2
(1 +βx)
1−βα+ (α−β)µx (1 +βµx)
1−βα+ (α−β)x
= max
0≤x≤X
1 2log2
1 +βx
1 +βµx×1 +αµx 1 +αx
(51)
= max
0≤x≤X
1
2log2αβµx2+ (αµ+β)x+ 1
αβµx2+ (α+βµ)x+ 1 , (52) where (51) has been proved in [18].
By comparing to (14), we conclude that (52) is actually an achievable rate. Thus, we have,
Cs(Pr) = max
0≤x≤X
1
2log2αβµx2+ (αµ+β)x+ 1
αβµx2+ (α+βµ)x+ 1 . (53) Considering the obtained results in (49) and (53), Theorem 1 is proved.
B. Decode and Forward
The secrecy capacity for DF relaying can be computed, by using max-flow min-cut theorem, as
Cs(Pr) =1 2min
CS−R, CsR−D , (54) whereCS−R is the capacity of source-to-relay andCsR−D is the secrecy capacity of the second hop that is given by [5],
CsR−D = (
max
|ω|2≤Pr p(xr)∈ρ
hI(xr;yd)−I(xr;ye)i )+
=
log2
1 +αPr
1 +βPr
+
. (55)
If CsR−D ≤CS−R, then we have Cs(Pr) = 12CsR−D, but if CS−Ris less thanCsR−D, the relay can adjust its power such that CsR−D =CS−R, i.e., it does not need to use all of its available power. According to (8), we will have3 ,
CS−R= log2
1 +α|ω|2 1 +β|ω|2
. (56)
By noting (5) and the definition ofµ, we get,
|ω|2= µ−1
α−βµ . (57)
3It is worth mentioning that1+α|ω|1+β|ω|22 is an increasing function with respect to|ω|forα > β, thus decreasing|ω|reduces the secrecy rate of the second hop.
Consequently, the secrecy capacity of DF strategy and the optimum relay’s power can be written, respectively, as,
Cs(Pr) =
0 α≤β
1 2log2
1+αPr 1+βPr
α > β and 1+αP1+βPr
r ≤µ
1
2log2µ α > β and 1+αP1+βPr
r > µ , (58) and
|ωopt|2=
0 α≤β
Pr α > β and 1+αP1+βPrr ≤µ
µ−1
α−βµ α > β and 1+αP1+βPr
r > µ .
(59)
IV. SIMULATIONRESULTS
In this section, we provide some numerical results for illustration of secrecy capacity versus the power budget for both AF and DF strategies. In these simulations, distribution of all channel coefficients are assumed to be Rayleigh. Also, all of received noises are assumed to be circularly symmetric complex Gaussian random variables with zero mean and unit variance. Moreover, we derive simulation results for different values of relay-destination channel strengthsσ2hd= 1,2,4and 8, while it is assumedσh2r =σh2e = 1. Also, source transmit power is set to Ps= 10dBW4.
In Fig. 3, the secrecy capacity of AF and DF cooperative wiretap channels versus power budget is depicted for various relay-destination channel strengths, which, as is expected, shows that the secrecy capacity of DF strategy is greater than that of AF strategy, since the received signal at the destination is a degraded version of the relay’s. Moreover, it is demonstrated that by increasing the relay-destination channel strength, more secrecy capacity is attained. Also, the secrecy capacity approaches to a constant value as the relay’s power tends to infinity. This is due to the fact that the capacity of the first hop acts as bottleneck.
V. CONCLUSION
This paper aims at comparing the secrecy capacity of cooperative wire-tap channel with the help of a relay in the middle of transmission incorporating AF and DF strategies.
Accordingly, the secrecy capacity of these two strategies are derived and numerically compared for Rayleigh channels.
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−200 −15 −10 −5 0 5 10 15 20 25 30
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Secrecy Capacity (bits/transmission)
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