1972 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 12, DECEMBER 2012
Transmission Policies for
Improving Physical Layer Secrecy Throughput in Wireless Networks
Rung-Hung Gau, Member, IEEE
Abstract—In this letter, we analyze the physical layer secrecy
throughput in wireless fading networks with independent eaves-droppers that do not collude. We study the impacts of the total number of eavesdroppers on the secrecy throughput. In addition, we propose two channel-adaptive transmission policies for improving the secrecy throughput. The proposed transmission policies have low computational complexity and therefore are feasible in practice. We use analytical results and simulation results to justify the usage of the proposed schemes.
Index Terms—Physical layer security, wireless fading
net-works, probability models, performance analysis.
I. INTRODUCTION
R
ECENTLY, physical layer security in wireless networksdraws a lot of attention. While cryptography-based se-curity is mainly based on computational complexity theory, physical layer security is mainly based on information theory. In particular, cryptography-based privacy is based on the fact that some problems such as prime factorization and discrete logarithm are computationally intractable, although in principle they can be solved by brute-force search. In contrast, information-theoretic secrecy assures that an eavesdropper is unable to decode the private information irrespective of the computational capability of the eavesdropper. Technologies of physical layer security and technologies of cryptography-based security are complementary rather than competing.
In this letter, we study the physical layer secrecy throughput for wireless fading networks containing independent eaves-droppers that do not collude. Gopala, Lai, and El Gamal [1] derived the secrecy capacity when the transmitter knows the channel gains of both the legitimate receiver and the eavesdropper. In addition, they showed that the secrecy ca-pacity can be achieved based on a water-filling algorithm for power control. However, the computational complexity of the water-filling algorithm is high, since it requires solving integral equations. For mobile devices, energy saving is very important. Typically, energy consumption is an increasing function of computational complexity. To strike a balance between computational complexity and secrecy throughput, we propose two channel-adaptive transmission policies and derive related analytical results. We use both analytical results and simulation results to justify the usage of the proposed channel-adaptive transmission policies.
Manuscript received July 22, 2012. The associate editor coordinating the review of this letter and approving it for publication was M. Tao.
R.-H. Gau is with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan (e-mail: [email protected]).
This work was supported in part by the National Science Council, Taiwan, R.O.C., under grant number NSC 100-2628-E-009-017-MY2.
Digital Object Identifier 10.1109/LCOMM.2012.100812.121625
II. RELATEDWORK
Wyner [2] introduced the so-called wiretap channel and the associated notion of secrecy capacity. In particular, an achiev-able secrecy rate is defined as a transmission rate at which the confidential message is secretly transmitted from the source node to the destination node, while keeping the eavesdropper from getting information of the confidential message. Leung-Yan-Cheong and Hellman [3] derived the secrecy capacity for Gaussian wiretap channels. Khisti, Tchamkerten, and Wornell [4] studied the problem of secure broadcasting over wireless fading channels. The secrecy capacity of the MIMO wiretap channel was characterized by Khisti and Wornell [5] [6]. Jeong, Kim, and Kim [7] proposed jointly optimizing the beamforming vector at the source node and the beamforming matrix at the relay node for amplify-and-forward MIMO relay networks. Bloch, Barros, Rodrigues, and McLaughlin [8] developed a secure communication protocol that uses a four-step procedure to ensure wireless information-theoretic secu-rity. Dong, Han, Petropulu, and Poor [9] proposed improving wireless physical layer security via cooperating relays. Zhou, Ganti, Andrews, and Hjorungnes [10] studied the throughput cost for achieving a certain level of security in random networks where the legitimate nodes and eavesdroppers are distributed according to independent two-dimensional Poisson point processes.
III. SYSTEMMODELS
We consider a wireless fading network that contains a source node, a destination node, and independent eavesdrop-pers. Each node has a single (omnidirectional) antenna for transmission or reception. As in [10], it is assumed that the eavesdroppers do not collude and do not transmit data. Let N be the total number of eavesdroppers. The destination node is also called node 0, while the ith eavesdropper is called node i, ∀1 ≤ i ≤ N . Let σ2 be the power spectral density of the additive white Gaussian noise. Let W be the bandwidth used for data transmission. Without loss of essential generality, it is assumed that W = 1. Time is partitioned into time slots. Let T be the length of a time slot. Typically, the value of T is smaller than the coherence time of the wireless channel. Let
FX(x) = P {X ≤ x} be the cumulative distribution function
of the random variable X. Let Gi,t be a continuous random
variable that represents the gain of the channel from the source node to node i in time slot t, ∀0 ≤ i ≤ N, t ≥ 1. By definition, Gi,t≥ 0 for sure. It is assumed that for each fixed i, Gi,t’s are
independent and identically distributed (IID) random variables. In addition, it is assumed that Gi,t1 and Gj,t2 are statistically
independent, ∀(i, t1) = (j, t2). Similar to [7], it is assumed that the source node knows the value of (G0,t, G1,t, .., GN,t)
GAU: TRANSMISSION POLICIES FOR IMPROVING PHYSICAL LAYER SECRECY THROUGHPUT IN WIRELESS NETWORKS 1973
through channel estimation at the beginning of time slot t. Designing game-theoretic mechanisms for providing eaves-droppers incentives to cooperate in channel estimation is beyond the scope of the letter. Let P be an upper bound for
the average transmission power of the source node. Let Pt
be the transmission power of the source node in time slot t. Then, limn→∞ 1n
n
t=1Pt ≤ P . Define [x]+ = max(x, 0).
Define C(x) = W log2(1 + P ·xσ2 ), ∀x ≥ 0. According to
[11], when N = 1, the secrecy capacity in time slot t equals [W log2(1+Pt·G0,t σ2 )−W log2(1+ Pt·G1,t σ2 )]+ = [C( Pt·G0,t P )−
C(Pt·GP1,t)]+. In a time slot, if the eavesdropper with the best eavesdropping channel cannot decode the private information sent by the source node, none of the N eavesdroppers can decode the private information. Thus, the secrecy capacity in time slot t is [C(Pt·GP0,t) − C(Pt·maxi:1≤i≤NP Gi,t)]+, which is equal to [C(Pt·GP0,t) − maxi:1≤i≤NC(Pt·GPi,t)]+. Let Rt
be the data transmission rate of the source node in time slot t. A transmission policy determines the value of (Pt, Rt)
at the beginning of time slot t. Let 1{condition} be the indicator function with value one if the condition is true or with value zero otherwise. Denote the secrecy throughput when the transmission policy θ is used by S(θ). In time slot t, the least upper bound of the set of achievable secrete rates is [C(Pt·G0,t P )−maxi:1≤i≤NC( Pt·Gi,t P )]+. Thus, S(θ) is defined as follows. S(θ) = lim n→∞ 1 n n t=1 Rt× 1{Rt≤ [C(Pt· G0,t P ) − max1≤i≤NC( Pt· Gi,t P )] +}. (1) When secrecy is not a concern, one may set Rt= C(Pt·GP0,t).
In this letter, to optimize the secrecy throughput, the value of Rtis set according to the following equation.
Rt = [C(Pt· G0,t
P ) − maxi:1≤i≤NC(
Pt· Gi,t
P )]
+. (2)
We focus on wireless channels with Rayleigh fading. Let μ1, μ2, .., μN be positive real numbers. It is assumed that Gi,t
is an exponentially distributed random variable with mean equals μ1
i, ∀i, t. Let di be the distance between the source
node and node i, ∀0 ≤ i ≤ N . Typically, μi depends on di.
Let θ1 be the transmission policy in which Pt= P , ∀t ≥ 1.
Whenever appropriate, Gi,1 is abbreviated by Gi.
IV. CHANNEL-ADAPTIVETRANSMISSIONPOLICIES
In this section, for improving the physical layer secrecy throughput, while keeping the computational complexity low, we propose two channel-adaptive transmission policies, de-noted by θ2 and θ3. When θ2 or θ3 is used, the long-term average transmission power equals P .
A. Probability-based power allocation
The transmission policy θ2 exploits the probability that
the source node could secretly transmit information to the
destination node in a time slot. Define q = P {C(G0) >
maxi:1≤i≤NC(Gi)}. Namely, q is the probability that eaves-droppers cannot decode in a time slot. When the transmission
policy θ2 is used, the value of (Pt, Rt) is set according to
the following rules. If G0,t > maxi:1≤i≤NGi,t, Pt = Pq.
Otherwise, Pt= 0.
Wang, Yu, and Zhang [12] derived a close-form
expres-sion for q when μi = μ, ∀i. We derive the value of q
when (μ1, μ2, .., μN) is arbitrary. For each i, where i ∈
{1, 2, 3, .., N}, define an event Ai = {G0 ≤ Gi}. Then,
P {Ai} = P {G0 ≤ Gi} = μ0μ+μ0 i [13]. In addition, ∀S ⊂
{1, 2, 3, .., N}, since mini:i∈SGi is exponentially distributed
with mean equals (i:i∈Sμi)−1,
P {∩i:i∈SAi} = P {G0≤ min i:i∈SGi} = μ0 μ0+i:i∈Sμi. (3) Furthermore, q = {G0> max i:1≤i≤NGi} = 1 − P {G0≤ max i:1≤i≤NGi} = 1 − P {∪N i=1Ai} = 1 − N i=1 P {Ai} + i<j P {Ai∩ Aj} + .. + (−1)NP {A 1∩ A2∩ .. ∩ AN}. (4)
The last equality is based on the de Morgan laws in probability theory [13].
B. Dynamic power allocation
We now introduce the transmission policy θ3that allows the
transmitter to use more transmission power when the quality of the legitimate channel is much better than the quality of the eavesdropping channel. Let λ be a positive real number. Let P∗ be a real number that satisfies the following equation.
P∗× E[[1 − e−λ(G0,t−maxi:1≤i≤NGi,t)] × 1{G0,t− max
i:1≤i≤NGi,t}] = P. (5)
When θ3 is used, the value of Pt is set according to the
following equation.
Pt = P∗× (1 − e−λ(G0,t−maxi:1≤i≤NGi,t)) ×
1{G0,t− max
i:1≤i≤NGi,t}. (6)
Note that when 0 < λ < ∞, Ptis an increasing function of
G0,t−maxi:1≤i≤NGi,t, as long as G0,t−maxi:1≤i≤NGi,t>
0. When λ = ∞, power allocation becomes independent of channel states.
V. ANALYTICALRESULTS
A. Secrecy throughput analysis
We derive the value of S(θ1) as follows. Abbreviate Gi,1
by Gi. Then, S(θ1) = lim n→∞ 1 n n t=1 [C(G0,t) − max i:1≤i≤NC(Gi,t)] +× 1 = E[[C(G0) − max i:1≤i≤NC(Gi)] +]. (7)
1974 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 12, DECEMBER 2012
The first equality is based on Equation (1), Equation (2), and Pt = P . Recall that for each fixed i, Gi,t’s are IID random
variables. Based on the law of large numbers [13], we have the second equality.
Based on the above equation, we have S(θ1) = E[[C(G0) − max i:1≤i≤NC(Gi)] +] = E[E[[C(G0) − max i:1≤i≤NC(Gi)] +|G 0]] = ∞
0 E[[C(G0) − maxi:1≤i≤NC(Gi)]
+|G
0= x] dFG0(x)
=
∞
0 E[[C(x) − maxi:1≤i≤NC(Gi)]
+] dF
G0(x). (8)
The second equality is based on E[X] = E[E[X|Y ]]. Note that the Riemann-Stieltjes integral is used in the right-hand side of the third equality.
Since Fmaxi:1≤i≤NGi(y) =
N
i=1P {Gi ≤ y} =
N
i=1(1 −
e−μiy), we have
dFmaxi:1≤i≤NGi(y)
dy = N i=1 μie−μiy j:j=i (1 − e−μjy). (9)
Then, based on Equation (8), we have S(θ1) = ∞ 0 x 0 [C(x) − C(y)] × [ N i=1 μie−μiy j:j=i 1 − e−μjy] ×μ0e−μ0·xdy dx. (10) Similarly, S(θ2) = lim n→∞ 1 n n t=1 [C(G0,t q ) − max1≤i≤NC( Gi,t q )] + = E[[C(G0 q ) − C( maxi:1≤i≤NGi q )] +] = ∞ 0 x 0 [C( x q) − C( y q)] × [ N i=1 μie−μiy× j:j=i (1 − e−μjy)] × μ 0e−μ0xdy dx. (11) Proposition 1: S(θ2) > S(θ1). Proof:
1. For each fixed (x, y), where 0 ≤ y ≤ x, define hx,y(z) =
C(xz) − C(yz), ∀z ∈ (0, 1]. By direct computation, dhx,ydz(z) =
( W ln(2)) × (σ 2z+P y σ2z+P x) × ( σ2P (y−x) (σ2z+P y)2).
2. Then, hx,y(z) < 0, ∀0 ≤ y < x, z ∈ (0, 1]. In addition,
hx,y(z) = 0, ∀0 ≤ y = x, z ∈ (0, 1]. Thus, since q ∈ (0, 1),
C(xq) − C(yq) ≥ C(x1) − C(y1), ∀0 ≤ y ≤ x, and the equality holds only when y = x.
3. Therefore, based on Equation (10) and Equation (11), S(θ2) > S(θ1).
QED.
B. A closed-form expression for P∗
We now derive a closed-form expression for P∗. Define
θ(y) =y∞(1 − e−λ(x−y))μ0e−μ0xdx. Then,
θ(y) = ∞ y (1 − e−λ(x−y))μ 0e−μ0xdx = ∞ y μ0e−μ0xdx − ∞ y μ0e−(λ+μ0)xeλydx = e−μ0y− ( μ0 λ + μ0) · e −μ0y = ( λ λ + μ0) · e −μ0y. (12)
For each pair (i, α), where i ∈ {1, 2, .., N } and α ∈ {1, 2, .., N − 1}, define the set BN,i,α as follows.
BN,i,α
= {(k1, k2, .., kα)|k1, k2, .., kα∈ {1, 2, 3, .., i − 1, i + 1,
i + 2, .., N }, k1< k2< .. < kα}. (13)
Then,
E[[1 − e−λ(G0,t−maxi:1≤i≤NGi,t)] × 1{G0,t− max i:1≤i≤NGi,t}] = ∞ 0 [ ∞ y (1 − e−λ(x−y))μ 0e−μ0xdx] × N i=1 μie−μiy j:j=i (1 − e−μjy) dy = ( λ λ + μ0) ∞ 0 e −μ0y N i=1 μie−μiy j:j=i (1 − e−μjy) dy = ( λ λ + μ0) N i=1 μi ∞ 0 e −(μ0+μi)y j:j=i (1 − e−μjy) dy = ( λ λ + μ0) N i=1 μi· [ 1 μ0+ μi + N −1 α=1 (−1)α× (k1,k2,..,kα)∈BN,i,α (μ0+ μi+ α β=1 μkβ)−1]. (14)
Based on Equation (9), the probability density function for maxi:1≤i≤NGi,t is Ni=1μie−μiy
j:j=i(1 − e−μjy). Thus,
we have the first equality. The second equality is based on Equation (12). The last equality is based on the fact that
∞
0 e−sydy = 1s,∀s > 0.
Therefore, based on Equation (5),
P∗ = P × {( λ λ + μ0) N i=1 μi· [ 1 μ0+ μi + N −1 α=1 (−1)α× (k1,k2,..,kα)∈BN,i,α (μ0+ μi+ α β=1 μkβ)−1]}−1. (15)
VI. SIMULATION ANDNUMERICALRESULTS
We wrote a C program to obtain the value of the secrecy throughput based on discrete event-driven simulations. Each simulation instance contains one million time slots. Let θ4
GAU: TRANSMISSION POLICIES FOR IMPROVING PHYSICAL LAYER SECRECY THROUGHPUT IN WIRELESS NETWORKS 1975 10−1 100 101 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 μ1 (S( θ )−S( θ2 ))/S( θ2 ) N=1, P=1, μ 0=1, σ 2 =1, W=1 θ=θ1 θ=θ3, λ=0.1 θ=θ3, λ=1 θ=θ3, λ=10 θ=θ4
Fig. 1. The impacts ofµ1on the secrecy throughput.
1 2 3 4 5 6 7 8 9 10 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
N: total number of eavesdroppers
S( θ ): secrecy throughput (bps/Hz) P=1, μ 0=1, μ1=μ2=..=μN=10, σ 2 =1, W=1, λ=1 θ=θ1 θ=θ2 θ=θ3 θ=θ4
Fig. 2. The impacts of the total number of eavesdroppers on the secrecy throughput.
be the transmission policy that is used in [1] for the case in which the transmitter knows the channel gains of the legitimate receiver and the eavesdroppers. In Figure 1, we show simulation results for S(θ)−S(θS(θ 2)
2) , when N = 1. We
find that S(θ1S(θ)−S(θ2)
2) is always negative, which is consistent
with Proposition 1. In contrast, S(θ3)−S(θ2)
S(θ2) ≥ 0, regardless of
the values of μ1 and P . However, though not shown in the
letter due to the limit of space, when P
σ2 ≥ 10, the difference
between S(θ3) and S(θ2) becomes negligible. In Figure 2, we
show the impacts of the total number of eavesdroppers on the secrecy throughput. In particular, regardless of the adopted transmission policy, as the total number of eavesdroppers increases, the value of the secrecy throughput decreases. In comparison with θ1, θ3 could increase the secrecy throughput
by more than 20%. In addition, S(θ3)
S(θ4)≥ 0.96. This means that
θ3 is near-optimal and it is unnecessary to find an optimal
value for λ by the exhaustive search. Unlike θ4, θ3 is based
on a close-form expression and does not require numerically
solving an integral equation in order to find the proper water level. We also used Maple to obtain equation-based numerical results. The numerical results are consistent with simulation
results. For example, when N = 2, P = 1.0, σ2 = 1.0,
μ0= 1.0, and μ1= μ2= 0.1, for S(θ1), the numerical result
is 0.006905, while the simulation result is 0.006852. When N = 2, P = 1.0, σ2= 1.0, μ0= 1.0, and μ1= μ2= 0.1, for
S(θ2), the numerical result is 0.011019, while the simulation
result is 0.011736.
VII. CONCLUSION
In this letter, we have studied the physical layer secrecy throughput in wireless fading networks with independent eavesdroppers that do not collude. We have analyzed the impacts of the total number of eavesdroppers on the secrecy throughput. In addition, to improve the secrecy throughput, we have proposed two channel-adaptive transmission policies. We have derived closed-form expressions for the proposed trans-mission policies. Furthermore, we have used analytical results and simulation results to show that the proposed channel-adaptive transmission policies could improve the physical layer secrecy throughput. Future work includes studying the case in which the gains of the eavesdropping channels are unknown. Another direction of future research is extending the analysis in the letter to cooperative communications systems.
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