On the Location of the Eavesdropper in Multi-Terminal ... - CiteSeerX

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On the Location of the Eavesdropper in Multi-Terminal Networks S. Anand and R. Chandramouli Department of ECE, Stevens Institute of Technology NJ 07030, USA {asanthan,mouli}@stevens.edu

Abstract We study the optimum location of an eavesdropper from a secrecy capacity perspective in multi-terminal networks with power control. We determine the logical location of an eavesdropper which a) results in zero secrecy capacity for maximum number of users in the network and b) results in zero secrecy capacity for the bottle-neck links. We then analyze the asymptotic secrecy capacity of the system and the asymptotic behavior of the optimum logical location of the eavesdropper. Results indicate that power control can make eavesdropping more difficult as it results in infeasible locations for the eavesdropper. Power control is also shown to provide scenarios which can result in positive asymptotic secrecy capacity for all the users.

Index Terms – Secrecy capacity, eavesdropper location, asymptotic. I. I NTRODUCTION Secrecy capacity is studied for systems with key-less security. Information transmitted by a source (or a transmitter) is not only received by the intended destination (or receiver), but also by an eavesdropper. Secrecy capacity is the maximum rate of transmission such that the bit error rate (BER) at the destination is zero while that at the eavesdropper is 1/2. For additive white Gaussian noise (AWGN) and multiple access interference (MAI) channels, secrecy capacity defined in [1] is the difference between the Shannon capacity of the channel between the source and destination and that between the source and eavesdropper. Let the signal-to-noise ratio (SNR) of the channel between the source and the destination be γd and that of the channel between the source and the eavesdropper be γe . If the channel bandwidth is W , then the secrecy capacity, S , is given by [1] 

S = W log2



1 + γd 1 + γe

+

,

(1)

where [θ]+ = max(θ, 0). From (1), it is observed that a zero secrecy capacity indicates that the eavesdropper obtains at least as much information as the receiver, and thus, an unsecured system. Wyner [1] first showed that positive secrecy capacity can be achieved without having a secret key of larger entropy than that of the message. Further studies evaluate the secrecy capacity for single-terminal networks without cryptographic keys [2], [3] and for multi-terminal networks both without cryptographic keys [4] as well as with cryptographic keys [5], [6]. Additional references on secrecy capacity and their detailed description are provided in [7] and the references therein. A reader would have a better understanding of the analysis we present in this paper after reading our study on the impact multi-terminal networks with pricing in [7], where we studied the impact of power control on the secrecy capacity of users (also termed transmit-receive pairs) in multi-terminal networks. The optimum power allocation to the transmitters was formulated as a non-cooperative game with non-linear and This work was funded by a research grant from the U. S. Government

a linear pricing. We used the theory of M− matrices [8] and obtained necessary and sufficient conditions for the existence of a unique Nash equilibrium. We then studied the impact of power control on the secrecy capacity of the users. Three scenarios were studied- a) co-located transmitters, b) co-located receivers and c) randomly placed transmitters and receivers. It was shown that power control improves the secrecy capacity of most user terminals for the scenarios with co-located receivers and randomly placed transmitters and receivers. Secrecy capacity of the users depend on the channel gains between all the transmitters and receivers and the location of the eavesdropper. It then becomes important to determine the location of the eavesdropper that makes the network vulnerable. This is the the location of the eavesdropper that results in zero secrecy capacity for maximum number of users. As mentioned earlier, zero secrecy capacity for a transmit-receive pair indicates that the eavesdropper obtains atleast as much information as the intended receiver and thus rendering the link, unsecured. In some scenarios, e. g., sensor networks, the information transfer between different transmit-receive pairs could be correlated and the eavesdropper could gain sufficient information by snooping any link effectively. This would typically be the link with least robustness (called as the “bottle-neck” link). Hence, it is also essential to study the location of the eavesdropper that makes the secrecy of the bottle-neck link, zero. In this paper, we extend our analysis in [7] to study the optimum logical location of the eavesdropper to result in zero secrecy capacity for a) maximum number of users in the network and b) the bottle-neck link in a multi-terminal network with power control. Results indicate that power control can make eavesdropping difficult as it results in infeasible locations for the eavesdropper. We then study the asymptotic secrecy capacity of a multi-terminal system with and without pricing. It is shown that asymptotically, the secrecy capacity of all the terminals in systems with no power control approach zero, whereas for systems with power control, the asymptotic secrecy capacity for each terminal can be positive. The findings in this paper show that the asymptotic secrecy capacity depends only on the relative channel gains from the transmitter to the receiver and the eavesdropper and is independent of the interference obtained from the other transmitters in the system. The rest of the paper is organized as follows. The system model is described in Section II. Section III details the analysis for obtaining the optimum location of the eavesdropper. In Section IV, we provide the analysis for the asymptotic secrecy capacity. Section V presents the conclusions. II. S YSTEM M ODEL We consider a multi-user system with bandwidth W and M transmitters/sources, M receivers/destinations and a single eavesdropper. Transmitter i transmits with power Pi . In [7] we obtained the optimal values of Pi that maximize the utility of the channel between the ith transmit-receive pair. We then studied the impact of such a power control mechanism on the secrecy capacity of the ith transmit-receive pair in the presence of an eavesdropper. Here, our objective is to determine the logical location of the eavesdropper so that the secrecy capacity at each receiver is zero. We define the optimal logical location of the eavesdropper to be the vector of channel gains from each of the transmitters. We consider the logical location (i. e., channel gain) instead of the physical co-ordinates of the eavesdropper because, although, the location (or the co-ordinates) of the eavesdropper affects the channel gains from the transmitters, it is not the only factor. Depending on the type of the environment, factors like fading and shadowing [9] also affect the channel gains. Thus, multiple physical locations could yield identical channel gains. The extension of our results to the actual physical location of the eavesdropper is a problem of geo-location 2

[10], [11], which is a separate topic of research and beyond the scope of this paper. We make the following assumptions in our analysis. •

The system has bandwidth W Hz.

•

Each transmitter can transmit at a maximum power Pmax .

•

Transmitter i transmits at rate ri bps to receiver i.

•

Receiver i has a gain Gi . This gain is due to spectrum spreading and given by Gi = W/ri .

•

The signal-to-interference ratio (SIR) experienced by receiver i is xi .

•

The channel gain from transmitter i to receiver j is hij . The channel gain matrix, H, is given by H = [hij ] 1≤i≤M . 1≤j≤M

•

The channel gain from transmitter i to the eavesdropper is hie . The channel gain vector from all transmitters to the eavesdropper, he , is given by he = [hie ]1≤i≤M .

•

The channel noise is additive white Gaussian noise (AWGN) with power spectral density N0 Watts/Hz. III. L OCATION

OF THE

E AVESDROPPER

A. Preliminaries Before we present the analysis for the optimal logical location of the eavesdropper, we present the definitions of some terms and the the main results obtained in [7], which we also use in this paper. For a power of Pi transmitted by transmitter i, the SIR experienced by receiver i, xi , is given by xi = P

Pi hii Gi . j6=i Pj hji + N0 W

(2)

The utility of the channel between transmitter i and receiver i, ui , is then given by ui = W ln(1 + xi ).

(3)

To optimally allocate powers to the users we solve the optimization problem maxp ui ∀i,

(4)

0 ≤ Pi ≤ Pmax ∀i.

(5)

subject to the constraints

In (4), p =

h

P1 P2 P3 . . . PM

iT

, where T represents the transpose of a matrix/vector. In [7], we formulated

the optimization problem mentioned above as an M − person non-cooperative game and argued that the Nash equilibrium1 of this game occurs when Pi = Pmax , ∀ i, thus resulting in large energy consumption and no power control. In order to address this issue, we proposed two pricing functions in [7] to price higher, those receivers that receive larger SIR’s. We proposed a non-linear and a linear pricing function (both, functions of the SIR, xi ) given by fi (xi ) = λ 1

ri xi Pi hii ri = λ PM xi + Gi j=1 Pj hji + N0 W

Definitions of the terms, “game” and “Nash equilibrium” are omitted here but provided in [7].

3

(6)

and fi (xi ) = λri xi = λ P

Pi hii ri Gi , j6=i Pj hji + N0 W

(7)

respectively, where λ is the pricing parameter. With the pricing functions, the power allocation problem was reformulated as maxp u ˆi = maxp [ui − fi (xi )] ∀i,

(8)

subject to the constraints in (5). In (8), fi (xi ) is either (6) or (7). This was also formulated as an M −person non-cooperative game and necessary and sufficient conditions were obtained for the existence of a unique Nash equilibrium. Let p∗ be the optimal power vector (i. e., the unique Nash equilibrium of the power control game with pricing) and let x∗ =

h

x∗i x∗2 x∗3 · · · x∗M

i

be the corresponding SIR vector. When all the transmitters transmit at

the optimal powers, the SIR experienced by the eavesdropper when it tries to listen to the signal transmitted by transmitter i, xie , is then given by xie = P

Pi∗ hie Gi . ∗ j6=i Pj hje + N0 W

(9)

The secrecy capacity of transmit-receive pair i (i. e., the channel between transmitter i and receiver i), Si , is obtained by evaluating the expression 

Si = W log2



1 + x∗i 1 + xie

+

.

(10)

B. Optimal Location We now present the analysis for obtaining the optimal logical location of the eavesdropper. Henceforth, throughout the paper, “location” of the eavesdropper implies the logical location unless explicitly mentioned otherwise. The optimal location of the eavesdropper is defined by the channel gain vector he = [hie ]1≤i≤M that results in minimum △

secrecy capacity for all transmit-receive pairs. Let xe = and

T

h

x1e x2e x3e · · · xM e

iT

, where xie is given by (9)

represents transpose of a matrix or a vector. The optimal location of the eavesdropper can then be obtained

by solving the optimization problem minhe Si ∀i,

(11)

0 ≤ hie ≤ 1 ∀i,

(12)

subject to the constraints

where Si is given by (10). The constraint 0 ≤ hie in (12) arises from the fact that the channel gains from all the transmitters to the eavesdropper are non-negative. The constraint hie ≤ 1 in (12) follows from the fact that the eavesdropper cannot obtain more power than what is transmitted by the transmitter. It is noted that the only parameter in Si that depends on he is xie . Hence, the optimization problem in (11) can be re-written as maxhe xie ∀i,

4

(13)

subject to the constraints in (12). The above optimization problem can be viewed as an M −person non-cooperative game. However, this game has only one player (the eavesdropper). This single eavesdropper can be looked upon as M virtual players (i. e., virtual eavesdroppers), each wanting to maximize its own xie irrespective of the xie ’s obtained by the other virtual players. Therefore, the strategy of the ith virtual player is the channel gain, hie . The utility of the ith virtual player is the SIR, xie . Since xie is an increasing function of hie , the value of hie that maximizes xie is hie = 1. In other words, the Nash equilibrium for the SIR maximization game specified by the optimization problem in (13) subject to the constraints in (12) is when hie = 1 ∀ i. This results in an impractical solution because it would mean that the eavesdropper is co-located with all the transmitters. The problem is therefore modified as follows. It is observed that the secrecy capacity is non-negative. Thus the minimum secrecy capacity that can be obtained by the ith transmit-receive pair is 0 and this occurs if xie ≥ x∗i . Hence, it suffices to ensure that xie = x∗i , ∀ i so that all the transmit-receive pairs obtain zero secrecy capacity. Thus, the optimum location of the eavesdropper is obtained by re-writing (9) with xie = x∗i and hence, solving the system of linear equations Pi∗ Gi x∗i hie

−

P

j6=i Pj hje

= N0 W

∀i.

(14)

1,

(15)

The above can be re-written in the form of the matrix equation 

he = N0 W D4 −1 D3 − cdT

where 1 is the column vector with all entries being unity,  

     D3 =     

G1 x∗1



+1 0

0 

0 .. . 0



   D4 =    

and

G2 x∗2



+1 0 .. . 0



G3 x∗3

−1

0

···

0

0

···

0



+1 .. . 0

0 P1∗ 0 ∗ 0 P2 0 0 0 P3∗ .. .. .. . . . 0 0 0

··· ··· ··· .. .



     ,     

··· 0 . .. .  .. GM ··· x∗ + 1 M

0 0 0 .. .

∗ · · · PM

(16)



   ,   

(17)

c = d = 1.

(18)

It is desired to obtain conditions on Pi∗ , Gi and x∗i that results in non-negative values of hie , ∀ i. Applying Theorem 

4.2 in [7], he is positive2 if and only if D3 − cdT



is an M−matrix3 . Therefore, the necessary and sufficient

conditions for the vector he , obtained by solving (15), to be positive, are determined by obtaining conditions under 

which D3 − cdT





is an M−matrix. In order to obtain necessary and sufficient conditions for D3 − cdT



to

2 A positive vector or a matrix is one in which all the elements are positive. A negative, non-positive and non-negative matrix or vector is defined similarly. 3 Appendix II in [7] provides a description of M−matrices.

5

be an M−matrix, we apply the Sherman-Morrison formula [12] for evaluating the inverse of rank-1 updates of matrices, which is as follows. Lemma 3.1: Consider any N × N non-singular matrix, B. Let a and b be any N × 1 column vectors such that 1 + bT B−1 a 6= 0. Then B−1 abT B−1 . (19) 1 + bT B−1 a The following Theorem provides necessary and sufficient conditions on x∗i and Gi so that is an M−matrix. the 

B + abT



matrix D3 − cdT





Theorem 3.1: The matrix D3 − cdT



−1

= B−1 −

is an M−matrix if and only if M X i=1

Proof: From Lemma 3.1, 

D3 − cdT

−1

x∗i < 1. x∗i + Gi

= D3 −1 +

(20)

D3 −1 cdT D3 −1 . 1 − dT D3 −1 c

(21)

Since D3 is a positive diagonal matrix, D3 −1 is also a positive diagonal matrix. Therefore, from (18) and (21), 

D3 − cdT

−1

is a positive matrix if and only if 1 − dT D3 −1 c > 0.

(22)

Eqn. (20) follows from (16), (18) and (22). 2

Thus, before computing the optimum location of the eavesdropper, it is essential to verify the condition in (20), which is very easy to apply. Theorem 3.1 provides a necessary and sufficient condition for he being positive. However, it still does not ensure the feasibility condition hie ≤ 1 ∀ i. This is addressed in the following Theorem. Theorem 3.2: Let △

αi =

x∗i

x∗i . + Gi

Then, an SIR vector xe results in 0 < he < 1 if and only if Pi∗ 1−

PM

j=1

αj

αi

Proof: From (15), (16), (17), (18) and (21), hie hie =




(23)

PM

j=1 αj



(24)

> N0 W ∀i. can be obtained as N0 W αi

Pi∗

< 1 and

1−

PM

j=1 αj

.

The feasibility condition, hie < 1, leads to Eqn. (24). The condition

(25) PM

j=1 αj

< 1 follows from Theorem 3.1. 2

The conditions mentioned in Theorem 3.2 are also easy to apply. It is observed that larger values of x∗i ’s result in larger values of the desired xie to make the secrecy capacity of the ith transmit-receive pair, zero. So, it becomes essential to determine if there exists an SIR vector x∗ such that if all transmit-receive pairs obtain SIR’s as specified by x∗ , then increasing the SIR of any user i would cause the hje to be infeasible for some j . In other words, it is 6

ˆ e that causes hje for some j to be unity. The following essential to determine whether there exists an SIR vector x ˆe. Theorem provides the existence of such an x 



ˆ e such that ∀ xe > x ˆ e 4 , D3 − cdT ceases to be an M−matrix and ∀ xe < x ˆe, Theorem 3.3: ∃ an SIR vector x 

D3 − cdT



is an an M−matrix.

The proof of Theorem 3.3 follows by applying Lemma 4.2 and Theorem 4.4 in [7]. As mentioned earlier, it is not necessary to determine he to maximize xie ∀ i. Instead it is only essential to

determine he such that xie = x∗i ∀ i. The eavesdropper would like to position itself such that the secrecy capacity ˆ e where x ˆ e as described in of maximum number of transmit-receive pairs obtain zero secrecy capacity. If xe < x

Theorem 3.3, then it is possible to find a feasible he (i. e., 0 < he < 1, where 0 is the M × 1 column vector ˆ e , then the of all zeroes) such that all transmit-receive pairs obtain zero secrecy capacity. However, if xe > x

optimum location of the eavesdropper such that maximum transmit-receive pairs obtain zero secrecy capacity can be determined as follows: ˜ = Let x

h

x ˜1 x ˜2 x ˜3 · · · x ˜M

iT

, where x ˜1 ≥ x ˜2 ≥ x ˜3 · · · ≥ x ˜M are the SIR’s x∗1 , x∗2 , · · ·, x∗M arranged in

the non-increasing order. Then (23) is applied each time for k = 1, 2, 3, · · ·, M by replacing x ˜k for x∗i till the condition in Theorem 3.2 is satisfied. If the condition is satisfied for x ˜∗k , then M − k + 1 transmit-receive pairs obtain zero secrecy capacity. If the condition is not satisfied for any value of x ˜k for k = 1, 2, 3, · · ·, M , then all transmit-receive pairs obtain positive secrecy capacity for all locations of the eavesdropper. The bottle-neck link in the system is the transmit-receive pair with the least utility (and hence, the least x∗i ). Without loss of generality, let transmit-receive pair k be the bottle-neck link. The location of the eavesdropper that results in zero secrecy capacity for the bottle-neck link is then determined by replacing x∗i = x∗k , ∀ i and then applying (25). The feasibility conditions specified by Theorem 3.2 still hold and if the conditions are violated, it implies that for any location of the eavesdropper, the bottle-neck link (and hence, all the links in the network) obtain positive secrecy capacity. C. Numerical Examples Some numerical examples for the optimum location of the eavesdropper are presented. Three scenarios for the locations of the transmitters and receivers are considered-1) transmitters are co-located, 2) receivers are co-located and 3) transmitters and receivers are randomly located, i.e., neither the transmitters nor the receivers are co-located. The H matrix was generated using the Jake’s wireless channel model [9]. We present the results for a system with M = 10, i.e., 10 transmit-receive pairs and consider two types of systems for each of the scenarios described

earlier- a) a low bandwidth system like a 3G Cellular system [13] with a bandwidth of 5MHz with transmitters transmitting at data rates less than 100 Kbps and b) a high bandwidth system like a wireless local area network (WLAN) [14] with a bandwidth of 20 MHz and transmitters transmitting at data rates up to 2 Mbps. We first compute the optimum location of the eavesdropper such that all transmit-receive pairs obtain zero secrecy capacity. Fig. 15 presents the optimum location of the eavesdropper for the system with 5 MHz bandwidth for the 4 Throughout this paper, we use the relational operator “>” and “ and element-wise