Outage Constrained Secrecy Rate Maximization Using Cooperative ...

Outage Constrained Secrecy Rate Maximization Using Cooperative Jamming

arXiv:1202.6597v1 [cs.IT] 29 Feb 2012

Shuangyu Luo, Jiangyuan Li, Athina Petropulu Rutgers University, Piscataway, NJ 08854

Abstract—We consider a Gaussian MISO wiretap channel, where a multi-antenna source communicates with a single-antenna destination in the presence of a single-antenna eavesdropper. The communication is assisted by multi-antenna helpers that act as jammers to the eavesdropper. Each helper independently transmits noise which lies in the null space of the channel to the destination, thus creates no interference to the destination. Under the assumption that there is eavesdropper channel uncertainty, we derive the optimal covariance matrix for the source signal so that the secrecy rate is maximized subject to probability of outage and power constraints. Assuming that the eavesdropper channels follow zero-mean Gaussian model with known covariances, we derive the outage probability in a closed form. Simulation results in support of the analysis are provided. Index Terms—Gaussian MISO wiretap channel, cooperative jamming, artificial noise, outage probability

I. I NTRODUCTION Physical layer secrecy exploits channel conditions to maximize the rate of reliable information delivered to the legitimate destination, with the eavesdropper being kept as ignorant of that information as possible. This line of work was pioneered by Wyner [1], who showed that when an eavesdropper’s channel is a degraded version of the source-destination channel, the source and destination can exchange secure messages perfectly at a non-zero rate, while the eavesdropper can learn almost nothing about the messages from its observations. The need for physical layer security in the context of wireless communications is motivated by challenges associated with classical cryptographic approaches, most notably the exchange and maintenance of private keys. The secrecy capacity for multiple antenna wiretap channels with perfect channel state information on the eavesdropper is established in [2], [3], [4] under sum power constraints, and in [5] and [6] under power-covariance constraints. For MISO wiretap channels, the optimal input covariance matrix that achieves the secrecy capacity under a sum power constraint was given in a closed form in [7]. As shown in [8], when the channel to the destination encounters more fading than the channel to the eavesdropper, a positive secrecy rate can not be guaranteed. One way to overcome this problem is to use helpers who amplify-and-forward, or decode-and-forward the source signal, or perform cooperative jamming (CJ). In the latter case, helpers do not need to receive nor relay the source message, but rather just transmit noise to degrade the channel to the eavesdropper, therefore increasing the secrecy rate. In [9], a multiple access wiretap channel was considered and it was shown that if the optimal power allocation policy does not allow a certain user to transmit, that particular user could increase the secrecy rate by transmitting artificial noise. In [10], multiple helpers were employed to transmit a weighted jamming signal that enforces nulling at the legitimate destination and maximize the secrecy rate. Following that work, [11] found the optimal weights by avoiding the nulling at the destination, which achieves higher secrecy rate. In [12], A MISO system with multi-antenna transmitter and singleantenna legitimate receiver and eavesdropper was studied. The trans-

mitter constructs a Gaussian distributed artificial noise that lies in the null space of its channel to the legitimate receiver, and sends a sum of information bearing signal and the artificial noise, which only confuses the eavesdropper but not the legitimate receiver. Since the transmitter does not know the channel to the eavesdropper, the power of the artificial noise is designed to uniformly spread along the null space. In [13], the use of artificial interference for a MIMO wiretap channel is studied. The transmitter sends a sum of signal and artificial noise. When the CSI of the eavesdropper is unknown, the artificial noise is designed to lie in the null space of the right singular vector associated with the largest singular value of its transmitter-receiver channel matrix, and the power of the noise is uniformly spread along the null space. In [14], the artificial noise is studied in a MISO channel overheard by multiple single-antenna eavesdroppers. The source transmits a mixture of message and artificial noise. Without the CSI of the eavesdroppers, the outage based artificial noise design is formulated, with a so-called safe convex approximation is used to find the solution. In this paper we consider a MISO wiretap channel with multiple multi-antenna helpers implementing cooperative jamming. A multiantenna source transmits the message, while each helper transmits jamming noise that lies in the subspace that is orthogonal to the helper-destination channel. Each helper generates jamming noise locally based on only its own link to the receiver. Without any other information, the power of the jamming noise is uniformly spread along the null space. We study the problem of determining the input covariance matrix so that the secrecy rate is maximized subject to a sum power constraint and also an outage probability constraint. The source can determine the optimal input covariance matrix solely based on statistical information about the source-eavesdropper channel, with the channel following a zero mean Gaussian model with identity covariance. Assuming that the helper-eavesdropper channels follow zero-mean Gaussian distributions with arbitrary covariance matrices, the outage probability isIt is as obtained in a closed form. Introducing the outage constraint not only provides quality control but also allows taking the uncertainty of the eavesdropper’s channel into consideration and simplifies the optimization problem with respect to the covariance matrix of the input. Notation - Throughout this paper, following notations are adopted. Upper case and lower case bold symbols denote matrices and vectors, respectively. Superscripts ∗, T and † denote respectively conjugate, transposition and conjugate transposition. Tr(A) denotes the trace of the matrix A. A  0 denotes that the matrix A is Hermitian positive semi-definite. |a| denotes absolute value of the complex number a. √ kak = a† a denotes Euclidean norm of the vector a. In denotes the identity matrix of order n (the subscript is dropped when the dimension is obvious). Cn denotes the set of all n √ × 1 complex vectors. E{·} denotes the expectation operator. i = −1. x ∼ y denotes x and y have identical distributions.

With this nulling noise, the received signal at Bob and Eve becomes respectively, √ (5) yb = Ps h†0 x + nb ,

hi h0

⋮

gi

ye =

⋮

√

Ps g0† x +

g0

Fig. 1.

System model.

ye =

√

Ps h†0 x + Ps g0† x +

N X

k=1 N X

(6)

The secrecy rate of this system equals to   Ps † h0 Qh0 C1 = log2 1 + N0   Ps g0† Qg0 − log 2 1 + PN . † 2 2 k=1 |wk | kEk gk k + N0

We consider a Gaussian MISO wiretap channel with N helpers, shown in Fig. 1. The transmitter Alice uses Nt antennas to send messages to the legitimate receiver, Bob, through the channel h∗0 ∈ CNt . The eavesdropper, Eve, intercepts messages through the channel g0∗ ∈ CNt . The transmitter is aided by N helpers; each helper has Nk antennas (Nk ≥ 2, k = 1, · · · , N ) and transmits noise to confound Eve. The channel from helper k to Bob is denoted by h∗k ∈ CNk , k = 1, · · · , N , and the channel from helper k to Eve is denoted by gk∗ ∈ CNk , k = 1, · · · , N . At Bob or Eve, the received signal is a combination of the source signal, jamming noise and additive white Gaussian noise (AWGN). The received signal at Bob and Eve can be expressed as: √

gk† (wk Ek tk ) + ne .

k=1

II. S YSTEM AND S IGNAL M ODELS

yb =

N X

(7)

To make C1 as large as possible, the maximum relay power should be used, i.e., it should hold that (Nk − 1)|wk |2 = Pk . With this, we have   C1 = log2 1 + ρ0 h†0 Qh0   ρ0 g0† Qg0 . (8) − log2 1 + PN † −1 kE g k2 + 1 k k k=1 ρk (Nk − 1)

where ρ0 = Ps /N0 , ρk = Pk /N0 , k = 1, · · · , N are the Signal-toNoise Ratio (SNR) at Alice and the kth helper. In the following section we study the problem of maximizing the secrecy rate with respect to Q using only statistical information about the eavesdropper channels. III. O UTAGE C ONSTRAINED S ECRECY R ATE M AXIMIZATION

h†k nk + nb

(1)

gk† nk + ne

(2)

k=1

where nk , k = 1, · · · , N represents the jamming noise that is generated by the helpers; and x is the Nt ×1 source signal vector with input covariance matrix Rx  0; nb and ne are the AWGN received at Bob and Eve with E[|nb |2 ] = E[|ne |2 ] = N0 . The source power is constrained as Tr(Rx ) ≤ Ps . We may represent Rx = Ps Q with Q  0 and Tr(Q) ≤ 1. The power available at helper k is Pk . In the following, we assume that each helper knows only its own link to the destination, hk , k = 1, · · · , N , and locally designs its jamming noise so that it delivers a null at Bob.

Let us assume that the nodes do not have any channel information with the following exceptions: • Alice knows h0 perfectly 2 • Alice has statistical information on g0 , i.e., g0 ∼ CN (0, σ I). • Helper k knows its own link to Bob, hk . The outage probability is defined as [15] Pout (R) =

min

Q0, Tr(Q)≤1

max R Pout (R) ≤ ǫ.

E{knk k2 } = E{Tr(nk n†k )} = |wk |2 E{Tr(Ek tk t†k E†k )}

(4)

= |wk |2 Tr(Ek E†k ) = (Nk − 1)|wk |2 .

The power constraint is hence (Nk − 1)|wk |2 ≤ Pk .

The problem of (10) is equivalent to the problem of maximizing the secrecy rate subject to QoS and power constraints as follows max R

(3)

It is clear that each helper should be equipped with Nk ≥ 2 antennas such that there are enough degrees of freedom to design nk based on (3). The general solution of (3) can be expressed as nk = Ek vk , where Ek (Nk × (Nk − 1)) is the null space of h†k , with E†k Ek = I and vk is any arbitrary (Nk − 1) × 1 vector. Let the jamming noise be nk = wk Ek tk , where wk is a weight that will be selected to meet the power constraint, and tk ∼ CN (0, I). The power of the jamming noise is

(10)

Q

s.t.

h†k nk = 0, k = 1, . . . , N

(9)

We will determine the maximum R such that the outage probability is below a prescribed small level, ǫ, which represents the qualify of service (QoS). Mathematically, the problem is formulated as

A. Nulling Noise Structure In order for the noise vector of helper k to cause nulling at Bob, it should hold that

Pr(C1 < R).

(11)

Q

s.t.

Q  0,

Tr(Q) ≤ 1,

Pr(C1 < R) ≤ ǫ.

(12)

A. Optimal Input Covariance Matrix Structure Lemma 1: For the problem of (11), and for g0 ∼ CN (0, σ 2 I), the optimal Q is given by Q⋆ = h0 h†0 /kh0 k2 , and the optimization problem can be written as max R R  s.t. Pr PN

(13)  ρ0 |g01 | 1+ρ0 kh0 k −1 ≤ ǫ. > −1 kE† g k2 +1 2R ρ (N −1) k k k k=1 k (14) 2

2

(m −1)

The proof is given in Appendix A. Next, we solve the problem of (13). Obviously, for the optimal R, the constraint of (14) holds with equality. Let us define the critical value χǫ so that   ρ0 |g01 |2 > χǫ = ǫ. (15) Pr PN † −1 kE g k2 + 1 k k k=1 ρk (Nk − 1)

Then, the optimal R is given by

Let us assume that g0 ∼ CN (0, σ 2 I) and gk ∼ CN (0, Σk ). In this case, the probability of outage can be found in a closed form as follows. The calculation of the outage probability involves calculating   ρ0 |g01 |2 Pr PN > χ (18) −1 kE† g k2 + 1 k k k=1 ρk (Nk − 1) (19)

We use the methods in [16] to calculate (19). Note that E†k gk ∼ CN (0, E†k Σk Ek ). Let E†k Σk Ek have eigen-decomposition Uk Dk U†k . Denote
ρ1 ρN D = diag D1 , · · · , DN , −σ 2 ρ0 /χ . (20) N1 − 1 NN − 1

After a few derivations, (19) is equivalent to

(21) †

where z ∼ CN (0, IN+1 ). Let Y = z Dz. Y is an indefinite quadratic form. The results in [16] give the expression for Pr(Y ≥ y). Denote ξ0 = σ 2 ρ0 /χ. Let ν1 , · · · , νK denote different positive diagonal entries of D, with multiplicity m1 , · · · , mK and m1 + · · · + mK = N . According to the result in [16, Eq. (32)] for the case z ∼ CN (0, IN+1 ), we have K X

k=1

exp (−y/νk ) (m −1) gˆ k (0, y) (−νk )mk −1 (mk − 1)! k

(22)

αkj = βkj =





K Y 1 1 , (1 + ξ0 /νk + ξ0 s) j=1 (αkj − νj s)βkj

1 − νj /νk −1 mj 1

j 6= k , j=k

j= 6 k , j=k

In simulation, Alice has three antennas, each helper has two antennas. Alice-Eve link g0 ∼ CN (0, I), and kth helper-Eve link gk ∼ CN (0, I). Alice has SNR ρ0 = 5 dB, and all N helpers have SNR ρk = 2 dB. Outage probability constraint, ǫ = 0.01, and Pout (R∗ ) = ǫ, where R∗ is given in (16). Results are averaged over 105 independent trials. For each trial, generate independent and identical distributed h0 , h0 ∼ CN (0, I). During one trial, h0 is kept as constants. For a R∗ > 0, lemma 2 has to be satisfied. When h0 is much more degraded than g0 , with small number of helpers, the secrecy rate can not be guaranteed to be positive. During each trial, for N = 5, · · · , 10 number of helpers, use bi-section method to search for a larger R that satisfying the outage constraint, until it converges to R∗ . In Fig. 2, the secrecy rate increases when the number of helpers increases. In Fig. 3, since Pr(R) is a function of R, fix R1 = 0.6 log2 (1 + Ps kh0 k2 ), plot the Pr(R1 ), the outage probability decreases when the number of helpers increases. Because Bob is not affected by the noise transmitted by the helpers, but Eve is, and the more helpers, the more confounded Eve will be.

1.5

1.4

1.3

1.2

1.1 5

Fig. 2.

6

7 8 Number of helpers

9

10

Secrecy Rate vs Number of Helpers. ρ0 = 5 dB, ρk = 2 dB.

V. C ONCLUSIONS

where gˆk (s, y) = e−sy

IV. S IMULATION AND A NALYSIS

(17)

B. Closed Form Outage Probability

Pr(Y ≥ y) = −

X (n − 1)!νjn (−1)n (n − 1)!ξ0n + , βkj n (1 + ξ0 /νk + ξ0 s) (αkj − νj s)n j=1

for n ≥ 2. (28)

To find the critical value χǫ , defined in (15), we can use the bisection method. The calculation of the outage probability is discussed the following subsection.

Pr(z† Dz < −1)

X βkj νj ξ0 + , (27) 1 + ξ0 /νk + ξ0 s j=1 αkj − νj s K

[lnˆ gk (s, y)](n) =

The condition for R > 0 is given in the following lemma.

which can be rewritten as  XN  ρ0 Pr ρk (Nk − 1)−1 kE†k gk k2 − |g01 |2 < −1 . k=1 χ

K

[lnˆ gk (s, y)](1) = −y −

(16)

⋆

Lemma 2: For a given ǫ, R⋆ > 0 if and only if   ρ0 |g01 |2 Pr PN > ρ0 kh0 k2 < ǫ. † −1 2 kEk gk k + 1 k=1 ρk (Nk − 1)

l=0

with

Secrecy Rate [bps/Hz]

R⋆ = log(1 + ρ0 kh0 k2 ) − log(1 + χǫ ).

and gˆk k (s, y) denotes the (mk − 1) order derivative of gˆk (s, y) with respect to s. Using the recursive computation method in [16, Eq. (21)], we have ! n−1 X n − 1 (l) (n) gˆk (s, y) = gˆk (s, y)[lnˆ gk (s, y)](n−l) , n ≥ 1 (26) l

(23) (24) (25)

We proposed a cooperative jamming scheme, where multiple helpers transmit nulling noise to maximize secrecy rate subject to an outage probability constraint, and a power constraint. Assuming that the transmitter only knows its channel to legitimate receiver and the statistical CSI of its channel to eavesdropper, each helper known its own link to receiver, we formulated and solved an outage constrained secrecy rate maximization problem. Simulations show that proposed design could guarantee a low outage probability.

0.07

constraint Pr(C1 < R) ≤ ǫ. Thus, it should hold that λ2 = · · · = λNt = 0. As a result, we have  ρ0 |g01 |2 Pr(C1 . (34) 2R λ1

Outage Probability

0.06 0.05 0.04

2

0.03 0.02 0.01 5

6

7 8 Number of helpers

9

10

Fig. 3. Outage Probability vs Number of Helpers when fix R = 0.6 log2 (1+ Ps kh0 k2 ). ρ0 = 5 dB, ρk = 2 dB.

A PPENDIX A P ROOF OF L EMMA 1

1− 1

0k From (34), we know that ρ0 kh − λ21R is increasing with λ1 but 2R decreasing with R, in other words, a larger λ1 will allow a larger R without violating the outage constraint Pr(C1 < R) ≤ ǫ. Since Tr(Q) = λ1 ≤ 1, it holds that λ1 = 1. As a result, we have  ρ0 |g01 |2 Pr(C1 2R Based on the result above, the problem of (11) is equivalent to the problem of (13). This completes the proof.

R EFERENCES

After a few derivations we get ρ0 g0† Qg0 −1 kE† g k2 + 1 k k k=1 ρk (Nk − 1)  1 + ρ0 h†0 Qh0 > − 1 . 2R

 Pr(C1 < R) = Pr PN

(29)

To determine the structure of the optimal Q, let Q have an eigendecomposition Q = UΛU† where Λ = diag(λ1 , · · · , λNt ) is the matrix of eigenvalues with λ1 ≥ · · · ≥ λNt and U is an unitary matrix. First, we notice that XNt g0† Qg0 = (U† g0 )† Λ(U† g0 ) ∼ g0† Λg0 = λi |g0i |2 (30) i=1

where we have used the fact that if g0 ∼ CN (0, σ02 I), then U† g0 ∼ g0 . According to (29) and (30), we have  ρ0 g0† Λg0 Pr(C1 < R) = Pr PN −1 kE† g k2 + 1 k k k=1 ρk (Nk − 1)  1 + ρ0 h†0 UΛU† h0 > −1 . (31) R 2 From (31), we know that for the optimal Λ, the optimal U should †

1+ρ h UΛU† h

0 0 0 maximize h†0 UΛU† h0 , since − 1 is increasing 2R with h†0 UΛU† h0 but decreasing with R, in other words, a larger h†0 UΛU† h0 will allow a larger R without violating the outage constraint Pr(C1 < R) ≤ ǫ. From this fact, we can reveal the structure of U. Let U† h0 = y = [y1 , · · · , yNt ]T . Then kyk2 = kh0 k2 . With this, we write

h†0 UΛU† h0 = y† Λy ≤ λ1 kyk2 = λ1 kh0 k2 .

(32)

Equality holds if y = [kh0 k, 0, · · · , 0]T . It follows from U† h0 = y that U = [h0 /kh0 k, u2 , · · · , uNt ]. According to (30), (31) and (32), we have P t 2  ρ0 N i=1 λi |g0i | Pr(C1 R 2 From (33), we know that for the optimal λ1 , smaller values of λ2 , · · · , λNt will allow a larger R without violating the outage

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