How to manage resources to provide physical layer security: Active versus passive adversary?

Physical Communication 27 (2018) 143–149

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How to manage resources to provide physical layer security: Active versus passive adversary? Mohammad Reza Abedi, Nader Mokari *, Hamid Saeedi Tarbiat Modares University, Islamic Republic of Iran

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Article history: Received 26 August 2017 Received in revised form 8 January 2018 Accepted 1 February 2018 Available online 6 February 2018 Keywords: Cooperative jamming Imperfect channel state information Physical layer security Active and passive adversary Semi-definite programming

a b s t r a c t This paper studies the required physical (e.g. relays and jammers) and radio resources (e.g. power) to provide physical layer security for relay and friendly jammer assisted multiple-input and single-output transmissions in the presence of multiple active and passive adversaries. The passive adversaries are half duplex and only able to overhear the transmissions from the legitimate transmitter to the legitimate receiver, while the active adversaries are full duplex and able to jam and eavesdrop simultaneously. Since the channel information between adversaries and other nodes are uncertain, robust optimization methods are considered. In this regard, the main aim is to maximize the worst case secrecy rate subject to the normalized transmit power constraints of legitimate transmitter, friendly jammer and relay, and channel state information uncertainty constraints. Through several examples, we then investigate the required increase in physical and radio resources to maintain secure communication when passive adversaries upgrades themselves to active adversaries. © 2018 Elsevier B.V. All rights reserved.

1. Introduction There has been a growing interest to provide security at the physical layer against potential adversary. The theoretical basis of physical layer security (PLS) was initiated by Wyner [1]. Accordingly, secrecy rate is defined as the achievable rate between the transmitter and receiver minus the rate from the transmitter to the eavesdropper. If the former rate is greater than the latter rate, the secrecy rate will be zero. Controlled interference, or artificial noise can be used to increase the secrecy rate and degrade the decoding ability of the adversaries. This is achieved with transmitters having multiple antennas [2–4]. On the other hand, the jamming signals of external relays can be used to degrade the adversary abilities [5–11]. In these works, it is assumed that the channels state information (CSI) between all nodes, including the CSI between adversaries and legitimate nodes, are perfectly known at which might not be a practical assumption. Accordingly, CSI uncertainty in PLS systems has been considered in [12–15]. It has to be mentioned that majority of the works on PLS assume a passive eavesdropper or adversary (PA) where the eavesdropper can only intercept the transmitted data. However, the adversary can upgrade itself to be able to also send jamming signals to destroy the transmitted data as in [16,17] or as in our prior works [18,19].

* Corresponding author.

E-mail addresses: [email protected] (M.R. Abedi), [email protected] (N. Mokari), [email protected] (H. Saeedi). https://doi.org/10.1016/j.phycom.2018.02.004 1874-4907/© 2018 Elsevier B.V. All rights reserved.

This case of active adversary (AA) will significantly degrade the secrecy rate and usually the secrecy rate will become zero.1 In this paper, we first assess the effect of such an upgrade on the adversary side on the secrecy rate. More importantly, in case of such an upgrade in the adversary side, we want to answer the following important question: Is it possible to maintain the secrecy rate at the level of the PA case at all and if so, what will be the required physical (i.e., number of friendly jammers and relays) and radio (i.e., the normalized transmit power) resources? To answer this question, a system model including a single legitimate transmitter (LT) (source), a single legitimate receiver (LR) (destination), multiple adversaries, jammers, and decode-andforward (DF) relays is assumed. To reduce the total throughput of network, AA can act as a jammer. This can create unfavorable conditions for secure communication. We then consider two scenarios. In the first one referred to as PA, the adversaries are eavesdropping the transmitted data from the LT to the LR. In the second one referred to as AA, the adversaries are eavesdropping the transmitted data from the LT to the LR while also sending jamming signals over it. In this paper, the CSI values between legitimate and adversary nodes are assumed to be uncertain and to take this issue 1 It is important to note that when the adversary becomes active, it can simultaneously overhear and jam the signal through its full duplex transceiver. In particular, we assume that the FD transceiver of the eavesdropper is equipped with an ideal self-interference canceler such that it can overhear the signal with the same quality as the PA case despite transmitting jamming signals towards the legitimate receiver.

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into account, robust resource allocation based on the worst-case approach are proposed. This paper is organized as follows. We present the notation and assumptions in the next section. In Sections 3 and 4, the corresponding optimization problems for the two proposed scenarios are provided. In Section 5, the simulation results of the two proposed scenarios are studied and we conclude the paper in Section 6. 2. Notations and assumptions In this paper, E{·} represents expectation. (·)H and ∥ · ∥ are the Hermitian transpose and the Euclidean norm, respectively. (·)† is the pseudo-inverse. tr(·) denotes the trace operator. I denotes an identity matrix. HN represents an N × N Hermitian matrix set. CM ×N denotes an M × N complex matrix set. In addition, A ≻ 0 denotes the positive definiteness of A. We let J , K and Q denote the set of jammers, relays and adversaries, respectively. The LT, jammer j ∈ J , relay k ∈ K and adversary q ∈ Q are equipped with Ns , Nj , Nk and Nq transmit antennas, respectively. For both AA and PA scenarios, we let hsd ∈ C1×Ns and hsq ∈ 1×Ns C represent the channel vectors from the LT to the destination and the qth adversary, respectively. In addition, hjd ∈ C1×Nj and hjq ∈ C1×Nj denote the channel vectors from the jth jammer to the LR and the qth adversary, respectively. Accordingly, hkd ∈ C1×Nk and hkq ∈ C1×Nk denote the channel vectors from the kth relay to the LR and the qth adversary, respectively. For the AA scenario, hqd ∈ C1×Nq and hqk ∈ C1×Nq represent the channel vectors from the qth AA to the LR and kth relay, respectively. gq ∈ C1×Nq denote the loop interference channel at the qth AA. The received noise at the LR, the kth relay and the qth adversary are assumed to be circular complex Gaussian random variables with zero-mean and ζd2 , ζk2 and ζq2 variances, respectively. For the sake of simplicity, it is assumed that ζd2 = ζk2 = ζq2 = ζ 2 . We model the channel vectors hsq , hjq , hkq , hqk and hqd as

˜ sq + ehsq , hsq = h

(1)

˜ jq + eh , hjq = h jq

(2)

˜ kq + eh , hkq = h kq ˜ hqk = hqk + eh ,

(3)

˜ qd + eh , hqd = h qd

(5)

(4)

qk

˜ sq , h˜ jq , h˜ kq , h˜ qk and h˜ qd represent the estimated value of the where h channels and ehsq , ehjq , ehkq , ehqk and ehqd denote the corresponding CSI errors, respectively. It is assumed that the channel mismatches lie in a bounded set [12], i.e., Ehsq = {ehsq : ∥ehsq ∥2 ≤ ϵh2sq }, Ehjq =

{ehjq : ∥ehjq ∥ ≤ ϵ }, Ehkq = {ehkq : ∥ehkq ∥ ≤ ϵ }, Ehqk = {ehqk : ∥ehqk ∥2 ≤ ϵh2qk }, Ehqd = {ehqd : ∥ehqd ∥2 ≤ ϵh2qd }, where ϵh2sq , ϵh2kq , ϵh2qk , ϵh2jq and ϵh2qd are known constants. 2

2 hjq

2

2 hkq

3. The PA scenario In this section, we consider a multiple-input and single-output (MISO) communication system with an LT, a set J = {1, 2, . . . , J } of J = |J | jammers, a set K = {1, 2, . . . , K } of K = |K| DF relays, an LR, and a set Q = {1, 2, . . . , Q } of Q = |Q| PAs. In this scenario, adversaries are only able to overhear the link between LT and LR. The data rate through the kth DF relay link assisted by friendly jammer j can be written as [20] RD kj

=

1 2

[

{

min log2

( 1+

hsk Gs hH sk

ζ 2 + hjk Gj hHjk

) ,

( log2

1+

H hkd Gk hH kd + hsd Gs hsd

)}] ,

ζ 2 + hjd Gj hHjd

(6)

where factor 21 appears because the relay transmission is divided into two stages. The transmitted signal by LT and its covariance matrix are denoted by zs and Gs = E{zs zH s }, respectively. The normalized transmit power constraint set for LT is represented by G s = {Gs : Gs ⪰ 0, tr(Gs ) ≤ Ps } where Ps is the maximum predefined normalized transmit power for it. The transmitted signal by jammer j and its covariance matrix are denoted by zj and Gj = E{zj zHj }, respectively. The normalized transmit power constraint set for jammer j is represented by G j = {Gj : Gj ⪰ 0, tr(Gj ) ≤ Pj } where Pj is the maximum predefined normalized transmit power for it. Gk is the covariance matrix of the signal transmitted by the kth relay, zk , which is given by Gk = E{zk zH k }. The power constraint is imposed such that Gk ∈ G k = {Gk : Gk ⪰ 0, tr(Gk ) ≤ Pk } where Pk is the maximum allowable transmission power for the kth relay. The qth PA overhears both hops, and its data rate is written as REkjq =

1 2

( log2

1+

Θ (Gs , ehsq ) + Θ (Gk , ehkq )

ζ 2 + Θ (Gj , ehjq )

) .

(7)

˜ sq + ehsq )Gs (h˜ sq + ehsq )H , Θ (Gk , eh ) = where Θ (Gs , ehsq ) = (h kq ˜ kq + eh )Gk (h˜ kq + eh )H and Θ (Gj , eh ) = (h˜ jq + eh )Gj (h˜ jq + (h kq kq jq jq ehjq )H . Accordingly, by exploiting relay k and friendly jammer j, the secrecy rate between LT and LR overheard by PA q can be obtained as E RSkjq = max 0, RD kj − Rkjq .

{

}

(8)

Therefore, the optimization problem can be formulated as follows Problem OPA : max Gs ∈G s ; Gk ∈G k ; Gj ∈G j

min

eh ∈Eh ; sq sq eh ∈Eh ; kq kq eh ∈Eh jq jq

s.t.

R˜ S ,

(9a)

hsk Gs hH sk

ζ + 2

hjk Gj hH jk

≤

H hkd Gk hH kd + hsd Gs hsd

ζ 2 + hjd Gj hHjd

, ∀k, j, (9b)

tr(Gs ) ≤ Ps ,

(9c)

tr(Gj ) ≤ Pj , ∀j,

(9d)

tr(Gk ) ≤ Pk , ∀k,

(9e)

∥ehsq ∥2 ≤ ϵh2sq , ∀q,

(9f)

∥ehjq ∥ ≤ ϵ , ∀j, q,

(9g)

∥ehkq ∥ ≤ ϵ

(9h)

2

2

Gs ⪰ 0,

2 hjq

2 hkq

, ∀k, q,

(9i)

Gj ⪰ 0, ∀j,

(9j)

Gk ⪰ 0, ∀k.

(9k)

where R˜ S = arg maxk∈K arg maxj∈J arg minq∈Q RSkjq . We remind that the difficulty in solving problem OPA comes from the inner minimization over ehsq , ehkq , and ehjq where it is a non-convex problem due to the non-convexity of the objective function and constraints. However, as shown in [12,18], through a proper transformation, problem OPA can be converted to a solvable quasiconvex optimization problem. Then, the conventional bisection method can be used to solve the problem. On the other hand, one can also use the Charnes–Cooper transformation [21] to transform

M.R. Abedi et al. / Physical Communication 27 (2018) 143–149

this problem into a single semidefinite programming problem. The optimization problem OPA is equivalent to Problem max

eh

Gs ∈G s ; Gk ∈G k ; Gj ∈G j

∈Ehsq ; sq eh ∈Eh ; kq kq eh ∈Eh jq jq

( 2 )( ) ζ + hjk Gj hHjk + hsk Gs hHsk ζ 2 + Θ (Gj , ehjq ) ( )( ), ζ 2 + hjk Gj hHjk ζ 2 + Θ (Gj , ehjq ) + Θ (Gs , ehsq ) + Θ (Gk , ehkq ) (10a) s.t. (9b), (9c), (9d), (9e), (9f ), (9g), (9h), (9i), (9j), (9k).

(10b)

At first, the jammers normalized transmit powers are obtained. Accordingly, the corresponding optimization problem is written as follows: Problem O˜ PA : Gj ∈G j ehjq ∈Ehjq

˜ jq + eh )Gj (h˜ jq + eh )H , (h jq jq hjd Gj hH jd H hjk Gj hjk

s.t.

˜ jq + eh )G∗j (h˜ jq + eh )H , (h jq jq

min

(11a)

The above optimization problem is a convex problem and thus strong duality holds for (16). The worst-case channel mismatch for (16) is given by

˜ jq G∗j (λI + G∗j )−1 , e∗hjq = −h γ, ] [ ˜ Hjq G∗j h λI + G∗j s.t. ˜ jq G∗j h˜ jq G∗j h˜ Hjq − λϵh2 − γ ⪰ 0. h jq

max

λ≥0,γ

(11b)

= 0.

(11c)

(18b)

min

˜ kq + eh )Gk (h˜ kq + eh )H , (h kq kq

min

Gk ∈G k ehkq ∈Ehkq

(19)

(9e), (9h), (9k).

s.t.

This optimization problem can be converted to

max ν,

Problem Oˆ 1PA :

(12a)

Gj ∈G j ,ν

ν,

min

˜ jq + eh )Gj (h˜ jq + eh )H ≥ ν, s.t. (h jq jq

(12b)

Gk ∈G k ,ν

s.t.

(9d), (9g), (9j), (11b), (11c),

)

+ h˜ jq Gj h˜ Hjq − ν ≥ 0,

− ehjq eHhjq + ϵh2jq ≥ 0.

,

(20b) (20c)

where constraints (20b) and (20c) can also be expressed as

(13b)

( ) − ehkq Gk eHhkq − 2Re h˜ kq Gk eHkq − h˜ kq Gk h˜ Hkq + ν ≥ 0,

(21a)

− ehkq eHhkq + ϵh2kq ≥ 0.

(21b)

(14)

Using S-procedure, there exists an ehkq ∈ CNk satisfying both of the above inequalities if and only if there exists a µ ≥ 0 such that

[ ] µINk − Gk −Gk h˜ Hkq ⪰ 0. −h˜ kq Gk −h˜ Hkq Gk h˜ kq − µϵh2kq + ν

Problem Oˆ 2PA :

( ) ψ + tr Gk h˜ Hkq h˜ kq + µϵh2kq , Gk ∈G k ,µ≥0,ψ≥0 [ ] µINk − Gk −Gk h˜ Hkq s.t. ⪰ 0, −h˜ kq Gk ψ min

O2PA :

Problem ˜

(15a)

Gj ∈G j ; µ≥0;ψ≥0

[ ] µINj + Gj Gj h˜ Hjq s.t. ⪰ 0, ˜ jq Gj h ψ

(9e), (9k), µ ≥ 0, ψ ≥ 0. (15b)

Problem ˜ is a semidefinite program (SDP) that consists of a linear objective function and LMI constraints. Therefore, we can solve this problem efficiently and obtain the optimal solution G∗j . Note that although ehjq does not explicitly appear in O˜ 2PA , the optimal robust covariance G∗j is already based on the hidden worstcase e∗h that can be expressed explicitly through the following

(22)

˜ H Gk h˜ kq − µϵ 2 + ν , where ψ ≥ 0, Oˆ HDR can Letting ψ = −h kq 3 hkq then be expressed as

jq

of LMI in (14), ψ ≥ 0, O˜ 1PA can then be expressed as

(9d), (9j), (11b), (11c).

≤ϵ

2 hkq

(9e), (9k),

˜ H Gj h˜ jq − µϵ 2 − ν , where because of the property Letting ψ = h jq h

˜ Hjq h˜ jq ) − µϵh2 , max − ψ + tr(Gj h jq

˜ kq + eh )Gk (h˜ kq + eh )H ≤ ν, (h kq kq

(13a)

To make the problem more tractable, (13a) and (13b) turn into linear matrix inequalities (LMIs), using S-procedure [22,23]. This type of S-procedure is called S-procedure in complex linear space that is explained in [23]. Using S-procedure, there exists an ehjq ∈ CNj satisfying both the above inequalities if and only if there exists a µ ≥ 0 such that

[ ] ˜ Hjq µINj + Gj Gj h ˜ jq Gj h˜ Hjq Gj h˜ jq − µϵh2 − ν ⪰ 0. h jq

(20a)

ehkq eH hkq

where the constraints (12b) and (9g) can also be expressed as

jq

(18a)

Now we obtain the relays transmission powers. Accordingly, the corresponding optimization problem is written as follows:

Problem O˜ 1PA :

O2PA

(17)

where λ is the solution of the following SDP problem:

The maximin problem O˜ PA can be transformed to

(

(16b)

Problem Oˆ PA :

= 0,

H ˜ ehjq Gj eH hjq + 2Re hjq Gj ehjq

(16a)

jq

s.t. ∥ehjq ∥2 ≤ ϵh2jq .

min

max min

problem: eh

O1PA :

145

(23a)

(23b)

Problem Oˆ 2PA is a SDP that consists of a linear objective function and LMI constraints. Therefore, we can obtain the optimal solution G∗k of Problem Oˆ 2PA . The mismatch between the CSI at the kth relay and qth adversary, e∗h can be expressed as kq

min eh

˜ kq + eh )Gk (h˜ kq + eh )H , (h kq kq

(24a)

kq

s.t. ∥ehkq ∥2 ≤ ϵh2kq .

(24b)

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M.R. Abedi et al. / Physical Communication 27 (2018) 143–149

Obviously, the strong duality holds for the above convex optimization problem. The worst-case channel mismatch for (24) is given as ∗ †

˜ kq Gk (λI + Gk ) , ehkq = −h ∗

∗

(25)

´ , Charnes–Cooper transformation, by letting µ = µ/γ ´ , ψ = ψ/γ PA ´ ˇ and Gs = Gs /γ for some γ > 0, and rewriting O2 as Problem Oˇ 3PA : min ´ s ,µ, ´ G ´ ψ,γ

where λ is the solution of the following SDP problem:

γ, ] [ ˜ Hkq G∗k h λI + G∗k s.t. ˜ kq G∗k h˜ kq G∗k h˜ Hkq − λϵh2 − γ ⪰ 0. h kq

max

(26b)

´ ≤ γ tr(Gk hkd hkd ) + ] [ µ ´ INs − G´ s −G´ s h˜ Hsq ⪰ 0, −h˜ sq G´ s ψ´

Problem Oˇ PA : max

( 2 )( ) ζ + hsk Gs hHsk ζ 2 + Θ (Gj , ehjq )

min

s.t.

(9b), (9c), (9f ), (9i).

(27b)

The optimization problem Oˇ PA can be transformed to Problem Oˇ 1PA : ( )( ) ζ 2 + Θ (G∗j , e∗gjq ) ζ 2 + hsk Gs hHsk

max

ν

Qs ∈G s ,ν

(

)

(28b) (9b), (9c), (9f ), (9i).

(32f) (32g)

˜ sq + ehsq )G∗s (h˜ sq + ehsq )H , (h

(33a)

∥ehsq ∥2 ≤ ϵh2sq .

(33b)

The above problem is a non-convex problem since we want to maximize a convex function. However, we can still obtain the global optimum by solving its dual problem given by

˜ sq G∗s (λI − G∗s )† , e∗hsq = h

(34)

γ, [ ] ˜ Hsq λI − G∗s G∗s h s.t. ˜ sq G∗s −h˜ sq G∗s h˜ Hsq − λϵh2 − γ ⪰ 0. h sq

max

(29a) (29b)

˜ sq Gs h˜ Hsq − where INs is an identity matrix of NS dimension. Letting −h µϵh2sq − ζ 2 − Θ (G∗j , e∗hjq ) − Θ (G∗k , e∗hkq ) + ζν2 = ψ , Oˇ 1PA can be converted into Problem Oˇ 2PA :

(35a) (35b)

Note that (35) is a SDP and hence can be solved efficiently using the interior-point method [22]. The pseudo-code of the proposed algorithm is illustrated in Fig. 1. To verify the convergence of the proposed algorithm to the sub-optimal solution of the proposed non-convex problem, we argue as follows: Let (15), (16), (23), (ϱ )

(ϱ+1)

(24), (32) and (33) be solvable. With a given Gs , ehsq solution of (33), we get that

(

(ϱ )

(ϱ ) while ehsq

(ϱ )

(ϱ+1)

R˜ S G(sϱ) , Gj , Gk , ehsq

is optimal

is only its feasible solution. Accordingly, (ϱ)

(ϱ )

, ehjq , ehkq

)

) ( (ϱ ) (ϱ ) (ϱ ) (ϱ ) (ϱ) ≤ R˜ S G(sϱ) , Gj , Gk , ehsq , ehjq , ehkq

(36)

Likewise, for (15), (16), (23), (24) and (32) the same argument is true. It is naturally concluded that

min

Gs ∈G s ,µ,ψ

(

( ) ζ ψ + µεh2sq + ζ 2 + tr(Gs h˜ Hsq h˜ sq ) + Θ (G∗j , e∗hjq ) + Θ (G∗k , e∗hkq ) ( )( , ) ζ 2 + Θ (G∗j , e∗hjq ) ζ 2 + hsk Gs hHsk 2

s.t.

(32e)

λ≥0,γ

Using S-procedure, there exists an ehsq satisfying both the above inequalities if and only if there exists a µ ≥ 0 such that ⎡ ⎤ −Gs h˜ Hsq µINs − Gs ⎣ ν ⎦ −h˜ sq Gs −h˜ sq Gs h˜ Hsq − µϵh2sq − ζ 2 − Θ (G∗j , e∗gjq ) − Θ (G∗k , e∗hkq ) + 2 ζ ⪰ 0, (30)

[ ] µINs − Gs −Gs h˜ Hsq ⪰ 0, −h˜ sq Gs ψ

(32d)

where λ is the solution of the following problem:

The constraints (28b) and (9f) can also be expressed as

( ) − ehsq Gs eHhsq − 2Re h˜ sq Gs eHhsq − h˜ sq Gs h˜ Hsq − ζ 2 ν − Θ (G∗j , e∗hjq ) − Θ (G∗k , e∗hkq ) + 2 ≥ 0, ζ − ehsq eHhsq + ϵh2sq ≥ 0.

(32c)

We will explicitly express ehsq under the norm-bounded constraint. The problem is formulated as

s.t.

s.t. ζ 2 ζ 2 + Θ (Gs , ehsq ) + Θ (G∗j , e∗hjq ) + Θ (G∗k , e∗hkq ) ≤ ν,

(32b)

∗

min

(28a)

,

G´s ⪰ 0, µ ´ ≥ 0, ψ´ ≥ 0.

ehsq ∈Ehsq

,

´

tr(Gs hH sd hsd )

´ s ) ≤ γ Ps , tr(G

), ( ζ 2 ζ 2 + Θ (Gj , ehjq ) + Θ (Gs , ehsq ) + Θ (Gk , ehkq ) (27a)

∗ H

tr(Gs hH sk hsk )

With solutions for G∗k , G∗j , e∗h and e∗h , we can formulate the jq kq optimization problem over Gs as

Gs ∈G s ehsq ∈Ehsq

(32a)

( ´ + µϵ s.t. ζ 2 ψ ´ h2sq + γ ζ 2 + tr(G´ s h˜ Hsq h˜ sq ) ) + γ Θ (G∗j , e∗hjq ) + γ Θ (G∗k , e∗hkq ) ≤ t , ) )( ( ζ 2 + Θ (G∗j , e∗hjq ) tr(G´ s hHsk hsk ) + γ ζ 2 = 1,

(26a)

λ≥0,γ

t,

(ϱ+1)

R˜ S G(sϱ+1) , Gj

(ϱ+1)

, Gk

(ϱ+1)

(ϱ+1)

(ϱ+1)

, ehsq , ehjq , ehkq ( ) (ϱ ) (ϱ ) (ϱ ) (ϱ ) (ϱ ) ≤ R˜ S G(sϱ) , Gj , Gk , ehsq , ehjq , ehkq

) (37)

(31a)

(ϱ ) (ϱ ) (ϱ) (ϱ ) (ϱ ) (ϱ ) Therefore, At each iteration, R˜ S (Gs , Gj , Gk , ehsq , eh , eh ) is

(31b)

monotonically decreasing. Consequently, the convergence of the proposed algorithm is verified.

jq

kq

(9b), (9c), (9i), ψ ≥ 0, µ ≥ 0.

4. The AA scenario

Problem Oˇ 2PA is quasi-convex that consists of a linear fractional objective function with a positive denominator. We use the

Here, a MISO communication system with an LT, J jammers, K DF relays, an LR, and Q AA’s is considered. In this scenario,

M.R. Abedi et al. / Physical Communication 27 (2018) 143–149

147

Fig. 1. The proposed resource allocation algorithm.

adversaries are able to overhear and jam the legitimate links simultaneously. The data rate between LT and LR assisted by friendly jammer j and relay k can be written as [20] RD kjq

=

1

[

{

(

min log2

1+

ζ 2 + Θ (Gq2 , ehqk ) + hjk Gj hHjk )}] ( H hrd Gk hH kd + hsd Gs hsd log2 1 + , ζ 2 + Θ (Gq1 , ehqd ) + hjd Gj hHjd 2

(38)

=

2

log2

1+

)

Θ (Gs , ehsq ) + Θ (Gk , ehkq )

ζ 2 + gq Gq1 gHq + gq Gq2 gHq + Θ (Gj , ehjq )

. (39)

Consequently, the secrecy rate of the legitimate link assisted by relay k and friendly jammer j can be written as E RSkjq = max 0, RD kjq − Rkjq .

{

}

(40)

Accordingly, the proposed optimization problem can be expressed as follows: Problem OAA : max Gs ∈G s ; Gk ∈G k ; Gj ∈G j

min

Gq ∈G q ,eh

∈Ehsq ; ∈Eh ;

R˜ S ,

(41a)

sq ,eh

eh Eh kq jq jq kq eh ∈Eh ,eh ∈Eh qk qk qd qd

∈

s.t.

hsk Gs hH sk

ζ 2 + Θ (Gq , ehqk ) + hjk Gj hHjk H hkd Gk hH kd + hsd Gs hsd

ζ 2 + Θ (Gq , ehqd ) + hjd Gj hHjd tr(Gq ) ≤ Pq , ∀q,

≤

(41f)

The solution of OAA is similar to the solution of OPA .

versary q to the destination and its covariance matrix are denoted by zq1 and Gq1 = E{zq1 zH q1 }, respectively. The normalized transmit power constraint set for this adversary is represented by G q1 = {Gq1 : Gq1 ⪰ 0, tr(Gq1 ) ≤ Pq } where Pq is the maximum tolerable transmission power for it. Moreover, the transmitted signal from adversary q to relay k and its covariance matrix are denoted by zq2 and Gq2 = E{zq2 zH q2 }, respectively. The normalized transmit power constraint set for this adversary is represented by G q1 = {Gq2 : Gq2 ⪰ 0, tr(Gq2 ) ≤ Pq } where Pq is the maximum tolerable transmission power for it. Thus, the data rate of adversary q overhearing both hops can be obtained as

(

(41e)

, ∀q,

Gq ⪰ 0,

,

= (h˜ qk + ehqk )Gq2 (h˜ qk + ehqk )H . The transmitted signal from ad-

1

∥ehqd ∥ ≤ ϵ

2 hqd

(9c), (9d), (9e), (9f ), (9g), (9h), (9i).

˜ qd + eh )Gq1 (h˜ qd + eh )H and Θ (Gq2 , eh ) where Θ (Gq1 , ehqd ) = (h qd qd qk

REkjq

(41d)

2

)

hsk Gs hH sk

∥ehqk ∥2 ≤ ϵh2qk , ∀q, k,

(41b)

, ∀k, j, q, (41c)

5. Simulation results In this section, the numerical results on the secrecy rate of the proposed systems are studied. The number of antennas for LT, each friendly jammer, each relay, and each AA is assumed to be four, i.e., Ns = Nj = Nk = 4. The obtained results are based on th Monte Carlo experiments consisting of 1000 independent trials. The normalized background noise power is set to 0 dB, for all legitimate and adversary nodes as in [12]. The normalized transmit power of LT, each friendly jammer, and each relay is assumed to be 5 dB, i.e., Ps = Pj = Pk = P = 5 dB, ∀j, k, as in [12]. The considered network topology, is illustrated in Fig. 2, in which the LT and LR are located at coordinates (−25,0) and (25,0) and relays, jammers, and adversaries are placed uniformly at random over a 50 m × 50 m square. The channel gain hab between any two nodes a −c /2 and b is given by hab = dab ejθab where dab is the distance between nodes a and b, c = 3.5 is the path loss coefficient, and θab indicates independent phase fading and is uniformly distributed over [0, 2π ) for any a, b. Moreover, we assume the channel mismatch is ϵh2sq =

ϵh2jq = ϵh2kq = ϵh2qk = ϵh2qd = 0.5, ∀j, k, q.

5.1. Effect of maximum allowable transmission power Fig. 3 shows the secrecy rate as a function of the maximum normalized transmit power, Ps = Pj = Pk = P /3, ∀j, k, under an individual power constraint. In general, by increasing the normalized transmit power P, the secrecy rate is also increased for both scenarios. We have considered 3 systems models where we set the number of relays and jammers to 1, 5 and 10. For similar system models, the PA scenario exhibits much higher secrecy rate than AA as expected. As an example, for AA with Pq = PQ = 5, ∀q dB, the secrecy rate is zero when the normalized transmit power is less than 10 dB. However, by increasing it, the secrecy rate of the AA scenario increases and can be maintained at the level of the PA scenario. Now consider the system model with 1 jammer and 1 relay. With the normalized transmit power of 10 dB, we can achieve a

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Fig. 4. Secrecy rate, R˜ S , vs. the channel mismatch, ϵh2sq = ϵh2

qd

ϵ

2 hqk

= ϵh2jq = ϵh2kq =

= ϵ , ∀j, k, q for AA and PA scenarios. Ps = Pj = Pk = P /3, ∀j, k, P = 20 dB, 2

Ns = Nq = Nj = Nk = 4, ∀j, k, q.

Fig. 2. The considered network topology.

Fig. 5. Secrecy rate, R˜ S , vs. number of relay, K for AA and PA scenarios. The adversaries and jammers. System parameters: Ps = Pk = Pj = P /3, ∀j, k dB. P = 25 dB. ϵh2sq = ϵh2 = ϵh2 = ϵh2 = ϵh2 = 0.1, ∀j, k, q, Ns = Nq = Nj = Nk = 4, ∀j, k, q.

qd

jq

kq

qk

secrecy rate as high as 0.7 bits/s/Hz. To maintain this rate for the AA scenario with PQ = 5 dB, we need to increase the normalized transmit power to 13 dB, i.e., we have to double the transmit power. For larger values of the normalized transmit power, the gap between the AA and PA performance decreases.

with different values of PQ . The results are reported in Fig. 5. As can be seen, for fixed normalized transmit power, when the adversary is upgraded to an active one, the resulting secrecy rate decreases. Nevertheless, we can maintain the original secrecy rate by increasing the number of relays and jammers. As an example, consider a system model with J = 5 and K = 1. In PA scenario, we reach a secrecy rate as high as 3.3 bits/s/Hz. Now assume that the adversary is upgraded to be able to send jamming signals with PQ = 10 dB. In this case to maintain the rate, if we want to keep J = 5 (the same as PA scenario), we have to increase the number of relays from 1 to 4. Alternatively, we can double the number of jammers, i.e., J = 10, and increase the number of relays from K = 1 to K = 2.

5.2. Effect of the channel mismatch

6. Conclusion

Fig. 3. Secrecy rate, R˜ S , vs. the maximum normalized transmit power constraint, Ps = Pj = Pk = P /3, ∀j, k, for AA and PA scenarios. P = 25 dB. ϵh2sq = ϵh2 = ϵh2 = qd

jq

ϵh2kq = ϵh2qk = 0.1, ∀j, k, q, Ns = Nq = Nj = Nk = 4, ∀j, k, q.

Fig. 4 shows the secrecy rate as a function of the channel mismatch, ϵh2sq = ϵh2 = ϵh2 = ϵh2 = ϵh2 = ϵ 2 , ∀j, k, q. As jq

kq

qk

qd

can be seen, the secrecy rate decreases as the channel mismatch increases. 5.3. Effect of multi jammer/relay In this part we set P = 25 dB and obtain the secrecy rate for different number of relays and jammers for PA and AA scenarios

In this paper, we investigated the required physical and radio resources to provide physical layer security when passive adversary upgrades itself to active adversary. To do so, we considered PA and AA scenarios where for both cases, we aimed at maximizing the worst case secrecy rate subject to the legitimate transmitter, friendly jammer and relay normalized transmit power constraints. Consequently, robust transmit covariance matrices were obtained by maximizing the worst-case secrecy rate. In this regard, the non-convex optimization problems are converted to quasi-convex

M.R. Abedi et al. / Physical Communication 27 (2018) 143–149

problems. Finally, we compared the resulting rate of AA and PA scenarios for different values of maximum normalized transmit power as well as different number of relays and jammers. Our results indicate that in majority of the cases, by a reasonable increase in the normalized transmit power and/or number of relays/jammers, we can maintain the same secrecy rate as the PA scenario for the AA scenario. Acknowledgments This work was supported by the Iran National Science Foundation under Grant 92028129. The associate editor coordinating the review of this paper and approving it for publication was A. Ozcelikkale. References [1] A.D. Wyner, The wire-tap channel, Bell Syst. Tech. J. 54 (8) (1975) 1355–1387. [2] S. Goel, R. Negi, Guaranteeing secrecy using artificial noise, IEEE Trans. Wirel. Commun. 7 (6) (2008) 2180–2189. [3] A.L. Swindlehurst, Fixed SINR solution for the MIMO wiretap channel, in: Proceeding of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP, Taipei, Taiwan, 2009, pp. 2437–2440. [4] Q. Li, W. Ma, A. So, Safe convex approximation to outage-based MISO secrecy rate optimization under imperfect CSI and with artificial noise, in: Proceeding of the Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, Pacific Grove, CA, USA, pp. 207–211, 2011. [5] L. Dong, Z. Han, A.P. Petropulu, H.V. Poor, Improving wireless physical layer security via cooperating relays, IEEE Trans. Signal Process. 58 (3) (2010) 1875– 1888. [6] I. Krikidis, J.S. Thompson, S. McLaughlin, Relay selection for secure cooperative networks with jamming, IEEE Trans. Wirel. Commun. 8 (10) (2009) 5003– 5011. [7] S. Gerbracht, C. Scheunert, E.A. Jorswieck, Secrecy outage in MISO systems with partial channel information, IEEE Trans. Inf. Theory Forensics Secur. 7 (2) (2012) 704–716. [8] S. Luo, J. Li, A. Petropulu, Outage constrained secrecy rate maximization using cooperative jamming, in: Statistical Signal Processing Workshop (SSP), Ann Arbor, MI, USA. [9] Z. Ding, M. Peng, H.-H. Chen, A general relaying transmission protocol for MIMO secrecy communications, IEEE Trans. Commun. 60 (11) (2012) 3461– 3471. [10] Y. Liu, J. Li, A.P. Petropulu, Destination assisted cooperativejamming for wireless physical-layer security, IEEE Trans. Inf. Forensics Secur. 8 (4) (2013) 682– 694. [11] N. Mokari, P. Azmi, H. Saeedi, Quantized ergodic radio resource allocation in OFDMA-based cognitive DF relay-assisted networks, IEEE Trans. Wirel. Commun. 12 (10) (2013) 5110–5123. [12] J. Huang, A.L. Swindlehurst, Robust secure transmission in MISO channels based on worst-case optimization, IEEE Trans. Signal Process. 60 (4) (2012) 1696–1707. [13] L. Zhang, Y.-C. Liang, Y. Pei, R. Zhang, Robust beamforming design: From cognitive radio MISO channels to secrecy MISO channels, in: Proceeding of the IEEE Global Telecommunications Conference, GLOBECOM, Hawaii, USA, 2009, pp. 1–5. [14] N. Mokari, S. Parsaeefard, H. Saeedi, P. Azmi, E. Hossain, Secure robust ergodic uplink resource allocation in relay-assisted cognitive radio networks, IEEE Trans. Signal Process. 63 (2) (2015) 291–304. [15] M.R. Abedi, N. Mokari, H. Saeedi, H. Yanikomeroglu, Secure robust resource allocation using full-duplex receivers, in: Proceeding of IEEE International Conference on Communication Workshop, ICCW, London, UK, 2015. [16] A. Chorti, S.M. Perlaza, Z. Han, H.V. Poor, On the resilience of wireless multiuser networks to passive and active eavesdroppers, IEEE J. Sel. Areas Commun. 31 (9) (2013) 1850–1863.

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Mohammad Reza Abedi received the M.Sc. degree in electrical engineering from the Amir Kabir University, Tehran, Iran. He is currently working as research assistant at Tarbiat Modares University, Tehran Iran. His research interests include wireless communications radio resource allocations, cooperative and spectrum sharing.

Nader Mokari Yamchi completed his Ph.D. studies in electrical Engineering at Tarbiat Modares University, Tehran, Iran in 2014. He joined the Department of Electrical and Computer Engineering, Tarbiat Modares University as an assistant professor in October 2015. He was also involved in a number of large scale network design and consulting projects in the telecom industry. His research interests include design, analysis, and optimization of communications networks.

Hamid Saeedi received the B.Sc. and M.Sc. degrees from the Sharif University of Technology, Tehran, Iran, in 1999 and 2001, respectively, and the Ph.D. degree from Carleton University, Ottawa, ON, Canada, in 2007, all in electrical engineering. From 2008 to 2009, he was a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA, USA. In 2010, he joined the Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran. His current research interests include coding and information theory, wireless communications, and resource allocation in conventional, heterogeneous, and cognitive radio networks. Dr. Saeedi has been a technical program committee member of some international conferences, including the Vehicular Technology Conference 2012, the Wireless Communication and Networking Conference 2014, 2015, and 2016, and the International Conference on Communications 2015. He was a recipient of some awards, including the Carleton University Senate Medal for Outstanding Academic Achievement, the Natural Sciences and Engineering Council of Canada Industrial Research and Development Fellowship, and the Ontario Graduate Scholarship. He is currently an Editor of the IEEE COMMUNICATIONS LETTERS.