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The Secure Degrees of Freedom of the MIMO

Multiple Access Channel with Multiple Unknown

Eavesdroppers

Mohamed Amir, Tamer Khattab, Tarek Elfouly, Amr Mohamed Qatar UniversityEmail: mohamed.amir, tkhattab, tarekfouly, [email protected]

AbstractWe investigate the secure degrees of freedom(SDoF) of a two-transmitter Gaussian multiple access wiretapchannel with multiple antennas at the transmitters, a legitimatereceiver and an unknown number of eavesdroppers eachwith a number of antennas less than or equal to a knownvalue NE . The channel matrices between the legitimatetransmitters and the receiver are available everywhere,while the legitimate pair have no information about theeavesdroppers channels. We provide the exact sum SDoF

for the considered system. We show that it is equal tomin{M1+ M2 NE,

1

2(max{M1, N}+ max{M2, N} NE), N}.

A new comprehensive upperbound is deduced and a newachievable scheme based on utilizing jamming is exploited. Weprove that Cooperative Jamming is SDoF optimal even withoutthe eavesdropper CSI available at the transmitters.

I. INTRODUCTION

The wiretap channel was first studied by Wyner [1], in

which a legitimate transmitter (Alice) wishes to send a

message to a legitimate receiver (Bob), and hiding it from

an eavesdropper (Eve). Wyner proved that Alice can send

positive secure rate using channel coding. He derived capacity-equivocation region for the degraded wiretap channel. Later,

Csiszar and Korner found capacity-equivocation the region for

the general wiretap channel [2], which was extended to the

Gaussian wiretap channel by Leung-Yan- Cheong and Hellman

[3].

A significant amount of work was carried thereafter to

study the information theoretic physical layer security for

different network models. The relay assisted wiretap channel

was studied in [4]. The secure degrees of freedom (SDoF)

region of multiple access channel (MAC) was presented in

[15]. The SDoF is the the pre-log of the secrecy capacity

region in the high-SNR regime. Using MIMO systems for

securing the message was an intuitive extension due to thespatial gain provided by multiple antennas. The MIMO wiretap

channel was studied in [5][13] and the secrecy capacity was

identified in [8].

All the aforementioned work assumes availability of either

partial or complete channel state information (CSI) at the

transmitter. Given that the eavesdropper is passive, it is of

high importance to study the case where the channel state

information is completely unknown to the legitimate users.

This work is supported by Qatar National Research Fund (QNRF) underthe National Priorities Research Program (NPRP) grant number NPRP 5-559-2-227. The statements herein are the sole responsibility of the authors.

The authors in [10], [11] study the secrecy capacity and

SDoF for different MIMO channels when the eavesdropper

channel is arbitrarily varying and known to the eavesdropper

only. In [10] the authors established the SDoF region for a

scenario comprising a single eavesdropper assuming a square

full-rank legitimate channel matrix (i.e. has zero null space).

Meanwhile, in [11] the authors addressed the SDoF of the

MAC assuming a single eavesdropper where all the trans-mitted symbols are intended for the legitimate receiver (no

exploitation of jamming).

Meanwhile, the idea of cooperative jamming was proposed

in [9], where some of the users transmit independent and iden-

tically distributed (i.i.d.) Gaussian noise towards the eaves-

dropper to improve the sum secrecy rate. Cooperative jamming

was used for deriving the SDoF for different networks. In [15],

cooperative jamming was used to jam the eavesdropper and

proved that the K-user MAC with single antenna nodes can

achieve K(K1)K(K1)+1 SDoF.

In this paper, we study the MIMO MAC with unknown

eavesdroppers CSI at the legitimate transmitters and receiver.

We provide a new upperbound for the achievable SDoF anddetermine the exact sum SDoF by providing an achievable

scheme. We show that our scheme is optimal and that the

achievable bound and the new upperbound are tight. Our

scheme requires neither cooperation between the legitimate

transmitters nor cooperation between any legitimate transmit-

ter and the legitimate receiver, and precoding matrices and the

jamming vectors are determined in distributed manner.

The major contributions of our work as compared to existing

literature can be summarized as follows: (i) our work incorpo-

rates the more general scenario of multiple eavesdroppers, (ii)

we assume no restrictions on the relation between the number

of antennas at transmitters and the receiver, (iii) we study

the more comprehensive case where all the eigenvalues of thelegitimate channel has non-zero vlaues1, (iv) we deduce a new

upperbound on the SDoF, and (v) we provide a scheme that

achieves the new upperbound. For the special case of a single

eavesdropper, our proposed scheme achieves a sum SDoF

superior to those reported in the literature such as in [11].

The paper is organized as follows. Section II defines the

system model and the secrecy constraints. The main results are

1The cases where some of the eigenvalues are equal to zero represent specialdegraded cases of the more general non-zero eigenvalues case, where theSDoF decreases for every zero eigenvalue till it collapses to the trivial caseof zero SDoF for all-zero eigenvalues.


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presented in Section III. In Section IV, the new upperbound is

derived and the achievable scheme is presented in Section V.

The paper is concluded in Section VI. We use the following

notation, a for vectors, A for matrices, A for the hermitian

transpose ofA, [A]+ for themax A, 0 and Null(A) to definethe nullspace ofA.

I I . SYSTEM MODEL

We consider a communication system of two transmitters

and a single receiver in vicinity of an unknown number of

passive eavesdroppers. Transmitters one and two are equipped

with M1 and M2 antennas, respectively. The legitimate re-

ceiver is equipped with N antennas, while the j th eavesdrop-

per is equipped with NEj NE antennas. Let xi denote theMi 1 vector of Gaussian coded symbols to be transmittedby transmitter i, where i {1, 2}. We can write the receivedsignal at the legitimate receiver at time (sample) k as

Y(k) =

2

i=1

HiVixi(k) + n(k) (1)

and the received signal at the jth eavesdropper as

Zj(k) =2i=1

Gi,j(k)Vixi(k) + nEj(k), (2)

where Hi is the N Mi matrix containing the channelcoefficients from transmitter i to the receiver, Gi,j(k) is theNEj Mi matrix containing the channel coefficients fromtransmitteri to the eavesdropperj , Viis the precoding unitary

matrix (i.e. ViVi = I) at transmitter i, n(k) and nEj(k) are

the N1 and the NEj 1 additive white Gaussian noisevectors with zero mean and variance 2 at the legitimate

receiver and the jth eavesdropper, respectively. We assumethat the transmitters know the maximum number of antennas

any eavesdropper can possess; namely, NE, but they do not

know any of the eavesdroppers channels Gi,j(k). We assumethat NE< M, where M=M1+ M2.

We define the Mi 1 channel input from legitimate trans-mitter i as

Xi(k) = Vixi(k). (3)

Fig. 1: System model

Each transmitter i intends to send a message Wi over n

channel uses (samples) to the legitimate receiver simultane-

ously while preventing the eavesdroppers from decoding its

message. The encoding occurs under a constrained power

given by2

iE

XiX

P (4)

Expanding the notations over n channel extensions

we have Hni = Hi(1),Hi(2), . . . ,Hi(n), Gni,j =

Gi,j(1),Gi,j(2), . . . ,Gi,j(n) and similarly the time extendedchannel input, Xni, time extended channel output at legitimate

receiver,Yn and time extended channel output at eavesdropper

j, Znj as well as noise at legitimate receiver, nn and noise at

eavesdroppers, nnEj .

At each transmitter, the message Wi is uniformly and

independently chosen from a set of possible secret messages

for transmitteri,Wi = {1, 2, . . . , 2nRi}. The rate for message

Wi is Ri 1n

log |Wi|, where | | denotes the cardinalityof the set. Transmitter i uses a stochastic encoding function

fi : Wi Xni to map the secret message into a transmittedsymbol. The receiver has a decoding function : Yn (W1, W2), where Wi is an estimate ofWi.

Definition 1. A secure rate tuple (R1, R2) is said to beachievable if for any > 0 there exist n-length codes suchthat the legitimate receiver can decode the messages reliably,

i.e.,

Pr{(W1, W2)= ( W1, W2)} (5)

and the messages are kept information-theoretically secure

against the eavesdroppers, i.e.,

H(W1, W2|Znj) H(W1, W2) (6)

H(Wi|Znj) H(Wi) i= 1, 2, (7)

whereH()is the Entropy function and(6) implies the secrecyfor any subset S {1, 2} of messages including individualmessages [15].

Definition 2. The sum SDoF is defined as

Ds= limP

supi

Ri12log P

, (8)

where the supremum is over all achievable secrecy rate tuples

(R1, R2), Ds = d1+ d2, andd1 andd2 are the secure DoFof transmitters one and two, respectively.

III. MAI NR ESULTS

Theorem 1. The sum SDoF of the two user MAC channel is

Ds=

M NE C11

2max(M1, N) +

1

2max(M2, N)

1

2NE C2

N C3

(9)


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where the conditions C1, C2 andC3 are given by:

C1 : M N

or M1 < N, M > N andNE 2(M N)

or M1 > N, M 2 < N and NE [M1 N+ 2M2]

C2 : M1 < N and NE < 2(M N)

or M1 > N, M 2 < N and M1 N NE < [M1 N+ 2M2]

or M1 > N, M 2 N and NE M 2N

C3 : NE < [M1 N]+ + [M2 N]

+

Proof. To prove the theorem, we deduce the converse in Sec-

tion IV and provide the achievable scheme in Section V.

IV. CONVERSE

Without loss of generality, we consider the case of only one

eavesdropper with NE antennas (maximum possible numberof antennas in any eavesdropper). The SDoF of the single

eavesdropper scenario is certainly an upperbound for the

multiple eavesdroppers case2. Accordingly, we omit the eaves-

dropper subscript for simplicity of notation. Starting from the

sum rates and following an approach similar to [15], we have:

n

2i=1

Ri I(W1, W2;Yn) I(W1, W2;Z

n) + nc1 (10)

I(W1, W2;Yn,Zn) I(W1, W2;Z

n) + nc1

= I(W1, W2;Yn|Zn) + nc1

I(Xn1 ,Xn2 ;Y

n|Zn) + nc1

= h(Yn|Zn) h(Yn|Zn,Xn1 ,Xn2 ) + nc1

= h(Yn|Zn) h(nn|Zn,Xn1 ,Xn2 ) + nc1

h(Yn|Zn) + nc2

= h(Yn,Zn) h(Zn) + nc2

= h(Xn1 ,Xn2 ,Y

n,Zn) h(Xn1 ,Xn2 |Y

n,Zn)

h(Zn) + nc2 (11)

h(Xn1 ,Xn2 ,Y

n,Zn) h(Zn) + nc2

= h(Xn1 ,Xn2 ) + h(Y

n,Zn|Xn1 ,Xn2 )

h(Zn) + nc2

h(Xn1 ,Xn2 ) h(Z

n) + nc3

=M1m=1

h(xn1m) +M2m=1

h(xn2m) h(Zn) + nc3

(12)

let x be a concatenating vector of x1 and x2, x =[x1

Tx2

T]T

.let B be a permutation matrix when multiplied

by Z results in a vector Z with the mth element (m {1, 2, . . . , M 1 + M2}) depending on xm and em, where em

2Increasing the number of eavesdroppers can only reduce the SDoF of thelegitimate users.

are constants depending on B and Gi: i{1,2}.

Z= BZ=

e1x1

e2x2

...

eNE1xNE1

eNE

xNE

+ .. + eM

1

+M2

xM

1

+M2

(13)

and,

h(Z) = h(Z) + log|B| (14)

Substituting (14) into (12),

n

2i=1

Ri nM1m=1

h(x1m) + n

M2m=1

h(x2m) nh(Z)

+ log |B| + nc32

i=1

Ri M1+M2

m=1

h(xm) NE1

m=1

h(emxm)

h

M1+M2NE+1

m=1

em+NE1xm+NE1

+ NEh(n) + c4

(M1+ M2 NE+ 1) log P

log

||[eNE. . . eM1+M2 ]||2

P

+ c5

Hence,

Ds M1+ M2 NE, (15)

where (11) comes from the perfect secrecy constraint [15],

[17], xim is the mth element of xi. All ci: i{1,2,...5} are

constants independent of P. Given that the receiver has onlyN antennas, then

Ds N (16)

Let We1 and We2 be the messages sent by transmitter one

and transmitter two, respectively, which the eavesdropper can

decode. Let d1e and d2e be the maximum degrees of freedom

for messages sent by transmitter one and two, respectively,

which can be decoded by the eavesdropper.

Suppose that we can set all coefficients ofG1 to zero to hide

all information sent by transmitter one from the eavesdropper,so the resulting channel becomes identical to the Z channel of

Figure 2(b). However, setting G1 coefficients to zero cannot

decrease the performance of the coding scheme. Therefore,

the DoF tuple for the modified channel is upperbounded by

d1+ d2+ d2e max(M2, N) (17)

Similarly, using the modified Z channel in Figure 2(a),

d1+ d2+ d1e max(M1, N) (18)


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Fig. 3: The intersection of signal spaces at the receiver

A. Achievability forMN

For this region, transmitters one and two send the jamming

signals using precoders VJ1 and VJ2 , with dimensions J1 and

J2, respectively, such that J1+ J2 = NE.

Random jamming: The jamming precoders and symbols

are randomly chosen. We call this method random jamming.The receiver zero-forces the jamming signal using the post

processing-matrixU as in (26). Accordingly, M NEsecureDoF can be sent. Since N M, the receiver can decodeM NE DoF after zero-forcing the jamming signal.

U = [I aa1] (26)

where

a = H1VJ1 +H2V

J2 (27)

B. Achievability forM > NFor this region we use three methods for jamming, aligned

jamming, nullspace jamming and random jamming. As ran-

dom jamming was described above, the other two will be

explained in the following.

Aligned jamming: The jamming signals of both transmitters

are aligned at the legitimate receiver signal space. Let I bethe jamming space at the receiver. Each transmitter aligns a

part or the whole of its jamming signal into this jamming

space. The total signal space of transmitter one and transmiter

two occupies only M1 and M2 dimensions, receptively, at

the receiver. These two spaces are distinct if M1 < N, so

a common space is needed to direct the jamming signal into.

LetA1and A2span the received signal spaces of transmitter

one and two at the receiver, Iis chosen to be the intersectionof these two spaces, i.e.,

I= A1

A2. (28)

Iwould have positive size only ifMN[18]. Without lossof generality, we design VJ1 and V

J2 such that,

H1VJ1 =I (29)

H2VJ2 =I (30)

While the system of equations in (29) (30) has more variables

than the number of equations, (28) ensures that the system has

a unique solution as I lies in the spans ofH1 and H2.Let

Hi=

H

i

H

i

i = 1, 2, (31)

where Hi contains the Mi rows ofHi and Hi contains theotherN Mi rows.

Let

I=

IiIi

i = 1, 2, (32)

where Ii contains the Mi rows of I and Ii contains the

otherN Mi rows.Therefore, we can choose the following design which sat-

isfies (29) and (30)

VJ1 = (H

1)1I1 (33)

VJ2 = (H

2)1I2 (34)

For the legitimate receiver to remove the jamming signal and

decode the legitimate message, it zero forces the jamming

signal using the post processing matrix U as in (26).

For the case NE is odd, each transmitter will align its

jamming signal into an NE2 dimensional half space usinglinear alignment. The remaining 1dimensional space willbe equally shared between the two transmitters jamming

signal using real interference alignment [16], yielding each

transmitters jamming signal to occupy NE2 .

Nullspace jamming: In nullspace jamming method, trans-

mitter one sends a jamming signal ofJ1 dimensions using theprecoder VJ1 which lies in the nullspace of the channel H1,

while transmitter two sends a jamming signal ofJ2dimensions

using the precoder VJ2 which lies in the nullspace of the

channel H2.

VJ1 =Null(H1) (35)

VJ2 =Null(H2) (36)

This blocks J1 + J2 dimensions at the eavesdropper andleaves N free dimensions at the legitimate receiver to attain

the legitimate signal in addition to other jamming if needed.

Case M1 < N andNE2(M N):In this region, aligned jamming and random jamming are

used. The first transmiter signal is divided into two parts

of sizes J1 and J2 , The first part uses aligned jamming

precoder with J1 = min(NE2 , M N), or J1 = (MN),

while the second part uses random jamming precoder with

J2 = [NE2J1]+. The second transmitter jamming signal

size is J1 and uses aligned jamming preocder. This scheme

wastes 2J1+ J2 = NE dimensions of the transmitters signalspace for jamming, so they can transmit at M NE DoF


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to the receiver. On the other hand, the jamming occupies

Js= J1+ J2 dimensions at the receiver.

Lemma 1. For the scheme proposed in the case M1 N , M 2 < N andNE [M1 N+ 2M2], thecondition N Js M NE is achieved, which impliesthat the legitimate signal and the jamming spaces are not

overlapping.

Proof.

J2 = M2

J3 = NE [(M1 N) + 2M2]Js = NE [(M1 N) + M2]

N Js = N NE+ M1 N+ M2

N Js = M NE

Case M1 > N , M 2 < N andM1 N NE < (M1 N+ 2M2):

In this region, two jamming methods are used, the first

transmitter jamming signal is divided into two parts of sizes

J1 = [M1 N]+, J2 = min(M2,

NEJ12 ), while the second

user jamming signal size is J2. The first transmitter uses

nullspace jamming for its first part and aligned jamming forthe second part, which is aligned to the second transmitters

jamming signal at the receiver. The jamming occupiesJs= J2dimensions at the receiver.

Lemma 3. For the proposed scheme under the case M1 >

N, M2 < N andM1 N NE < (M1 N+ 2M2), theachievable secure degrees of freedom is given by

d1+ d2 max(M1, N) +max(M2, N) NE

2 (38)

Proof.

Js =

NE [M1 N]

2

N Js = NNE [M1 N]

2

N Js = 2N NE+ [M1 N]

2

N Js = N NE+ M1

2

Thus,

d1+ d2 min(M NE,N NE+ M1

2 )

d1+ d2 N NE+ M1

2 ,

which can be rewritten as,


2 .

Case M1 > N, M2 N andNEM 2N:In this region, two jamming methods are used. Each trans-

mitter jamming signal is divided into two parts of sizes

J1,i= Mi N, i {1, 2}

J2 =NE

i{1,2} J1,i

2 .


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Both transmitters use nullspace jamming for their first part and

aligned jamming for the second part. The jamming occupies

Js= J2 dimensions at the receiver.

Lemma 4. For the proposed scheme under case M1 >

N, M2 N andNE M 2N, the achievable securedegrees of freedom is

d1+ d2

max(M1, N) +max(M2, N) NE2 (39)

Proof.

Js = NE (M1 N+ M2 N)

2

N Js = NNE [M 2N]

2

N Js = 2N NE+ M 2N

2

N Js = M NE

2 .

Thus,

d1+ d2 M N

E2 , (40)

which can be rewritten as


2

VI . CONCLUSION

We studied the two-transmitter Gaussian multiple access

wiretap channel with multiple antennas at the transmitters,

legitimate receivers and eavesdroppers. Generalizing new up-

perbound was established and a new achievable scheme was

provided. We used the new optimal scheme to derive the sum

secure DoF of the channel. We showed that the our scheme

meets the upperbound or all M1, M2, NE combinations. We

showed that Cooperative Jamming is SDoF optimal even

without the eavesdropper CSI available at the transmitters by

showing that jamming signal independent of the eavsdropper

channel and only dependes on the signal transmitted power

make the eavsdropper decoded DoF. Finally we showed that

if any eavsdropper has more antennas that the sum of the

transmiting antennas or the receiving antennas the SDoF is

zero.

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[13] R. Liu, T. Liu, and H. V. Poor. multiple-input multiple-output Gaussianbroadcast channels with confidential messages. IEEE Trans. Inf. Theory,vol. 56, no. 9, pp. 4215227, Sept. 2010.

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