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MIMO Multiple Access Channel with an Arbitrarily Varying Eavesdropper: Secrecy Degrees of Freedom Xiang He, and Ashish Khisti, Member, IEEE, and Aylin Yener, Member, IEEE

Abstract—A two-transmitter Gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated. The channel matrices of the legitimate users are fixed and revealed to all the terminals, whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper. The secrecy degrees of freedom (s.d.o.f.) region under a strong secrecy constraint is characterized. A transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver, and uses a single-user wiretap code for each user, is shown to achieve the s.d.o.f. region. The converse involves establishing an upper bound on a weightedsum-rate expression. This is accomplished by using induction, where at each step one combines the secrecy and multipleaccess constraints associated with an adversary eavesdropping a carefully selected group of sub-channels.

I. I NTRODUCTION Information theoretic security was first introduced by Shannon in [1], which studied the problem of transmitting confidential information in a communication system in the presence of an eavesdropper with unbounded computational power. Since then, an extensive body of work has been devoted to studying this problem for different network models by deriving fundamental transmission rate limits [2]–[4] and designing low-complexity schemes to approach these limits in practice [5], [6]. Secure communication using multiple antennas was extensively studied as well, see e.g., [7]–[18], [30]. These works investigated efficient signalling mechanisms using the spatial degrees of freedom provided by multiple antennas to limit an eavesdropper’s ability to decode information. The underlying information theoretic problem, the multi-antenna wiretap channel, was studied and the associated secrecy capacity was identified. We note that these works assumed that the eavesdropper’s channel state information is available either Manuscript received on February 24, 2012; revised on January 17, 2013; accepted on March 18, 2013. This work was presented in part at the 49th Annual Allerton Conference on Communication, Control, and Computing, September, 2011. This work is supported in part by NSF Grant 0964362. A. Khisti’s work was supported by an NSERC Discovery Grant. The ordering of c authors is alphabetical. Copyright ⃝2013 IEEE. Xiang He was with the Department of Electrical Engineering at the Pennsylvania State University, University Park, PA 16802. He is now with Microsoft (email: [email protected]). Ashish Khisti is with the Department of Electrical and Computer Engineering at University of Toronto, Toronto, ON, M5S 3G4, Canada (email: [email protected],ca). Aylin Yener is with the Department of Electrical Engineering at the Pennsylvania State University, University Park, PA 16802 (email: [email protected]).

completely or partially, although such an assumption may not be justified in practice. As a more pessimistic but stronger assumption, references [19]–[21] study secrecy capacity when the eavesdropper channel is arbitrarily varying and its channel states are known to the eavesdropper only. Reference [20] studies the single-user Gaussian multi-input-multi-output (MIMO) wiretap channel and characterizes the secrecy degrees of freedom (s.d.o.f.). The same paper extended the single user analysis to the two user Gaussian MIMO multiple access (MIMO-MAC) channel and characterized the s.d.o.f. region when all the legitimate terminals had equal number of antennas. However the MIMOMAC with arbitrary number of antennas at the terminals was left as an open problem. Our main contribution is to fully characterize the s.d.o.f. region of the two-transmitter MIMO MAC channel when the eavesdropper channel is arbitrarily varying. We show that the s.d.o.f. region can be achieved by a scheme that orthogonalizes the transmit signals of the two users at the intended receiver. Moreover, it suffices to use a single-user wiretap channel code [20] and no coordination between the users is necessary except for synchronization and sharing the transmit dimensions. To establish the optimality of this scheme, our converse proof decomposes the MIMO MAC channel into a set of parallel and independent channels using the generalized singular value decomposition (GSVD). A set of eavesdroppers, each monitoring a subset of links, is selected using an induction procedure and the resulting secrecy constraints are combined to obtain an upper bound on a weighted sum-rate expression. The outer bound matches the achievable rate in terms of the s.d.o.f. region, thus settling the open problem raised in [20] for the case of two transmitters. The scalar multiple-access channel when the eavesdropper channel is perfectly known, has been studied extensively e.g., [22]–[29]. If the channel model has real inputs and output, Gaussian signalling is in general suboptimal and user cooperating strategies, as well as signal alignment techniques, are necessary [26]. It was shown in [28, sec. 5.16] that the individual s.d.o.f. of this model could not exceed 2/3. Recently [29] improves this result and shows that the sum s.d.o.f. of this model cannot exceed 2/3 and shows this is achievable for almost all channel gains by using a scheme that transmits a superposition of information and noise symbols. Interference alignment is used to align the noise symbols at the legitimate receiver, and simultaneously mask the information symbols at the eavesdropper.

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NR antennas P¯1

NT1 antennas R X1

W1

T2

NT2 antennas

ˆ1 W ˆ2 W

T1 P¯2

W2

Y

NE antennas ˜ Y E

X2

Eavesdropper

W1 W2

Fig. 1. The MIMO MAC wiretap channel where NT1 = NT2 = 2, NR = 3, NE = 1.

The remainder of this paper is organized as follows. In Section II we describe the system model. The main result is stated as Theorem 1 in Section IV. The proof of the theorem is divided into two parts. First, we establish the result for the case of parallel channels in Section V. Subsequently, in Section VI we establish the result for the general case by decomposing the MIMO-MAC channel into a set of independent parallel channels. Such a reduction is used both in the proof of the converse as well as the coding scheme. Section VII concludes the paper. We use the following notation throughout the paper: For a set A, VA denotes the set of random variables {Vj , j ∈ A} and similarly Vi,A denotes the set of variables {Vi,j , j ∈ A}. We use {δn } to denote a non-negative sequence of n that converges to 0 when n goes to ∞. We use bold upper-case font for matrices and vectors. The distinction between matrices and vectors will be clear from the context. For a set A, |A| denotes its cardinality and a short hand notation xn is used for the sequence {x1 , x2 , . . . , xn }. Finally, ϕ denotes the empty set. II. S YSTEM M ODEL

˜ Y(i) =

2 ∑ k=1 2 ∑

1∑ |Xk (i)|2 ≤ P¯k , n→∞ n i=1 n

lim

k = 1, 2.

(3)

for each message W1 ∈ W1 and W2 ∈ W2 . (c) Secrecy Constraint: We consider the strong secrecy constraint [20]: ( ) ˜ n , k=1,2 = 0. (4) ˜n = H ˜ n |H lim sup I W1 , W2 ; Y k k n→∞ ˜ n {Hk ,k=1,2} We say that the rate-pair (Rs,1 , Rs,2 ) is achievable if there exists a sequence of codes Q with |W|k = 2nRs,k that satisfies the above conditions. The associated secrecy degrees of freedom (s.d.o.f.) are defined as [30], [31]: Rs,k (P¯k ) dk = lim sup (5) ¯ . P¯ →∞ log2 Pk k

As shown in Figure 1, we consider a discrete-time channel model where two transmitters communicate with one receiver in the presence of an eavesdropper. We assume transmitter i has NTi antennas, i = 1, 2, the legitimate receiver has NR antennas whereas the eavesdropper has NE antennas. The channel model is given by Y(i) =

legitimate parties and the eavesdropper and remain constant during the period of communication. We assume NE , the number of eavesdropper antennas, is known to the legitimate parties and the eavesdropper. We define a length n code C (n) for our setup as follows. User k, k = 1, 2, wishes to transmit a confidential message Wk , k = 1, 2, to the receiver over n channel uses, while both messages, W1 and W2 , must be kept confidential from the eavesdropper. The message W1 and W2 are uniformly distributed over the sets W1 and W2 respectively. We assume that |Wk | = 2nRs,k . User k transmits an input sequence Xnk = fk,n (Wk ) where fk,n : Wk → Cn is the encoding function at ˆ 1, W ˆ 2 ) = gn (Yn ) user k. The decoder outputs an estimate (W n of the transmitted messages where gn : C → W1 × W2 is the decoding function. The error probability is defined as, ˆ 1 ̸= W1 ∪ W ˆ 2 ̸= W2 ). Pe(n) = Pr(W A sequence of codes Q = {C (n) } is said to be feasible if the following conditions are satisfied: (a) Reliability Constraint: limn→∞ Pe(n) = 0. (b) Power Constraint: The sequence Xnk must satisfy the power constraint

Hk Xk (i) + Z(i)

(1)

˜ k (i)Xk (i) H

(2)

k=1

where i ∈ {1, . . . , n} denotes the time-index, Hk , k = 1, 2, are channel matrices and Z is the additive Gaussian noise observed by the intended receiver, which is composed of independent rotationally invariant complex Gaussian random variables with zero mean and unit variance. The sequence ˜ k (i), k = 1, 2}, is an of eavesdropper channel matrices {H arbitrary sequence of length n and only revealed to the eavesdropper. In contrast, Hk , k = 1, 2 are revealed to both the

The set of all achievable s.d.o.f. constitutes secrecy degrees of freedom. We note the use of lim sup in (5) implies that for any coding scheme one must consider a subsequence of powers that attains the lim sup. For the upper and lower bounds we consider, the limit actually existis, and hence lim sup and lim are the same. We make the following additional remark about the channel model. Remark 1: An arbitrarily varying channel (AVC) is defined by a stochastic mapping W n : X n → Y n , where X and Y are the alphabets of the channel input and output symbols respectively and W n (y n |xn , sn ) =

n ∏

W (yi |xi , si ).

i=1

Here W (y|x, s) is the transition probability that the channel output symbol y is observed when a channel input symbol x is transmitted and the channel state equals s. The sequence sn = (s1 , s2 , . . . , sn ) denotes the sequence of channel states that can vary in an arbitrary manner. There is a large variety

HE AND KHISTI AND YENER: MIMO MULTIPLE ACCESS CHANNEL WITH AN ARBITRARILY VARYING EAVESDROPPER: SECRECY DEGREES OF FREEDOM 3

x1

transmitted by user 2 to the intended receiver and assuming that the eavesdropper monitors either x1 or x2 we have that d1 ≤ 1. Similarly we argue that d2 ≤ 1. To obtain an upper bound on the sum-rate we let the two transmitters to cooperate and reduce the system to a 3 × 3 MIMO link. The s.d.o.f. of this channel [20] yields d1 + d2 ≤ 2. This outer bound, illustrated in Figure 2(b), does not match with the achievable region given by (9).

d2 y1

x2

A Simple Outer Bound

1 y2

x3

d1 1

y3

Achievable Region

x4

(b)

(a)

Fig. 2. (a) A special case of MIMO MAC wiretap channel where NT1 = NT2 = 2, NR = 3, NE = 1, (b) Comparison between achievable s.d.o.f. region and a simple outer bound derived by considering one eavesdropper at a time.

of problems on AVC channels depending on the nature of the error-criteria used (average or maximal error) and the permissible coding strategies (deterministic vs random coding). We refer the reader to [32, Chap. 12] for a comprehensive treatment of AVC channel models. In the present work we assume that the eavesdropper’s channel is an AVC channel, where the state variable s(i) = ˜ 1 (i), H ˜ 2 (i)) (c.f. (2)). We assume that no common random(H ness is shared between the legitimate users. The encoders are allowed to use private randomness in their encoding functions. Furthermore we assume that the state sequence be independent of Xn : ˜ n, H ˜ n ) = Pr(Xn = xn ). ˜n = H ˜n = H Pr(Xn = xn |H 1 1 2 2

(6)

The evaluation of e.g., (4) is based on this condition. Remark 2: For proving our converse it is sufficient to consider a (weaker) compound channel model [30] — the channel ˜ k are selected at the start of the communication matrices H from a certain set, say H, and remain fixed for the entire codeword. By a judicious choice of the set H, it is possible to obtain a matching upper bound for the s.d.o.f. III. M OTIVATION We discuss a simple example that illustrates why the problem considered is non-trivial. As illustrated in Figure 2(a), in this example, each transmitter has 2 antennas and the intended receiver has 3 antennas, while the eavesdropper has only 1 antenna. Let x1 , x2 , x3 , x4 denote the transmitted signals from the two users and y1 , y2 , y3 denote the signals observed by the intended the receiver. And the main channel is given by y1 = x1 + z1 ,

y3 = x4 + z3

y2 = x2 + x3 + z2

(7) (8)

where zi , i = 1, 2, 3 denote additive channel noise. As shown in [20], a secrecy degree of freedom min(NTk , NR ) − NE = 1 is achievable for a user if the other user remains silent. Time sharing between these two users leads to the following achievable s.d.o.f. region: d1 + d2 ≤ 1,

dk ≥ 0,

k = 1, 2

(9)

A. Cut-Set Upper Bound For the converse, we begin by considering a simple “cut-set” like upper bound, which reduces each channel to a singleuser MIMO wiretap channel. First, by revealing the signals

B. Proposed Upper Bound As we shall show in Theorem 1, (9) is indeed the s.d.o.f. capacity region and hence a new converse is necessary to prove this result. It can be readily seen that the outer bound in Section III-A only considers one eavesdropper at a time. For example, when deriving d1 ≤ 1, we assume there is only one eavesdropper which is monitoring either x1 or x2 . When deriving d2 ≤ 1, we assume there is only one eavesdropper which is monitoring either x3 or x4 . Similarly when deriving d1 +d2 ≤ 2 we again assume that there is one eavesdropper on either of the links. Our key observation is that a tighter upper bound is possible to find if we consider the simultaneous effect of two eavesdroppers. In our system model, because of the arbitrarily varying channel model, there are infinitely many possible eavesdroppers, each corresponding to a different channel state sequence. The challenge is to find a finite number of eavesdroppers, whose joint effect leads to a tight converse. Our choice of eavesdroppers is based on the following intuition: When an eavesdropper chooses which links to monitor, it should give precedence to those links over which only one user can transmit. This is because these links are the major contributor to the sum s.d.o.f. d1 + d2 since they are dedicated links to a certain user. Based on this intuition, we consider the following two eavesdroppers: one monitors y1 for W1 and the other monitors y3 for W2 . As we shall show later in Lemma 1, the first eavesdropper implies the following upper bound on R1 : ) ( (10) n(R1 − δn ) ≤ I xn2 ; y2n |y1n , xn{3,4} and the second eavesdropper implies the following upper bound on R2 : ( ) n(R2 − δn ) ≤ I y1n , xn{3,4} ; y2n (11) Their joint effect can be captured by adding (10) and (11) [33], which leads to: ( ) n(R1 + R2 − 2δn ) ≤ I xn2 , y1n , xn{3,4} ; y2n (12) n Since there is only one term, ( which is y2 , at )the right side of the mutual information I xn2 , y1n , xn{3,4} ; y2n , we observe the sum s.d.o.f. can not exceed 1, thereby justifying that (9) is indeed the largest possible s.d.o.f. region for Figure 2(a). As captured by (10) and (11), a simultaneous selection of two different eavesdroppers for the two users reduces the effective signal dimension at the receiver from three to one, thus leading to a tighter converse. As we shall show later in Section V-C, in generalizing this example, we are required

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d2 p2

d2 p4

d2

p2 = p4

d2 p2 = p4

p4

p2

p3 p3 d1

d1 p1 (b)

p3

d1 p1 = p3

p1

(a)

(c)

d1

Fig. 3. The secrecy degrees of freedom (s.d.o.f.) region in Theorem 1: (a) 0 ≤ NE ≤ min{r0 − r1 , r0 − r2 }, (b) min{r0 − r1 , r0 − r2 } ≤ NE ≤ max{r0 − r1 , r0 − r2 }, (c) max{r0 − r1 , r0 − r2 } ≤ NE

to systematically select a sequence of eavesdroppers using induction.

p1 Fig. 4. Interpretation of the s.d.o.f. region as a convex hull of rectangles: (d1 , d2 ) : 0 ≤ di ≤ [ti − NE ]+ , i = 1, 2, where ti is the number of degrees of freedom occupied by user i. To achieve reliable transmission, we must have (20) and (21).

IV. M AIN R ESULT

W1

In this section, we state the main result of this work. To express our result, we define rt as the rank of Ht , t = 1, 2 and r0 as the rank of [ H1 | H2 ]. We will refer to rt as the number of transmit dimensions at user t = 1, 2 and r0 as the number of dimensions at the receiver. Theorem 1: The secrecy degrees of freedom region of the MIMO multiple access channel with arbitrarily varying eavesdropper channel is given by the convex hull of the following five points of (d1 , d2 ): p0 = (0, 0) ( ) + p1 = [r1 − NE ] , 0 ( ) + p2 = 0, [r2 − NE ] ( ) + + p3 = [r1 − NE ] , [r0 − r1 − NE ] ( ) + + p4 = [r0 − r2 − NE ] , [r2 − NE ]

A 1 2

B

3 4

W2

(13) Fig. 5.

C

Definition of the set A, B, C, where |B| = 4.

(14) (15) (16) (17)

∆

where we use [x]+ = max{x, 0}. Fig. 3 illustrates the structure of the s.d.o.f. region as a function of the number of eavesdropping antennas. In Fig. 3 (a) we have NE ≤ min(r0 − r1 , r0 − r2 ). In this case the s.d.o.f. region is a polymatroid [34, Definition 3.1] described by di ≤ ri −NE and d1 +d2 ≤ r0 −2NE . Fig. 3 (b) illustrates the shape of the s.d.o.f. region when min{r0 − r1 , r0 − r2 } ≤ NE ≤ max{r0 −r1 , r0 −r2 }. In Fig. 3 (b), without loss of generality, we assume r1 < r2 and the s.d.o.f. region is bounded by the lines di ≥ 0, d1 ≤ r1 − NE and (r1 + r2 − r0 )d1 + (r1 − NE )d2 ≤ (r1 − NE ) × (r2 − NE ).

(d1 , d2 ) 0 ≤ d1 ≤ [t1 − NE ]+ 0 ≤ d2 ≤ [t2 − NE ]+ t1 + t2 ≤ r0 0 ≤ ti ≤ ri , i = 1, 2

(18)

When min(r1 , r2 ) > NE ≥ max(r0 − r1 , r0 − r2 ), the s.d.o.f. region, as illustrated in Fig. 3 (c) is bounded by di ≥ 0 and the line d2 d1 + ≤ 1. (19) r1 − NE r2 − NE The s.d.o.f. region in Theorem 1 allows the following simple interpretation: The region can be expressed as a convex hull of a set of rectangles shown by Figure 4 (illustrated

for Figure 3 (a)). Each rectangle is parameterized by the dimensions of the subspace occupied by the transmission signals from the two users, denoted by (t1 , t2 ), where ti indicates the dimension of user i, i = 1, 2. Then in order for the signals from both transmitters to be received reliably by the receiver, we must have t1 + t2 ≤ r 0

(20)

0 ≤ ti ≤ ri , i = 1, 2

(21)

Each user then transmits confidential messages with 0 ≤ di ≤ [ti − NE ]+ over the available ti dimensions, where the −NE term is an effect of the secrecy constraint (4). It is clear that p3 , p4 given by (16) and (17) are in one of these rectangles. Hence the convex hull of these rectangles yields the s.d.o.f. region stated in Theorem 1. Finally we note that if NE ≥ max(NT1 , NT2 ) the s.d.o.f. region reduces to (0, 0) and this implies that secure communication is not possible in this regime. V. P ROOF FOR THE PARALLEL C HANNEL M ODEL In this section, we establish Theorem 1 for the case of parallel channels. As illustrated in Fig. 5, the receiver observes yi = x1i + zi ,

i ∈ A,

(22)

yi = x1i + x2i + zi , yi = x2i + zi ,

i ∈ B, i ∈ C,

(23) (24)

HE AND KHISTI AND YENER: MIMO MULTIPLE ACCESS CHANNEL WITH AN ARBITRARILY VARYING EAVESDROPPER: SECRECY DEGREES OF FREEDOM 5

where the noise random variables across the sub-channels are independent and each is distributed according to CN (0, 1) and {x1i }i∈A∪B and {x2i }i∈B∪C denote the transmit symbols of user 1 and user 2 respectively. The parallel channel model is a special case of (1) with     I|A| O|A| , H2 =  , I|B| I|B| H1 =  O|C| I|C| (25) where I|A| , I|B| and I|C| denote the identity matrices of size |A|, |B| and |C| respectively, and O|A| and O|B| denote the matrices, all of whose entries are zeros. Note that we do not make any assumption on the eavesdropper’s channel model (2). A. Achievability It suffices to establish the achievability of points p3 and p4 in (16) and (17) respectively. The rest of the region follows through time-sharing between these points. Note that for the proposed parallel channel model ( ) + + p3 = [|A| + |B| − NE ] , [|C| − NE ] (26) ( ) + + p4 = [|A| − NE ] , [|B| + |C| − NE ] (27) To prove the achievability of p3 we restrict user 2 to transmit only on the last |C| components of in (24) and allow user 1 to transmit over all of the components of A ∪ B in (22) and (23). Note that in this case, the signals of these two users do not interfere with each other at the intended receiver. From [20], user 1 can transmit W1 such that d1 = [|A| + |B| − NE ]+ and ˜ n) = 0 ˜ n1 = H ˜ n1 Xn1 |H lim sup I(W1 ; H 1

n→∞ ˜ n H1

(28)

and user 2 can transmit W2 such that d2 = [|C| − NE ]+ and ˜ n) = 0 ˜ n2 Xn2 |H ˜ n2 = H lim sup I(W2 ; H 2

n→∞ ˜ n H2

(29)

˜ n Xn to denote the sequence where we use H k k ˜ {Hk (i)Xk (i), i = 1, ..., n}. Furthermore since (W1 , Xn1 ) is independent of (W2 , Xn2 ) we have that ˜ n, ˜ nk = H ˜ n1 Xn1 , H ˜ n2 Xn2 |H lim sup I(W1 ; H k

k=1,2

) = 0 (30)

˜ n, ˜ nk = H ˜ n1 Xn1 , H ˜ n2 Xn2 |H lim sup I(W2 ; H k

k=1,2

) = 0 (31)

n→∞ ˜ n H1,2 n→∞ ˜ n H1,2

˜ n , k = 1, 2, we have ˜n = H Note that for H k k ( ) ˜ n1 Xn1 , H ˜ n2 Xn2 |W2 I W1 ; H ( ) ˜ n1 Xn1 , H ˜ n2 Xn2 ≤I W1 ; W2 , H (32) ( ) ( ) ˜ n1 Xn1 , H ˜ n2 Xn2 + I W1 ; W2 |H ˜ n1 Xn1 , H ˜ n2 Xn2 =I W1 ; H (33) ( ) ( ) n n n n n n n n ˜ 1 X1 , H ˜ 2 X2 + I W1 , H ˜ 1 X1 ; W2 , H ˜ 2 X2 ≤I W1 ; H (34) ( ) n n ˜ 1 X1 =I W1 ; H (35)

˜ n2 Xn2 ) is where the last step follows from the fact that (W2 , H n n ˜ independent from (W1 , H1 X1 ). Therefore from (32)-(35) we observe that (28) implies ( ) ˜ n , k=1,2 = 0. ˜ n1 Xn1 , H ˜ n2 Xn2 |W2 , H ˜ nk = H lim sup I W1 ; H k n→∞ ˜ n H1,2

(36) Using (36) and (31), we obtain ( ˜ n, ˜ n1 Xn1 , H ˜ n2 Xn2 |H ˜ nk = H lim sup I W1 , W2 ; H k n→∞ ˜ n H1,2

) k=1,2

=0 (37)

and the secrecy constraint (4) follows from the data-processing inequality. Hence we have proved the point p3 is achievable. The achievability of p4 is proved by repeating the argument above by exchanging user 1 with user 2. Remark 3: As is evident from (37), the secrecy guarantee achieved by one user is not affected by the transmission strategy of the other user. B. Converse : NE ≤ min(|A|, |C|) We need to show that the s.d.o.f. region is contained within d1 ≤ |A| + |B| − NE d2 ≤ |C| + |B| − NE d1 + d2 ≤ |A| + |B| + |C| − 2NE

(38) (39) (40)

Since (38) and (39) directly follow from the single user case in [20], we only need to show (40). Let Ek be the set of links such that an eavesdropper is monitoring for Wk , k = 1, 2. |E1 | = |E2 | = NE . A ⊇ E1 , C ⊇ E2 . We establish the following upper bound on the achievable rate pairs. Lemma 1: ( ) ) ( n n n n n(Rs,1 − δn ) ≤ I X1,A\E ; Y A\E1 + I X1,B ; YB |M 1 (41) n n n n(Rs,2 − δn ) ≤ I(X2,C\E ; Y ) + I (M ; Y ) (42) B C\E2 2 ( n n ) where M = YA , X2,B∪C . Proof: The proof is provided in Appendix A. The proof is completed upon adding (41) and (42) so that n(Rs,1 + Rs,2 − 2δn ) n n n n ≤ (X1,A\E ; YA\E ) + I(X2,C\E ; YC\E ) 1 1 2 2 n + I(M, X1,B ; YBn )

and using

) 1 n n I(X1,A\E1 ; YA\E1 ) ≤ |A| − NE d n ( ) 1 n n d I(X2,C\E ; Y ) ≤ |C| − NE C\E2 2 n ( ) 1 n d I(M, X1,B ; YBn ) ≤ |B| n

(43)

(

∆

(44) (45) (46)

x(P ) where d(x) = limP →∞ log characterizes the pre-log scaling 2P of x with respect to P .

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(a)

As shown in Figure 6, by our construction, the vector F1 spans the first |F| elements of G ⊗ , the vector F2 spans the next |F| elements of G ⊗ etc. The constant ci denotes the index number of copies of the G vector necessary to cover Fi . When i = |G| the row-vector Fi terminates exactly at the end of the last G vector in G ⊗ . Hence,

V1

F1 1

2

1

2 3

3 4

5

6 7

G F1 1 (b)

1

2 2

F3

F2

V4

F4

3

4

5

6

7

1

3

4

5

6

7

1

2

3

4

5

2

3

4

5

G

c|G| = |F|, 6

7

G

Fig. 6. The set Fk , G, and Vk when |F| = 3, |G| = 7 and |B| = 8. (a) Case I, i = 1, c1 = 1. (b) Case II, i = 4, H5 = {1}, F5 = {6, 7, 1}, V5 = {2, 3, 4, 5, 6, 7}, c4 = 2, c5 = 3.

C. Converse : NE > max(|A|, |C|) Without loss of generality, we assume |C| ≥ |A|. Let Ek be the set of links such that an eavesdropper is monitoring for Wk , k = 1, 2. Let |E1 | = |E2 | = NE , A ⊂ E1 , and C ⊂ E2 . Define the set F, G such that F = B\E1 , G = B\E2 . Since |C| ≥ |A|, we have |G| ≥ |F|. Then Theorem 1 reduces to dk ≥ 0, k = 1, 2 and |G|d1 + |F|d2 ≤ |F| × |G|

(47)

which we now show. We first introduce the following lemma: Lemma 2: For any choice of F ⊆ B and G ⊆ B with appropriate cardinalities the rates Rs,1 and Rs,2 are upper bounded by ( ) n n n(Rs,1 − δn ) ≤ I X1,F ; YFn |M, X1,B\F (48) ( ) n n(Rs,2 − δn ) ≤ I M, X1,B\G ; YGn (49) { } n where M = YAn , X2,B∪C . Proof: The proof is provided in Appendix B. For the remainder of the proof we assume without loss of generality that B = {1, . . . , |B|}. We fix G = {1, . . . , |G|} while choosing |G| different sets of |F| elements: F1 , . . . , F|G| , the sets V0 , . . . , V|G| and a sequence of ci in the following recursive manner. Definition 1: Let V0 = G, c0 = 1. For i ≥ 1 recursively construct Fi as follows. 1) Case I: |Vi−1 | ≥ |F| Let Fi = {Vi−1 (1), . . . , Vi−1 (|F |)}, where Vi−1 (k) denotes the kth smallest element in Vi−1 . Let Vi = Vi−1 \Fi , and ci = ci−1 . This case is illustrated in Figure 6(a) for i = 1. 2) Case II: |Vi−1 | < |F| Let Fi = Vi−1 ∪ Hi , and Vi = G\Hi , and ci = ci−1 + 1, where Hi = {1, 2, . . . , |F| − |Vi−1 |}. This case is illustrated in Figure 6(b) for i = 4. To interpret the above construction, we note that the set G is a row-vector with |G| elements and let G ⊗ be obtained by concatenating |F| identical copies of the G vector i.e., G ⊗ = [G | G | . . . G] {z } | |F| copies

(50)

V|G| = ϕ.

(51)

By going through the above recursive procedure and invoking Lemma 2 repeatedly, each time by setting F in (48) and (49) to be Fi , we establish the following upper bound on the rate region. Lemma 3: For each i = 0, 1, . . . , |G| and the set of channels F1 , F2 , . . . , F|G| defined in Def. 1, the rate pair (Rs,1 , Rs,2 ) satisfies the following upper bound i · n(Rs,1 − δn ) + ci · n(Rs,2 − δn ) ≤

i ∑ n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni ).

(52)

j=1

Before providing a proof, we note that (47) follows from (52) as described below. Evaluating (52) with i = |G|, ˜ s,i = Rs,i − δn , using (51) and letting R ˜ s,1 + n|F|R ˜ s,2 ≤ n|G|R

|G| ∑

n I(M, X1,B ; YFnj )

(53)

j=1

=

|G| { } ∑ n ) h(YFnj ) − h(YFnj |M, X1,B j=1

(54) = n {|G| · |F| · log2 P + Θ(1)} , (55) where the last step uses the fact that ∑ h(Ykn ) ≤ n{|F| log2 P + O(1)}, h(YFnj ) ≤

(56)

k∈Fj

and n n n ) = h(YFnj |X1,F , X2,F ) = n · O(1). (57) h(YFnj |M, X1,B j j

Dividing each side of (55) by log2 P and taking the limit P → ∞ yields (47). Proof of Lemma 3: We use induction over the variable i to establish (52). For i = 0, note that c0 = 0 and V1 = G and hence (52) is simply (49). This completes the proof for the base case. For the induction step, we assume that (52) holds for some t = i, we need to show that (52) also holds for t = i + 1, i.e., (i + 1) · n(Rs,1 − δn ) + ci+1 · n(Rs,2 − δn ) ≤ i+1 ∑

n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni+1 ) (58)

j=1

holds. For our proof we separately consider the cases when |F| ≤ |Vi | and when |Vi | < |F| holds. When |F| ≤ |Vi |, from Definition 1 Fi+1 ⊆ Vi ,

Vi+1 = Vi \Fi+1 ,

ci+1 = ci

(59)

HE AND KHISTI AND YENER: MIMO MULTIPLE ACCESS CHANNEL WITH AN ARBITRARILY VARYING EAVESDROPPER: SECRECY DEGREES OF FREEDOM 7

holds. Then (58) follows by combining (52) with (48) as we show below. Note that n n I(M, X1,B\G ; YVni ) = I(M, X1,B\G ; YFni+1 |YVni \Fi+1 )

≤

n + I(M, X1,B\G ; YVni+1 ) n I(M, X1,B\G , YVni \Fi+1 ; YFni+1 )

holds. From (49) and (58) we have that i · n(Rs,1 − δn ) + (ci + 1) · n(Rs,2 − δn ) =

(60) +

≤

+

n I(M, X1,B\G ; YVni+1 )

(62) n n n n n ≤ I(M, X1,B\G , X1,G\Fi+1 ; YFi+1 ) + I(M, X1,B\G ; YVi+1 ) (63) n n n n = I(M, X1,B\Fi+1 ; YFi+1 ) + I(M, X1,B\G ; YVi+1 ) (64) where (60) follows from the chain rule of the mutual information and the definition of Vi+1 in (59), while (62) follows from the Markov condition n n n YVni \Fi+1 ↔ (X1,V , X2,V ) ↔ (M, YFni+1 , X1,B\G ) i \Fi+1 i \Fi+1 (65) n and the fact that M = (X2,B∪C , YAn ) already includes n X2,Vi \Fi+1 , (63) follows from the fact that Vi ⊆ G, while (64) follows from the fact that {B\G} ∪ {G\Fi+1 } = {B\Fi+1 }. Substituting (64) into the last term in (52) we get

i · n(Rs,1 − δn ) + ci · n(Rs,2 − δn ) ≤

i ∑

n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni )

j=1 n + I(M, X1,B\G ; YGn )

n I(M, X1,B\G ; YVni+1 )

(61) n n I(M, X1,B\G , X1,V ; YFni+1 ) i \Fi+1

i ∑

=

i ∑

(70)

n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni )

j=1 n n n ) + I(M, X1,B\G ; YVni+1 ) + I(M, X1,B\G ; YHni+1 |YG\H i+1 (71)

As we will show subsequently, n n n I(M, X1,B\G ; YVni ) + I(M, X1,B\G ; YHni+1 |YG\H ) i+1 n ; YFni+1 ). (72) ≤ I(M, X1,B\F i+1

Combining (48), (71) and (72) and using ci+1 = ci + 1 we get that (i + 1) · n(Rs,1 − δn ) + ci+1 · n(Rs,2 − δn ) ≤

i ∑

n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni+1 )

j=1 n n + I(M, X1,B\F ; YFni+1 ) + I(XFni+1 ; YFni+1 |M, X1,B\F ) i+1 i+1 (73)

=

i ∑

n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni+1 )

j=1 n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni )

n ; YFni+1 ), + I(M, X1,B

(74)

j=1

≤

i ∑

n I(M, X1,B ; YFnj )

+

n I(M, X1,B\F ; YFni+1 ) i+1

j=1 n + I(M, X1,B\G ; YVni+1 ).

(66)

which establishes (58). It only remains to establish (72) which we do now. First, since Fi+1 ⊆ G it follows that {B\G} ⊆ {B\Fi+1 } and hence we bound the first term in the left hand side of (72) as n n I(M, X1,B\G ; YVni ) ≤ I(M, X1,B\F ; YVni ). i+1

Finally combining (66) with (48) and using ci+1 = ci (c.f. (59)) we have (i + 1) · n(Rs,1 − δn ) + ci+1 · n(Rs,2 − δn ) ≤

i ∑

= {|F| − |Vi | + 1, . . . , |G| − |Vi |} ∪ {|G| − |Vi | + 1, . . . , |G|} = {G\(Hi+1 ∪ Vi )} ∪ Vi

j=1

+ + =

i+1 ∑

n n I(M, X1,B ; YFnj ) + I(M, X1,B\G ; YVni+1 )

(67)

= {G\Fi+1 } ∪ Vi

(68)

where the last relation follows from the definition of Fi+1 (c.f. (69)). Using (76) we can bound the second term in (72) as follows.

j=1

as required. When |F| > |Vi |, as stated in Definition 1 we introduce Hi+1 = {1, 2, . . . , |F| − |Vi |} and recall that Fi+1 = Vi ∪ Hi+1 ,

Vi+1 = G\Hi+1 ,

Next, since the set Hi+1 = {1, . . . , |F| − |Vi |} constitutes the first |F|−|Vi | elements of G and Vi = {|G|−|Vi |+1, . . . , |G|} constitutes the last |Vi | elements of G and |F| ≤ |G| we have that {G\Hi+1 } = {|F| − |Vi | + 1, . . . , |G|}

n n I(M, X1,B ; YFnj ) + I(M, X1,B\F ; YFni+1 ) i+1 n I(M, X1,B\G ; YVni+1 ) n n I(X1,F ; YFni+1 |M, X1,B\F ) i+1 i+1

(75)

ci+1 = ci + 1 (69)

(76)

n n I(M, X1,B\G ; YHni+1 |YG\H ) i+1 n n = I(M, X1,B\G ; YHni+1 |YG\F , YVni ) i+1

≤ ≤ ≤

n n I(M, X1,B\G , YG\F ; YHni+1 |YVni ) i+1 n n I(M, X1,B\G , X1,G\F ; YHni+1 |YVni ) i+1 n I(M, X1,B\F ; YHni+1 |YVni ), i+1

(77) (78) (79) (80)

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where in (79), we use the Markov relation

we have that

n n n n YG\F ↔ (X1,G\F , X2,G\F ) ↔ (M, X1,B\G , YFni+1 ) i+1 i+1 i+1 (81) n and the fact that M = (X2,B∪C , YAn ) already contains n X2,G\F . Combining (75) and (80) gives i+1 n n n I(M, X1,B\G ; YVni ) + I(M, X1,B\G ; YHni+1 |YG\H ) i+1 n ; YFni+1 ), (82) ≤ I(M, X1,B\F i+1

thus establishing (72). This completes the proof.

n n |B|n(Rs,1 − δn ) + |F|n(Rs,2 − δn ) ≤ |F |I(X2,C\E ; YC\E ) 2 2

+

|B| ∑

n I(M, X1,B ; YFnj ). (91)

j=1

Finally substituting ( ) 1 n n ≤ |C| − NE d I(M, X2,C\E ; Y C\E2 2 n ( ) 1 n d I(M, X1,B ; YFnj ) ≤ |F|, n

(92) (93)

in (91) we obtain (85). D. Converse: min(|A|, |C|) ≤ NE ≤ max(|A|, |C|) We assume without loss of generality that |C| ≥ |A| and as before let Ek be the set of links such that an eavesdropper is monitoring for message Wk . Since |E1 | = |E2 | = NE and |A| ≤ NE ≤ |C| holds, we select the sets such that the relations A ⊆ E1 ⊆ A ∪ B and C ⊇ E2 are both satisfied. Define F = B\E1 and note that |F| = |A| + |B| − NE . Theorem 1 reduces to the following region : 0 ≤ d1 ≤ |F| 0 ≤ d2 ≤ |B| + |C| − NE

(83) (84)

|B|d1 + |F|d2 ≤ (|B| + |C| − NE ) × |F |

(85)

Since (83) and (84) directly follow from the single user case [20], we only need to establish (85). As in earlier cases we begin by establishing the following bounds on the rate pair (Rs,1 , Rs,2 ): n n n(Rs,1 − δn ) ≤ I(X1,F ; YFn |M, X1,B\F ) ( ) n n n n(Rs,2 − δn ) ≤ I (M ; YB ) + I X2,C\E2 ; YC\E 2

(86) (87)

( n ) where M = X2,B∪C , YAn . Proof: The proof for (86) is identical to (48) in Lemma 2 since the proof does not depend on the choice of E2 . The proof for (87) is identical to (42) in Lemma 1. To establish (83)-(85), note that by defining ) 1 ( n ′ n = Rs,2 − I X2,C\E Rs,2 ; YC\E , (88) 2 2 n we have from (87) that ′ n(Rs,2 − δn ) ≤ I (M ; YBn )

(89)

′ and the bounds on Rs,1 and Rs,2 in (86) and (89) are identical to the bounds (48) and (49) in Lemma 2 with ′ G = B. Applying Lemma 3 to Rs,1 and Rs,2 for each i = 0, 1, · · · , |G|, it follows that ′ i · n(Rs,1 − δn ) + ci · n(Rs,2 − δn )

≤

i ∑

n I(M, X1,B ; YFnj ) + I(M, YVni ).

(90)

j=1

where the sets Vi , Fi and the sequence ci are as in Definition 1. Substituting (89) into (90) and evaluating the bound for i = |B|

VI. G ENERAL MIMO-MAC The result for the general MIMO case (1) follows by a transformation that reduces the model to the case of parallel independent channels in the previous section while preserving the secrecy degrees of freedom region. As we discuss next, this transformation involves the generalized singular value decomposition (GSVD) [35] and a channel enhancement argument. For an analogous application of GSVD to broadcast channels see e.g., [21], [36], [37]. We note that channel enhancement techniques are used in many different problems in multiuser information theory; see e.g. [12]. A. GSVD Transformation Theorem 2: [35] Given a pair of matrices H1 and H2 such that the rank of Hi is ri , i = 1, 2, and the rank of [ H1 | H2 ] is r0 , there exists unitary matrices U1 , U2 , W, Q and nonsingular upper triangular matrix R such that for s = r1 + r2 − r0 , r˜1 = r1 − s, r˜2 = r2 − s, [ H ] H (94) UH 1 H1 Q = Σ1(NT1 ×r0 ) W R(r0 ×r0 ) , 0 (r0 ×NR ) [ ] H H UH (95) 2 H2 Q = Σ2(NT2 ×r0 ) W R(r0 ×r0 ) , 0 (r0 ×NR )   I1(˜r1 טr1 )  (96) S1(s×s) Σ1 =  O1((NT1 −˜r1 −s)טr2 )   O2((NT2 −˜r2 −s)טr1 )  (97) Σ2 =  S2(s×s) I2(˜r2 טr2 ) where Ii , i = 1, 2 are r˜i × r˜i identity matrices, Oi , i = 1, 2 are zero matrices, and Si , i = 1, 2 are s × s diagonal matrices with positive real elements on the diagonal line that satisfy S21 +S22 = Is , and r˜1 +s+˜ r2 = r0 . For clarity, the dimension of each matrix is shown in the parenthesis in the subscript. I1 has the same number of columns as O2 . I2 has the same number of columns O1 . However, Oi , i = 1, 2 are not necessarily square matrices and can be empty, i.e., having zero number of rows. For convenience in notation we define A = WH R and observe that A is a square and non-singular matrix. Then from Theorem 2, we have: [ H ] A H Q Ht Ut = ΣH (98) t , t = 1, 2. 0

HE AND KHISTI AND YENER: MIMO MULTIPLE ACCESS CHANNEL WITH AN ARBITRARILY VARYING EAVESDROPPER: SECRECY DEGREES OF FREEDOM 9

Without loss of generality, we can cancel Q and Ut and rewrite (1) as: [ H ] Ar0 ×r0 Y= ΣH X 0(NR −r0 )×r0 N ×r 1 1 0 R [ H ] Ar0 ×r0 ΣH X + Z. (99) + 0(NR −r0 )×r0 N ×r 2 2 0

R

Since Q and Ut are unitary matrices, the components of Z are independent from each other and the power constraints of each transmitter remains the same as P¯i , i = 1, 2. Because the components of Z are independent, the intended receiver can discard the last NR − r0 components in Y without affecting the secrecy capacity region of this channel. This means that we only need to consider the case where NR = r0 and rewrite (1) as: H H Y = AH r0 ×r0 (Σ1 X1 + Σ2 X2 ) + Z.

(100)

B. Converse For establishing the converse, we further enhance the channel model in (100) to the following H ′ Y = ΣH 1 X1 + Σ2 X2 + σ+ Z

(101)

2 where σ+ ≤ 1 is any sufficiently small constant such that, σ+ H times the maximal eigenvalue of Ar0 ×r0 Ar0 ×r0 , is smaller than 1 and Z′ is a circularly symmetric unit-variance Gaussian noise vector. To establish (101), note that we can express

Z = σ+ · AH Z′ + Z′′

(102)

where Z′′ is a Gaussian random vector, independent of Z′ and with a covariance matrix 2 AH Ir0 ×r0 − σ+ r0 ×r0 Ar0 ×r0

(103)

which is guaranteed to be positive semi-definite by our choice of σ+ . Upon substituting (102) into (100), we have ( H ) H ′ ′′ Y = AH (104) r0 ×r0 Σ1 X1 + Σ2 X2 + σ+ Z + Z . We consider an enhanced receiver that is revealed Z′′ . Clearly this additional knowledge can only increase the rate and serves as an upper bound. It is also clear that since Z′′ is independent of (X1 , X2 , Z′ ), it suffices to use this information to cancel Z′′ in (104) and then discard it. Furthermore since the matrix A is invertible, upon canceling it, we obtain (101). H We further enhance the receiver by replacing ΣH 1 and Σ2 H H ¯ 1 and Σ ¯ 2 so that the model reduces to with Σ ′ ¯H ¯H Y=Σ 1 X1 + Σ2 X2 + σ+ Z

where



¯H  Σ 1 =

(105) 

Ir˜1 טr1



I1(s×s) 01(r˜2 ×(NT

1 −r1

))

r0 ×NT1

(106)

 ¯H  Σ 2 =

02(r˜1 ×(NT

2



−r2 ))



I2(s×s) I2(˜r2 טr2 )

r0 ×NT2

(107) are obtained by replacing each diagonal Si by the identity matrix. The model (105) can only have a higher capacity, since each diagonal entry in Si is between (0, 1). We observe that in the resulting channel model is identical to (22)-(24) |A| = r0 − r2 |B| = s = r1 + r2 − r0 |C| = r0 − r1

(108) (109) (110)

2 except that the noise variance is reduced by a factor of σ+ . Since a fixed scaling in the noise power does not affect the secure-degrees of freedom, an outer bound on the s.d.o.f. for the parallel channel model (22)-(24) with A, B and C defined via (105), continues to be an outer bound on the s.d.o.f. region for the general MIMO-MAC channel. Substituting (108)-(110) in the upper bounds in section V-B, V-C and V-D we establish the converse in Theorem 1.

C. Achievability To establish the achievability for the general MIMO case we further use a suitable degradation mechanism to reduce the model (100) to H ′′ Y = ΣH 1 X1 + Σ2 X2 + σZ

(111)

where σ ≥ 1 is any sufficiently large constant such that, σ 2 times the minimum eigenvalue of AH r0 ×r0 Ar0 ×r0 , is greater than 1 and Z′′ is a circularly symmetric unit-variance Gaussian noise vector. Since A is non-singular we are guaranteed that all the singular values of A are non-zero and hence a σ < ∞ exists. To establish (111), let Z′ be a Gaussian noise vector with covariance σ 2 AH r0 ×r0 Ar0 ×r0 − Ir0 ×r0

(112)

independent of Z and consider a degraded version of (100) ( H ) H ′ Y = AH (113) r0 ×r0 Σ1 X1 + Σ2 X2 + Z + Z which can be simulated at the receiver by adding additional noise Z′ to its output. Since Z + Z′ ∼ CN (0, σ 2 AH A), we can express Z + Z′ = σAH Z′′ . Substituting into (113) and canceling the non-singular matrix A, we arrive at (111). Let s¯ > 0 denote the minimum element on the diagonals of S1 and S2 in (96) and (97) respectively. By appropriately scaling down the transmit powers on each of the sub-channels we can further reduce (104) to σ ′′ ¯H ¯H Y=Σ Z (114) 1 X1 + Σ2 X2 + s¯ ¯ k are defined in (106) and (107) respectively. The where Σ model (114) is identical to the parallel channel model (22)(24) with the size of sets A, B and C in (108)-(110) and with a noise power that is larger by a factor of σ 2 /¯ s2 . Since a constant factor in the noise power does not affect the secrecy degrees of freedom, the coding schemes described in section V-A achieves the lower bound in Theorem 1.

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VII. C ONCLUSION

independent from each other and hence

In this work we have studied the two-transmitter Gaussian complex MIMO-MAC wiretap channel where the eavesdropper channel is arbitrarily varying and its state is known to the eavesdropper only, and the main channel is static and its state is known to all nodes. We have completely characterized the s.d.o.f. region for this channel for all possible antenna configurations. We have proved that this s.d.o.f. region can be achieved by a scheme that orthogonalizes the transmit signals of the two users at the intended receiver, in which each user achieves secrecy guarantee independently without cooperation from the other user. The converse was proved by carefully changing the set of signals available to the eavesdropper through an induction procedure in order to obtain an upper bound on a weighted-sum-rate expression. We note that the scope of this paper is limited to the case of two-transmitters. Our proof involves simultaneously decomposing the channel matrices of the two users into parallel channels using the Generalized Singular Decomposition (GSVD). Then a set of eavesdropper channels is carefully constructed for the parallel-channel model to obtain an upper bound, tighter than the usual cut-set bound. Since the GSVD transform does not easily extend to more than two matrices, we did not pursue the case of more than two transmitters and leave this extension as a future work. We also note that our setup assumes that the channel matrices of the legitimate receivers are static i.e., fixed for the entire period of communication. Our core ideas readily extend to the case when the channel gains of the legitimate users change over time, but are revealed to all the terminals. As suggested by this work, the optimal strategy for a communication network where the eavesdropper channel is arbitrarily varying can potentially be very different from the case where the eavesdropper channel is fixed and its state is known to all terminals. This is also observed for example in the MIMO broadcast channel [21] and the two-way channel [38], [39]. Finally we note that the proposed setup allows the eavesdropper terminals to perfectly emulate the legitimate receiver’s channel if sufficiently many antennas are available. Such an assumption may be unavoidable if the environment is uncontrolled and an eavesdropper could be placed right where the intended receiver is located. In controlled environments where the eavesdropper must maintain a certain physical separation, our proposed setup may still be realistic if not pessimistic. A PPENDIX A P ROOF OF L EMMA 1 For Rs,1 , from Fano’s inequality, we have n n(Rs,1 − δn ) ≤ I(W1 ; YA∪B∪C ) − I(W1 ; YEn1 ) ( ) n ≤ I W1 ; YA∪B∪C |YEn1 ( ) n n ≤ I W1 ; YA∪B∪C , X2,B∪C |YEn1 ( ) n n = I W1 ; YA∪B , X2,B∪C |YEn1

(115) (116) (117) (118)

where the last step (118) relies on the fact that the additive noise at each receiver end of each sub-channel in Figure 5 is

n n n YCn → X2,C → (W1 , YA∪B , YEn1 , X2,B ) ) ( n n holds. Since X2,C , X2,B is independent from W1 , and E1 ⊆ A, (118) can be written as: ( ) n n I W1 ; YA∪B |YEn1 , X2,B∪C ( ) n n n =I W1 ; Y(A\E |Y , X (119) E1 2,B∪C 1 )∪B ( ) ( ) n n n =I W1 ; YA\E |YEn1 , X2,B∪C + I W1 ; YBn |YAn , X2,B∪C 1 (120)

where the last step (120) follows from the fact E1 ⊆ A and hence A = (A\E1 ) ∪ E1 . We separately bound each of the two terms above. ) ( n n n |Y , X I W1 ; YA\E E 2,B∪C 1 1 ) ( n n n ≤ I W1 , YE1 , X2,B∪C ; YA\E (121) 1 ) ( n n n ≤ I W1 , YEn1 , X2,B∪C , X1,A\E ; YA\E (122) 1 1 ( ) n n = I X1,A\E ; YA\E (123) 1 1 where the last step follows from the Markov chain relation n n n YA\E ↔ XA\E ↔ (W1 , YEn1 , X2,B∪C ), We upper bound the 1 1 second term in (120) as follows ( ) n I W1 ; YBn |YAn , X2,B∪C ) ( n n ; YBn |YAn , X2,B∪C ≤ I X1,A∪B ( n ) n = I X1,B ; YBn |YAn , X2,B∪C ) ( n n n , X1,B + I X1,A ; YBn |YAn , X2,B∪C ( n ) n = I X1,B ; YBn |YAn , X2,B∪C

(124) (125) (126)

n where we use the Markov relation W1 ↔ X1,A∪B ↔ n n (YA , X2,B∪C ) in step (124) and (126) follows from the fact Markov relation n n n , YAn ). ) ↔ (X2,C , X2,B YBn ↔ (X1,B

(127)

Note that (41) follows upon substituting (123) and (126) into (120). For Rs,2 , from Fano’s inequality and the secrecy constraint, we have: n n n(Rs,2 − δn ) ≤ I(W2 ; YA∪B∪C ) − I(W2 ; X2,E ) (128) 2 ( ) n n ≤I W2 ; YA∪B∪C |X2,E2 (129) ( ) n n n =I W2 ; YB∪C |YA , X2,E2 (130) ( ) n n =I W2 ; Y(C\E |YAn , X2,E (131) 2 2 )∪B ( ) ( ) n n n n n n n =I W2 ; YC\E |Y , X + I W ; Y |Y , X , Y 2 A 2,E B A 2,E C\E 2 2 2 2 (132)

where (130) follows from the fact that YAn is independent of n (W2 , X2,B∪C ) and (131) follows from the fact that YEn2 → n n X2,E2 → (YB∪C\E2 , W2 , YAn ) holds. We separately bound each term in (132). ( ) ( ) n n n n I W2 ; YC\E |YAn , X2,E ≤ I W2 , YAn , X2,E ; YC\E 2 2 2 2 (133)

HE AND KHISTI AND YENER: MIMO MULTIPLE ACCESS CHANNEL WITH AN ARBITRARILY VARYING EAVESDROPPER: SECRECY DEGREES OF FREEDOM11

( ) n n n n ≤ I X2,C\E , W , Y , X ; Y 2 A 2,E C\E 2 2 2 ( ) n n = I X2,C\E2 ; YC\E2 ,

(134) (135)

where the justification for establishing (135) is identical to (123) and hence omitted. We finally bound the second term in (132). ( ) n n I W2 ; YBn |YAn , X2,E , Y (136) C\E2 2 ) ( n n n (137) ≤I X2,B∪C ; YBn |YAn , X2,E , YC\E 2 2 ( ) n n n ≤I YAn , X2,B∪C , X2,E , YC\E ; YBn (138) 2 2 ( n n ) n =I YA , X2,B∪C , X2,E ; YBn 2 ( ) n n n + I YC\E ; YBn |YAn , X2,B∪C , X2,E (139) 2 2 ( n n ) n =I YA , X2,B∪C ; YB (140) where the justification for arriving at (140) is similar to (126) and hence omitted. Substituting (135) and (140) into (132) we establish (42). A PPENDIX B P ROOF OF L EMMA 2 n Assume the eavesdropper monitors YAn and X1,E for W1 . 1 \A Then for Rs,1 , from Fano’s inequality, we have:

n(Rs,1 − δn )

( ) n n ≤I (W1 ; YA∪B∪C ) − I W1 ; YAn , X1,E 1 \A ) ( n n n ≤I W1 ; YA∪B∪C |YA , X1,E1 \A ( ) n n =I W1 ; YB∪C |YAn , X1,E 1 \A ( ) n n n ≤I W1 ; YB∪C , X2,B∪C |YAn , X1,E \A 1 ( ) n n n n =I W1 ; YB∪C |YA , X1,E1 \A , X2,B∪C ( ) n n =I W1 ; YFn |YAn , X1,E , X 2,B∪C 1 \A ) ( n n n n =I W1 ; YF |YA , X1,B\F , X2,B∪C

(141) (142)

where the last step uses the fact that the second term in (150) n n involves conditioning on (X1,F , X2,F ) and hence is zero. This establishes (48). For Rs,2 , we assume the eavesdropper is monitoring n n X2,C , X2,E for W2 . Using Fano’s inequality and the secrecy 2 \C constraint, we have: ( ) n n n(Rs,2 − δn ) ≤ I (W2 ; YA∪B∪C ) − I W2 ; X2,E 2 ) ( n n ≤I W2 ; YA∪B∪C |X2,E 2 ) ( n n n , X1,E |X2,E ≤I W2 ; YA∪B∪C 2 ∩B 2 ( ) n n n =I W2 ; YB∪C |X2,E , YAn , X1,E 2 2 ∩B ( n ) n n n ≤I X2,B∪C ; YB∪C |X2,E , YAn , X1,E 2 2 ∩B ( n ) n n n =I X2,B∪C ; YG∪E |X2,E , YAn , X1,E 2 2 2 ∩B ( n ) n n =I X2,B∪C ; YGn |X2,E , YAn , X1,E 2 2 ∩B ( n ) n n n + I X2,B∪C ; YEn2 |X2,E , YA∪G , X1,E 2 2 ∩B ( n ) n n =I X2,B∪C ; YGn |X2,E , YAn , X1,E 2 2 ∩B ( n ) n ≤ I X2,B∪C , YAn , X1,E ;Y n ( ) 2 ∩B G n ≤I M, X1,B\G ; YGn

(152) (153) (154) (155) (156) (157) (158) (159) (160) (161)

n where (155) follows from the fact that (X1,E , YAn ) are 2 ∩B the transmitted signals from user 1 and independent of n (W2 , X2,E ) and (157) follows from the fact that C ⊆ E2 ⊆ 2 B∪C and G = B\E2 and hence E2 ∪G = B∪C holds. Eq. (159) follows from the fact that since the noise on each channel is n n n n Markov, we have YEn2 ↔ (X2,E , X1,E ) ↔ (YA∪G , XB∪C ) 2 2 ∩B and hence the second term in (158) is zero. Hence we have proved Lemma 2.

(143) (144) (145) (146) (147)

n where (145) follows from the fact that X2,B∪C is independent n of (W1 , YAn , X1,E ). while (146) follows from the fact that 1 \A since the noise across the channels is independent the Markov condition n n n (YEn1 \A , YCn ) ↔ (X1,E , X2,B∪C ) ↔ (W1 , YB\E , YAn ) 1 \A 1

holds and furthermore we have defined F = B\E1 . Since the channel noise is independent of the message, n n n W1 ↔ X1,A∪B ↔ (YFn∪A , X1,B\F , X2,B∪C ) holds. Hence ( ) n n I W1 ; YFn |YAn , X1,B\F , X2,B∪C (148) ( ) n n n ≤I X1,A∪B ; YFn |YAn , X1,B\F , X2,B∪C (149) ( ) n n n =I X1,F ; YFn |YAn , X1,B\F , X2,B∪C ( ) n n n + I X1,A∪B\F ; YFn |YAn , X1,B , X2,B∪C (150) ( ) n n n =I X1,F ; YFn |YAn , X1,B\F , X2,B∪C (151)

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IEEE TRANSACTIONS ON INFORMATION THEORY

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Xiang He (S’08, M’10) received B.S. and M.S. degrees in Electrical Engineering from Shanghai Jiao Tong University, Shanghai, China in 2003 and 2006 respectively. His master study is about high speed FPGA implementation of channel encoder, decoder and MIMO detectors. He received his Ph.D. degree in 2010 from the Department of Electrical Engineering at the Pennsylvania State University and joined Microsoft in the that year. In 2010, he received Melvin P. Bloom Memorial Outstanding Doctoral Research Award from the Department of Electrical Engineering at the Pennsylvania State University and the best paper award from the Communication Theory Symposium in IEEE International Conference on Communications (ICC). In 2011, he was named as one of the exemplary reviewers by IEEE Communication Letters. His research interests include information theoretic secrecy, coding theory, queuing theory, optimization techniques, distributed detection and estimation.

Ashish Khisti (M’08) is an assistant professor in the Electrical and Computer Engineering (ECE) department and a Canada Research Chair (Tier II) in Network Information Theory at the University of Toronto, Toronto, Ontario Canada. He received his BASc degree in Engineering Sciences from University of Toronto in 2002 and his S.M and Ph.D. degrees from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA in 2004 and 2008 respectively. He has been with the University of Toronto since 2009. His research interests span the areas of information theory, wireless physical layer security and streaming communication systems. During his graduate studies, Professor Khisti was a recipient of the NSERC postgraduate fellowship, Harold H. Hazen Teaching award and the Morris Joseph Levin Masterworks award. At the University of Toronto he is a recipient of the Ontario Early Researcher Award (2012) and a Hewlett-Packard IRP award (2011, 2012). He is an associate editor of IEEE TRANSACTIONS ON COMMUNICATIONS. Professor Khisti co-organized a workshop on Physical Layer Security at IEEE GLOBECOM Conference (2011) and a workshop on Interactive Information Theory at the Banff International Research Station (2012).

Aylin Yener (S’91-M’00) received the B.Sc. degree in electrical and electronics engineering, and the B.Sc. degree in physics, from Bo˘gazic¸i University, Istanbul, Turkey; and the M.S. and Ph.D. degrees in electrical and computer engineering from Wireless Information Network Laboratory (WINLAB), Rutgers University, New Brunswick, NJ. Commencing fall 2010, for three semesters, she was a P.C. Rossin Assistant Professor at the Electrical Engineering and Computer Science Department, Lehigh University, PA. In 2002, she joined the faculty of The Pennsylvania State University, University Park, PA, where she was an Assistant Professor, then Associate Professor, and is currently Professor of Electrical Engineering since 2010. During the academic year 2008-2009, she was a Visiting Associate Professor with the Department of Electrical Engineering, Stanford University, CA. Her research interests are in information theory, communication theory and network science, with recent emphasis on green communications and information security. She received the NSF CAREER award in 2003.

HE AND KHISTI AND YENER: MIMO MULTIPLE ACCESS CHANNEL WITH AN ARBITRARILY VARYING EAVESDROPPER: SECRECY DEGREES OF FREEDOM13

Dr. Yener served as the student committee chair for the IEEE Information Theory Society 2007-2011, and was the co-founder of the Annual School of Information Theory in North America co-organizing the school in 2008, 2009 and 2010. She currently serves on the board of governors as the treasurer of the IEEE Information Theory Society.