On the Secure Degrees of Freedom of the K-user MAC and 2-user Interference Channels
aa r X i v : . [ c s . I T ] M a r On the Secure Degrees of Freedom of the K-userMAC and 2-user Interference Channels
Mohamed Amir , Tamer Khattab and Tarek Elfouly Electrical Engineering, Qatar University Computer Science & Computer Engineering, Qatar UniversityEmail: [email protected], [email protected], [email protected]
Abstract —We investigate the secure degrees of freedom (SDoF)of the K -user MIMO multiple access (MAC) and the two userMIMO interference channel. An unknown number of eavesdrop-pers are trying to decode the messages sent by the transmitters.Each eavesdropper is equipped with a number of antennas lessthan or equal to a known value N E . The legitimate transmittersand receivers are assumed to have global channel knowledge.We present the sum SDoF of the two user MIMO interferencechannel. We derive an upperbound on the sum SDoF of the K -user MAC channel and present an achievable scheme thatpartially meets the derived upperbound. I. I
NTRODUCTION
The noisy wiretap channel was first studied by Wyner [1],in which a legitimate transmitter (Alice) wishes to send amessage to a legitimate receiver (Bob), and hide it froman eavesdropper (Eve). Wyner proved that Alice can sendpositive secure rate to Bob using channel coding. He de-rived capacity-equivocation region for the degraded wiretapchannel. A significant amount of work was carried thereafterto study the information theoretic physical layer security fordifferent network models. The relay assisted wiretap channelwas studied in [2]. The secure degrees of freedom (SDoF)region of multiple access (MAC) channel was presented in[3]. The SDoF is the the pre-log of the secrecy capacity regionin the high-SNR regime. Using MIMO systems for securingthe message was an intuitive extension due to the spatialgain provided by multiple antennas. MIMO wiretap channelsecrecy capacity was identified in [4]. Meanwhile, the idea ofcooperative jamming was proposed in [5], where some of theusers transmit independent and identically distributed (i.i.d.)Gaussian noise towards the eavesdropper to improve the sumsecrecy rate of the legitimate parties.In this paper, we study the K -user MIMO MAC and the twouser MIMO interference channel, each with unknown numberof eavesdroppers. We assume that the legitimate pair has globalchannel knowledge. We present the sum SDoF of the twouser MIMO interference channel. We derive an upperboundon the the sum SDoF of the K -user MAC channel and presentan achievable scheme that partially meets the upperbounddepending on the relations between the nodes’ number ofantennas. We use the following notation, a for vectors, A formatrices, A † for the hermitian transpose of A , [ A ] + for the This research was made possible by NPRP 7-923-2-344 grant from theQatar National Research Fund (a member of The Qatar Foundation). Thestatements made herein are solely the responsibility of the authors. max A, and Null( A ) to define the nullspace of A , while a C b is used to define the b -combination of a set a II. S
YSTEM MODEL
We consider two communication systems, the K -userMIMO MAC and the two user MIMO interference channel.The K -user MIMO MAC consists of K transmitters, eachis equipped with M antennas and one legitimate receiverequipped with N antennas. The two user MIMO interferencechannel consists of two transmitters and two receivers, eachis equipped with M antennas. Both systems are studied invicinity of an unknown number of passive eavesdroppers. The j th eavesdropper is equipped with N Ej ≤ N E antennas, where N E is a constant known to all transmitters. Let x i denote the M × vector of symbols to be transmitted by transmitter i ,where i ∈ { , , ..., K } . We can write the received signal atthe j th legitimate receiver at time (sample) k as Y j ( k ) = q X i =1 H i,j V i x i ( k ) + n j ( k ) , (1)where i ∈ { , , ..., K } , j = 1 and q = K for the MACchannel, i, j ∈ { , } and q = 2 for the interference channeland the received signal at the j th eavesdropper is Z j ( k ) = q X i =1 G i,j ( k ) V i x i ( k ) + n Ej ( k ) , (2)where H i,j is the matrix containing the channel coefficientsfrom transmitter i to the legitimate receiver j , G i,j ( k ) is thematrix containing the channel coefficients from transmitter i to the eavesdropper j , V i is the precoding unitary matrix(i.e. V i V † i = I ) at transmitter i , n j ( k ) and n Ej ( k ) are theadditive white Gaussian noise vectors with zero mean andvariance σ at the legitimate receiver and the j th eavesdropper,respectively. We assume that the transmitters have globalchannel knowledge. We assume that N E < M . We definethe M × channel input from legitimate transmitter i as X i ( k ) = V i x i ( k ) . (3)Each transmitter i intends to send a message W i over n chan-nel uses (samples) to the legitimate receiver simultaneouslywhile preventing the eavesdroppers from decoding its message.The encoding occurs under a constrained power given byE n tr ( X i X † i ) o ≤ P ∀ i = 1 , ..., q (4)xpanding the notations over n channel extensions we get H ni = { H i (1) , H i (2) , . . . , H i ( n ) } . Similarly we can define G ni,j , X ni , Y n , Z nj . At each transmitter, the message W i isuniformly and independently chosen from a set of possiblesecret messages for transmitter i , W i = { , , . . . , nR i } .The rate for W i is R i , n log |W i | , where | · | denotes thecardinality of the set. A secure rate tuple ( R , ..., R q ) is saidto be achievable if for any ǫ > there is an n -length codessuch that the legitimate receiver decode the messages reliably,i.e., Pr { ( W , ..., ˆ W q )) = ( ˆ W , ..., ˆ W q ) } ≤ ǫ (5)and the messages are kept information-theoretically secureagainst the eavesdroppers, i.e., lim n −→∞ n H ( W , ..., W q | Z nj ) ≥ lim n −→∞ n H ( W , ..., W q ) − ǫ (6)where H ( · ) is the Entropy function and (6) implies the secrecyfor any subset S ⊂ { , } of messages including individualmessages [6]. The sum SDoF is defined as D s = lim P →∞ sup X i R i log P , (7)where the supremum is over all achievable secrecy rate tuples ( R , ..., R q ) , D s = d + ... + d q , and d i is the secure DoF oftransmitter i . III. K U SER
MIMO MAC
Theorem 1.
The number of SDoF of the K user MAC channelis upperbound as, D s ≤ min( KM − N E , N − N E K ) if M < NM − N E K if N ≤ M < N + N E K N if M ≥ N + N E K (8) Proof:
The first bound for D s ≤ KM − N E represent the DoFloss caused by the number of eavesdroppers’ antennas on thetransmitter side. Without loss of generality, we provide anupperbound for the case of existence of only one eavesdropperwith N E antennas. The SDoF of the single eavesdropper sce-nario is certainly an upperbound for the multiple eavesdropperscase, as increasing the number of eavesdroppers can onlyreduce the SDoF of the legitimate users. Accordingly, we omitthe eavesdropper subscript for simplicity of notation. Supposethat we can added | M − N | + antennas to the receiver sidethat won’t decrease the SDoF, the sum rate is upperboundedby the capacity of an equivalent MIMO wiretap channel with ( M + M ) transmit antennas and H = [ H H ] , x =[ x x ... x K ] T and precoding matrix V . The secrecy capacity( C s )for the MIMO wiretap channel with one eavesdropper andfixed known eavesdropper channel was presented in [4], andis an upperbound for all cases studied in this paper. It is easyto see that if the eavesdropper channel is unknown and timevarying the SDoF is also upperbounded by the fixed channelcase. The secrecy capacity ( C s ) was found to be equal to, C s = ( X , Y ) − I ( X , Z ) (9) = max K x log( | ( I + H , K x H † , ) | − log | ( I + G K x G † ) | ) (10) where K x is the covariance matrix of the transmitted signal.As H † H and G † G are hermitian, they can be diagonalizedas G † G = U G Λ G U † G , H † H = U H Λ H U † H , where U G U † G = I and U G U † G = I . Without loss of generality, Let V = [ V L V N ] , where V N contains the N E orthonormal basisof G , while V L contains the M − N E basis of the orthogonalcomplement of V N , and K x = VΛ K x V † . Therefore, D s ≤ lim P →∞ P (cid:0) max Λ Kx ) (log | I + U H Λ H U † H VΛ K x V † |− log | I + U G Λ G U † G VΛ K x V † | ) (cid:1) ( a ) ≤ lim P →∞ P (cid:0) max Λ Kx (log | Λ H Λ K x | − log | Λ G Λ K x | (cid:1) ( b ) ≤ lim P →∞ P (cid:0) max Λ Kx (log KM Y i =1 λ iH λ iK x − log N E Y i =1 λ iG λ iK x ) (cid:1) ≤ lim P →∞ P (cid:0) max Λ Kx ( KM X i =1 log λ iH λ iK x − N E X i =1 log λ iG λ iK x ) (cid:1) ≤ KM − N E (11) where λ iK x is the i th diagonal value of Λ K x and λ iG , λ iH aredefined similarly. (a) is because log | I + AB | = log | I + BA | for the above matrices, (b) is because lim P →∞ log | I + B | log P =lim P →∞ log | B | log P for any matrix B , and because | AB | = | A || B | for square matrices, and | V K x | , | V H | , | U | are independentof P .The second bound M − N E K represents the DoF loss ofeach transmitter due to the number of eavesdroppers antennasavailable. Let d ie be degrees of freedom of the parts of themessages sent by transmitter i , which can be decoded bythe eavesdropper. For the receiver to be able to decode thesecure messages with inter-message interference and achievethe designated SDoF of each transmitter, the receiver mustbe able to project the i th secure signal into an interferencefree space of dimension d i . On the other hand, the non secureparts of the messages can overlap at the receiver or even doesnot reach the receiver because the receiver is not interestedin decoding them and treated as interference. Let α i be thenumber of degrees of freedom of the non secure part of themessage i that reaches the receivers while β i be the number ofdegrees of freedom of the part that does not reach the receiver.Accordingly, d ie = α i + β i and the number of degrees offreedom of the message i is equal to ( d i + α i + β i ) . Since,the receiver is not interested in decoding the non secure partswith sizes { α i ; i = 1 , , ..., K } , and the non secure messagesoccupy at least max( α j ; j = 1 , , ..., K ) DoF, then K X i =0 d i + max( α j ; j = 1 , , ..., K ) ≤ N (12)while, β i ≤ M − N ∀ i = 1 , , ..., K (13)Moreover, since the eavesdropper has N E < M antennas,i.e the Dof of the transmitted messages is larger than N E then K X i =0 d ie = N E (14)The secure DoF is then upperbounded as s ≤ maximize { d j ; j = 1 , , ..., K } K X i =0 d i (15)subject to, K X i =0 d i + max( α j ; j = 1 , , ..., K ) ≤ N (16) K X i =0 ( α i + β i ) = N EK X i =0 β i ≤ K ( M − N ) The sum SDoF is maximized by minimizing max( α i ; j =1 , , ..., K ) . Combining the second and third constraint wehave K X i =0 α i ≥ N E − K ( M − N ) (17)and minimizing max( α i ; j = 1 , , ..., K ) is achieved bychoosing all { α i ; j = 1 , , ..., K } to be equal, and P Ki =0 α i = N E − K ( M − N ) . Accordingly, max( α j ; j = 1 , , ..., K ) ≤ N E K − ( M − N ) (18)and, D s ≤ M − N E K (19)Similarly, we can prove that for M ≤ N , the SDoF isupperbounded as, D s ≤ N − N E K (20)The Third bound D s ≤ N is the due to limited number ofantennas at the receiver which limits the SDoF. Achievable scheme :For securing the legitimate messages, the transmitters uses atwo-step noise injection by simultaneously sending a jammingsignal and using a stochastic encoder as follows,1) The transmitters send a jamming signal with power P J = αP that guarantees that all eavesdropper have aconstant rate ( o ( logP ) ) for all legitimate signal powervalues, where α is a constant controlled the transmittersto adjust the jamming.2) A stochastic encoder is built using random binning. Theencoder randomness rate is designed to be larger thanof the post-jamming eavesdroppers leakage, hence alleavesdroppers would have zero rate with the code lengthgoes to infinity meeting the secrecy constraints in (6).The jamming signal transmitted is a N E vector r =[ r r ... r K ] T with random symbols using { V J , V J , , V JK } as jamming precoders . Hence, the transmitted coded signalcan be broken into legitimate signal, s i , and jamming signal, r i , the precoder, V i can be also broken into legitimate For the special case N E = 1 , only one user sends a single jammingsymbol. precoder, V Li , and jamming precoder, V Ji such that V i x i = (cid:2) V Li V Ji (cid:3) " s i r i , ∈ { , , ..., K } . Choosing V J to be the unitary matrix, the jamming powerbecomes P J = E { tr ( r i r † i ) } = αP , where α is a constantcontrolled by the transmitter. Proposition 1.
The jamming signal, r , overwhelms all eaves-droppers’ signal space, and all eavesdroppers end up decodingzero DoF of the legitimate messages. The transmitter then usesa stochastic encoder to satisfy the secrecy constraint in (6)Let ¯ R e = I ( Z ; s , s , ..., s K ) be the rate of the eavesdropperwith the best channel assuming in worst case scenario that italso has N E antennas. Let R e = I ( Z ; W , W , ..., W K ) bethe legitimate message rate of the same eavesdropper, where R e < ¯ R e because of the stochastic encoder used. let ¯ R e j bethe rate of the j th eavesdropper. Then ¯ R e j ≤ ¯ R e ∀ j ∈ L ,where L is the unknown number of eavesdroppers. Proof: n ¯ R e ≤ I ( Z n ; s n , s n , ..., s nK )= h ( Z n ) − h ( Z n | s n , s n , ..., s nK )¯ R e = h ( Z ) − h ( Z | s , s , ..., s K ) ≤ N E log P − N E logP J + o (log P ) ≤ N E log P − N E log αP + o (log P ) ≤ o (log P ) = C E (21) where C E is a constant that does not depend on P and knownto the transmitters. Remark 1.
The constant eavesdropper post-jamming ratecomes from the fact that P J is controlled by the transmitter.Hence, setting P J = αP , a constant SNR is guaranteed at theeavesdroppers and a constant rate independent of P . For thecase of the constant known eavesdropper channel or unknownfading channel with known statistics, the constant C is knowntransmitter. The transmitters use the post-jamming rate difference totransmit perfectly secure messages using a stochastic en-coder similar to the one described in [8] according to thestrongest eavesdropper’s rate, C , in worst case scenario toachieve the secrecy constraint in 6 . The Wyner code C i ∈ C ( R ti , R i , n ) ∀ i = 1 , , ..., K of size nR ti is used to encode aconfidential message set W i = { , , ..., nR i } of transmitter i , R ti is the transmitted total rate and R i the secure messagerate (i.e. R ti ≥ R i ), and n is the codeword length. As aresult, the rate R l = R ti − R i is the cost of secrecy orthe rate lost to secure the legitimate message. For a Wynercode, if ˆ R e = R l , then the eavesdropper cannot decode thesecure message sent (i.e lim n −→∞ n R e ≤ ǫ ) . The Wyner code C ( R ti , R i , N ) is built using random binning [9]. We gener-ate nR ti codewords s ni ( w i , v i ) , where w i = 1 , , ..., nR i ,and v i = 1 , , ..., n ( R ti − R i ) , by choosing the nR ti symbols s i ( w i , v i ) independently at random according to the inputdistribution p ( s i ) . Then we distribute them randomly into nR i bins such that each bin contains n ( R ti − R i ) codewords.The stochastic encoder of C ( R ti , R i , N ) is described by aatrix of conditional probabilities so that, given w i ∈ W i ,we randomly and uniformly select a codeword to transmitfrom the bin w i or in other words, we select v i from { , , ..., n ( R ti − R i ) } and transmit s ni ( w i , v i ) . We assume thatthe legitimate receiver employs a typical-set decoder. Giventhe received signal y n , the legitimate receiver tries to find apair ( ˆ w, ˆ v ) so that s n ( ˆ w, ˆ v ) and y n are jointly typical [9]. Weset R i = I ( s i , Y ) − I ( s i , Z ) − ǫ and R ti = I ( s i , Y ) − ǫ .The error probability and equivocation calculations are straightforward extensions of similar Wyner random binning encoders[9], H ( W i n ) = I ( s ni ; Y n ) − I ( s ni ; Z n ) − mǫ (22) H ( W i n | Z n ) = I ( s ni ; Y n | Z n ) − I ( s ni ; Z n | Z n ) − nǫ (23) = I ( s ni ; Y n , Z n ) − I ( s ni , Z n ) − nǫ (24) ≥ H ( W i n ) − nǫ (25)and, H ( W n , ..., W K n | Z n ) = K X i H ( W i n | Z n ) (26) ≥ K X i H ( W i n ) − Knǫ (27) ≥ H ( W n , ..., H ( W K n ) − Knǫ (28).
Let U be the post-processing matrix that projects the signalinto a jamming free space at the legitimate receiver. The securemessages sum rate is then, K X i =1 R i ≥
12 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I + K X i =1 ( UH i V Li s i s † i V L † i H † i U † ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − R e (29) As lim n −→∞ n R e ≤ ǫ for all values of G i and P , a positivesecrecy rate, which is monotonically increasing with P , isachieved. Computing the secrecy degrees of freedom boilsdown to calculating the degrees of freedom for the first term inthe right hand side of (29), which represents the receiver DoFafter jamming is applied. Next we will calculate the SDoF andshow how jamming is designed to Maximize the achievableSDoF. A. Achievability for M ≥ N + N E K For this region, the transmitters send the jamming signalsusing precoders V Ji , using Nullspace jamming , respectively.All the precoders have N E K jamming streams such that the totalnumber of jamming streams reaching each eavesdropper equal N E . Nullspace jamming:
In nullspace jamming method, thetransmitter i sends a jamming signal of J i dimensions usingthe precoder V Ji which lies in the nullspace of the channel H i , V Ji = Null ( H i ) i ∈ , , ...., K (30)This blocks N E dimensions at each eavesdropper and leaves N free dimensions at the legitimate receiver to attain thelegitimate signal, thus the following sum SDoF is achievable, D s ≤ N (31) B. Achievability for
M < N
For this region we use aligned jamming for blocking theeavesdropper where jamming is aligned at the receiver tominimize the wasted space and maximize the SDoF
Aligned jamming:
The jamming signals of both transmittersare aligned at the legitimate receiver signal space. Each group j of transmitters of size L j ≤ K aligns portions of its jammingsignal together at the receiver. There are K C L j groups ofsize L , while the total number of groups P a = Ka =2 ,b ≤ a i C j ,the number of jamming of streams assigned to each group ( J g /N E ) depends on the relation between ( M, N, N E ) . Let I j be the jamming space at the receiver designated for group j . Each transmitter aligns a part or the whole of its jammingsignal into this jamming space. The total signal space oftransmitter i occupies only M < N dimensions at the receiver.This make the received signal spaces of different transmittersdistinct at the receiver. So a common space is needed todirect the jamming signal into. Let A i span the receivedsignal space of transmitter i , i.e span the space including allpossible received vectors at the receiver, I is chosen to be theintersection of these spaces, i.e., I j = L j \ i =1 A i . (32) I j would have positive size only if M ≥ N . Without loss ofgenerality, we design ( V Ji , i = 1 , , ..., K ) such that, H , V J = H , V J = ... = H L j , V JL j = I j (33)While the system of equations in ((33)) has more variablesthan the number of equations, (32) ensures that the systemhas a unique solution as I j lies in the spans of ( H i, ; i =1 , , ..., K .Let H i, = " H ′ i, H ′′ i, I = " I ′ j I ′′ j ∀ i = 1 , , ..., K, j = 1 , , .., L j (34)where H ′ i, contains the M rows of H i, and H ′′ i, containsthe other N − M rows, and I ′ i contains the M rows of I and I ′′ i contains the other N − M rows. Therefore, we can choosethe following design which satisfies (33) V Ji = ( H ′ i, ) − I ′ j ∀ i = 1 , , ..., K, j = 1 , , .., L j (35)For the legitimate receiver to remove the jamming signal anddecode the legitimate message, it zero forces the jammingsignal using the post-processing matrix U . For the case N E is odd, each transmitter will align its jamming signal intoan ⌊ N E ⌋ –dimensional half space using linear alignment. Theremaining dimensional space will be equally shared betweenthe two transmitters’ jamming signal using real interferencealignment [7], yielding each transmitter’s jamming signal tooccupy N E .The jamming alignment is possible for group j the size ofintersection of j is greater than zero or L j ( M − N ) + M ≥ (36)where the number of jamming streams J j that can be senty each group is constrained to J j ≤ L j ( L j ( M − N ) + M ) (37)where the J j streams wastes J j L j dimensions at the receiverfor jamming. For maximizing the achievable SDoF, the groupwith the smaller ratio J j L j is used for jamming first.The alignment process begins with assigning the maximumnumber of jamming streams to the largest possible group asit can align the largest number of jamming streams per onedimension wasted at the receiver. D s ≤ min( KM − N E, N − N E L ) , (38)where L is the result of the following optimization,maximize L (39)subject to, MN − M ≤ L ≤ K (40) N E ≤ L ( L ( M − N + M )) (41)The previous scheme meets the upperbound in 8 at (( L = K or MN − M ≤ K ) and N E ≤ K ( K ( M − N )+ M ) ) and at ( KM − N E < N − NN E ) achieving D s = min( KM − N E , N − NN E ) . C. Achievability for N + N E K > M > N In this region the transmitters uses both the aligned andNullspace jamming methods, each transmitter sends M − N jamming streams using Nullspace jamming and sends N E K − ( M − N ) jamming streams using aligned jamming.The achievable SDoF is then D s ≤ N − N E − K ( M − N ) L (42) D s ≤ N − N E − K ( M − N ) L (43)where L is defined as in (39), and the achievable regionmeets the upperbound at ( L = K and N E ≤ K (( K + 1)( M − N ) + M ) ) and at ( K (2 M − N ) − N E < N − NN E ) . D s ≤ N − N E − K ( M − N ) K (44) ≤ M − N E K (45)IV. T HE TWO USER M × M INTERFERENCE CHANNEL
Theorem 2.
The number of SDoF of the two user M × M interference channel is upperbound as, d + d ≤ M − N E (46) Proof:
Let d e and d e be the maximum degrees of freedomthat the eavesdropper can decode out of the transmitters oneand two signals, respectively. Suppose that we added M − N antennas to receiver one, this can only improve the codingscheme rate. As receiver one fully receive the signal sentby transmitter two to receiver two X with modified noise,and X can be decoded by receiver one with no interference by definition. Then d and X occupies two distinct spaces atreceiver one, the SDoF is upperbounded then as d + max( d e , d + d e ) ≤ M (47)and for both results of max( d e , d + d e ) , the following is true d + d + d e ≤ M (48)Similarly, by adding M − N antennas to receiver two, we have d + d + d e ≤ M (49)Moreover, since the eavesdropper has N E antennas then d e + d e = N E (50)Combining (13), (49), (50) d + d ≤ M − N E (51) Theorem 3.
For the two user M × M interference channel,the following number of SDoF is achievable d + d ≤ M − N E (52) Proof:
For this channel, the jamming is aligned using basic inter-ference alignment method combined with a stochastic encodersimilar to the one used in the MAC, H V J = H V J (53) H V J = H V J (54)Using this method N E jamming streams are aligned at N E dimensions at each receiver. This leaves N − N E dimensionsfree of Jamming at each receiver. As both receivers fullyreceives both messages and for each receiver to decode its ownmessage the interfering message should occupy an orthogonalspace, then d + d ≤ M − N E (55)V. C ONCLUSION
We studied the K -user MAC channel and the two userinterference channel with multiple antennas at the transmit-ters, legitimate receivers and eavesdroppers. Generalizing newupperbound was established and a new achievable schemewas provided. We showed that our scheme is optimal for theinterference channel and partially optimal for the MAC.R EFERENCES[1] A. D. Wyner.
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