K-user Interference Channels: Achievable Secrecy Rate and Degrees of Freedom
aa r X i v : . [ c s . I T ] M a y K -user Interference Channels: Achievable SecrecyRate and Degrees of Freedom Xiang He Aylin Yener
Wireless Communications and Networking LaboratoryElectrical Engineering DepartmentThe Pennsylvania State University, University Park, PA 16802 [email protected] [email protected]
Abstract —In this work, we consider achievable secrecy ratesfor symmetric K -user ( K ≥ ) interference channels withconfidential messages. We find that nested lattice codes andlayered coding are useful in providing secrecy for these channels.Achievable secrecy rates are derived for very strong interference.In addition, we derive the secure degrees of freedom for a rangeof channel parameters. As a by-product of our approach, wealso demonstrate that nested lattice codes are useful for K-usersymmetric interference channels without secrecy constraints inthat they yield higher degrees of freedom than previous results. I. I
NTRODUCTION
In a wireless environment, interference is always present.Traditionally, interference is viewed as a harmful physicalphenomenon that should be avoided. Yet, from the secrecyperspective, if interference is more harmful to an eavesdropper,it can be a resource to protect confidential messages. To fullyappreciate and evaluate the potential benefit of interferenceto secrecy, the fundamental model to study is the interferencechannel with confidential messages. This model with two usershas been investigated extensively up to date, e.g., [1]–[3].The K -user ( K ≥ ) interference channel, when all linkcoefficients are i.i.d. fading, has been studied both with andwithout secrecy constraints [4], [5]. In these references, thekey ingredient for achievability is interference alignment intemporal domain. For the case without secrecy constraints,reference [4] proves the degree of freedom characterization tobe K/ for the sum rate.For the static channel without secrecy constraints, [6] showsthe degrees of freedom can not exceed K/ , though whetherthis bound is achievable remains elusive except for when thechannel gains of the intended links are algebraic irrational andthe other channel gains are rational numbers [7]. References[8], [9] show K/ can be approached asymptotically for astatic K -user symmetric channel if the channel gain of theinterfering link goes to or ∞ . Both [8] and [9] employ theidea of interference alignment in the signal space: Reference[8] uses the Q -bit expansion and reference [9] uses the latticecode with a sphere as the shaping set [10].For the static channel with confidential messages, the prob-lem of finding the secure degrees of freedom has largelyremained unaddressed so far. In this paper, we focus onthe K -user ( K ≥ ) interference channel with confidentialmessages, where each receiver is an eavesdropper with respect bX X X Z Z Z S S S D D D W W W ˆ W ˆ W ˆ W Y Y Y bb Fig. 1. K -User Symmetric Interference Channel, K = 3 to messages not intended for it. We first derive achievable ratesusing nested lattice codes for very strong interference. We theninvestigate the secure degrees of freedom of the sum rate forthis channel. We show that positive secure degrees of freedomare achievable, made possible by the fact that users can protecteach other via cooperative jamming [11]. Inspired by [9], alayered encoding and decoding scheme is used. The achievedsecure degrees of freedom is roughly half of the achievabledegrees of freedom in the model without secrecy constraintsand is achievable for both weak and strong interference regime.The key ingredient is a tool first introduced in [12] whichallows us to bound the secrecy rates under nested lattice codes.As a by-product of our approach, we also show that for thecase without secrecy constraints, a degree of freedom higherthan found in [8], [9] is achievable. The main reason leading tothis improvement is the use of the nested lattice codes insteadof sphere-shaped lattice codes as in [9]. This leads to differentdecodability conditions and power allocation among differentlayers.The rest of the paper is organized as follows: In SectionII, we describe the system model. In Section III, we derivethe very strong interference condition and the correspondingachievable secrecy rates. Section IV presents the achievabledegrees of freedom for the sum rate and the sum secrecy rateand compares it with previous results. Section V concludesthe paper. II. S YSTEM M ODEL
We consider the Gaussian interference channel shown inFigure 1 for K = 3 . The average power constraint for eachsource node S i is P . Z i , i = 1 , ..., K are independent Gaussianandom variables with zero mean and unit variance. Thechannel gain coefficient between S i and D i is b , while thechannel gain coefficient between S i and D j , i = j is 1.Node S i tries to send a secret message W i to node D i , whilekeeping it secret from all the other receiving nodes D j , j = i . Hence, for W , ..., W K , node D is viewed as a potentialeavesdropper. Let the signal received by D over N channeluses be Y N . The corresponding secrecy constraint is given by: lim N →∞ N H (cid:0) W , ..., W K | Y N (cid:1) = lim N →∞ N H ( W , ..., W K ) (1)The secrecy constraints due to node D , ..., D K are defined ina similar fashion.III. A CHIEVABLE S ECRECY R ATES U NDER V ERY S TRONG I NTERFERENCE
In this section, we summarize several key steps of theachievability proof and derive the very strong interferencecondition. For clarity, we focus on K = 3 . The scheme isapplicable to K > as well.We note that the achievable scheme is similar to that ofthe many-to-one interference channel [13]. However, becauseof the increased connections in the network, the very stronginterference condition shall differ from that of [13]. A. Source Node
Let (Λ , Λ c ) be a nested lattice structure in R N , where Λ c isthe coarse lattice. The modulus operation x mod Λ c is definedas x mod Λ c = x − arg min y ∈ Λ c d ( x, y ) , where d ( x, y ) is theEuclidean distance between x and y . The fundamental region V (Λ c ) of the lattice Λ c is defined as the set { x : x mod Λ c = x } .The i th source node constructs its input to the channel over N channel uses, X Ni , as follows: Let t i ∈ Λ ∩ V (Λ c ) . Let d i be the dithering noise that is uniformly distributed over V (Λ c ) .Then X Ni = ( t Ni + d Ni ) mod Λ c .We assume the dithering noise d i is known by all destinationnodes. B. Destination Node
Because of the symmetry of the channel, without loss ofgenerality, we focus on the first destination node D . Thedestination first decodes the modulus sum of the interference,and then decodes its intended message.The signal received by D over N channel uses is: Y N = bX N + ( X N + X N ) + Z N (2)Node tries to decode t N + t N mod Λ c . Although X N is not Gaussian, it can be approximated by a Gaussian distri-bution as N → ∞ , as shown in [14, (82)] or [13, (15)-(21)].Hence we can apply the analysis in [14, Theorem 5], that theprobability of decoding error will go to as N → ∞ when R ≤ . (cid:18)
12 + Pb P + 1 (cid:19) (3) With the knowledge of t N + t N mod Λ c , node can recon-struct X N + X N mod Λ c . After subtracting this term from Y N mod Λ c , the rest part of the interference signal is (cid:0) bX N + Z N (cid:1) mod Λ c (4)Then, it can be shown [14, (89)] [13, (27)] that if b P + 1 < P (5)then this signal can be approximated by bX N + Z N (6)That is to say: lim N →∞ Pr( bX N + Z N = bX N + Z N mod Λ c ) = 0 (7)Finally, the destination tries to decode t from (6). Basedon [14, Theorem 5], the probability of decoding error will goto zero as N → ∞ , if R < C ( b P ) (8)In summary, if (3), (5) and (8) hold, then the decoding errorprobability at node should vanish as N → ∞ . C. Equivocation Rate
The computation of the equivocation rate is the same as[13], as shown below: H (cid:0) t N , t N | Y N , d Ni , i = 1 , , (cid:1) (9) ≥ H (cid:0) t N , t N | Y N , X N , Z N , d Ni , i = 1 , , (cid:1) (10) = H (cid:0) t N , t N | X N + X N , d Ni , i = 1 , , (cid:1) (11)In [12, Theorem 1], it is proved that we can find an integer T , ≤ T ≤ N , such that X N + X N is uniquely determinedby { X N + X N mod Λ c , T } . Using this result, (11) equals H (cid:0) t N , t N | X N + X N mod Λ c , T , d Ni , i = 1 , , (cid:1) (12) = H (cid:0) t N , t N | t N + t N mod Λ c , T , d Ni , i = 1 , , (cid:1) (13) = H (cid:0) t N , t N | t N + t N mod Λ c , T (cid:1) (14) = H (cid:0) t N , t N | t N + t N mod Λ c (cid:1) + H (cid:0) T | t N , t N (cid:1) − H (cid:0) T | t N + t N mod Λ c (cid:1) (15) ≥ H (cid:0) t N , t N | t N + t N mod Λ c (cid:1) − H ( T ) (16)The first term in (16) can be bounded as follows: H (cid:0) t N , t N | t N + t N mod Λ c (cid:1) (17) = H (cid:0) t N | t N + t N mod Λ c (cid:1) + H (cid:0) t N | t N , t N + t N mod Λ c (cid:1) = H (cid:0) t N (cid:1) = N R (18)where R is the rate of the codebook computed as R = N log k Λ ∩ V (Λ c ) k .Hence the mutual information leaked to the eavesdropper isbounded as: I (cid:0) t N , t N ; Y N , d Ni , i = 1 , , (cid:1) ≤ N ( R + 1) (19)Intuitively, this means each pair of users have to pay R + 1 inrate to confuse the eavesdropper. Under a symmetric setting,ach user loses . R + 0 . in rate. This leaves room of . R − . for each user to send the secret message, which leads tothe following theorem: Theorem 1:
For any
R, P, b such that (3), (5) and (8) hold,a secrecy rate of [0 . R − . + is achievable for each user. If b ≤ min { P − P , q P + − P } (20)then R = C ( b P ) . Remark 1:
Under this condition on b , it can be verified(3) and (5) become redundant. Hence the secrecy rate is givenwhen R is selected to be C ( b P ) . Remark 2:
Reference [15] considers the 3 user symmetricinterference channel without secrecy constraints. A differentlattice structure is used [10], where V (Λ c ) is replaced by asphere or a sphere shell. After power normalization, the verystrong interference condition of [15] can be expressed as b ≤ √ P − P (21)Comparing (21) with (20), we notice (20) is slightly looser.Hence, using a nest lattice structure allows a slightly widerrange of channel parameter under which the channel has verystrong interference. D. K > Theorem 1 can be extended to the case with more than 3users. In this case, The achievable rate becomes (cid:20) R − RK − − log ( K − K − (cid:21) + (22)Equation (3) becomes R ≤ . (cid:18) K − Pb P + 1 (cid:19) (23)Hence, the very strong interference condition (20) becomes b ≤ min { P − P , q P − c + ( c +1) − c +12 P } (24)where c = K − K − . Remark 3:
It is then interesting to look at the behavior ofthe secrecy rate when the number of users K → ∞ in thevery strong interference channel. From (22), the secrecy ratewill converge to R . This means the cost of secrecy per uservanishes. A similar phenomenon is also observed in [13] forthe many-to-one interference channel.IV. S ECURE D EGREES OF F REEDOM
In this section, we derive the achievable secure degreesof freedom for a given channel gain b . Like [9], a layeredlattice structure is used. However, instead of the sphericalcode, the nested lattice code is used in order to leverage therepresentation theorem [12] to bound the secrecy rate. A. Source Node
Due to symmetry, we focus on source node . The trans-mitted signal is the sum of signals from different layers. Thesignal from the i th layer over N channel uses, X N , is givenby X N = M X i =1 X N ,i (25)where, like [9], the total number of levels M is to bedetermined by total power. X ,i is the signal for the i th level,which is given by: X N ,i = ( t N ,i + d N ,i ) mod Λ c,i (26)where d N ,i is the dithering noise uniformly distributed over V (Λ c,i ) . We assume the dithering noise for each level at eachsource node is independent from each other. t N ,i is taken fromthe Voronoi code book Λ i ∩ V (Λ c,i ) , where the variance of V (Λ c,i ) is chosen to be P i . Let the rate of this codebook be R k,i for the k th user. B. Destination Node1) Strong Interference Regime:
Like [9], we first examinethe case where the destination node decodes the interferencefirst, and then decodes the intended signals. The case wherethe destination decodes the intended signals first can beanalyzed in the similar fashion. Due to symmetry, we focuson destination node D . For the i th layer, the destination nodedecodes the modulus sum of the interference, subtracts it, thendecodes the signal from source node S . Suppose decoding forall layers j , j > i , are successful, and the modulus operation atlayer j incurs negligible distortion for signals at lower layers.Then the remaining signal after subtracting the decoded signalscan be approximated by: Y N ,i = bX N ,i + X N ,i + X N ,i + X ≤ j
12 + P i b P i + A i (cid:19) (31)fter decoding ( t N ,i + t N ,i ) mod Λ c,i , node D subtracts ( t N ,i + d N ,i + t N ,i + d N ,i ) mod Λ c,i from Y N ,i mod Λ c,i . Thesignal after the subtraction is given by: ˆ Y N ,i = ( bX N ,i + X ≤ j P I,i (33) ˆ Y N ,i can be approximated by bX N ,i + X ≤ j
12 + P i b P i + A i = 1 + b P i A i (38)It is easy to check that (38) leads to: P i = 2 − b + √ − b + b b A i (39)For P i to be a real number, we require − b + b > .This, along with the fact that A i > , means b < − √ (40)which is about . . Define α = 2 − b + √ − b + b b , β = b + 2 (41)Then P = α and P i = α β X ≤ j (42) Therefore P i = α ( αβ + 1) i − (43)The power expended by each user is given by P = M X i =1 P i = ( αβ + 1) M − β (44)Since αβ > , we have lim M →∞ P = ∞ .Under this power allocation, R ,i is given by R ,i = 0 .
52 log (cid:18)
12 + P i b P i + A i (cid:19) + 0 .
52 log (cid:18) b P i A i (cid:19) (45) = 12 (cid:18) . (cid:18) A i +1 A i (cid:19) − . (cid:19) (46)Therefore M P i =1 R ,i = (0 . ( A M +1 ) − . M ) .Let R k denote the rate of the k user. Hence, R k = M P i =1 R k,i , k = 1 , , . If there is no secrecy constraints, thedegree of freedom is given by lim P →∞ P k =1 R k log (cid:18) P i =1 P i (cid:19) =1 . − lim M →∞ . M log P (47) =1 . − . ( αβ + 1) (48)Let R e,k denote the rate of the k user. When there are secrecyconstraints, each layer can support a secrecy rate of [0 . R ,i − . + . The secure degrees of freedom is given by lim P →∞ P i =1 R e,i log (cid:18) P i =1 P i (cid:19) ≥ lim M →∞ × (cid:18) M P i =1 (0 . R ,i − . (cid:19) log (3 P ) (49) = 34 − . ( αβ + 1) (50)We still need to check if the condition (33) are met. Underthe current power allocation, we have A i = ( αβ + 1) i − (51)(33) means P i ≥ b P i + A i (52)Hence we require (1 − b ) α ≥ (53)which holds when (40) holds. In summary, we have thefollowing theorem: Theorem 2: If b < − √ , the following degrees offreedom for the sum rate is achievable: . − . ( αβ + 1) (54) −5 −0.200.20.40.60.811.21.41.6 b DOF using nested lattice codeDOF using sphere shapedlattice code DOF using TDMADOF using Q−bitexpansion Secure DOF Secure DOF
Fig. 2. Degrees of freedom (DOF)
Moreover, the following degrees of freedom for the sum secrecy rate is achievable: (cid:20) − . ( αβ + 1) (cid:21) + (55)
2) Weak Interference Regime:
When the intended signalis strong enough, the destination should decode it first, andthen decode the interference later. In this case, (30) and (36)become P I,i = 2 P i + A i , P S,i = b P i (56) P ′ I,i = A i , P ′ S,i = 2 P i (57)Equation (38) becomes b P i P i + A i = 12 + P i A i (58)This means P i is given by (43) with α given below α = 0 . (cid:16) b + p b + 4 (cid:17) (59)and β remains as b + 2 .In order for the modulus operation to introduce negligibledistortion to lower layers, we require P i > P ′ I,i (60)This translates into α > . This means b > / . Hence, wehave the following theorem: Theorem 3: If b > / , then the degrees of freedom givenby (54) and secure degrees of freedom given by (55) areachievable, where α is given by (59) and β = b + 2 .
3) Numerical Results:
As shown in Figure 2, when b →∞ or b → , the secure degrees of freedom converge to ,which is half the secure degrees of freedom achievable in themodel without secrecy constraints.Also compared in Figure 2 are the degrees of freedom whenthere are no secrecy constraints. The black dashed lines showthe degrees of freedom from [9] using a sphere shaped lattice code. The blue dotted line denotes the degrees of freedomachieved by the Q -bit expansion method in [8], which is K (1 − log b (2 K )) , where K = 3 in Figure 2. The blue linesare the degrees of freedom achieved by our proposed schemeusing the nested lattice code. We see that it consistentlyoutperforms the scheme from [9] when b < − √ orwhen / < b < . V. C ONCLUSION
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