Secure Degrees of Freedom for Gaussian Channels with Interference: Structured Codes Outperform Gaussian Signaling
aa r X i v : . [ c s . I T ] M a y Secure Degrees of Freedom for Gaussian Channelswith Interference: Structured Codes OutperformGaussian Signaling
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 prove that a positive secure degreeof freedom is achievable for a large class of Gaussian channelsas long as the channel is not degraded and the channel is fullyconnected. This class includes the MAC wire-tap channel, the 2-user interference channel with confidential messages, the 2-userinterference channel with an external eavesdropper. Best knownachievable schemes to date for these channels use Gaussiansignaling. In this work, we show that structured codes outperformGaussian random codes at high SNR when channel gains are realnumbers.
I. I
NTRODUCTION
Information theoretic security, originally proposed by Shan-non [1], seeks the fundamental limits of reliable transmis-sion rates when the messages must be kept secret from acomputation-power unlimited adversary whose observation ofthe transmitted signals contain some uncertainty. By now, it iswell known that introducing interference into the channel in aproper manner may increase the uncertainty observed by theadversary and hence allow for a higher rate of secret messages[2]–[4]. The interference should be introduced in a way suchthat it is more harmful to the adversary than it is to the intendedreceiver of the messages. Hence, the key is to achieve a finebalance between secrecy against the adversary and the levelof harmful interference to the system.For these channel models, the achieved rate obtained sofar is far from the outer bounds. For example, the genieouter bound from [4] increases with power P at the speed of . ( P ) [5, (69)]. The achievable secrecy rate convergesto a constant when P goes to ∞ [4, Theorem 2]. This meansthe gap between the achievable rate and the outer bound isunbounded and the trade-off between secrecy and interferenceis still not well-understood. In fact, once the channel model issuch that the intended receiver is not harmed by the introducedinterference, the achieved secrecy rate immediately comeswithin . bits/channel use of the capacity region, as wasshown for the one sided interference channel in [6] and theorthogonal MAC wire-tap channel in [7].In this work, we consider the more general case whereintroducing interference will both confuse the eavesdropperand harm the intended receiver simultaneously. We show, for a large class of Gaussian channels with confidentialmessages where introducing interference effect both the in-tended receiver and the eavesdropper, that signaling usingstructured codes can out-perform signaling with i.i.d. Gaussiancodebooks in high SNR. This class includes the GaussianMAC-wiretap channel [2], the Gaussian interference channelwith confidential messages [3] and the Gaussian interferencechannel with an external eavesdropper [7]. It has been afolk conjecture that the achievable rate regions with Gaussiancodebooks in these works were likely optimal and efforts [4],[5], [7] have been made to find outer bounds to prove this. Adirect consequence of the result we report in this paper is thatthis is not so.This insight comes from studying the secure degree offreedom of the interference assisted wire-tap channel [4],which falls under the three channels mentioned above whenonly one source node has confidential message to send. Asmentioned before, reference [4] shows that the achieved rateusing Gaussian codebooks converges to a constant as powerincrease, which implies the obtained secure degree of freedomfor this channel is . In contrast, we find that a positive degreeof freedom is actually achievable for all channel gains as longas the channel is not degraded. The key to getting this resultis the use of different types of structured codes for appropriatechannel gains rather than Gaussian signaling.We note that a positive secure degree of freedom is knownto be achievable for the fading channel [8], which requirescoding over different fading states and hence does not implythe result here.The result here provides another example that structuredcodes are useful in proving information theoretic results. A listof examples that structured codes outperform simple randomcoding arguments in non-secrecy problems can be found in [9].Using structured code in secrecy problems was first proposedby the authors in [10]. Up to date structured codes are foundto be useful for relay channels due to the possibility ofcompute-and-forward [9], [10], or for interference channelswith more than two users due to the possibility of interferencealignment [11]–[13]. The result here provides the first examplethat structured codes are indeed useful for two user Gaussian ˜ X Z Z S S D D W ˆ W Y Y X ± √ a √ b Fig. 1. Interference-assisted Wire-tap Channel channels as well. II. S
YSTEM M ODEL
Consider the Gaussian interference-assisted wire-tap chan-nel [4] shown in Figure 1. In this model, node S sends a secretmessage W via ˜ X to node D , which must be kept secretfrom node D . We assume the channel is fully connected,which means that no link’s channel gain equals zero. Thisassumption is obviously valid for a wireless medium. Thenafter normalizing the channel gains of the two intended linksto , the received signals at the two receiving node D and D can be expressed as ˜ Y = ˜ X + √ aX + Z Y = √ b ˜ X ± X + Z (1)where Z i , i = 1 , is a zero-mean Gaussian random variablewith unit variance. For now, we assume √ a and √ b are realnumbers. The case with complex numbers will be brieflyexplained in Section II-B.Since W must be kept secret from D , we require lim n →∞ n H ( W ) = lim n →∞ n H ( W | Y n ) (2)The achieved secrecy rate R e is defined as lim n →∞ n H ( W ) such that the condition (2) is fulfilled and W can be reliablyreceived by D .Let X = √ b ˜ X and Y = √ b ˜ Y . Then from (1), we have Y = X + √ abX + √ bZ Y = X ± X + Z (3)In the sequel, we will focus on this scaled model instead, aswe find it more convenient to use it to explain our results.Let the average power constraint of node S i on X i be ¯ P i .The secure degree of freedom of the secrecy rate is defined as lim sup ¯ P i →∞ ,i =1 , R e log (cid:18) P i =1 ¯ P i (cid:19) (4)It is clear that the secure degree does not change, whether themodel is described via (1) or (3). A. Relationship with Other Channels
The significance of the interference-assisted wiretap channelis that it can be considered as a special case of a large class ofchannel models with confidential messages, as shown below:1) If node S has a confidential message W for D , whichmust be kept secret from D , then the channel is theMAC-wiretap channel considered in [2]. W ˜ X Z Z S S D D W ˆ W Y Y X ± √ a √ b W Z D ˆ W W Fig. 2. Interference-assisted Wire-tap Channel as a Special Case of theInterference Channel with an External Eavesdropper
2) If node S has a confidential message W for D ,and the message must be kept secret from D , thenthe channel is the interference channel with confidentialmessage considered in [3].3) As shown in Figure 2, we can add another receivingnode D to Figure 1, to which node S wants to sent aconfidential message W . Again W must be kept secretfrom D . Then the channel becomes the interferencechannel with an external eavesdropper considered in [7,Section VI].Hence, any secrecy rate achieved in the the interference-assisted wire-tap channel is an achievable individual rate forall the three multi-user channels mentioned above. Remark 1:
The results here also strengthen a result theauthors derived previously for the K -user interference channelwith confidential messages, where K ≥ [11]. In [11], it isnot known if, for sum secrecy rate, a positive secure degree offreedom is achievable for arbitrary channel gains. Since theinterference-assisted wire-tap channel is also a special case ofthe K -user channel, we see the answer to this question is yesunless the channel is degraded for any pair of the K users. B. Complex Channel Gains
More general than the channel with real channel gains isthe channel with complex channel gains. The reason that wefocus on the real case in the sequel is that the complex caseis actually easier in terms of achieving positive secure degreeof freedom, as explained below.Since the channel is fully connected, after normalization ofthe channel gains and variable substitution, the received signalsat nodes D and D can be expressed as [14]: Y = X + √ abe jψ X + √ bZ Y = X + X + Z (5)where Z i , i = 1 , are rotational invariant complex Gaussianrandom variables with unit variance. Then we have: Theorem 1:
A secure degree of freedom of is achievableif ψ = 0 or π mod 2 π . Proof Outline:
Let Im X i = 0 , i = 1 , . Let cot x =cos x/ sin x . Then since Im Y = Im Z , Im Y does not provideany information about W to the eavesdropper. Hence wecan assume the eavesdropper receives Re Y only. Node D computes g ( Y ) = Re Y − cot ψ Im Y . Then the channel cane expressed as g ( Y ) = Re X + √ b (Re Z − cot ψ Im Z )Re Y = Re X + Re X + Re Z (6)The channel then becomes an one-sided interference channel.By transmitting a i.i.d Gaussian noise via Re X , the channelis equivalent to a Gaussian wire-tap channel. It is known thatthe following secrecy rate is achievable [15]: C (cid:18) P ( b csc ψ ) / (cid:19) − C (cid:18) P P + 1 / (cid:19) (7)where C ( x ) = log (1 + x ) . P i is the average powerconstraint on X i . Hence a secure degree of freedom of isachievable for this channel. C. Gaussian Signaling
In [4], an achievable rate is derived with Gaussian code-books and power control. One implication of this achievablerate is the high SNR behavior as described by Theorem 2therein. We re-state this result below:
Theorem 2:
With Gaussian codebooks, the achievable se-crecy rate R converges to a constant when the power con-straint of node D and D goes to ∞ .This means the achieved secure degree of freedom by thecoding scheme in [4] is .III. T HE A CHIEVABLE S CHEME
A. Results on Structured Codes1) Nested Lattice:
A nested lattice code is defined as anintersection of N -dimensional fine lattice Λ and the fundamen-tal region of an N -dimensional “coarse” lattice Λ c , denotedby V (Λ c ) . We require that Λ c ⊂ Λ . Let u Ni be uniformlydistributed over Λ ∩ V (Λ c ) . Let the dithering noise d Ni bea continuous random vector which is uniformly distributedover V (Λ c ) . Define modulus operation such that x mod Λ c = x − arg min u ∈ Λ c k x − u k . Then the values of X i over N channel uses are computed as X Ni = ( u Ni + d Ni ) mod Λ c (8) u Ni , d Ni , i = 1 , are independent. We also assume d Ni , i = 1 , are known by all receiving nodes. Hence they are not used toenhance secrecy.As will be shown later, we are interested in lower-boundingthe expression I ( u N ; X N ± X N , d N , d N ) , which correspondsto the rate of information leaked to the eavesdropper. To dothat, we need the following result. Its proof follows from therepresentation theorem introduced in [10] and is given in [14]. Theorem 3:
There exists an random integer T , such that ≤ T ≤ N , and X N ± X N is uniquely determined by { T, X N ± X N mod Λ c } .Using Theorem 3, we have I ( u N ; X N ± X N , d N , d N )= I ( u N ; X N ± X N mod Λ c , T, d N , d N ) (9) ≤ I ( u N ; X N ± X N mod Λ c , d N , d N ) + H ( T ) (10) = I ( u N ; u N ± u N mod Λ c ) + H ( T ) (11) = H ( T ) ≤ N (12)This means at most N bit per channel use is leaked to theeavesdropper over N channel uses.
2) Integer Lattice:
An integer lattice code with parameter Q is composed of points in the set [0 , Q ) ∩ Z where Z is theset of all integers. As will be shown later, in this case, the rateof information leaked to the eavesdropper is given by f ( Q ) defined as: f ( Q ) = I ( X ; X ± X ) (13)where X i , i = 1 , is uniformly distributed over [0 , Q ) ∩ Z . f ( Q ) can be lower bounded by the following lemma: Lemma 1:
For a positive integer Q , f ( Q ) ≤
12 log (2 πe ( 16 − Q )) <
12 log ( πe < . (14)The proof follows from [13, Lemma 12] and is given in [14].We next use these results to derive achievable secure degreeof freedom for the interference assisted wire-tap channel. B. When √ ab is algebraic irrationalTheorem 4: A secure degree of freedom of / is achiev-able when √ ab is an algebraic irrational number. Proof:
We use the lattice codebook used in [13, Theorem1]. Let Λ P,ε be the scalar lattice defined as: Λ P,ε = n x : x = P / ε z, z ∈ Z o (15)The codebook C P,ε is given by: C P,ε = Λ
P,ε ∩ h −√ P , √ P i (16)where P = min { ¯ P , ¯ P } . It then can be verified that, for largeenough P , we have log |C P,ε | ≥ log (cid:16) P / − ε − (cid:17) ≥ log (cid:16) P / − ε (cid:17) (17)The codebook is used for both node S and node S . Thecodeword transmitted by node S is chosen based on the secretmessage W . The codeword transmitted by node S is chosenindependently according to a uniform distribution.Since the input from S is i.i.d., the channel is thenequivalent to a memoryless wire-tap channel [16]. Accordingto [16], any secrecy rate R such that R < I ( X ; Y ) − I ( X ; Y ) (18)is achievable. Hence we need to find a lower bound to theright hand side of (18).According to [13, Theorem 1], p ( X ) is chosen to be auniform distribution over C P,ε . Under this input distribution,following a similar derivation to [13, Theorem 1], it can beshown that when
P > α β , we have I ( X ; Y ) ≥ (cid:18) − (cid:18) − P ε b (cid:19)(cid:19) log ( |C P,ε | ) − (19)or I ( X ; Y ) , we have I ( X ; Y ) ≤ I ( X ; Y , Z ) = I ( X ; X ± X ) ≤ . (20)where (20) follows from Lemma 1. Using (19) (20), and (17),we find (18) is lower bounded by (cid:18) − (cid:18) − P ε b (cid:19)(cid:19) (cid:18) − ε (cid:19) log ( P ) − . (21)for sufficiently large P . From (17), ε can take any valuebetween (0 , / . Hence we have completed the proof. Remark 2:
When √ ab = 1 and all channel gains arepositive, the channel is degraded and from the outer boundin [4], the secure degree of freedom is 0. Since algebraicirrational numbers are dense on the real line, it follows thatthe secure degree of freedom is discontinuous at √ ab = 1 .The result in Section III-B only applies when √ ab isalgebraic irrational, which is a set of measure on the realline. In the sequel we consider the case where √ ab is eitherrational or transcendental. C. When √ ab ≥ or / √ ab ≥ / Here we use the Q -bit expansion scheme similar to the onein [17]. Let Q = √ ab if √ ab ≥ . Otherwise, let Q = 1 / √ ab .Let ⌊ Q ⌋ denote the largest integer ≤ Q . Theorem 5:
The following secure degree of freedom isachievable:
12 log ⌊ Q ⌋ log Q − f ( ⌊ Q ⌋ )2 log Q (22)where f ( Q ) is defined in (13). (22) is lower bounded by
12 log ⌊ Q ⌋ log Q − log (cid:16) πe (cid:0) (cid:1) − ⌊ Q ⌋ (cid:17) ( Q ) (23)For Q = 2 , (22) equals . . Proof:
We begin by considering the case when √ ab ≥ . X k = p P M − X i =0 a k,i Q i , k = 1 , (24)where P is a constant scaling factor. a k,i is uniformly dis-tributed over [0 , ⌊ Q ⌋− ∩ Z , hence a k,i is uniquely determinedby X k .The signal received by node D is given by Y = p P ( M − X i =0 a ,i Q i + M − X i =0 a ,i Q i +1 ) + √ bZ (25)We then derive a lower bound to I ( X ; Y ) − I ( X ; Y ) aswe did for Theorem 4.Following a similar derivation to [13, Theorem 1], withFano’s inequality, it can be shown that I ( X ; Y ) is lowerbounded as: I ( X ; Y ) ≥ (cid:18) − (cid:18) − P b (cid:19)(cid:19) H ( X ) − (26) −3 −2 −1 S e c u r e D o F Lower Bound Lower Bound Actual Performance
Fig. 3. Secure degree of freedom
For I ( X ; Y ) , we have: I ( X ; Y ) ≤ I ( X ; X ± X ) (27) ≤ M − X i =0 I ( a ,i ; a ,i ± a ,i ) = M f ( ⌊ Q ⌋ ) (28)Therefore, the following secrecy rate is achievable R e = M (1 − − P b ))(log ⌊ Q ⌋ ) − − M f ( ⌊ Q ⌋ ) (29)It can be verified that the transmission power is given by: V ar [ X i ] = P ⌊ Q ⌋ − ! Q M − Q − i = 1 , (30)The secure degree of freedom is hence given by by: lim M →∞ (cid:0)(cid:0) − (cid:0) − P b (cid:1)(cid:1) log ⌊ Q ⌋ − f ( ⌊ Q ⌋ ) (cid:1) M log ( Q M ) (31) = 12 (cid:18) − (cid:18) − P b (cid:19)(cid:19) log ⌊ Q ⌋ log Q − f ( ⌊ Q ⌋ )2 log ( Q ) (32)which can be made arbitrarily close to (22) by choosing alarge enough P . (23) then follows from (22) via Lemma 1.When Q = 2 , it can be verified that f ( ⌊ Q ⌋ ) = 0 . , and(32) can be made to be arbitrarily close to / .The case of / √ ab ≥ can be proved in a similar fashion.Details can be found in [14].In Figure 3, we plot the secure degree of freedom achievedby Theorem 5. We notice as √ ab moves away from , thelower bound given by (23) becomes tighter, and the securedegree of freedom converges to . . Remark 3:
A coding scheme similar to the one describedin this section can be constructed with a nested lattice code.However, the provable secure degree of freedom turns out tobe smaller [14]. In case it is desired for X k to have zero mean, we can simply shift X k by a constant, which will not change the secrecy rate. . When √ ab = 1 When Y = X + X + Z , the channel is degraded. Thesecure degree of freedom is known to be [4]. When Y = X − X + Z , we have the following result: Theorem 6:
The secure degree of freedom of . isachievable. Proof outline:
Here we let X k = √ P M − P i =0 a k,i Q i . Q =2 . The difference is that a k,i is not uniformly distributed over { , } . Instead we choose to pick its distribution to maximize I ( a ,i ; a ,i + a ,i ) − I ( a ,i ; a ,i − a ,i ) (33)which is about . when Pr( a ,i = 1) = 0 . , Pr( a ,i = 1) = 0 . . The theorem follows by deriving alower bound to I ( X ; Y ) − I ( X ; Y ) . The details are providedin [14]. E. When < √ ab < or / < √ ab < Let √ ab = p/q + γ/q , where p, q are coprime positiveintegers, and − < γ < , γ = 0 . In this case, the channelcan be expressed as: qY = qX + ( p + γ ) X + q √ bZ (34) Y = X ± X + Z (35) Theorem 7:
The following secure degree of freedom isachievable when | γ | < / √ : log (cid:0) − γ (cid:1) − log (cid:0) γ (cid:1) − ( f ( γ )) − ( γ ) (36)where f ( γ ) = (1 − γ )( q + ( p + γ ) ) + γ (37) Proof:
Here we use a layered coding scheme similarto [11]. Let the signal send by user k , X Nk , be the sum ofcodewords from M layers: X Nk = M P i =1 X Nk,i , k = 1 , . For the i th layer, we use the nested lattice code described in SectionIII-A1. Let Λ i be the fine lattice and Λ c,i be the coarse latticeused in layer i . Hence the signal X Nk,i is computed as X Nk,i = (cid:0) u Nk,i + d Nk,i (cid:1) mod Λ c,i (38)where d Nk,i is the dithering noise and u Nk,i is the lattice point: u Nk,i ∈ V (Λ c,i ) ∩ Λ i , k = 1 , (39)Define R i as the rate of the codebook for the i th layer. Define P i as the average power per dimension of the i th layer. Atlayer i , node D first decodes qu N ,i + pu N ,i mod Λ c,i , thendecodes u N ,i . The decoder first computes ˆ Y i =[ (cid:0) qu N ,i + pu N ,i (cid:1) + γX N ,i + i − X t =1 (cid:0) qX N ,t + ( p + γ ) X N ,t (cid:1) + q √ bZ N ] mod Λ c,i (40) Define A i as A i = i − P t =1 (cid:16) q + ( p + γ ) (cid:17) P t + q b . In order fornode D to correctly decode qu N ,i + pu N ,i mod Λ c,i from ˆ Y i ,we require [14]: R i ≤
12 log (cid:18) P i γ P i + A i (cid:19) (41)Node D is then left with the following: [ γX N ,i + i − X t =1 (cid:0) qX N ,t + ( p + γ ) X N ,t (cid:1) + q √ bZ N ] mod Λ c,i (42)As long as P i > γ P i + A i (43)(42) can be approximated with high probability [14] by thefollowing: γX N ,i + i − X t =1 (cid:0) qX N ,t + ( p + γ ) X N ,t (cid:1) + q √ bZ N (44)Otherwise a decoding error is said to occur. Node D thencomputes: [ γu N ,i + i − X t =1 (cid:0) qX N ,t + ( p + γ ) X N ,t (cid:1) + q √ bZ N ] mod γ Λ c,i (45)In order for node D to decode u N ,i from this signal correctly,we require [14]: R i ≤
12 log (cid:18) γ P i A i (cid:19) (46)Then node D can recover the following signal from (44): i − X t =1 (cid:0) qX N ,t + ( p + γ ) X N ,t (cid:1) + q √ bZ N (47)which will be fed to the decoder at lower layers.Let the right hand side of (41) equal to the right hand sideof (46): P i γ P i + A i = γ P i A i (48)Then, we have: P i = α ( αβ + 1) i − q b (49)where α = − γ γ , β = q + ( p + γ ) . Under this powerallocation, A i is given by A i = ( αβ + 1) i − q b (50) R i follows from substituting (49) and (50) into (46): R i = 12 log (cid:18) − γ γ (cid:19) (51) .5 1 1.5 2 2.5 3 3.5 400.050.10.150.20.250.30.350.40.450.5 (ab) Integer Lattice
Fig. 4. Secure degree of freedom (43) requires the term inside the log in (51) to be greater than1. This means | γ | < √ . The total power of D i is given by M X t =1 P t = ( αβ + 1) M − β q b (52)From (12), each layer leaks at most N bit to the eavesdropperover N channel uses. Hence the secrecy rate R e,i contributedby layer i is related to R i as R e,i ≥ R i − . Hence a secrecyrate of R e = P Mi =1 R i − M is achievable. The secure degreeof freedom is therefore given by lim P →∞ R e log P = lim M →∞ M P i =1 R i − M log P = log (cid:16) − γ γ (cid:17) − log ( αβ + 1) (53)which equals (36) in Theorem 7. Remark 4:
In Figure 4, we plot the achieved secure degreeof freedom by the nested lattice coding scheme in this section. q ≤ and p is chosen to be positive integers smaller thanand coprime with q . The figure shows that the secure degreeof freedom is positive when . < √ ab < or < √ ab < .This, along with the results in the previous sections, provesthat the secure degree of freedom is positive everywhere exceptwhen the channel is degraded . Remark 5:
The scheme described in this section also ap-plies to the case where √ ab > or √ ab < / . Plottedwith dashed lines in Figure 4 is the performance of theinteger lattice from previous section. Comparing it with theperformance of the scheme in this section, we find that neitherscheme dominates the other in performance. Remark 6:
It is possible to construct a coding schemesimilar to the one described in this section using an integerlattice, which may yield a higher secure degree of freedom.However, it is difficult to find a uniform description of such code for all p and q . Hence, instead we use a nested lattice codeto prove that a positive secure degree of freedom is achievable.IV. C ONCLUSION
In this work, we have proved that a positive secure degreeof freedom is achievable for the fully connected interferenceassisted wire-tap channel when the channel is not degraded.As a consequence of this high SNR result, we are able to claimthat, in contrast to common belief, Gaussian signaling is notoptimal for a large class of two user Gaussian channels.An added practical value of our result is that it implies thatthe cooperation of just one node is sufficient to achieve anarbitrarily large secrecy rate given enough power. Since largescale cooperation involving multiple nodes is not essential,this fact enhances the robustness of the network in an adverseenvironment. R
EFERENCES[1] C. E. Shannon. Communication Theory of Secrecy Systems.
Bell SystemTechnical Journal , 28(4):656–715, September 1949.[2] E. Tekin and A. Yener. The General Gaussian Multiple Access and Two-Way Wire-Tap Channels: Achievable Rates and Cooperative Jamming.
IEEE Trans. on Info. Theory , 54(6):2735–2751, June 2008.[3] R. Liu, I. Maric, P. Spasojevic, and R. D. Yates. Discrete Memory-less Interference and Broadcast Channels with Confidential Messages:Secrecy Rate Regions.
IEEE Trans. on Info. Theory , 54(6):2493–2507,June 2008.[4] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor. The Gaussian WiretapChannel With a Helping Interferer.
IEEE International Symposium onInfo. Theory , July 2008.[5] X. He and A. Yener. A New Outer Bound for the Gaussian InterferenceChannel with Confidential Messages.
Annual Conf. on Info. Sciencesand Systems , March 2009.[6] Z. Li, R. D. Yates, and W. Trappe. Secrecy Capacity Region of a Classof One-Sided Interference Channel.
IEEE International Symposium onInfo. Theory , July 2008.[7] E. Ekrem and S. Ulukus. On the Secrecy of Multiple Access WiretapChannel.
Allerton Conf. on Communication, Control, and Computing ,September 2008.[8] O. Koyluoglu, H. El-Gamal, L. Lai, and H. V. Poor. InterferenceAlignment for Secrecy. submited to IEEE Trans. on Info. Theory,October, 2008.[9] B. Nazer and M. Gastpar. The Case for Structured Random Codes inNetwork Capacity Theorems.
European Trans. on Telecommunications ,19(4):455–474, June 2008.[10] X. He and A. Yener. Providing Secrecy with Lattice Codes.
AllertonConf. on Communication, Control, and Computing , September 2008.[11] X. He and A. Yener. K -user Interference Channels: Achievable SecrecyRate and Degrees of Freedom. To appear, IEEE Info. Theory Workshop,June, 2009.[12] S. Sridharan, A. Jafarian, S. Vishwanath, S. A. Jafar, and S. Shamai.A Layered Lattice Coding Scheme for a Class of Three User GaussianInterference Channels. Allerton Conf. on Communication, Control, andComputing , September 2008.[13] R. Etkin and E. Ordentlich. On the Degrees-of-Freedom of the K-User Gaussian Interference Channel. Submitted to IEEE Trans. on Info.Theory, June, 2008.[14] X. He and A. Yener. Providing Secrecy With Structured Codes: Toolsand Applications to Gaussian Two-user Channels. Submitted to IEEETrans. on Info. Theory, April, 2009.[15] S. Leung-Yan-Cheong and M. Hellman. The Gaussian Wire-tap Channel.
IEEE Trans. on Info. Theory , 24(4):451–456, July 1978.[16] I. Csiszar and J. Korner. Broadcast Channels with Confidential Mes-sages.
IEEE Trans. on Info. Theory , 24(3):339–348, May 1978.[17] V. R. Cadambe, S. A. Jafar, and S. Shamai. Interference Alignment onthe Deterministic Channel and Application to Fully Connected AWGNInterference Networks.