Secure Degrees of Freedom of the Gaussian Wiretap Channel with Helpers and No Eavesdropper CSI: Blind Cooperative Jamming
aa r X i v : . [ c s . I T ] F e b Secure Degrees of Freedom of the GaussianWiretap Channel with Helpers and NoEavesdropper CSI: Blind Cooperative Jamming
Jianwei Xie Sennur Ulukus
Department of Electrical and Computer EngineeringUniversity of Maryland, College Park, MD 20742 [email protected] [email protected]
Abstract —We consider the Gaussian wiretap channel with M helpers, where no eavesdropper channel state information (CSI)is available at the legitimate entities. The exact secure d.o.f. ofthe Gaussian wiretap channel with M helpers with perfect CSIat the transmitters was found in [1], [2] to be MM +1 . One of thekey ingredients of the optimal achievable scheme in [1], [2] is toalign cooperative jamming signals with the information symbolsat the eavesdropper to limit the information leakage rate. Thisrequired perfect eavesdropper CSI at the transmitters. Motivatedby the recent result in [3], we propose a new achievable schemein which cooperative jamming signals span the entire space of theeavesdropper, but are not exactly aligned with the informationsymbols. We show that this scheme achieves the same secured.o.f. of MM +1 in [1], [2] but does not require any eavesdropperCSI; the transmitters blindly cooperative jam the eavesdropper. I. I
NTRODUCTION
Wyner introduced the wiretap channel in which a legiti-mate transmitter wants to have secure communications witha legitimate receiver in the presence of an eavesdropper, anddetermined its capacity-equivocation region for the degradedcase [4]. Csiszar and Korner extended this result to the general,not necessarily degraded, wiretap channel [5]. Leung-Yan-Cheong and Hellman determined the capacity-equivocationregion of the Gaussian wiretap channel [6]. This line ofresearch has been subsequently extended to many multi-usersettings. Here, we are particularly interested in models withmultiple independent legitimate transmitters, e.g., interferencechannel with confidential messages [7], [8], interference chan-nel with external eavesdroppers [9], multiple access wiretapchannel [10]–[14], wiretap channel with helpers [15], andrelay-eavesdropper channel with deaf helpers [16].Since in most multi-user scenarios it is difficult to obtainthe exact secrecy capacity region, recently, there has been asignificant interest in studying the asymptotic performance ofthese systems at high signal-to-noise ratio (SNR) in terms oftheir secure degrees of freedom (d.o.f.) regions. Achievablesecure d.o.f. has been studied for several channel structures,such as the K -user Gaussian interference channel with confi-dential messages [17], [18], K -user interference channel withexternal eavesdroppers [19] in ergodic fading setting [17], This work was supported by NSF Grants CNS 09-64632, CCF 09-64645,CCF 10-18185 and CNS 11-47811. X M +1 X X W ˆ Wh W X g N N Y Y Fig. 1. The Gaussian wiretap channel with M helpers. [20], Gaussian wiretap channel with helpers [1], [2], [21]–[23], Gaussian multiple access wiretap channel [24] in er-godic fading setting [25], multiple antenna compound wiretapchannel [26], and wireless X network [27]. The exact sumsecure d.o.f. was found for a large class of one-hop wirelessnetworks, including the wiretap channel with M helpers, two-user interference channel with confidential messages, and K -user multiple access wiretap channel in [2], and for all two-unicast layered wireless networks in [28], [29].In this paper, we revisit the Gaussian wiretap channel with M helpers, see Fig. 1. The secrecy capacity of the Gaussianwiretap channel with no helpers is the difference between theindividual channel capacities of the transmitter-receiver andthe transmitter-eavesdropper pairs. This difference does notscale with the SNR, and hence the secure d.o.f. of the Gaussianwiretap channel with no helpers is zero, indicating a severepenalty due to secrecy. It has been known that the secrecyrates can be improved if there are helpers which can transmitindependent signals [10], [11], however, if the helpers transmiti.i.d. Gaussian signals, then the secure d.o.f. is still zero [25].It has been also known that positive secure d.o.f. could beachieved if the helpers sent structured signals [21]–[23], butthe exact secure d.o.f. was unknown. References [1], [2] deter-mined the exact secure d.o.f. of the Gaussian wiretap channelwith M helpers to be MM +1 . This result was derived under V V U U V V U U V V U U X X Y Y h g Fig. 2. Illustration of the alignment scheme for the Gaussian wiretap channel with M helpers with eavesdropper’s CSI available at all transmitters. the assumption that the eavesdropper’s CSI was available atthe transmitters. In the present paper, we show that the samesecure d.o.f. can be achieved even when the eavesdropper’sCSI is unknown at the legitimate transmitters. This resultis practically significant because, generally, it is difficult orimpossible to obtain the eavesdropper’s CSI. Since the upperbound developed in [1], [2] is valid for this case also, wethus determine the exact secure d.o.f. of the Gaussian wiretapchannel with M helpers with no eavesdropper CSI to be MM +1 .The achievable scheme in the case of no eavesdropper CSIhere is significantly different than the achievable scheme witheavesdropper CSI developed in [1], [2].In particular, in [1], [2], the legitimate transmitter dividesits message into M sub-messages and sends them on M different irrational dimensions . Each one of the helpers sendsa cooperative jamming signal. The message signals and thecooperative jamming signals are sent in such a way that: 1)the cooperative jamming signals are aligned at the legitimatereceiver in the same irrational dimension, so that they oc-cupy the smallest possible space at the legitimate receiverto enable the decodability of the message signals, and 2)each cooperative jamming signal is aligned exactly in thesame irrational dimension with one of the message signalsat the eavesdropper to protect it. This scheme is illustratedin Fig. 2 for M = 2 helpers. In [1], [2], we used insightsfrom [21]–[23] to show that, when a cooperative jammingsignal is aligned with a message signal in the same irrationaldimension at the eavesdropper, this alignment protects themessage signal, and limits the information leakage rate tothe eavesdropper by a constant which does not depend onthe transmit power. Meanwhile, due to the alignment of thecooperative jamming signals in a small space at the legitimatereceiver, the information rate to the legitimate receiver canbe made to scale with the transmit power. We use this realinterference alignment [30], [31] based approach to achieve asecure d.o.f. of MM +1 for almost all channel gains , and developa converse to show that it is in fact the secure d.o.f. capacity.The achievable scheme in the present paper again dividesthe message into M sub-messages. Each one of the helperssends a cooperative jamming signal. As a major difference from the achievable scheme in [1], [2], in this achievablescheme, the legitimate transmitter also sends a cooperativejamming signal. This scheme is illustrated in Fig. 3 for M = 2 helpers. In this case, the message signals and thecooperative jamming signals are sent in such a way that:1) all M + 1 cooperative jamming signals are aligned atthe legitimate receiver in the same irrational dimension, and2) all cooperative jamming signals span the entire space at the eavesdropper to limit the information leakage to theeavesdropper. We use insights from [3], which developed anew achievable scheme that achieved the same secure d.o.f. asin [26] without eavesdropper CSI, to show that the informationleakage to the eavesdropper is upper bounded by a function,which can be made arbitrarily small. On the other hand, sincethe cooperative jamming signals occupy the smallest space atthe legitimate receiver, the information rate to the legitimatereceiver can be made to scale with the transmit power. In thisachievable scheme, we let the legitimate transmitter and thehelpers blindly cooperative jam the eavesdropper. Because ofthe inefficiency of blind cooperative jamming, here, we had touse more cooperative jamming signals than in [1], [2], i.e., in[1], [2] we use a total of M cooperative jamming signals fromthe helpers, while here we use M + 1 cooperative jammingsignals, one of which coming from the legitimate transmitter.II. S YSTEM M ODEL AND D EFINITIONS
The Gaussian wiretap channel with M helpers, see Fig. 1,is defined by Y = h X + M +1 X j =2 h j X j + N (1) Y = g X + M +1 X j =2 g j X j + N (2)where Y is the channel output of the legitimate receiver, Y is the channel output of the eavesdropper, X is the channelinput of the legitimate transmitter, X i , for i = 2 , . . . , M + 1 ,are the channel inputs of the M helpers, h i is the channelgain of the i th transmitter to the legitimate receiver, g i is thechannel gain of the i th transmitter to the eavesdropper, and N U V U V V V U U U U U V V U U X X Y Y h g Fig. 3. Illustration of the alignment scheme for the Gaussian wiretap channel with M helpers with no eavesdropper CSI. and N are two independent zero-mean unit-variance Gaussianrandom variables. All channel inputs satisfy average powerconstraints, E (cid:2) X i (cid:3) ≤ P , for i = 1 , . . . , M + 1 .Transmitter intends to send a message W , uniformlychosen from a set W , to the legitimate receiver (receiver ). The rate of the message is R △ = n log |W| , where n isthe number of channel uses. Transmitter uses a stochasticfunction f : W → X to encode the message, where X △ = X n is the n -length channel input. We use boldfaceletters to denote n -length vector signals, e.g., X △ = X n , Y △ = Y n , Y △ = Y n , etc. The legitimate receiver decodesthe message as ˆ W based on its observation Y . A secrecyrate R is said to be achievable if for any ǫ > there exists an n -length code such that receiver can decode this messagereliably, i.e., the probability of decoding error is less than ǫ ,Pr h W = ˆ W i ≤ ǫ (3)and the message is kept information-theoretically secureagainst the eavesdropper, n H ( W | Y ) ≥ n H ( W ) − ǫ (4)i.e., that the uncertainty of the message W , given the observa-tion Y of the eavesdropper, is almost equal to the entropy ofthe message. The supremum of all achievable secrecy rates isthe secrecy capacity C s , and the secure d.o.f., D s , is definedas D s △ = lim P →∞ C s log P (5)Note that D s ≤ is an upper bound. To avoid trivial cases,we assume that h = 0 and g = 0 . Without the independenthelpers, i.e., M = 0 , and with full knowledge of all channelgains, the secrecy capacity of the Gaussian wiretap channel isknown [6] C s = 12 log (cid:0) h P (cid:1) −
12 log (cid:0) g P (cid:1) (6)and from (5) the secure d.o.f. is zero. Therefore, we assume M ≥ . If there exists a j ( j = 2 , . . . , M + 1 ) such that h j = 0 and g j = 0 , then a lower bound of secure d.o.f. canbe obtained for this channel by letting this helper jam theeavesdropper by i.i.d. Gaussian noise of power P and keepingall other helpers silent. This lower bound matches the upperbound, giving the secure d.o.f. On the other hand, if thereexists a j ( j = 2 , . . . , M +1 ) such that h j = 0 and g j = 0 , thenthis helper can be removed from the channel model withoutaffecting the secure d.o.f. Therefore, in the rest of the paper,we assume that M ≥ and h j = 0 and g j = 0 for all j = 1 , · · · , M + 1 .III. A CHIEVABLE S CHEME WITH NO E AVESDROPPER
CSIIn this section, we propose an achievable scheme to achievethe secure d.o.f. of MM +1 with no eavesdropper CSI at anyof the transmitters. The only assumption we make is that thelegitimate transmitter knows an upper bound of P M +1 k =1 g k ≤ ¯ c on the eavesdropper channel gains.Let { V , V , · · · , V M +1 , U , U , U , · · · , U M +1 } be mutu-ally independent discrete random variables, each of whichuniformly drawn from the same PAM constellation C ( a, Q ) C ( a, Q ) = a {− Q, − Q + 1 , . . . , Q − , Q } (7)where Q is a positive integer and a is a real number used tonormalize the transmission power, and is also the minimumdistance between the points belonging to C ( a, Q ) . Exactvalues of a and Q will be specified later. We choose the inputsignal of the legitimate transmitter as X = 1 h U + M +1 X k =2 α k V k (8)where { α k } M +1 k =2 are rationally independent among themselvesand also rationally independent of all channel gains. The inputsignal of the j th helper, j = 2 , , · · · , M + 1 , is chosen as X j = 1 h j U j (9)Note that, neither the legitimate transmitter signal in (8) northe helper signals in (9) depend on the eavesdropper CSI g k } M +1 k =1 . With these selections, observations of the receiversare given by, Y = M +1 X k =2 h α k V k + M +1 X j =1 U j + N (10) Y = M +1 X k =2 g α k V k + M +1 X j =1 g j h j U j + N (11)The intuition here is as follows: We use M independentsub-signals V k , k = 2 , , · · · , M + 1 , to represent the originalmessage W . The input signal X is a linear combination of V k s and a jamming signal U . At the legitimate receiver, allof the cooperative jamming signals, U k s, are aligned suchthat they occupy a small portion of the signal space. Since { , h α , h α , · · · , h α M +1 } are rationally independent forall channel gains, except for a set of Lebesgue measurezero, the signals n V , V , · · · , V M +1 , P M +1 j =1 U j o can be dis-tinguished by the legitimate receiver. In addition, we ob-serve that n g h , · · · , g M +1 h M +1 o are rationally independent, andtherefore, { U , U , · · · , U M +1 } span the entire space at theeavesdropper; see Fig. 3. Here, by the entire space , we meanthe maximum number of dimensions that the eavesdropper iscapable of decoding, which is M + 1 in this case. Since the entire space at the eavesdropper is occupied by the cooperativejamming signals, the message signals { V , V , · · · , V M +1 } aresecure, as we will mathematically prove in the sequel.Since, for j = 1 , X j is an i.i.d. sequence and is independentof X , the following secrecy rate is achievable [5] C s ≥ I ( V ; Y ) − I ( V ; Y ) (12)where V △ = { V , V , · · · , V M +1 } .First, we use Fano’s inequality to bound the first term in(12). Note that the space observed at receiver consists of (2 Q + 1) M (2 M Q + 2 Q + 1) points in M + 1 dimensions , andthe sub-signal in each dimension is drawn from a constellationof C ( a, ( M + 1) Q ) . Here, we use the property that C ( a, Q ) ⊂ C ( a, ( M + 1) Q ) . By using the Khintchine-Groshev theoremof Diophantine approximation in number theory [30], [31], wecan bound the minimum distance d min between the points inreceiver 1’s space as follows: For any δ > , there exists aconstant k δ such that d min ≥ k δ a (( M + 1) Q ) M + δ (13)for almost all rationally independent { , h α , h α , · · · , h α M +1 } , except for a set of Lebesguemeasure zero. Then, we can upper bound the probability ofdecoding error of such a PAM scheme by considering theadditive Gaussian noise at receiver ,Pr h V = ˆ V i ≤ exp (cid:18) − d min (cid:19) (14) ≤ exp (cid:18) − a k δ M + 1) Q ) M + δ ) (cid:19) (15) where ˆ V is the estimate of V by choosing the closest pointin the constellation based on observation Y . For any δ > ,if we choose Q = P − δ M +1+ δ ) and a = γP /Q , where γ is aconstant independent of P , thenPr h V = ˆ V i ≤ exp (cid:18) − k δ γ ( M + 1) P M + 1) Q ) M + δ )+2 (cid:19) (16) = exp (cid:18) − k δ γ ( M + 1) P δ M + 1) M +1+ δ ) (cid:19) (17)and we can have Pr h V = ˆ V i → as P → ∞ . To satisfy thepower constraint at the transmitters, we can simply choose γ ≤ min " | h | + M +1 X k =2 | α k | − , | h | , | h | , · · · , | h M +1 | (18)By Fano’s inequality and the Markov chain V → Y → ˆ V ,we know that H ( V | Y ) ≤ H ( V | ˆ V ) (19) ≤ (cid:18) − k δ γ ( M + 1) P δ M + 1) M +1+ δ ) (cid:19) log(2 Q + 1) M (20) = o (log P ) (21)where δ and γ are fixed, and o ( · ) is the little- o function. Thismeans that I ( V ; Y ) = H ( V ) − H ( V | Y ) (22) ≥ H ( V ) − o (log P ) (23) = log(2 Q + 1) M − o (log P ) (24) ≥ log P M (1 − δ )2( M +1+ δ ) − o (log P ) (25) = M (1 − δ ) M + 1 + δ (cid:18)
12 log P (cid:19) − o (log P ) (26)Next, we need to bound the second term in (12), I ( V ; Y ) = I ( V, U ; Y ) − I ( U ; Y | V ) (27) = I ( V, U ; Y ) − H ( U | V ) + H ( U | Y , V ) (28) = I ( V, U ; Y ) − H ( U ) + H ( U | Y , V ) (29) = h ( Y ) − h ( Y | V, U ) − H ( U ) + H ( U | Y , V ) (30) = h ( Y ) − h ( N ) − H ( U ) + H ( U | Y , V ) (31) ≤ h ( Y ) − h ( N ) − H ( U ) + o (log P ) (32) ≤
12 log 2 πe (1 + ¯ cP ) −
12 log 2 πe − log(2 Q + 1) M +1 + o (log P ) (33) ≤
12 log P − ( M + 1)(1 − δ )2( M + 1 + δ ) log P + o (log P ) (34) = ( M + 2) δM + 1 + δ (cid:18)
12 log P (cid:19) + o (log P ) (35)here U △ = { U , U , · · · , U M +1 } and ¯ c is the upper bound on P M +1 k =1 g k defined at the beginning of this section, and (32)is due to the fact that given V and Y , the eavesdropper candecode U with probability of error approaching zero since n g h , · · · , g M +1 h M +1 o are rationally independent for all channelgains, except for a set of Lebesgue measure zero. Then, byFano’s inequality, H ( U | Y , V ) ≤ o (log P ) similar to the stepin (21).Combining (26) and (35), we have C s ≥ I ( V ; Y ) − I ( V ; Y ) (36) ≥ M (1 − δ ) M + 1 + δ (cid:18)
12 log P (cid:19) − ( M + 2) δM + 1 + δ (cid:18)
12 log P (cid:19) − o (log P ) (37) = M − (2 M + 2) δM + 1 + δ (cid:18)
12 log P (cid:19) − o (log P ) (38)where again o ( · ) is the little- o function. If we choose δ arbi-trarily small, then we can achieve MM +1 secure d.o.f. for thismodel where there is no eavesdropper CSI at the transmitters.IV. C ONCLUSIONS
We studied the Gaussian wiretap channel with M helperswithout any eavesdropper CSI at the transmitters. We proposedan achievable scheme that achieves a secure d.o.f. of MM +1 ,which is the same as the secure d.o.f. reported in [1], [2]when the transmitters had perfect eavesdropper CSI. The newachievability scheme is based on real interference alignmentand blind cooperative jamming. While [1], [2] aligned co-operative jamming signals with the information symbols atthe eavesdropper to protect the information symbols, whichrequired eavesdropper CSI, here we used one more cooperativejamming signal to span the entire space at the eavesdropperto protect the information symbols. As in [1], [2], here also,we aligned all of the cooperative jamming signals in thesame dimension at the legitimate receiver, in order to occupythe smallest space at the legitimate receiver to allow forthe decodability of the information symbols. Therefore, wealigned the cooperative jamming signals carefully only atthe legitimate receiver, which required only the legitimatereceiver’s CSI at the transmitters.R EFERENCES[1] J. Xie and S. Ulukus. Secure degrees of freedom of the Gaussianwiretap channel with helpers. In , Monticello, IL, October 2012.[2] J. Xie and S. Ulukus. Secure degrees of freedom of one-hop wirelessnetworks. Submitted to
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