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Providing Secrecy With Structured Codes: Tools and Applications to TwoUser Gaussian Channels
, 2009
"... Recent results have shown that structured codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. For Gaussian channels with secrecy constraints, however, efforts to date rely on random codes. In this work, we advocate that structured c ..."
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Cited by 45 (17 self)
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Recent results have shown that structured codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. For Gaussian channels with secrecy constraints, however, efforts to date rely on random codes. In this work, we advocate that structured codes are useful for providing secrecy, and show how to compute the secrecy rate when structured codes are used. In particular, we solve the problem of bounding equivocation rates with one important class of structured codes, i.e., nested lattice codes. Having established this result, we next demonstrate the use of structured codes for secrecy in twouser Gaussian channels. In particular, with structured codes, we prove that a positive secure degree of freedom is achievable for a large class of fully connected Gaussian channels as long as the channel is not degraded. By way of this, for these channels, we establish that structured codes outperform Gaussian random codes at high SNR. This class of channels include the twouser multiple access wiretap channel, the twouser interference channel with confidential messages and the twouser interference wiretap channel. A notable consequence of this result is that, unlike the case with Gaussian random codes, using structured codes for both transmission and cooperative jamming, it is possible to achieve an arbitrary large secrecy rate given enough power.
Secure Degrees of Freedom of the Gaussian Wiretap Channel with Helpers
"... Abstract — The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the secure d.o.f. of the Gaussian wiretap channel with no helpers is zero. It has been known that a strictly positive secure d.o.f. can be obtained in the Gaussian wiretap cha ..."
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Cited by 22 (16 self)
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Abstract — The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the secure d.o.f. of the Gaussian wiretap channel with no helpers is zero. It has been known that a strictly positive secure d.o.f. can be obtained in the Gaussian wiretap channel by using a helper which sends structured cooperative signals. We show that the exact secure d.o.f. of the Gaussian wiretap channel with a helper is 1. Our achievable scheme is based on 2 real interference alignment and cooperative jamming, which renders the message signal and the cooperative jamming signal separable at the legitimate receiver, but aligns them perfectly at the eavesdropper preventing any reliable decoding of the message signal. Our converse is based on two key lemmas. The first lemma quantifies the secrecy penalty by showing that the net effect of an eavesdropper on the system is that it eliminates one of the independent channel inputs. The second lemma quantifies the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. We extend this result to the case of M helpers, and show that the exact secure d.o.f. in this case is M
Secure Degrees of Freedom of Onehop Wireless Networks
, 2012
"... We study the secure degrees of freedom (d.o.f.) of onehop wireless networks by considering four fundamental wireless network structures: Gaussian wiretap channel, Gaussian broadcast channel with confidential messages, Gaussian interference channel with confidential messages, and Gaussian multiple a ..."
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Cited by 19 (12 self)
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We study the secure degrees of freedom (d.o.f.) of onehop wireless networks by considering four fundamental wireless network structures: Gaussian wiretap channel, Gaussian broadcast channel with confidential messages, Gaussian interference channel with confidential messages, and Gaussian multiple access wiretap channel. The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the secure d.o.f. of the Gaussian wiretap channel with no helpers is zero. It has been known that a strictly positive secure d.o.f. can be obtained in the Gaussian wiretap channel by using a helper which sends structured cooperative signals. We show that the exact secure d.o.f. of the Gaussian wiretap channel with a helper is 1 2. Our achievable scheme is based on real interference alignment and cooperative jamming, which renders the message signal and the cooperative jamming signal separable at the legitimate receiver, but aligns them perfectly at the eavesdropper preventing any reliable decoding of the message signal. Our converse is based on two key lemmas. The first lemma quantifies the secrecy penalty by showing that the net effect of an eavesdropper on the system is that it eliminates one of the independent channel inputs. The second lemma quantifies the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. We extend this result to the case of M helpers, and show that the exact secure d.o.f. in this case is M M+1. We then generalize this approach to more general network structures with multiple messages. We show that the sum secure d.o.f. of the Gaussian broadcast channel with confidential messages and M helpers is 1, the sum secure d.o.f. of the twouser interference channel with confidential messages is 2 3, the sum secure d.o.f. of the twouser interference channel with confidential messages and M helpers is 1, and the sum secure d.o.f. of the Kuser multiple access wiretap channel is
MIMO Multiple Access Channel with an Arbitrarily Varying Eavesdropper: Secrecy degrees of freedom
 IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY
, 2013
"... A twotransmitter 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 t ..."
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Cited by 13 (5 self)
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A twotransmitter 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 singleuser wiretap code for each user, is shown to achieve the s.d.o.f. region. The converse involves establishing an upper bound on a weightedsumrate 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 subchannels.
On the Role of Feedback in Two Way Secure Communication
 In 42nd Annual Asilomar Conference on Signals, Systems, and Computers
, 2008
"... Abstract—For bidirectional communication, the most general form of encoders should consider the signals received in the past as inputs. However, in practice, it would also be highly desirable if feedback could be ignored for encoding purposes since this would lead to a simple system design. In this ..."
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Cited by 12 (5 self)
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Abstract—For bidirectional communication, the most general form of encoders should consider the signals received in the past as inputs. However, in practice, it would also be highly desirable if feedback could be ignored for encoding purposes since this would lead to a simple system design. In this work, we investigate the question of whether and how much loss in secrecy rate would be incurred, if such an approach were taken. To do so, we investigate the role of feedback in secrecy for two threenode twoway channel models. First, we show that feedback is indeed useful for a class of fullduplex twoway wiretap channels. In this case, when feedback is ignored, the channel is equivalent to a Gaussian degraded relay channel with confidential messages to the relay. The usefulness of feedback is demonstrated by deriving an upper bound for this channel when feedback is ignored, and then proving that, when feedback is used, a secrecy rate higher than this upper bound is achievable. Secondly, we consider the halfduplex Gaussian twoway relay channel where there is an eavesdropper colocated with the relay node, and find that the impact of feedback is less pronounced compared to the previous scenario. Specifically, the loss in secrecy rate, when ignoring the feedback, is quantified to be less than 0.5 bit per channel use when the power of the relay goes to infinity. We also show that this rate region is achievable under a simple time sharing scheme with cooperative jamming, which, with its simplicity and nearoptimum performance, is a viable alternative to an encoder using feedback. I.
Secure Degrees of Freedom of the Gaussian Multiple Access Wiretap Channel
"... Abstract—We show that the sum secure degrees of freedom (d.o.f.) of the Kuser Gaussian multiple access (MAC) wiretap channel is K(K−1). Our achievability is based on real interferK(K−1)+1 ence alignment and structured cooperative jamming. Each user divides its message into K − 1 submessages, and ..."
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Cited by 10 (6 self)
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Abstract—We show that the sum secure degrees of freedom (d.o.f.) of the Kuser Gaussian multiple access (MAC) wiretap channel is K(K−1). Our achievability is based on real interferK(K−1)+1 ence alignment and structured cooperative jamming. Each user divides its message into K − 1 submessages, and sends a linear combination of signals carrying these submessages together with a structured cooperative jamming signal. All cooperative jamming signals are aligned in a single dimension at the legitimate receiver allowing for reliable decoding of the message carrying signals by the legitimate receiver. Each cooperative jamming signal is aligned with K − 1 message signals at the eavesdropper limiting the information leakage rate to the eavesdropper. We provide a matching converse establishing the exact sum secure d.o.f. of the Gaussian MAC wiretap channel as
On the Sum Secure Degrees of Freedom of TwoUnicast Layered Wireless Networks
"... Abstract—In this paper, we study the sum secure degrees of freedom (d.o.f.) of twounicast layered wireless networks. Without a secrecy constraint, the sum d.o.f. of this class of networks was studied by [1] and shown to take only one of three possible values: 1, 3/2 and 2, for all network configura ..."
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Cited by 7 (4 self)
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Abstract—In this paper, we study the sum secure degrees of freedom (d.o.f.) of twounicast layered wireless networks. Without a secrecy constraint, the sum d.o.f. of this class of networks was studied by [1] and shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where the message of each sourcedestination pair must be kept informationtheoretically secure from the unintended receiver. We show that the sum secure d.o.f. can take 0, 1, 3/2, 2 and at most countably many other positive values, which we enumerate. s1 u1 u2 u3 t1 t2 s2 w1 w2 w3
Secure Degrees of Freedom of KUser Gaussian Interference Channels: A Unified View
, 2013
"... We determine the exact sum secure degrees of freedom (d.o.f.) of the Kuser Gaussian interference channel. We consider three different secrecy constraints: 1) Kuser interference channel with one external eavesdropper (ICEE), 2) Kuser interference channel with confidential messages (ICCM), and 3) ..."
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Cited by 5 (4 self)
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We determine the exact sum secure degrees of freedom (d.o.f.) of the Kuser Gaussian interference channel. We consider three different secrecy constraints: 1) Kuser interference channel with one external eavesdropper (ICEE), 2) Kuser interference channel with confidential messages (ICCM), and 3) Kuser interference channel with confidential messages and one external eavesdropper (ICCMEE). We show that for all of these three cases, the exact sum secure d.o.f. is K(K−1) 2K−1. We show converses for ICEE and ICCM, which imply a converse for ICCMEE. We show achievability for ICCMEE, which implies achievability for ICEE and ICCM. We develop the converses by relating the channel inputs of interfering users to the reliable rates of the interfered users, and by quantifying the secrecy penalty in terms of the eavesdroppers’ observations. Our achievability uses structured signaling, structured cooperative jamming, channel prefixing, and asymptotic real interference alignment. While the traditional interference alignment provides some amount of secrecy by mixing unintended signals in a smaller subspace at every receiver, in order to attain the optimum sum secure d.o.f., we incorporate structured cooperative jamming into the achievable scheme, and intricately design the structure of all of the transmitted signals jointly.
Secure degrees of freedom region of the Gaussian multiple access wiretap channel
 In Asilomar Conference
, 2013
"... Abstract — [1] showed that the sum secure degrees of freedom (s.d.o.f.) of the Kuser Gaussian multiple access (MAC) wiretap channel is K(K−1) K(K−1)+1. In this paper, we determine the entire s.d.o.f. region of the Kuser Gaussian MAC wiretap channel. The converse follows from a middle step in the c ..."
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Cited by 3 (2 self)
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Abstract — [1] showed that the sum secure degrees of freedom (s.d.o.f.) of the Kuser Gaussian multiple access (MAC) wiretap channel is K(K−1) K(K−1)+1. In this paper, we determine the entire s.d.o.f. region of the Kuser Gaussian MAC wiretap channel. The converse follows from a middle step in the converse of [1]. The achievability follows from exploring the polytope structure of the converse region, determining its extreme points, and then showing that each extreme point can be achieved by an muser MAC wiretap channel with K−m helpers, i.e., by setting K−m users ’ secure rates to zero and utilizing them as pure (structured) cooperative jammers. A byproduct of our result is that the sum s.d.o.f. is achieved only at one corner point of the s.d.o.f. region. I.
Degrees of Freedom of the Single Antenna Gaussian Wiretap Channel with a Helper Irrespective of the Number of Antennas at the Eavesdropper
"... Abstract—A Gaussian wiretap channel with a helper, i.e., a cooperative jammer, is considered and its secure degrees of freedom (s.d.o.f.) is computed. Previous work showed that the s.d.o.f. is upper bounded by 1 2 in this model when all parties are equipped with one antenna each. In this paper, the ..."
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Cited by 2 (0 self)
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Abstract—A Gaussian wiretap channel with a helper, i.e., a cooperative jammer, is considered and its secure degrees of freedom (s.d.o.f.) is computed. Previous work showed that the s.d.o.f. is upper bounded by 1 2 in this model when all parties are equipped with one antenna each. In this paper, the more challenging scenario where the eavesdropper has multiple antennas is tackled. Relying on structured signaling and cooperative jamming, specifically, by real interference alignment, it is shown that s.d.o.f. of 1 2 is achievable irrespective of the number of antennas the eavesdropper may have as long as the cooperative jammer has the same number of antennas as the eavesdropper. The design insight revealed is that the price to pay for the increase in the number of antennas at the eavesdropper is an equivalent increase in the number of antennas at the cooperative jammer in order to maintain the same s.d.o.f. I.