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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
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.
Unified Secure DoF Analysis of KUser Gaussian Interference Channels
"... Abstract—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 ..."
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Abstract—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)
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.
Secure degrees of freedom of the multiple access wiretap channel with no eavesdropper CSI,’’ presented at the
 IEEE Int. Symp. Inf. Theory, Hong Kong
, 2015
"... Abstract—We consider the Kuser Gaussian multiple access wiretap channel (MACWT), where no eavesdropper channel state information (CSI) is available at the transmitters. We show that the exact sum secure degrees of freedom (s.d.o.f.) of this channel model is K−1 K. This result shows that, under the ..."
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Abstract—We consider the Kuser Gaussian multiple access wiretap channel (MACWT), where no eavesdropper channel state information (CSI) is available at the transmitters. We show that the exact sum secure degrees of freedom (s.d.o.f.) of this channel model is K−1 K. This result shows that, under the condition of no eavesdropper CSI, the MACWT acts like a singletransmitter K − 1 helper wiretap channel. We further show that, when a subset of the transmitters have eavesdropper CSI, then higher sum s.d.o.f. can be achieved, and the system can be operated as a MACWT for the users with eavesdropper CSI, with the remaining users acting as helpers. In particular, if m of the K transmitters have eavesdropper CSI, we show that m(K−1) m(K−1)+1 sum s.d.o.f. can be achieved, showing the benefits of having the eavesdropper CSI at the transmitters. 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.
Secure Degrees of Freedom Regions of Multiple Access and Interference Channels: The Polytope Structure∗
, 2014
"... The sum secure degrees of freedom (s.d.o.f.) of two fundamental multiuser network structures, the Kuser Gaussian multiple access (MAC) wiretap channel and the Kuser interference channel (IC) with secrecy constraints, have been determined recently as K(K−1)K(K−1)+1 [1,2] and K(K−1) 2K−1 [3,4], res ..."
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The sum secure degrees of freedom (s.d.o.f.) of two fundamental multiuser network structures, the Kuser Gaussian multiple access (MAC) wiretap channel and the Kuser interference channel (IC) with secrecy constraints, have been determined recently as K(K−1)K(K−1)+1 [1,2] and K(K−1) 2K−1 [3,4], respectively. In this paper, we determine the entire s.d.o.f. regions of these two channel models. The converse for the MAC follows from a middle step in the converse of [1,2]. The converse for the IC includes constraints both due to secrecy as well as due to interference. Although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the 4user case, i.e., the 2 or 3user cases do not illustrate the generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. In both MAC and IC cases, we develop explicit schemes that achieve the extreme points of the polytope region given by the converse. Specifically, the extreme points of the MAC region are 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. The extreme points of the IC region are achieved by a (K −m)user IC with confidential messages, m helpers, and N external eavesdroppers, for m ≥ 1 and a finite N. A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetricrate extreme point, for both MAC and IC channel models.
Secure Degrees of Freedom of Onehop Wireless Networks with No Eavesdropper CSIT∗
, 2015
"... We consider three channel models: the wiretap channel withM helpers, the Kuser multiple access wiretap channel, and the Kuser interference channel with an external eavesdropper, when no eavesdropper's channel state information (CSI) is available at the transmitters. In each case, we establish ..."
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We consider three channel models: the wiretap channel withM helpers, the Kuser multiple access wiretap channel, and the Kuser interference channel with an external eavesdropper, when no eavesdropper's channel state information (CSI) is available at the transmitters. In each case, we establish the optimal sum secure degrees of freedom (s.d.o.f.) by providing achievable schemes and matching converses. We show that the unavailability of the eavesdropper's CSIT does not reduce the s.d.o.f. of the wiretap channel with helpers. However, there is loss in s.d.o.f. for both the multiple access wiretap channel and the interference channel with an external eavesdropper. In particular, we show that in the absence of eavesdropper's CSIT, the Kuser multiple access wiretap channel reduces to a wiretap channel with (K − 1) helpers from a sum s.d.o.f. perspective, and the optimal sum s.d.o.f. reduces from K(K−1)K(K−1)+1 to K−1 K. For the interference channel with an external eavesdropper, the optimal sum s.d.o.f. decreases from K(K−1)2K−1 to K−1 2 in the absence of the eavesdropper's CSIT. Our results show that the lack of eavesdropper's CSIT does not have a signicant impact on the optimal s.d.o.f. for any of the three channel models, especially when the number of users is large. This implies that physical layer security can be made robust to the unavailability of eavesdropper CSIT at high signal to noise ratio (SNR) regimes by careful modication of the achievable schemes as demonstrated in this paper. 1