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Interference alignment with asymmetric complex signaling  settling the HostMadsenNosratinia conjecture
 IEEE TRANSACTION ON INFORMATION THEORY
, 2009
"... It has been conjectured by HøstMadsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degreeoffreedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, th ..."
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Cited by 65 (17 self)
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It has been conjectured by HøstMadsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degreeoffreedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degreeoffreedom is achievable. In this paper, we settle the HøstMadsenNosratinia conjecture in the negative. We show that at least 1.2 degreesoffreedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees of freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees of freedom we introduce the novel idea of asymmetric complex signaling i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian pointtopoint, multipleaccess and broadcast channels where circularly
Real Interference Alignment with Real Numbers
, 2009
"... A novel coding scheme applicable in networks with single antenna nodes is proposed. This scheme converts a single antenna system to an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. Interference can be aligned along these dimensions and higher Multiplexing gains ..."
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Cited by 46 (2 self)
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A novel coding scheme applicable in networks with single antenna nodes is proposed. This scheme converts a single antenna system to an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. Interference can be aligned along these dimensions and higher Multiplexing gains can be achieved. Tools from the field of Diophantine approximation in number theory are used to show that the proposed coding scheme in fact mimics the traditional schemes used in MIMO systems where each data stream is sent along a direction and alignment happens when several streams arrive at the same direction. Two types of constellation are proposed for the encoding part, namely the single layer constellation and the multilayer constellation. Using the single layer constellation, the coding scheme is applied to the twouser X channel and the threeuser Gaussian Interference Channel (GIC). In case of the twouser X channel, it is proved that the total DegreesofFreedom (DOF), i.e. 4, of 3 the channel is achievable almost surely. This is the first example in which it is shown that a time invariant single antenna system does not fall short of achieving its total DOF. For the threeuser GIC, it is shown that the DOF of 4 is achievable almost surely. 3 Using the multilayer constellation, the coding scheme is applied to the symmetric threeuser GIC. Achievable DOFs are derived for all channel gains. As a function of the channel gain, it is observed that the DOF is everywhere discontinuous. In particular, it is proved that for the irrational channel gains the achievable DOF meets the upper bound 3. For the rational gains, 2 the achievable DOF has a gap to the available upper bounds. By allowing carry over from multiple layers, however, it is shown that higher DOFs can be achieved.
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
Ergodic Secret Alignment
, 2012
"... In this paper, we introduce two new achievable schemes for the fading multiple access wiretap channel (MACWT). In the model that we consider, we assume that perfect knowledge of the state of all channels is available at all the nodes in a causal fashion. Our schemes use this knowledge together wit ..."
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Cited by 21 (17 self)
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In this paper, we introduce two new achievable schemes for the fading multiple access wiretap channel (MACWT). In the model that we consider, we assume that perfect knowledge of the state of all channels is available at all the nodes in a causal fashion. Our schemes use this knowledge together with the timevarying nature of the channel model to align the interference from different users at the eavesdropper perfectly in a onedimensional space while creating a higher dimensionality space for the interfering signals at the legitimate receiver, hence allowing for better chance of recovery. While we achieve this alignment through signal scaling at the transmitters in our first scheme (scalingbased alignment), we let nature provide this alignment through the ergodicity of the channel coefficients in the second scheme [ergodic secret alignment (ESA)] [1], [2]. For each scheme, we obtain the resulting achievable secrecy rate region. We show that the secrecy rates achieved by both schemes in the twouser fading MACWT scale with signaltonoise ratio (SNR) as 1 2 log(SNR). Hence, we show the suboptimality of the independent identically distributed (i.i.d.) Gaussian signalingbased schemes with and without cooperative jamming by showing that the secrecy rates achieved using i.i.d. Gaussian signaling with cooperative jamming do not scale with SNR. In addition, we introduce an improved version of our ESA scheme where we incorporate cooperative jamming to achieve higher secrecy rates. Moreover, we derive the necessary optimality conditions for the power control policy that maximizes the secrecy sum rate achievable by our ESA scheme when used solely and with cooperative jamming. Finally, we discuss the extension of the proposed schemes to the case where there are more than two users and show that, for theuser fading MACWT, each of the two schemes achieves secrecy sum rate that scales with SNR as 1 log(SNR).
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
Principles of Physical Layer Security in Multiuser Wireless Networks: A Survey
"... This paper provides a comprehensive review of the domain of physical layer security in multiuser wireless networks. The essential premise of physical layer security is to enable the exchange of confidential messages over a wireless medium in the presence of unauthorized eavesdroppers, without rely ..."
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Cited by 18 (1 self)
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This paper provides a comprehensive review of the domain of physical layer security in multiuser wireless networks. The essential premise of physical layer security is to enable the exchange of confidential messages over a wireless medium in the presence of unauthorized eavesdroppers, without relying on higherlayer encryption. This can be achieved primarily in two ways: without the need for a secret key by intelligently designing transmit coding strategies, or by exploiting the wireless communication medium to develop secret keys over public channels. The survey begins with an overview of the foundations dating back to the pioneering work of Shannon and Wyner on informationtheoretic security. We then describe the evolution of secure transmission strategies from pointtopoint channels to multipleantenna systems, followed by generalizations to multiuser broadcast, multipleaccess, interference, and relay networks. Secretkey generation and establishment protocols based on physical layer mechanisms are subsequently covered. Approaches for secrecy based on channel coding design are then examined, along with a description of interdisciplinary approaches based on game theory and stochastic geometry. The associated problem of physical layer message authentication is also briefly introduced. The survey concludes with observations on potential research directions in this area.
Real interference alignment for the Kuser Gaussian interference compound wiretap channel
 In 48th Annual Allerton Conference on Communication, Control and Computing
, 2010
"... Abstract — We study the Kuser Gaussian interference wiretap channel with N external eavesdroppers. All the transmitters, receivers and eavesdroppers have a single antenna each. We propose an achievable scheme to lower bound the secure degrees of freedom (d.o.f.) for each transmitterreceiver pair. ..."
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Cited by 11 (10 self)
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Abstract — We study the Kuser Gaussian interference wiretap channel with N external eavesdroppers. All the transmitters, receivers and eavesdroppers have a single antenna each. We propose an achievable scheme to lower bound the secure degrees of freedom (d.o.f.) for each transmitterreceiver pair. Our approach is based on the (real) interference alignment technique. Our achievable scheme not only aligns the interference at each receiver to prevent the d.o.f. from vanishing, but also aligns the signals observed by the eavesdroppers to reduce the secrecy penalty. The achievable secure d.o.f. of each transmitterreceiver pair is shown to be 1 2 channel gains.
Opportunistic Interference Alignment for MIMO Interfering MultipleAccess Channels
 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION
, 2013
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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 MIMO XChannels With Output Feedback and Delayed CSIT
 IEEE Transactions on Information Forensics and Security
, 2013
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