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arXiv:0905.2638 · Globecom

Secure Degrees of Freedom for Gaussian Channels with Interference: Structured Codes Outperform Gaussian Signaling

Xiang He, Aylin Yener

Published Nov 1, 2009• 25 citations
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Abstract

In this work, we prove that a positive secure degree of freedom is achievable for a large class of real Gaussian channels as long as the channel is not degraded and the channel is fully connected. This class includes the MAC wiretap channel, the 2-user interference channel with confidential messages, the 2-user interference channel with an external eavesdropper. Best known achievable schemes to date for these channels use Gaussian signaling. In this work, we show that structured codes outperform Gaussian random codes at high SNR when channel gains are real numbers.

DOI
10.1109/glocom.2009.5425429

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Secure Degrees of Freedom for Gaussian Channels with Interference: Structured Codes Outperform Gaussian Signaling

Xiang He, Aylin Yener

Abstract—In this work, we prove that a positive secure degree of freedom is achievable for a large class of real Gaussian channels as long as the channel is not degraded and the channel is fully connected. This class includes the MAC wiretap channel, the 2-user interference channel with confidential messages, the 2-user interference channel with an external eavesdropper. Best known achievable schemes to date for these channels use Gaussian signaling. In this work, we show that structured codes outperform Gaussian random codes at high SNR when channel gains are real numbers. I.

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SECURE DEGREES OF FREEDOM OF WIRELESS NETWORKS

Jianwei Xie

This dissertation studies the security of wireless interference networks from an information-theoretic point of view. In this setting, several transmitter-receiver pairs wish to have secure communication against the eavesdropper(s). The central goal of this dissertation is to develop a framework based on information-theoretic principles to determine the complete solutions for the signaling schemes in different wireless interference networks with large transmit powers, and derive the corresponding fundamental limits in terms of the secure degrees of freedom (s.d.o.f.). Next, we study one-hop wireless networks by considering four fundamental wireless network structures: Gaussian wiretap channel with helpers, Gaussian broadcast channel (BC) with confidential messages, Gaussian interference channel (IC) with confidential messages, and Gaussian multiple access (MAC) wiretap channel. The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the s.d.o.f. of the Gaussian wiretap channel with no helpers is zero. We show that the exact s.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 s.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 s.d.o.f. of the Gaussian BC with confidential messages and M helpers is 1, the sum s.d.o.f. of the two-user IC with confidential messages is 2/3, the sum s.d.o.f. of the two-user IC with confidential messages and M helpers is 1, and the sum s.d.o.f. of the K-user MAC wiretap channel is K(K-1)/(K(K-1)+1). Then, we study the sum s.d.o.f. of multi-receiver networks. In this dissertation, we determine the exact sum s.d.o.f. of the K-user Gaussian IC. We consider three different secrecy constraints: 1) K-user IC with one external eavesdropper (IC-EE), 2) K-user IC with confidential messages (IC-CM), and 3) K-user IC with confidential messages and one external eavesdropper (IC-CM-EE). 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 IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. 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 sub-space at every receiver, in order to attain the optimum sum s.d.o.f., we incorporate structured cooperative jamming into the achievable scheme, and intricately design the structure of all of the transmitted signals jointly. Next, we study the entire s.d.o.f. regions of multi-user network structures. In this dissertation, we determine the entire s.d.o.f. regions of the K-user MAC wiretap channel and the K-user IC with secrecy constraints. The converse for the MAC follows from a middle step in the converse of the sum s.d.o.f. 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 4-user case, i.e., the 2 or 3-user 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 m-user 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. As a byproduct of determining the entire s.d.o.f. regions of MAC and IC channels, we show that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point, for both MAC and IC channel models. Then, we determine the sum s.d.o.f. of two-unicast layered wireless networks. Without any secrecy constraints, the sum d.o.f. of this class of networks was shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where, in addition to being reliably transmitted, each message is required to be kept information-theoretically secure from the unintended receiver. We show that the sum s.d.o.f. can only take one of five possible values: 0, 2/3, 1, 3/2,2, for all network configurations. To determine the sum s.d.o.f., we divide the class of two-unicast layered networks into several sub-classes, and propose an achievable scheme based on the specific structure of the networks in each sub-class. Our achievable schemes are based on real interference alignment, cooperative jamming, interference neutralization and cooperative jamming neutralization techniques. Next, we consider the Gaussian wiretap channel with M helpers, where no eavesdropper channel state information (CSI) is available at the legitimate entities. One of the key ingredients of our optimal achievable scheme with perfect CSI is to align cooperative jamming signals with the information symbols at the eavesdropper to limit the information leakage rate. This requires perfect eavesdropper CSI at the transmitters. We propose a new achievable scheme in which cooperative jamming signals span the entire space of the eavesdropper, but are not exactly aligned with the information symbols. We show that this scheme achieves the same s.d.o.f. of M/(M+1) but does not require any eavesdropper CSI; the transmitters blindly cooperative jam the eavesdropper. Then, we study the separability of the parallel MAC wiretap channel. Separability, when exists, is useful as it enables us to code separately over parallel channels, and still achieve the optimum overall performance. It is well-known that the parallel single-user channel, parallel MAC and parallel BC are all separable, however, the parallel IC is not separable in general. In this dissertation, we show that, while MAC is separable MAC wiretap channel is not separable in general. We prove this via a specific linear deterministic MAC wiretap channel. We then show that even the Gaussian MAC wiretap channel is inseparable in general. Finally, we show that, when the channel gains are drawn from continuous distributions, and when the s.d.o.f. region is considered, then the Gaussian MAC wiretap channel is almost surely separable. Finally, we study the two-user one-sided IC with confidential messages. In this IC, in addition to the usual selfishness of the users, the relationship between the users is adversarial in the sense of both receivers' desires to eavesdrop on the communication of the other pair. We develop a game-theoretic model for this setting. We start with a model where each pair's payoff is their own secrecy rate. We then propose a refinement for the payoff function by explicitly accounting for the desire of the receiver to eavesdrop on the other party's communication. This payoff function captures the adversarial relationship between the users better. We determine the Nash equilibria for the binary deterministic channel for both payoff functions.

•Jan 1, 2014•0 citations

Security in the Gaussian interference channel: Weak and moderately weak interference regimes

Parisa Babaheidarian, Somayeh Salimi, Panos Papadimitratos

We consider a secure communication scenario through the two-user Gaussian interference channel: each transmitter (user) has a confidential message to send reliably to its intended receiver while keeping it secret from the other receiver. Prior work investigated the performance of two different approaches for this scenario; i.i.d. Gaussian random codes and real alignment of structured codes. While the latter achieves the optimal sum secure degrees of freedom (s.d.o.f.), its extension to finite SNR regimes is challenging. In this paper, we propose a new achievability scheme for the weak and the moderately weak interference regimes, in which the reliability as well as the confidentiality of the transmitted messages are maintained at any finite SNR value. Our scheme uses lattice structure, structured jamming codewords, and lattice alignment in the encoding and the asymmetric compute-and-forward strategy in the decoding. We show that our lower bound on the sum secure rates scales linearly with log(SNR) and hence, it outperforms i.i.d. Gaussian random codes. Furthermore, we show that our achievable result is asymptotically optimal. Finally, we provide a discussion on an extension of our scheme to K > 2 users.

arXiv:1604.07386•IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2016•3 citations

Secure degrees of freedom of the Gaussian multiple access wiretap channel

Jianwei Xie, Şennur Ulukuş

We show that the sum secure degrees of freedom (d.o.f.) of the K-user Gaussian multiple access (MAC) wiretap channel is K(K-1)/K(K-1)+1. Our achievability is based on real interference alignment and structured cooperative jamming. Each user divides its message into K - 1 sub-messages, and sends a linear combination of signals carrying these sub-messages 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 K(K-1)/K(K-1)+1.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2013•36 citations

Secure Degrees of Freedom of $K$ -User Gaussian Interference Channels: A Unified View

Jianwei Xie, Şennur Ulukuş

We determine the exact sum secure degrees of freedom (d.o.f.) of the K-user Gaussian interference channel. We consider three different secrecy constraints: 1) K-user interference channel with one external eavesdropper (IC-EE); 2) K-user interference channel with confidential messages (IC-CM); and 3) K-user interference channel with confidential messages and one external eavesdropper (IC-CM-EE). 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 IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. Our converse is based on developing a direct relationship between the differential entropies of the channel inputs and the rates of the users, and quantifying the effect of eavesdropping on the rates in terms of the differential entropies of the eavesdroppers' observations. Our achievability is based on 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.

arXiv:1305.7214•IEEE Transactions on Information Theory•Mar 9, 2015•81 citations

Providing Secrecy With Structured Codes: Two-User Gaussian Channels

Xiang He, Aylin Yener

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 Gaussian random codes. In this paper, we advocate that structure in random code generation is useful for providing secrecy as well. In particular, a Gaussian wiretap channel in the presence of a cooperative jammer is studied. Previously, the achievable secrecy rate for this channel was derived using Gaussian signaling, which saturated at high signal-to-noise ratio (SNR), owing to the fact that the cooperative jammer simultaneously helps by interfering with the eavesdropper, and hurts by interfering with the intended receiver. In this paper, a new achievable rate is derived through imposing a lattice structure on the signals transmitted by both the source and the cooperative jammer, which are aligned at the eavesdropper but remain separable at the intended receiver. We prove that the achieved secrecy rate does not saturate at high SNR for all values of channel gains except when the channel is degraded.

IEEE Transactions on Information Theory•Jan 30, 2014•68 citations
In this work, we prove that a positive secure degree of freedom is achievable for a large class of real Gaussian channels as long as the channel is not degraded and the channel is fully connected. This class includes the MAC wiretap channel, the 2-user interference channel with confidential messages, the 2-user interference channel with an external eavesdropper. Best known achievable schemes to date for these channels use Gaussian signaling. In this work, we show that structured codes outperform Gaussian random codes at high SNR when channel gains are real numbers.
Abstract—In this work, we prove that a positive secure degree of freedom is achievable for a large class of real Gaussian channels as long as the channel is not degraded and the channel is fully connected. This class includes the MAC wiretap channel, the 2-user interference channel with confidential messages, the 2-user interference channel with an external eavesdropper. Best known achievable schemes to date for these channels use Gaussian signaling. In this work, we show that structured codes outperform Gaussian random codes at high SNR when channel gains are real numbers. I.
This dissertation studies the security of wireless interference networks from an information-theoretic point of view. In this setting, several transmitter-receiver pairs wish to have secure communication against the eavesdropper(s). The central goal of this dissertation is to develop a framework based on information-theoretic principles to determine the complete solutions for the signaling schemes in different wireless interference networks with large transmit powers, and derive the corresponding fundamental limits in terms of the secure degrees of freedom (s.d.o.f.). Next, we study one-hop wireless networks by considering four fundamental wireless network structures: Gaussian wiretap channel with helpers, Gaussian broadcast channel (BC) with confidential messages, Gaussian interference channel (IC) with confidential messages, and Gaussian multiple access (MAC) wiretap channel. The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the s.d.o.f. of the Gaussian wiretap channel with no helpers is zero. We show that the exact s.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 s.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 s.d.o.f. of the Gaussian BC with confidential messages and M helpers is 1, the sum s.d.o.f. of the two-user IC with confidential messages is 2/3, the sum s.d.o.f. of the two-user IC with confidential messages and M helpers is 1, and the sum s.d.o.f. of the K-user MAC wiretap channel is K(K-1)/(K(K-1)+1). Then, we study the sum s.d.o.f. of multi-receiver networks. In this dissertation, we determine the exact sum s.d.o.f. of the K-user Gaussian IC. We consider three different secrecy constraints: 1) K-user IC with one external eavesdropper (IC-EE), 2) K-user IC with confidential messages (IC-CM), and 3) K-user IC with confidential messages and one external eavesdropper (IC-CM-EE). 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 IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. 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 sub-space at every receiver, in order to attain the optimum sum s.d.o.f., we incorporate structured cooperative jamming into the achievable scheme, and intricately design the structure of all of the transmitted signals jointly. Next, we study the entire s.d.o.f. regions of multi-user network structures. In this dissertation, we determine the entire s.d.o.f. regions of the K-user MAC wiretap channel and the K-user IC with secrecy constraints. The converse for the MAC follows from a middle step in the converse of the sum s.d.o.f. 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 4-user case, i.e., the 2 or 3-user 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 m-user 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. As a byproduct of determining the entire s.d.o.f. regions of MAC and IC channels, we show that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point, for both MAC and IC channel models. Then, we determine the sum s.d.o.f. of two-unicast layered wireless networks. Without any secrecy constraints, the sum d.o.f. of this class of networks was shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where, in addition to being reliably transmitted, each message is required to be kept information-theoretically secure from the unintended receiver. We show that the sum s.d.o.f. can only take one of five possible values: 0, 2/3, 1, 3/2,2, for all network configurations. To determine the sum s.d.o.f., we divide the class of two-unicast layered networks into several sub-classes, and propose an achievable scheme based on the specific structure of the networks in each sub-class. Our achievable schemes are based on real interference alignment, cooperative jamming, interference neutralization and cooperative jamming neutralization techniques. Next, we consider the Gaussian wiretap channel with M helpers, where no eavesdropper channel state information (CSI) is available at the legitimate entities. One of the key ingredients of our optimal achievable scheme with perfect CSI is to align cooperative jamming signals with the information symbols at the eavesdropper to limit the information leakage rate. This requires perfect eavesdropper CSI at the transmitters. We propose a new achievable scheme in which cooperative jamming signals span the entire space of the eavesdropper, but are not exactly aligned with the information symbols. We show that this scheme achieves the same s.d.o.f. of M/(M+1) but does not require any eavesdropper CSI; the transmitters blindly cooperative jam the eavesdropper. Then, we study the separability of the parallel MAC wiretap channel. Separability, when exists, is useful as it enables us to code separately over parallel channels, and still achieve the optimum overall performance. It is well-known that the parallel single-user channel, parallel MAC and parallel BC are all separable, however, the parallel IC is not separable in general. In this dissertation, we show that, while MAC is separable MAC wiretap channel is not separable in general. We prove this via a specific linear deterministic MAC wiretap channel. We then show that even the Gaussian MAC wiretap channel is inseparable in general. Finally, we show that, when the channel gains are drawn from continuous distributions, and when the s.d.o.f. region is considered, then the Gaussian MAC wiretap channel is almost surely separable. Finally, we study the two-user one-sided IC with confidential messages. In this IC, in addition to the usual selfishness of the users, the relationship between the users is adversarial in the sense of both receivers' desires to eavesdrop on the communication of the other pair. We develop a game-theoretic model for this setting. We start with a model where each pair's payoff is their own secrecy rate. We then propose a refinement for the payoff function by explicitly accounting for the desire of the receiver to eavesdrop on the other party's communication. This payoff function captures the adversarial relationship between the users better. We determine the Nash equilibria for the binary deterministic channel for both payoff functions.
We consider a secure communication scenario through the two-user Gaussian interference channel: each transmitter (user) has a confidential message to send reliably to its intended receiver while keeping it secret from the other receiver. Prior work investigated the performance of two different approaches for this scenario; i.i.d. Gaussian random codes and real alignment of structured codes. While the latter achieves the optimal sum secure degrees of freedom (s.d.o.f.), its extension to finite SNR regimes is challenging. In this paper, we propose a new achievability scheme for the weak and the moderately weak interference regimes, in which the reliability as well as the confidentiality of the transmitted messages are maintained at any finite SNR value. Our scheme uses lattice structure, structured jamming codewords, and lattice alignment in the encoding and the asymmetric compute-and-forward strategy in the decoding. We show that our lower bound on the sum secure rates scales linearly with log(SNR) and hence, it outperforms i.i.d. Gaussian random codes. Furthermore, we show that our achievable result is asymptotically optimal. Finally, we provide a discussion on an extension of our scheme to K > 2 users.
We show that the sum secure degrees of freedom (d.o.f.) of the K-user Gaussian multiple access (MAC) wiretap channel is K(K-1)/K(K-1)+1. Our achievability is based on real interference alignment and structured cooperative jamming. Each user divides its message into K - 1 sub-messages, and sends a linear combination of signals carrying these sub-messages 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 K(K-1)/K(K-1)+1.
Secure Degrees of Freedom of KKK -User Gaussian Interference Channels: A Unified View
We determine the exact sum secure degrees of freedom (d.o.f.) of the K-user Gaussian interference channel. We consider three different secrecy constraints: 1) K-user interference channel with one external eavesdropper (IC-EE); 2) K-user interference channel with confidential messages (IC-CM); and 3) K-user interference channel with confidential messages and one external eavesdropper (IC-CM-EE). 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 IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. Our converse is based on developing a direct relationship between the differential entropies of the channel inputs and the rates of the users, and quantifying the effect of eavesdropping on the rates in terms of the differential entropies of the eavesdroppers' observations. Our achievability is based on 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.
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 Gaussian random codes. In this paper, we advocate that structure in random code generation is useful for providing secrecy as well. In particular, a Gaussian wiretap channel in the presence of a cooperative jammer is studied. Previously, the achievable secrecy rate for this channel was derived using Gaussian signaling, which saturated at high signal-to-noise ratio (SNR), owing to the fact that the cooperative jammer simultaneously helps by interfering with the eavesdropper, and hurts by interfering with the intended receiver. In this paper, a new achievable rate is derived through imposing a lattice structure on the signals transmitted by both the source and the cooperative jammer, which are aligned at the eavesdropper but remain separable at the intended receiver. We prove that the achieved secrecy rate does not saturate at high SNR for all values of channel gains except when the channel is degraded.