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Allerton Conference on Communication, Control, and Computing

Compound wire-tap channels

Yingbin Liang, Gerhard G. Kramer, H. Vincent Poor, Shlomo Shitz Shamai

Published Jan 1, 2007• 107 citations
OpenAlex

Abstract

The compound wire-tap channel is studied, which is based on the classical wire-tap channel with the channel from the source to the destination and the channel from the source to the wire-tapper taking a number of states, respectively. This channel can also be viewed as the wire-tap channel with mul-tiple destinations and multiple wire-tappers, i.e., multicast with multiple wire-tappers. The source wishes to transmit information to all destinations and wants to keep the information secret from all wire-tappers. For the discrete memoryless compound wire-tap channel, lower and upper bounds on the secrecy capacity are derived and are shown to match for the degraded channel. The parallel Gaussian compound wire-tap channel is further studied, for which the secrecy capacity and the characterization of an optimal power allocation are obtained. The secrecy degree of freedom (s.d.o.f.) is also derived, which connects the secure communication rate in the high SNR regime to secure networking coding for deterministic networks. Finally, the multi-antenna (i.e., MIMO) compound wire-tap channel is studied. The secrecy capacity is established for the degraded MIMO compound wire-tap channel and an achievable s.d.o.f. is given for the general MIMO compound wire-tap channel.

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Compound Wiretap Channels

Yingbin Liang, Gerhard G. Kramer, H. Vincent Poor +1 more

This paper considers the compound wiretap channel, which generalizes Wyner's wiretap model to allow the channels to the (legitimate) receiver and to the eavesdropper to take a number of possible states. No matter which states occur, the transmitter guarantees that the receiver decodes its message and that the eavesdropper is kept in full ignorance about the message. The compound wiretap channel can also be viewed as a multicast channel with multiple eavesdroppers, in which the transmitter sends information to all receivers and keeps the information secret from all eavesdroppers. For the discrete memoryless channel, lower and upper bounds on the secrecy capacity are derived. The secrecy capacity is established for the degraded channel and the semideterministic channel with one receiver. The parallel Gaussian channel is further studied. The secrecy capacity and the secrecy degree of freedom ( ) are derived for the degraded case with one receiver. Schemes to achieve the for the case with two receivers and two eavesdroppers are constructed to demonstrate the necessity of a prefix channel in encoder design. Finally, the multi-antenna (i.e., MIMO) compound wiretap channel is studied. The secrecy capacity is established for the degraded case and an achievable is given for the general case.

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Recent results on compound wire-tap channels

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The compound wire-tap channel is studied, which is based on Wynerpsilas wire-tap model with both the channel from the source to the destination and the channel from the source to the wire-tapper taking a number of states. No matter which states occur for the two channels, the source wishes to guarantee that the destination decodes its message successfully and that the wire-tapper does not obtain the source message. The semideterministic compound wire-tap channel is first studied, in which the channel from the source to the destination is deterministic and has only one state. The secrecy capacity is obtained. An example parallel Gaussian compound wire-tap channel is then studied, in which both channels have two states. Three schemes are studied, and it is shown that introducing randomness either into the source message or into the encoder achieves the maximal secrecy degree of freedom. Both channels studied in this paper demonstrate that creating an auxiliary input, and hence adding a prefix channel from this auxiliary input to the actual channel input, improves the secrecy rate.

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The fading wire-tap channel is investigated, where the source-to-destination channel and the source-to-wire-tapper channel are corrupted by multiplicative fading gain coefficients in addition to additive Gaussian noise terms. The channel state information is assumed to be known at both the transmitter and the receiver. The parallel wire-tap channel with independent subchannels is first studied, which serves as an information-theoretic model for the fading wire-tap channel. The secrecy capacity of the parallel wire-tap channel is established. This result is then specialized to give the secrecy capacity of the fading wire-tap channel, which is achieved with the source node dynamically changing the power allocation according to the channel state realization. An optimal source power allocation is obtained to achieve the secrecy capacity.

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MIMO Wiretap Channels

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This chapter considers securing wireless communications at the physical layer using multiple antennas. In particular, the multiple antenna (MIMO) wiretap channel is presented and its secrecy capacity is provided. Further investigation of the high-SNR characterization of the secrecy capacity, i.e., secure degrees of freedom, reveals that the secrecy capacity of the MIMO wiretap channel does not scale with the transmit power when the eavesdropper has an advantage over the legitimate transmitter in the number of antennas. An external multi-antenna cooperative jammer terminal is introduced to the channel model in order to improve its secrecy capacity scaling with power. The secure degrees of freedom for this new multi-terminal multi-antenna channel is characterized. While for these models the eavesdropper channel state information is known at the legitimate terminals, the chapter next removes this assumption and presents results that utilize multiple antennas in order to provide secure communications irrespective of the eavesdropper channel state. An achievable strong secrecy rate and the secure degrees of freedom characterization for the MIMO wiretap channel when the eavesdropper channel is arbitrary, varying, and unknown at the legitimate terminals are presented. Finally, the extension of this model to the two-user multiple access MIMO wiretap channel is considered and its secure degrees of freedom region is characterized. Introduction The wiretap channel and the notion of secrecy capacity, introduced in [1] and generalized by [2], have provided the framework for physical layer design for wireless channels with information theoretic guarantees against eavesdropping. A natural model in wireless communications is one which utilizes multiple antennas for transmission and reception at each node. The improvement in secrecy rate that can be obtained using multiple antennas in a point-to-point channel with an eavesdropper has been studied extensively in several references, including [3, 4]. Specifically, the secrecy capacity of the Gaussian model consisting of a multiple antenna legitimate transmitter, a multiple antenna legitimate receiver, and a multiple antenna eavesdropper, termed the multiple-input multiple-output (MIMO) wiretap channel, has been identified in full generality in [5, 6].

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Secrecy capacity of the 2-2-1 gaussian MIMO wire-tap channel

Shabnam Shafiee, Nan Liu, Şennur Ulukuş

We find the secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.

International Symposium on Communications, Control and Signal Processing•Mar 1, 2008•12 citations
The compound wire-tap channel is studied, which is based on the classical wire-tap channel with the channel from the source to the destination and the channel from the source to the wire-tapper taking a number of states, respectively. This channel can also be viewed as the wire-tap channel with mul-tiple destinations and multiple wire-tappers, i.e., multicast with multiple wire-tappers. The source wishes to transmit information to all destinations and wants to keep the information secret from all wire-tappers. For the discrete memoryless compound wire-tap channel, lower and upper bounds on the secrecy capacity are derived and are shown to match for the degraded channel. The parallel Gaussian compound wire-tap channel is further studied, for which the secrecy capacity and the characterization of an optimal power allocation are obtained. The secrecy degree of freedom (s.d.o.f.) is also derived, which connects the secure communication rate in the high SNR regime to secure networking coding for deterministic networks. Finally, the multi-antenna (i.e., MIMO) compound wire-tap channel is studied. The secrecy capacity is established for the degraded MIMO compound wire-tap channel and an achievable s.d.o.f. is given for the general MIMO compound wire-tap channel.
This paper considers the compound wiretap channel, which generalizes Wyner's wiretap model to allow the channels to the (legitimate) receiver and to the eavesdropper to take a number of possible states. No matter which states occur, the transmitter guarantees that the receiver decodes its message and that the eavesdropper is kept in full ignorance about the message. The compound wiretap channel can also be viewed as a multicast channel with multiple eavesdroppers, in which the transmitter sends information to all receivers and keeps the information secret from all eavesdroppers. For the discrete memoryless channel, lower and upper bounds on the secrecy capacity are derived. The secrecy capacity is established for the degraded channel and the semideterministic channel with one receiver. The parallel Gaussian channel is further studied. The secrecy capacity and the secrecy degree of freedom ( ) are derived for the degraded case with one receiver. Schemes to achieve the for the case with two receivers and two eavesdroppers are constructed to demonstrate the necessity of a prefix channel in encoder design. Finally, the multi-antenna (i.e., MIMO) compound wiretap channel is studied. The secrecy capacity is established for the degraded case and an achievable is given for the general case.
The compound wire-tap channel is studied, which is based on Wynerpsilas wire-tap model with both the channel from the source to the destination and the channel from the source to the wire-tapper taking a number of states. No matter which states occur for the two channels, the source wishes to guarantee that the destination decodes its message successfully and that the wire-tapper does not obtain the source message. The semideterministic compound wire-tap channel is first studied, in which the channel from the source to the destination is deterministic and has only one state. The secrecy capacity is obtained. An example parallel Gaussian compound wire-tap channel is then studied, in which both channels have two states. Three schemes are studied, and it is shown that introducing randomness either into the source message or into the encoder achieves the maximal secrecy degree of freedom. Both channels studied in this paper demonstrate that creating an auxiliary input, and hence adding a prefix channel from this auxiliary input to the actual channel input, improves the secrecy rate.
The fading wire-tap channel is investigated, where the source-to-destination channel and the source-to-wire-tapper channel are corrupted by multiplicative fading gain coefficients in addition to additive Gaussian noise terms. The channel state information is assumed to be known at both the transmitter and the receiver. The parallel wire-tap channel with independent subchannels is first studied, which serves as an information-theoretic model for the fading wire-tap channel. The secrecy capacity of the parallel wire-tap channel is established. This result is then specialized to give the secrecy capacity of the fading wire-tap channel, which is achieved with the source node dynamically changing the power allocation according to the channel state realization. An optimal source power allocation is obtained to achieve the secrecy capacity.
This chapter considers securing wireless communications at the physical layer using multiple antennas. In particular, the multiple antenna (MIMO) wiretap channel is presented and its secrecy capacity is provided. Further investigation of the high-SNR characterization of the secrecy capacity, i.e., secure degrees of freedom, reveals that the secrecy capacity of the MIMO wiretap channel does not scale with the transmit power when the eavesdropper has an advantage over the legitimate transmitter in the number of antennas. An external multi-antenna cooperative jammer terminal is introduced to the channel model in order to improve its secrecy capacity scaling with power. The secure degrees of freedom for this new multi-terminal multi-antenna channel is characterized. While for these models the eavesdropper channel state information is known at the legitimate terminals, the chapter next removes this assumption and presents results that utilize multiple antennas in order to provide secure communications irrespective of the eavesdropper channel state. An achievable strong secrecy rate and the secure degrees of freedom characterization for the MIMO wiretap channel when the eavesdropper channel is arbitrary, varying, and unknown at the legitimate terminals are presented. Finally, the extension of this model to the two-user multiple access MIMO wiretap channel is considered and its secure degrees of freedom region is characterized. Introduction The wiretap channel and the notion of secrecy capacity, introduced in [1] and generalized by [2], have provided the framework for physical layer design for wireless channels with information theoretic guarantees against eavesdropping. A natural model in wireless communications is one which utilizes multiple antennas for transmission and reception at each node. The improvement in secrecy rate that can be obtained using multiple antennas in a point-to-point channel with an eavesdropper has been studied extensively in several references, including [3, 4]. Specifically, the secrecy capacity of the Gaussian model consisting of a multiple antenna legitimate transmitter, a multiple antenna legitimate receiver, and a multiple antenna eavesdropper, termed the multiple-input multiple-output (MIMO) wiretap channel, has been identified in full generality in [5, 6].
Abstract—We find the secrecy capacity of the 2-2-1 Gaussian MIMO wiretap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary. Index Terms—Information-theoretic security, multiple-input multiple-output (MIMO), multiple antennas, secrecy capacity,
We find the secrecy capacity of the 2-2-1 Gaussian MIMO wire-tap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian signalling in the form of beam-forming is optimal, and no pre-processing of information is necessary.