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IEEE Journal of Selected Topics in Signal Processing

Achieving Undetectable Communication

Seonwoo Lee, Robert J. Baxley, Mary Ann Weitnauer, Brett T. Walkenhorst

Published Apr 10, 2015• 278 citations
Publisher

Abstract

In this paper we consider the problem of achieving a positive error-free communications rate without being detected by an eavesdropper-we coin this the privacy rate. Specifically, we analyze the privacy rate over additive white Gaussian Noise (AWGN) channels with finite and infinite number of samples and Rayleigh single input-single (SISO) and multiple input-multiple output (MIMO) channels with infinite samples when an eavesdropper employs a radiometer detector and has uncertainty about his noise variance. Leveraging recent results on the phenomenon of a signal-to-noise ratio (SNR) wall when there is eavesdropper noise power measurement uncertainty, we show that a nonzero privacy rate is possible. We also show that in this scenario, the detector should not necessarily take as many samples as possible.

DOI
10.1109/jstsp.2015.2421477

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In this paper we consider the problem of achieving a positive error-free communications rate without being detected by an eavesdropper-we coin this the privacy rate. Specifically, we analyze the privacy rate over additive white Gaussian Noise (AWGN) channels with finite and infinite number of samples and Rayleigh single input-single (SISO) and multiple input-multiple output (MIMO) channels with infinite samples when an eavesdropper employs a radiometer detector and has uncertainty about his noise variance. Leveraging recent results on the phenomenon of a signal-to-noise ratio (SNR) wall when there is eavesdropper noise power measurement uncertainty, we show that a nonzero privacy rate is possible. We also show that in this scenario, the detector should not necessarily take as many samples as possible.
In this paper we consider the problem of achieving a positive error-free communications rate without being detected by an eavesdropper - we coin this the privacy rate. Specifically, we present the privacy rate over Additive White Gaussian Noise (AWGN) and Rayleigh channels when an eavesdropper employs a radiometer detector and has uncertainty about her noise variance. Leveraging recent results on the phenomenon of an Signal to Noise Ratio (SNR) wall when there is eavesdropper noise power measurement uncertainty, we show that a non-zero privacy rate is possible. We also find the SNR wall under Rayleigh fading. We conclude by presenting realizable privacy rates for several practical scenarios.
Uncertainty about noise variance is unavoidable. Many existing works exploited the noise uncertainty for covert communication and meanwhile assumed that the adversary third party Willie adopts a radiometer detector, in which case a positive covert and reliable rate O(N) bits/N channel uses is achievable. Nonetheless, the radiometer is not optimal in general. The limit covert and reliable rate based on noise uncertainty against the optimal detector remains unknown. To shed light on this problem, this letter proves that when Willie suffers from noise uncertainty but has two or more antennas for detection, the limit covert and reliable rate over additive white Gaussian noise channels is O(N) bits/N channel uses. This rate is the same as the limit rate without the noise uncertainty, which means that the covert performance gain stemming from the noise uncertainty is indeed fragile. The key of the proof is the proposed correlation detector for Willie, which is easy to deploy and does not require any knowledge about channel coefficients or transmitter’s codebook.
Gaussian wireless channels are studied under two forms of security attacks: (i) the jamming attack, where the adversary is active and transmits corruptive signal, but is not interested in the content of the communication, (ii) the eavesdropping attack, where the adversary is passive and overhears the communication and tries to obtain the message that has been transmitted. The active attack is studied in a multi-user setting. The behavior of two users and one jammer in an additive white Gaussian channel (AWGN) with and without fading is investigated, when they participate in a non-cooperative zero-sum game, with the channel's input/output mutual information as the objective function. We assume that the jammer can eavesdrop the channel and can use the information obtained to perform correlated jamming. We also differentiate between the availability of perfect and noisy information about the user signals at the jammer. Under various assumptions on the channel characteristics, and the extent of channel state information available at the users and the jammer, we show the existence, or otherwise non-existence of a simultaneously optimal set of strategies for the users and the jammer, and characterize those strategies whenever they exist. For the passive eavesdropping attack, we study multiple-input multiple-output (MIMO) AWGN channels. We first consider a multiple-input single-output (MISO) channel, where the transmitter has multiple antennas, while the receiver and the eavesdropper have single antennas each. We find achievable rates for this channel. With the channel input restricted to Gaussian signalling with no pre-processing of information, optimal transmission strategies that maximize the achievable secrecy rates are found, in terms of the input covariance matrices. It is shown that, under the optimal communication strategy, the system reduces to a single-input single-output (SISO) channel. We then extend the achievability results to fading Gaussian MISO channels. Finally, as a step toward generalizing the problem to one with multiple antennas at the receiver, we discuss the Gaussian 2-2-1 channel with a transmitter and a receiver with two antennas each, and a single antenna eavesdropper. We develop an achievability scheme similar to those of the SISO and MISO channels, and further show that in fact, it achieves the secrecy capacity of this 2-2-1 channel.
We consider transmission of confidential data over multiple-input single-output multiple-eavesdropper (MISOME) Rayleigh-fading wiretap channels. The transmitter has access to full channel state information (CSI) of a legitimate link but only partial CSI of an eavesdropper link in forms of the average received signal-to-noise power ratio (SNR). In this location-aware scenario, we develop the optimal power allocation strategy that maximizes the secrecy rate achievable by beamforming. The optimal power control is regulated in an on-off fashion with the threshold depending only on the numbers of antennas and the average SNRs. By rescuing the vanishing high-SNR degree of freedom for secure communication from this optimal on-off beamforming with the artificial noise scheme, we further derive the achievable ergodic secrecy rate of MISOME wiretap channels in closed form.
Orthogonal frequency division multiplexing (OFDM) systems have enjoyed widespread adoption in high data rate wired and wireless networks, due to their ability to efficiently cope with slowly varying dispersive channels. This paper considers the information theoretic secrecy rates that are achievable by an OFDM transmitter/receiver pair in the presence of an eavesdropper that might either use an OFDM structure or choose a more complex receiver architecture. The analysis is performed through modeling of the OFDM system with an eavesdropper as a special case of a high dimensional multiple-input multiple-output (MIMO) wiretap channel, which allows the secrecy loss due to the OFDM structure constraints, and the information gain for an eavesdropper that uses a more complex receiver to be quantified. The results are expressed in terms of both ergodic rates and outage probabilities for multipath Rayleigh fading channels, and in terms of dependence on the signal to noise ratio (SNR) ratio between the main and eavesdropper channels.
This paper investigates the power allocation problem for artificial noise (AN) secure precoding systems, and proposes closed-form solutions for maximizing the achievable secrecy rate. It is assumed that the transmitter knows the full channel information at the legitimate receiver, and knows only the statistics of the channel information at the eavesdropper. Lower bounds are derived for the secrecy rates in multiple-input single-output channels with single or multiple eavesdroppers and multiple-input multiple-output channels with multiple eavesdroppers. When the number of transmit antennas is sufficiently large, the bounds are tight, and closed-form solutions can be derived from these bounds. The analytical results suggest simple and yet informative solutions as follows: Let the numbers of receive antennas at the legitimate receiver and at the eavesdropper beNrN_{r}Nr​andNr,eN_{r,e}Nr,e​, respectively. The system should distributeNr,e/(Nr+Nr,e)N_{r,e}/(N_{r}+N_{r,e})Nr,e​/(Nr​+Nr,e​)of the power to AN in the high SNR regime, and distribute zero power to AN in the low SNR regime; the rate loss due to the eavesdropper is−Nrlog⁡(Nr/(Nr+Nr,e))−Nr,elog⁡(Nr,e/(Nr+Nr,e))-N_{r}\log(N_{r}/(N_{r}+N_{r,e}))-N_{r,e}\log(N_{r,e}/(N_{r}+N_{r,e}))−Nr​log(Nr​/(Nr​+Nr,e​))−Nr,e​log(Nr,e​/(Nr​+Nr,e​))bits/sec/Hz in the high SNR regime and nearly negligible in the low SNR regime. The derived results also show that equal power and water-filling power allocations lead to similar solutions and rate loss. Simulation results corroborate the theoretical results.