IEEE Transactions on Wireless Communications

Secret Information of Wireless Multi-Dimensional Gaussian Channels

Gianni Pasolini, Davide Dardari

Published Feb 23, 2015• 17 citations

Abstract

Wireless channel reciprocity can be exploited by two users willing to achieve confidential communications over a public channel as a common source of randomness for the generation of a secret key. This paper presents an analytical framework for investigating the amount of secret information that characterizes wireless multi-dimensional Gaussian channels used as source of randomness. The intrinsic secrecy content of wide-sense stationary wireless channels in frequency, time and spatial domains is derived through asymptotic analysis. Some significant case studies are presented, where single and multiple antenna eavesdroppers are considered. In the numerical results, the role of signal-to-noise ratio, spatial correlation, frequency and time selectivity, sampling frequency and imperfect channel reciprocity is investigated.

DOI: 10.1109/twc.2015.2406320

Related papers

Secret key generation in correlated multi-dimensional Gaussian channels

Gianni Pasolini, Davide Dardari

Wireless channel reciprocity can be successfully exploited as a common source of randomness for the generation of a secret key by two legitimate users willing to achieve confidential communications over a public channel. This paper presents an analytical framework to investigate the theoretical limits of secret-key generation when wireless multi-dimensional Gaussian channels are used as source of randomness. The intrinsic secrecy content of wide-sense stationary wireless channels in frequency, time and spatial domains is derived through asymptotic analysis as the number of observations in a given domain tends to infinity. Some significant case studies are presented where single and multiple antenna eavesdroppers are considered. In the numerical results, the role of signal-to-noise ratio, spatial correlation, frequency and time selectivity is investigated.

International Conference on CommunicationsJun 1, 20143 citations

Information-theoretic Secrecy in Multi-user Channels

Ersen Ekrem

Inherent openness of the wireless medium imposes stronger challenges on the security of wireless communications. Information-theoretic security addresses these challenges at the physical layer by using tools from wireless communication theory, signal processing and information theory. In information-theoretic security, physical layer communication is intelligently designed to exploit the characteristics of the wireless medium, such as fading, interference, cooperation, and multi-dimensional signaling, in order to provide or improve security. In this dissertation, we study the security of several fundamental wireless network configurations from an information-theoretic perspective. First, we study the Gaussian multiple-input multiple-output (MIMO) wiretap channel. In this channel, the transmitter sends a common message to both the legitimate user and the eavesdropper. In addition to the common message, a private message is sent only to the legitimate user, which needs to be kept hidden as much as possible from the eavesdropper. We obtain the entire capacity-equivocation region for this channel model. In particular, we show the sufficiency of jointly Gaussian auxiliary random variables and channel input to evaluate the existing single-letter description of the capacity-equivocation region due to Csiszar-Korner. Next, we study the secure broadcasting problem, where a transmitter wants to have secure communication with multiple legitimate users in the presence of an external eavesdropper. We study several special cases of the secure broadcasting problem. First, we consider the degraded multi-receiver wiretap channel, and establish its secrecy capacity region. Second, we consider the parallel less noisy multi-receiver wiretap channel, and obtain its common message secrecy capacity and sum secrecy capacity. Third, we consider the parallel degraded multi-receiver wiretap channel for the two-user and two-sub-channel case, and obtain its entire secrecy capacity region. Finally, we consider a parallel channel model with two sub-channels, where the transmitter can use only one of the subchannels at any time, and characterize its secrecy capacity region. Then, we study the two-user Gaussian MIMO broadcast channel with common and confidential messages. In this channel model, the transmitter sends a common message to both users, and a confidential message to each user which needs to be kept perfectly secret from the other user. We obtain the entire capacity region of this channel. We also explore the connections between this channel model and its non-confidential counterpart, i.e., the Gaussian MIMO broadcast channel with common and private message. Next, we consider the Gaussian MIMO multi-receiver wiretap channel and obtain its secrecy capacity region for the most general case. We first show that even for the single-input single-output (SISO) case, existing converse techniques fall short of proving the secrecy capacity region, to emphasize the need for a new proof technique, which we develop by using the relationships between the Fisher information and the differential entropy. Using this new proof technique, we obtain the secrecy capacity region of the degraded MIMO channel. We then establish the secrecy capacity region of the general MIMO channel by using the channel enhancement technique in conjunction with the capacity result we obtained for the degraded MIMO channel. For the general MIMO channel, we show that dirty-paper coding (DPC) combined with stochastic encoding attains the entire secrecy capacity region. Then, we study the multi-receiver wiretap channel for a more general scenario, where, in addition to confidential messages, the transmitter sends public messages to the legitimate users, on which there are no secrecy constraints. First, we consider the degraded discrete memoryless channel, and obtain inner and outer bounds for the capacity region. These inner and outer bounds match for certain cases, providing the capacity region. Second, we obtain an inner bound for the general discrete memoryless channel by using Marton's inner bound. Third, we consider the degraded Gaussian MIMO channel, and show that jointly Gaussian auxiliary random variables and channel input are sufficient to exhaust the inner and outer bounds. Finally, we provide an inner bound for the capacity region of the general Gaussian MIMO channel. Next, we focus on the multiple access wiretap (MAC-WT) channel whose capacity region is unknown. We consider a special class of MAC-WT channels which we call the weak eavesdropper class, where each user's link to the legitimate receiver is stronger than its link to the eavesdropper. For this class of channels, we develop an outer bound for the secrecy capacity region, which partially matches the achievable region in an n-letter form. We evaluate a looser version of our outer bound for the Gaussian case, and show that our outer bound is within 0.5 bits/channel use of the achievable rates along the individual secrecy rates for all weak eavesdropper Gaussian MAC-WT. Then, we investigate the effects of user cooperation on the secrecy of broadcast channels by considering the cooperative relay broadcast channel (CRBC). We propose an achievable scheme that combines Marton's coding scheme for broadcast channels and Cover and El Gamal's compress-and-forward (CAF) scheme for relay channels. For the Gaussian CRBC, we show that both users can have positive secrecy rates, which is not possible for scalar Gaussian broadcast channels without cooperation. We further investigate the effects of user cooperation on secrecy by considering the multiple access channel with generalized feedback (MAC-GF), which can be viewed as the MAC-dual of the CRBC. We propose a CAF-based achievable secrecy rate region for the MAC-GF. Specializing our results to a Gaussian MAC-GF, we present numerical results which demonstrate that cooperation can improve secrecy for the MAC-GF. Next, we study the two-user one-eavesdropper discrete memoryless compound wiretap channel, and provide the best known lower bound for the secrecy capacity of this compound channel. We evaluate this achievable secrecy rate for the Gaussian MIMO case by using DPC. We show that this achievable secrecy rate achieves at least half of the secrecy capacity of this Gaussian MIMO compound wiretap channel, and also attains the secrecy capacity when the eavesdropper is degraded with respect to one of the two users. Then, we study the degraded compound multi-receiver wiretap channel (DCMRWC), which, in addition to a group of eavesdroppers, has two groups of users, namely the stronger group and the weaker group. We study two different communication scenarios for this channel. In the first scenario, there is only one eavesdropper, and the transmitter sends a confidential message to each group of legitimate users while keeping both messages secret from the eavesdropper. In the second scenario, we study the DCMRWC with layered messages without any restriction on the number of eavesdroppers. For both scenarios, we obtain the secrecy capacity region for the discrete memoryless channel, the parallel channel, and the Gaussian parallel channel. For the Gaussian MIMO channel, we obtain the secrecy capacity region when there is only one user in the second group. Next, we study the two-user fading broadcast channel and obtain its ergodic secrecy capacity region. We show that, thanks to fading, both users can have simultaneous secure communication with the transmitter, although this is not possible in the scalar non-fading Gaussian broadcast channel where only one user can have secure communication. This simultaneous secrecy of both users is achieved by an opportunistic communication scheme, in which, at each time instant, the transmitter communicates with the user having a better channel gain. Then, we study the secure lossy transmission of a vector Gaussian source to a legitimate user in the presence of an eavesdropper, where both the legitimate user and the eavesdropper have vector Gaussian side information. We obtain an outer bound for the rate, equivocation and distortion region. Moreover, we obtain the maximum equivocation at the eavesdropper when there is no constraint on the transmission rate. By using this maximum equivocation result, we show two facts. First, for this problem, in general, Wyner-Ziv scheme is suboptimal, although, it is optimal in the absence of an eavesdropper. And, second, even when there is no transmission rate constraint, an uncoded transmission scheme is suboptimal; the presence of an eavesdropper necessitates the use of a coded scheme to attain the maximum equivocation. Finally, we revisit the secure lossy source coding problem. In all works on this problem, either the equivocation of the source at the eavesdropper or the equivocation of the legitimate user's reconstruction of the source at the eavesdropper is used to measure secrecy. We first propose the relative equivocation of the source at the eavesdropper with respect to the legitimate user as a new secrecy measure. We argue that this new secrecy measure is the one that corresponds to the natural generalization of the equivocation in a wiretap channel to the context of secure lossy source coding. Under this new secrecy measure, we provide a single-letter description of the rate, relative equivocation and distortion region, as well as its specializations to degraded and reversely degraded cases. We investigate the relationships between the optimal scheme that attains this region and the Wyner-Ziv scheme.

Jan 1, 20120 citations

On information-theoretic secrecy for wireless networks

Etienne Perron, Suhas N. Diggavi, Emre Telatar

The aim of information-theoretic secrecy is to ensure that an eavesdropper who listens to the wireless transmission of a message can only collect an arbitrarily small number of information bits about this message. In contrast to cryptography, there are no assumptions on the computational power of the eavesdropper. Information-theoretically secret communication has been studied for many particular wireless network topologies. In the main part of this thesis, we consider such communication for arbitrary acyclic wireless network topologies. We provide lower and upper bounds on the strong perfect secrecy capacity for the case when the channels of the network are either Gaussian or deterministic. These results are based on the recent understanding of the capacity of wireless networks (without secrecy constraints) by Avestimehr, Diggavi and Tse. As a side result, we give inner and outer bounds on the capacity region for the multisource problem in arbitrary wireless networks with Gaussian or deterministic signal interaction. For linear deterministic signal interaction, we find the exact capacity region. For Gaussian signal interaction, we are able to bound the gap between the two bounds on the capacity region. This gap depends only on the network topology, but not on the signal-to-noise ratio (SNR), which leads to an approximation of the capacity region for the high SNR regime. We further consider a particular network topology, called the fan-network, in which we assume that an eavesdropper has physical access to every node in a subset of the relay nodes. We give a general upper bound on the perfect secrecy capacity, and we characterize the perfect secrecy capacity for two special cases. In the second part of the thesis, we consider interactive secrecy, i.e., secrecy in the presence of a public feedback link from the destination to the source. We focus on the problem of secret key generation rather than secret communication. The benefit of public discussion for secret key generation in a broadcast channel was first shown by Maurer. We extend his ideas to a relay network called the line network, leading to a lower bound on the strongly secret key capacity for this network topology. Finally, we introduce a new channel coding setup called the interference-multiple access (IMA) channel. This channel is a variant of the interference channel where one of the receivers is required to decode the messages from both transmitters. We derive an inner bound on the capacity region of the IMA channel, as well as an outer bound for the so-called structured IMA channel. In a semi-deterministic version of the structured IMA channel, the bounds match, providing a characterization of the capacity region. In the Gaussian case, we obtain a 1 bit-approximation of the capacity region. We also show an inner bound on the equivocation-capacity region for the IMA channel, where we require that part of the private message for one receiver is kept information-theoretically secret from the other receiver.

Jan 1, 201026 citations

Secret key establishment using wireless channels as common randomness in time-variant MIMO systems

Chan Chen

Encryption of confidential data with a secret key has become a widespread technique for securing wireless transmissions. However, existing key distribution methods that either deliver the secret key with a key distribution center or exchange the secret key using public-key cryptosystems are unable to establish perfect secret keys necessary for symmetric encryption techniques. This research considers secret key establishment, under the broad research area of information theoretic security, using the reciprocal wireless channel as common randomness for the extraction of perfect secret keys in multiple-input multiple-output (MIMO)communication systems. The presentation discusses the fundamental characteristics of the time-variant MIMO wireless channel and establishes a realistic channel simulation model useful for assessing key establishment algorithms. Computational examples show the accuracy and applicability of the model. The discussion then turns to an investigation of the influence of the spatial and temporal correlation of the channel coefficients on the bound of the key size generated from the common channel, and it is found that a sampling approach exists that can generate a key using the minimum sampling time. A practical key generation protocol is then developed based on an enhancement of a published channel coefficient quantization method that incorporates flexible quantization levels, public transmission of the correlation eigenvector matrix and low-density parity-check (LDPC) coding to improve key agreement. This investigation leads to the development of improved channel quantization techniques that dynamically shift the quantization boundaries at one node based on the information provided by the other node. Analysis based on a closed-form bound for the key error rate (KER) and simulations based on the channel model and measurement data show that the enhanced algorithms are able to dramatically reduce key mismatch and asymptotically approach the KER bound. Finally, other secret key generation algorithms based on channel-encryption rather than quantization are discussed, leading to a new concept for secret key generation using the common wireless channel.

Jan 1, 20103 citations

Opportunistic secret communication in wireless systems

Li, Zang

This thesis examines the challenges of information-theoretic secret communication that exploits the temporal and spatial variations of the wireless medium to improve secret communication rates. We first examine the secrecy capacity of a system consisting of independent parallel channels with one transmitter, one intended receiver and one eavesdropper. We show that the secrecy capacity of the system is the sum of the secrecy capacities of the individual subchannels. We further derive the optimal power allocation strategy for a system of parallel AWGN channels subject to a total power constraint, and also extend the results to random fading channels with additive Gaussian noise. We then study the achievable secrecy rate with Gaussian random codes for the situation where the channel of the intended receiver is a constant AWGN channel, while the eavesdropper's channel is fast Rayleigh fading with unknown realizations but known statistics to the transmitter. The proposed method with artificial noise and bursting provides ways to achieve positive secrecy rate even when Bob's channel is worse than Eve's channel on average. We also examine the achievable secrecy rate for a multiple antenna system, and the optimal input structure needed to achieve this rate. For the multiple input single output case, an analytical solution is derived. Multiple antenna systems provide extra degrees of freedom to the transmitter so that a beamforming-like approach can be used to provide advantage to the intended receiver against the eavesdropper. Next we derive a secrecy capacity outer bound region for a class of one-sided interference channels. The outer bound is shown to be tight for a class of binary deterministic one-sided interference channels, and can be achieved within one bit for some Gaussian one-sided interference channels.Finally, as Gaussian random codes are impractical, we evaluate achievable secrecy rates with discrete signaling. We observe that with discrete signaling, there exists an optimal power that maximizes the achievable secrecy rate. For the AWGN channel, larger constellation is always better. While for fading channel, the optimal constellation size varies with the power constraint, and discrete signaling can perform better than random Gaussian coding.

Jan 1, 20090 citations

Robust secret key generation for frequency-selective channels

Magnus Sandell

One effective way of generating and distributing secret keys for secure communication is to exploit the randomness of wireless channels. Since a potential eavesdropper has a different propagation channel than two legitimate users, it can be used to generate unique and secure keys. After reciprocal channel sounding, a frequency-selective channel can be quantised to produce the secret keys. Due to different sampling times at the two legitimate devices, the phase of the frequency response will not be equal and hence compromise the secret key generation. In this paper, we show how to overcome this problem and verify the results through both simulations and hardware measurements. We also compare scalar and vector quantisation, and show how the latter can be designed to be robust to synchronisation offsets.

IEEE Vehicular Technology ConferenceJun 1, 20221 citation