The Sound and the Fury: Hiding Communications in Noisy Wireless Networks with Interference Uncertainty

12/14/2017 ∙ by Zhihong Liu, et al. ∙ Xidian University 0

Covert communication can prevent the opponent from knowing that a wireless communication has occurred. In the additive white Gaussian noise channels, a square root law is obtained and the result shows that Alice can reliably and covertly transmit O(√(n)) bits to Bob in n channel uses. If additional "friendly" node near the adversary can produce artificial noise to aid in hiding the communication, the covert throughput can be improved, i.e., Alice can covertly transmit O({n,λ^α/2√(n)}) bits to Bob over n uses of the channel (λ is the density of friendly nodes and α is the path loss exponent). In this paper, we consider the covert communication in noisy wireless networks, where potential transmitters form a stationary Poisson point process. Alice wishes to communicate covertly to Bob without being detected by a warden Willie. In this scenario, Bob and Willie not only experience the ambient noise, but also the aggregated interference. Our results show that uncertainty in interference experienced by Willie is beneficial to Alice. When the distance between Alice and Willie d_a,w=ω(n^δ/4) (δ=2/α is stability exponent), Alice can reliably and covertly transmit O(_2√(n)) bits to Bob in n channel uses, and there is no limitation on the transmit power of transmitters. Although the covert throughput is lower than the square root law and the friendly jamming scheme, the spatial throughput is higher. From the network perspective, the communications are hidden in "the sound and the fury" of noisy wireless networks, and what Willie sees is merely a "shadow" wireless network where he knows for certain that some nodes are transmitting, but he cannot catch anyone red-handed.

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I Introduction

Traditional cryptography methods for network security can not solve all security problems. In wireless networks, if a user wishes to communicate covertly without being detected by other detectors, encryption to preventing eavesdropping is not enough [1]. Even if a message is encrypted, the metadata, such as network traffic pattern, can reveal some sensitive information [2]. Furthermore, if the adversary cannot detect the transmission, he has no chance to launch the “eavesdropping and decoding” attack even if he has boundless computing and storage capabilities. On other occasions, such as in a battlefield, soldiers hope to hide their tracks and communicate covertly. Another occasion, such as defeating “Panda-Hunter” attack [3], also needs to prevent the adversary from detecting the transmission behavior of users to protect the location privacy.

Consider the scenario where a transmitter Alice would like to communicate with a receiver Bob covertly over a wireless channel in order to not being detected by a warden Willie. In [4], Bash et al. found a square root law in additive white Gaussian noise (AWGN) channels, that is, Alice can transmit bits reliably and covertly to Bob over uses of wireless channels. The square root law implies pessimistically that the asymptotic privacy rate approaches zero. If Willie does not know the time of the transmission attempts of Alice, Alice can reliably transmit bits to Bob while keeping the Willie’s detector ineffective with a slotted AWGN channel model containing slots [5]. To improve the performance of covert communication, Lee et al. [6] found that, Willie has measurement uncertainty about its noise level due to the existence of SNR wall [7], then they obtained an asymptotic privacy rate which approaches a non-zero constant. Following Lee’s work, He et al. [8] defined new metrics to gauge the covertness of communication. They took the distribution of noise measurement uncertainty into consideration. Wang et al. [9] considered the covert communication over the discrete memoryless channels (DMC), and found that the privacy rate scales like the square root of the blocklength. Bloch et al. [10] discussed the covert communication problem from a resolvability perspective. He developed an alternative coding scheme such that, if the warden’s channel statistics are known, on the order of reliable covert bits may be transmitted to Bob over channel uses with only on the order of bits of secret key. Soltani et al. [11] studied the covert communications on renewal packet channels. They introduced some information-theoretic limits for covert communication over packet channels where the packet timings of legitimate users are governed by a Poisson point process.

Although the research on covert wireless communication focuses on the transmission capability, it is quite different from the works that measure the performance of wireless networks [12][13]. In general, the covertness of communication is due to the existence of noise that the adversary cannot accurately distinguish between the signal and noise. If we can increase the measurement uncertainty of the adversary, the performance of covert communication can be improved. Take the following occasion as an example,

“One day morning you walked in the woods. A lark with beautiful tail feathers was singing. You closed your eyes, listening …Although a little breeze was rustling and tumbling in the woods, you could still hear the sweet lark sing in the clear air of the day. All of a sudden, a crowd of larks flew here, you was drowned in the noisy twitters …You no longer knew whether the lark with beautiful tail feathers was still singing or not …”

Now the lark’s song is submerged in the interference and is difficult to be detected. Interference or jamming is usually considered harmful to wireless communications, but it is also a useful security tool. Cooperative jamming is regarded as a prevalent physical-layer security approach [14][15] in wireless communication environment. Jammers inject additional interferences when the transmitter sends messages in order to interfere the potential eavesdroppers [16][17][18][19]. Sobers et al. [20][21] employed cooperative jamming to obtain covert communication. To achieve the transmission of bits covertly to Bob over uses of the channel, they added a “jammer” to the environment to help Alice for security objectives. Soltani et al. [22][23] considered a network scenario where there are multiple “friendly” nodes that can generate interference to hide the transmission from multiple adversaries. They assumed that the friendly nodes are in collusion with Alice and can determine the closest node to each warden.

Fig. 1: System configuration of covert wireless communication. Alice wishes to transmit reliably and covertly to Bob. The interferers, or other transmitters (represented by black circles) are distributed according to a two-dimensional PPP in the presence of warden Willie (represented by red cross).

In this work, we consider a large-scale wireless network, where the locations of potential transmitters form a stationary Poisson point process (PPP), and their transmission decisions are made independently (as depicted in Fig. 1). In this scenario, Bob and Willie not only experience noise, but also interference signal from other transmitters simultaneously. Since the measure uncertainty of aggregated interference is greater than the background noise, the uncertainty of Willie will increase along with the increase of interference. Although the other transmitters do not collaborate with Alice, and Bob’s noise increases as well (multiuser interference cancellation technique [24] is not used), we find that the covert communication between Alice and Bob is still possible. Alice can reliably and covertly transmit bits to Bob in channel uses when the distance between Alice and Willie (

is the stability exponent). Although the covert throughput is lower than the square root law and the friendly jamming scheme, its spatial throughput is higher, and Alice does not presuppose the location knowledge of Willie. From the perspective of network, all transmitters in the network can achieve the same covert throughput with the same transmit power, and the larger transmit power level does not increase the probability of being detected since Willie will also experience a stronger interference. Willie cannot determine which node is transmitting except he can approach very close to a certain node (in this occasion the node will find Willie and stop transmitting). “The sound and the fury” of the noisy wireless channels make the network a “shadow” network to Willie.

Contributions. This paper makes the following contributions:

  1. We considered covert wireless communications in a network scenario, and established the bound on reliable covert bits that may be transmitted. We found that the random interference in a large-scale wireless network makes the network a “shadow network” to Willie, and can achieve a high spatial throughput.

  2. Leveraging on analysis and simulation results, we proposed practical methods to improve the performance of covert communications in noisy wireless networks.

The rest of the paper is structured as follows. We formulate the problem and system model in Section II. Next, we study the covert communication with interference uncertainty in Section III. We then present the discussions in Section IV and conclude our work in Section V.

Ii Problem Formulation and System Model

In this section, prior to presenting the system model, we give a running example to illustrate the problem of covert wireless communications discussed in this paper.

Ii-a Motivating Scenario

Covert communication has a very long history. It is always related with steganography [25] which conceals messages in audio, visual or textual content. However, steganography is an application layer communication technique and is not suitable in physical-layer covert communication. The well-known physical-layer covert communication is spread spectrum which is using to protect wireless communication from jamming and eavesdropping [26]. Another kind of covert communications is network covert channels [27][28] in computer networks. While steganography requires some form of content as cover, the network covert channels require network protocols as carrier. In this paper, we consider physical-layer covert communication that employs the background noise and the aggregated interference in wireless channels to hide transmission attempts.

Let us take the source location privacy protection in the Panda-Hunter game [3] as an example. In the Panda-Hunter Game, a sensor network with a large number of sensors has been deployed to monitor the habitat of pandas. As soon as a panda is observed by a sensor, this sensor will store the observation data, and then report the observations to a sink via multi-hop wireless channels. However, there is a hunter (the adversary Willie) in the network who tries to capture the panda. The hunter does not care the readings of sensors, what he really cares is the location of the message originator. To find the message originator near the panda, he listens to a sensor in his vicinity to determine whether this sensor is transmitting message. If he finds a transmitter, he then searches for the next sensor who is communicating with this transmitter. Via this method, he can trace back the routing path until he reaches the message originator and catches the panda. As a result, the source location information becomes critical and must be protected in this occasion.

To tackle this problem, Kamat et al. proposed phantom routing techniques to provide source-location privacy from the perspective of network routing [3]. Phantom routing techniques achieve location privacy by combining flooding and single-path routing together. From another point of view, the physical-layer covert communication can provide another kind of solution to the Panda-Hunter game. If we can hide the transmission from the hunter in noise and interference of the noisy wireless channels, the hunter will not able to determine which sensor is transmitting, and therefore cannot trace back to the source. What the hunter sees is a noisy and a shadow wireless network.

Ii-B Channel Model

Consider a wireless communication scene where Alice (A) wishes to transmit a message to the receiver Bob (B). Right next to them, a warden Willie (W) is eavesdropping over the wireless channel and trying to find whether or not Alice is transmitting.

We adopt the wireless channel model similar to [4][23], and throughout this paper we use the similar notations. We consider a time-slotted system where the time is divided into successive slots with equal duration. All wireless channels are assumed to suffer from discrete-time AWGN with real-valued symbols. Alice transmits real-valued symbols

. The receiver Bob observes the vector

, where , and is the noise Bob experiences which can be expressed as , where

are independent and identically distributed (i.i.d.) random variables (RVs) representing the background noise of Bob with

, and are i.i.d. RVs characterizing the aggregated interference from other transmitters in the wireless network.

As to Willie, he observes the vector , where , and is the noise Willie experiences which can be expressed as , where are i.i.d. RVs representing the background noise of Willie with , and are i.i.d. RVs characterizing the aggregated interference Willie experiences.

Suppose each node in the network is equipped with one antenna, and Bob and Willie experience the same background noise power, i.e., . Besides, different from the occasion discussed in [23], no location information of Willie and other transmitters is available in our model.

Ii-C Network Model

Consider a large-scale wireless network, where the locations of transmitters form a stationary Poisson point process (PPP)[29] on the plane . The density of the PPP is represented by , denoting the average number of transmitters per unit area. Suppose each potential transmitter has an associated receiver, the transmission decisions are made independently across transmitters and independent of their locations for each transmitter, and the transmission power employed for each node are constant power . Any other channel models with power control or threshold scheduling will have similar results with some scale factors. Suppose the wireless channel is modeled by large-scale fading with path loss exponent (). Let the Euclidean distance between node and node is denoted as . For simplicity, let the channel gain of channel between and is static over the signaling period, and all links experience unit mean Rayleigh fading. Then, the aggregated interference seen by Bob and Willie are the functional of the underlying PPP and the channel gain,

(1)
(2)

where each is a Gaussian random variable which represents the signal of the -th transmitter in -th channel use, and

(3)
(4)

are shot noise (SN) process, representing the powers of the interference that Bob and Willie experience, respectively. The Rayleigh fading assumption implies

is exponentially distributed with

.

The powers of aggregated interferences, and , are RVs which are determined by the randomness of the underlying PPP of transmitters and the fading of wireless channels. Therefore they are difficult to be predicted. Besides, the closed-form distribution of the interference is hard to obtain and we have to bound it.

Ii-D Hypothesis Testing

To find whether Alice is transmitting or not, Willie has to distinguish between the following two hypotheses,

(5)
(6)

Based on the received vector , Willie should make a decision on whether the received signal is noise+interference or signal plus noise+interference. We assume that Willie employs a radiometer as his detector, and does the following statistic test

(7)

where denotes Willie’s detection threshold and is the number of samples.

Let and be the events that the received signal of Willie is noise+interference and Alice’s signal plus noise+interference, respectively, then the probability of false alarm and missed detection can be denoted as and , respectively. Willie wishes to minimize his probability of error , but Alice’s ultimate objective is to guarantee that the average probability of error for an arbitrarily small positive .

First of all, Willie has to estimate the power level of noise+interference. The noise

not only comes from the thermal noise in his receiver but also the environmental noise from his surroundings. Besides, the aggregated interference he sees is a random variable which is determined by the randomness of the underlying PPP of transmitters and the channel gains. The only way for Willie to estimate the noise+interference level is to gather samples. However, he cannot determine definitely whether the samples he collected contain Alice’s transmission signal or not.

Besides, Alice should guarantee that the transmission is reliable, i.e., the desired receiver (Bob) can decode her message with arbitrarily low average probability of error at long block lengths. For any , Bob can achieve as .

In this paper, we use standard Big-, Little-, and Big- notations to describe bounds on asymptotic growth rates. The parameters and notation used in this paper are illustrated in Table I.

Transmit power
Number of channel use
Path loss exponent
Stability exponent
PPP of potential transmitters
Intensity of PPP
Alice’s signal in -th channel use
Signal of node in -th channel use
(Bob’s, Willie’s) background noise in -th channel use
Power of noise (Bob, Willie) observes
Interference (Bob, Willie) observes in -th channel use
Power of interference (Bob, Willie) observes
Power of noise plus interference (Bob, Willie) observes
Distance between and
Channel gain of channel between and
is exponentially distributed with
Gaussian distribution with mean

and variance

Probability of false alarm
Probability of missed detection
Mean of random variable
Variance of random variable
Outage probability for a typical receiver
Spatial throughput of successful transmissions
TABLE I: parameters and notation

Iii Covert Communication With Interference Uncertainty in Noisy Wireless Networks

In this section, we first present a theorem on the amount of information that can be transmitted covertly and reliably over AWGN channels in a noisy wireless network, then present its achievability and converse proof.

Theorem 1. Suppose a large-scale wireless network, where transmission decisions of nodes are made randomly, and the locations of transmitters form a PPP on the plane . When the distance between Alice and Willie , Alice can covertly and reliably transmit bits to Bob in channel uses in the case that ( is the stability exponent). Conversely, if the distance , and Alice attempts to send bits to Bob in channel uses, then, as , either Willie can detect her transmission with arbitrarily low probability of error , or Bob cannot decode Alice’s message with arbitrarily low error probability .

Iii-a Achievability

To transmit messages to Bob reliably, Alice should encode her messages. In this paper, we use the classical encoder scheme used in [4] and suppose that Alice and Bob have a shared secret of sufficient length. At first, Alice and Bob leverage the shared secret and random coding arguments to generate a secret codebook. Then Alice’s channel encoder takes as input message of length bits and encodes them into codewords of length at the rate of bits/symbol. Each codeword is a zero-mean Gaussian random where is the transmit power.

Iii-A1 Covertness

Alice’s objective is to hide her transmission attempts from being detected by Willie. If Willie’s probability of error for an arbitrarily small positive , then we can say that the covertness is satisfied.

Different from the cases studied in [4][23], Alice and Bob are located in a noisy wireless network. No location information of Willie and other potential transmitters is available, and Alice cannot collude with other “friendly” nodes. Willie not only experiences the background noise, but also the aggregated interference from other transmitters in the network. Therefore the power of noise and interference Willie experiences can be expressed as

(8)

where is the power of the background noise, is the power of the aggregated interference from other transmitters (defined in Equ. (4)). In general, the interference is more difficult to be predicted than the background noise, since the randomness of aggregated interference comes from the randomness of PPP and the fading channels, especially in a mobile wireless network.

Let

be the joint probability density function (PDF) of

when is true, be the joint PDF of when is true. Using the same analysis methods and the results from [4][23], if Willie employs the optimal hypothesis test to minimize his probability of detection error , then

(9)

where is the relative entropy between and , and the lower bound can be expressed as follows

(10)

The last step is due to , since in a dense and large-scale wireless network, the background noise is negligible compared to the aggregated interference from other transmitters [30]. Then the mean of is

(11)

for all links experience unit mean Rayleigh fading.

To estimate , we should have the closed-form expression of the distribution of . However, is an RV whose randomness originates from the random positions in PPP

and the fading channels. It obeys a stable distribution without closed-form expression for its PDF or cumulative distribution function (CDF). To address wireless network capacity, Weber

et al. [31] employed tools from stochastic geometry to obtain asymptotically tight bounds on the distribution of the signal-to-interference (SIR) level in a wireless network, yielding tight bounds on its complementary cumulative distribution function (CCDF). Next we leverage the bounds on CCDF to estimate the expectation .

Define a random variable

(12)

then, the lower bound on the CCDF of RV , , can be expressed as [31],

(13)

where , is the intensity of attempted transmissions in PPP , and . When , .

Therefore the upper bound of CDF of can be represented as

(14)

Next we can get the upper bound of CDF of as

(15)

where . For simplicity, we assume the channel gain of channel between Alice and Willie is static and constant, . Then can be denoted as .

Therefore the upper bound of PDF of can be represented as

(16)

where we set to normalize the function so that it describes a probability density.

Given the upper bound of PDF of , we can upper bound as follows

(17)

Thus, (11) and (17) yield the lower bound of as

(18)

Suppose for any , then we should set

(19)

Let , we have

(20)

Therefore, as long as , we can get for any . This implies that there is no limitation on the transmit power of Alice and other potential transmitters, the critical factor is the distance between Alice and Willie. This result is different from the works of Bash [4] and Soltani [23], in which Alice’s symbol power is a decreasing function of the codeword length . While this may appear counter-intuitive, the result in fact is explicable. We believe the reasons are two folds. First, higher transmission signal power will create larger interference which will make Willie more difficult to judge. Secondly, more close to the transmitter will give Willie more accurate estimation. This theoretical result is also verified using the experimental results in Section IV.

Iii-A2 Reliability

Next, we estimate Bob’s decoding error probability, denoted by . Let the noise power that Bob experiences be

(21)

where is the power of background noise Bob observes, is the power of the aggregated interference from other transmitters in the network. By utilizing the same approach in [4], Bob’s decoding error probability can be lower bounded as follows,

(22)

where the last step is obtained by the following inequality [23]

(23)

Hence the upper bound of Bob’s average decoding error probability can be estimated as follows

(24)

where is the upper bound of PDF of which obeys the similar distribution as (Equ. (16)),

(25)

where .

Although the interference Bob and Willie observe obey the similar distribution, they are correlative random variables. This is because the interference is caused by common randomness of the PPP [32]. When Bob is far away from Willie, the correlation between and is almost zero, which implies that the interferences seen by Bob and Willie are approximately independent. When Bob and Willie are very close to each other, they experience almost the same interference. In this occasion, and are approximately identical random variables.

Let , the path loss exponent , then . The Equ. (24) can be calculated as follows

As , for , when is large enough, we have

(27)

and

(28)
(29)
(30)

Therefore we have

(31)

where for .

Let for any , we have

(32)

which implies that Bob can receive

(33)

reliably in channel uses in the case that . This may be a pessimistic result at first glance since it is much lower than the bound derived in the work of Bash [4], i.e., Bob can reliably receive bits in channel uses. This is reasonable because Bob experiences not only the background noise but also the aggregated interference, resulting lower transmit throughput. However, in the work of Bash, Alice’s symbol power is a decreasing function of the codeword length , i.e., her average symbol power . When Bob use threshold-scheduling scheme to receive signal, Bob will have higher outage probability as . This is because Alice’s symbol power will become very lower to ensure the covertness as . If we hide communications in noisy wireless networks, the spatial throughput is higher than the work of Bash in which only background noise is considered. This will be discussed in Section IV.

Iii-B Converse

In this subsection we present the converse of the Theorem. Suppose Willie make a decision on whether the received signal includes Alice’s signal based on the received vector . He computes , and employs a radiometer as his detector to do the following statistical test with as his detection threshold,

(34)

where is the power of noise Willie experiences (defined in Equ. (8)), and we assume that Willie knows .

When is true, , where is the background noise, and represents the aggregated interference from other transmitters (defined in Equ. (2)). The transmitter sends codewords in the -th channel use. Willie observes

(35)

which contains readings of mean-shifted noise.

Next we estimate the mean and variance of . At first, we have to compute the mean and variance of . Because the RV , its mean and variance are 1 and 2, respectively. Hence,

(36)

yields . Given this, the mean of can be computed as

(37)

The last equation comes from the fact that where .

Because RVs and are uncorrelated random variables, the variance of can be computed in the same method as follows

(38)

and . Hence the variance of can be estimated as follows