Jamming is a well known adversarial attack on wireless communication , , , ,  wherein the attacker overpowers the communication between a transmitter and a receiver by injecting high-powered noise signals. Standard ways to mitigate jamming include frequency-hopping (FH) ,  and direct sequence spread spectrum (DSSS) schemes . In the case of DSSS, a narrowband signal is spread across a wide band of frequencies by using a spreading code so that an attacker, which does not possess the spreading code, will have its jamming signal rejected by the receiver. In the case of FH, which is the subject matter of this paper, the transmitter and the receiver synchronously hop across several carrier-frequencies so that the hopping pattern appears non-deterministic to the adversary. As a result, narrowband jamming, i.e., jamming a specific carrier-frequency, cannot guarantee performance degradation due to randomness in the hopping pattern. On the other hand, with wideband jamming, i.e., jamming all the carrier-frequencies, the effective noise power injected on each carrier-frequency would be too weak to induce significant degradation in the performance. Both DSSS and FH are effective under the assumption that the attacker is power-constrained. Citing these benefits, DSSS and FH have found extensive applications in military communication systems, and recently in many cyber-physical systems. While DSSS and FH introduce randomness in the choice of the spreading code and carrier-frequencies, respectively, introducing randomness over spatial orientation of the antennas has also been explored as a viable anti-jamming technique in communication involving highly-directional antennas .
With wireless communication being an integral part of most cyber-physical systems, e.g. urban transportation, smart-grid and other IOT systems , it is imperative to envision new attacks  on such systems and provide suitable countermeasures against them. Over the past decade, wireless communication technology has witnessed enormous progress in bandwidth-efficient physical-layer techniques that have helped wireless devices achieve high data-rate. One of the prominent areas rising in this space is full-duplex communication , , , wherein a radio device can simultaneously transmit and receive signals in the same frequency band. While efficient hardware- and software-architectures have helped full-duplex radios to achieve near-perfect self-interference cancellation, there have been concurrent developments in hardware implementation for low-latency processing of radio frequency (RF) signals  in the field of systems security. Aggregating the latest developments in the above areas, we believe that next-generation cyber-physical systems ought to assume strong attack models that employ state-of-the-art wireless techniques.
In this paper, we are interested in threat models arising out of full-duplex radios that operate as hidden relays between a transmitter and a receiver, as depicted in Fig. 1. Loosely speaking, this threat comes under the well known framework of man-in-the-middle attacks, wherein the attacker can manipulate the transmitted symbols before they reach the legitimate receiver. Although instantaneous modification of transmitted symbols has been addressed to mitigate interference in wireless networks [12, 13, 22], such ideas have not been studied as a threat to wireless security. An important question to answer along that direction is: Are the current-day wireless systems resilient to instantaneous manipulation of transmitted symbols in the air? From the standpoint of practicality, major challenges to instantaneous modification include (i) additional processing-delay and (ii) additional path-delay, introduced by the attacker. The processing-delay constraint restricts the attacker not to make dynamic decisions based on baseband signal processing of the received signals. On the other hand, the path-delay constraint resulting from the attacker’s position restricts it to be appropriately placed so that the forwarded components reach the receiver well within the delay of one symbol-period relative to the signals received from the main path. In a nutshell, if the above two constraints are respected, then, in principle, it is possible for the modified signals to arrive at the legitimate receiver within the symbol-period. We refer to such an attacker as Cognitive Radio from Hell (CRFH). The proposed adversarial model comes under the class of correlated jamming  wherein the jammer has full or partial information about the transmitter’s signals.
I-B CRFH Attacks on Frequency-Hopping
Our discussion in the preceding section indicates that the processing-delay and the path-delay constraints may preclude the attacker from executing instantaneous modification on wideband communication due to small symbol-periods. However, on narrowband communication systems, such as FH (wherein the bandwidth around the chosen carrier-frequency is small), the attacker can potentially execute instantaneous modification “in the air” due to relatively large symbol-periods compared to the path-delay on the main path. This has motivated us to study the effect of CRFH attacks on FH systems. Particularly, in the context of FH, we note that instantaneous modification of symbols is crucial to degrade the error-performance at Bob, otherwise, any unintentional delay introduced by the attacker, will result in a delay of one symbol-period or more. This allows the legitimate receiver to evade the attack by hopping to the next carrier-frequency before the delayed components arrive. Thus, to enforce degraded performance from instantaneous modification, the CRFH attacker on FH must respect the delay constraints on instantaneous modification.
I-C Related Work
To help translate the idea of CRFH attack to practice, recent advances in the field of full-duplex radios [15, 16, 17, 18, 19, 20] have shown that radios can be designed to cancel their self-interference while instantaneously forwarding the transmitted signals. In particular,  has showcased the possibility of building radios with the capability of instantaneous processing and relaying. Recently,  has demonstrated the effectiveness of instantaneous modification by full-duplex radios to achieve co-existence in interference channels. Also,  has showed that radio-frequency signals can be processed and retransmitted in the analogue domain with a delay of few nano-seconds.
The notion of modifying symbols in the air is also known under the framework of reactive jamming , which refers to the process of targeting selected packets in the air as it allows the attacker to destroy specific packets and yet go undetected. The authors of  have studied the feasibility of reactive jamming by designing and implementing a reactive jammer against 802.15.4 networks. Through the use of Universal Software Radio Peripherals (USRPs),  has demonstrated jamming attack with reaction time of the order of microseconds in indoor environments. In , the authors have addressed reactive jamming attacks where the adversary is capable of picking packets based on real-time classification of packets at the physical-layer. They have also proposed countermeasures to prevent real-time packet classification based on both cryptographic as well as physical-layer ideas. In , the authors have proposed a technique to detect reactive jammers on DSSS. The basic idea is to use statistics on attack-free packets and then identify packets attacked from those lost due to bad channel condition. In , reactive jamming on Orthogonal Frequency Division Multiplexing (OFDM) is considered, and an effective countermeasure based on Multiple-Input Multiple-Output (MIMO) systems has been proposed. Overall, inspired by the above works, particularly that of  and , we believe that it is imperative for existing cyber-physical systems to envision attacks that could arise out of full-duplex radios capable of instantaneous modification of transmitted symbols.
The contributions of this paper are summarized below:
We introduce a new adversarial attack, referred to as the convolution attack (CA), on FH based wireless communication. In this attack, the adversary, which is strategically positioned between the transmitter and the receiver, instantaneously multiplies the received passband signal by a random baseband signal, and then forwards it to the receiver within the symbol-period. Subsequently, the forwarded signals will combine with the signals directly received from the transmitter, thereby modifying the information symbols in the air. One of the highlights of the attack is that the attacker is able to perturb the transmitted symbols despite not knowing the active narrowband of communication. We show that the proposed attack forces the legitimate receiver to incorporate the forwarded signals in the decoding process; otherwise, discarding them as noise is shown to result in consequences akin to jamming. (see Section III). We also show that the CA forces the equivalent channel between the legitimate users to experience frequency-selectivity and rapid-fading, in such a way that neither pilot-based coherent detection techniques, nor traditional non-coherent and differential encoding/decoding detection techniques can mitigate the attack.
As a countermeasure against CA, we study the performance of an FH system with non-coherent On-Off Keying (OOK) as the underlying modulation scheme (see Section IV). This mitigation strategy, although a traditional communication scheme, is tailor-made to handle the threat model because switching-off the transmission forbids the attacker from perturbing the communication, while switching-on the transmission helps the receiver to collect energy despite the attack. With large number of antennas at the receiver, we show that the receiver can opportunistically use the attacker’s signal to its advantage to gather more energy for detection (see Section IV-A). We show that OOK is an effective countermeasure if the attacker executes the convolution attack on both the pilot symbols and the data symbols with the same attack parameters; this is because the threshold for energy detection is computed using the received energy distributions on the pilot symbols. However, if the attacker decides to selectively attack only the data symbols and not the pilots, then OOK is no longer an effective countermeasure.
As a second countermeasure against CA, we study the performance of non-coherent Binary Frequency Shift Keying (BFSK), a widely used modulation scheme with FH in military application. In this form of CA, the adversary instantaneously modifies the signal “in the air” so that the receiver witnesses comparable energy levels on the tones carrying bit- and bit-. As a result, this attack significantly degrades the error performance at the receiver when it uses threshold-based energy detection (see Section V). One the fundamental causes for this attack is that although the carrier-frequency is randomly hopped based on a shared secret-key between the legitimate users, the locations of the tones carrying bit- and bit- are deterministic upto one bit randomness when the attacker observes the signal in the air. We first show that this form of CA, when appropriately executed, introduces error-floor behaviour in the error performance. Subsequently, we propose a mitigation strategy, referred to as Enhanced BFSK, wherein unlike the standard frequency-hopping technique, the tones carrying bit- and bit- are also randomized based on an additional secret-key shared between the legitimate users. As a consequence, upon observing the transmitted signal in the air, the attacker continues to have uncertainty about the location of tone carrying the complementary bit, thereby forcing it to execute wideband jamming. Unlike the case of OOK, we show that BFSK is resilient even if the attacker selectively executes the CA on the data symbols and not on the pilots; this is because BFSK detection does not rely on the distribution of received energy on the pilots.
Henceforth, throughout the paper, we refer to the transmitter, the receiver, and the attacker as Alice, Bob and Eve, respectively. To model a power-constrained attacker, we assume that Eve has times more energy than Alice, i.e., , for some . This implies that as Alice increases her energy to improve the performance, Eve can also proportionately increase her energy. Furthermore, out of , Eve may use only a fraction of it, say , for some , on the CA. Thus, the key attack parameters of this paper are and .
Ii Convolution Attack on Frequency-Hopping
Consider an FH based amplitude-modulated communication scheme between Alice and Bob, wherein the carrier-frequency of the transmitted narrowband signal is randomly chosen from one of the tones, denoted by the set . Let Alice use a root raised cosine (RRC) waveform , denoted by , as the baseband signal of bandwidth Hz and symbol rate seconds. Furthermore, let , with , denote the sequence of complex symbols, where takes value from a -dimensional finite complex constellation, e.g. quadrature amplitude modulation. The corresponding train of baseband signals is given by
where is the average transmit energy by Alice assuming that and are appropriately normalized. After modulating on carrier-frequency , the transmitted passband signal is of the form
where denotes the real part of a complex number. The set is chosen such that , for , thereby leaving sufficient guard-band to mitigate inter-carrier interference. Alice employs a carrier-frequency for seconds, for some integer , before hopping to another value in . Meanwhile, Bob synchronously hops across the same sequence of carrier-frequencies every seconds, as the hopping pattern is generated using a shared secret key. Throughout the paper, we use small values of by assuming that Alice and Bob are capable of quickly switching the carrier-frequencies with minimal losses due to transients in the transmit and receive RRC filters. Otherwise, with , a sophisticated attacker can sense the narrowband of communication and subsequently inject jamming energy on the detected band, akin to traditional jamming. Thus, small values of helps the legitimate users to mitigate standard jamming attacks on frequency-hopping. The discrete-time version of the received baseband signal at Bob is given by
for , where is the complex channel gain on the tone , and is the additive noise at Bob, distributed as
. A complex random variable
is said to be circularly symmetric Gaussian distributed when the real and imaginary components ofare Gaussian and i.i.d. with mean
and variance. We assume that the channel remains fixed within the hopping interval, i.e., . For brevity, henceforth, we drop the reference to the carrier-frequency from the channel model in (3).
We assume that Eve is positioned somewhere between Alice and Bob, and she is not aware of the secret key used to generate the hopping pattern. At any point in time, due to the non-deterministic hopping pattern, Eve cannot successfully guess the active
carrier-frequency with probability one, and therefore narrowband jamming on a random carrier-frequency indoes not guarantee degraded error performance.
Keeping in view practical hurdles in executing wideband jamming by a power-constrained attacker, we envision a new attack, referred to as the convolution attack, which is as depicted in Fig. 1. We assume that Eve is capable of receiving and transmitting signals in the entire wideband covering all the carrier-frequencies. Through the CA, we show that Eve can modify the transmitted narrowband signal despite not knowing the active carrier-frequency. We draw our inspiration from the fact that analogue processing of narrowband signals is feasible in negligible amount of time . In the proposed attack, Eve multiples the received passband signal with a random baseband signal, denoted by , in the analogue domain, and then forwards it to Bob. This multiplication operation in the time-domain is equivalent to convolving the Fourier transform of the received signal with that of the random signal. Because of this operation, Eve can modify the narrowband signal without knowing the carrier-frequency. Subsequently, this modified version of the signal is added to the signal arriving directly from Alice, thus corrupting the overall received signal in the narrowband of interest. We make the following assumptions for executing the CA: (i) Negligible processing-delay at Eve, (ii) Negligible path-delay through Eve, and (iii) Full-duplex architecture with perfect cancellation at Eve.
In the next subsection, we mathematically describe how the forwarded signals from Eve affect the received signal at Bob.
Ii-a Signal Model with Convolution Attack
Since Eve does not know the active carrier-frequency, she receives signals in the entire band, covering all the carrier-frequencies. Specifically, the received signal is given by
where denotes the -th multipath component from Alice to Eve, and and , respectively denote the corresponding amplitude and delay associated with the multipath component. Once Alice and Bob are locked onto a carrier-frequency, we assume that the rest of the narrowbands are unused by other users in the network, and therefore, the non-signal component of constitutes only the additive noise at Eve. In general, when the narrowbands are shared among several users in the network, the received signal also constitutes interference from other users. Although Eve receives over a wide band of frequencies, we assume that the channel from Alice to Eve is frequency-flat over the active narrowband. Upon receiving , Eve multiplies it by a real random signal (of unit average-energy over the symbol-period), and then transmits
where is the gain introduced by Eve for some and . The product operation can be viewed as a way of introducing Doppler shifts to the passband signal by various frequency components of . With transmitted from Eve, the received signal at Bob is given by
where the first part is contributed by Eve, the second part comes directly from Alice, and the last part is the ambient noise generated at Bob’s receiver. In (5), and , respectively denote the amplitude and the delay associated with the -th multipath component from Eve to Bob. Similarly, and , respectively denote the amplitude and the delay associated with the -th multipath component from Alice to Bob. Also, note that is the processing-delay introduced by Eve when multiplying the two signals. Among the multipath components from Alice to Bob, let denote the first significant multipath component. Similarly, among the multipath components from Alice to Eve, and Eve to Bob, let and denote the first significant multipath components, respectively. In the proposed attack, Eve positions herself such that the following condition on delay is satisfied:
If the timing constraint in (6) is satisfied, then it is straightforward to verify that Eve’s signal can modify the current symbol in the air. After downconverting the received signal from the carrier-frequency , and then sampling and filtering, we obtain the discrete-time version of the baseband received signal, given by
for , where are the complex channels contributed by Eve, is the noise component forwarded by Eve, and is the additive noise at Bob. The channel contributed by Eve is possibly frequency-selective, where the number of taps of the channel, denoted by , depends on the chosen waveform . Intuitively, as depicted in Fig. 1, although the channel from Alice to Eve, and Eve to Bob are frequency-flat within a narrowband of Hz, the convolution operation in the frequency domain can disrupt the frequency-flat structure, thereby giving rise to a frequency-selective channel.
Observe that Eve is not injecting noise into the narrowband of interest, instead she is instantaneously modifying the transmitted symbols by a random quantity, which is some complex function of (i) the channel from Alice to Eve, (ii) the signal , and (iii) the channel from Eve to Bob. If the timing constraint in (6) is not satisfied, then the signal forwarded by the attacker does not modify the current symbol in the air, instead it reaches Bob in the subsequent symbol-periods. This implies that in (II-A). Although this form of attack continues to affect the signal-to-noise-ratio of subsequent symbols, the current symbol in the air does not get modified. A straightforward way for Alice and Bob to evade this attack is by locking to a given carrier-frequency for just one symbol before hopping to another carrier-frequency in . Thus, satisfying the timing constraint in (6) is crucial for Eve to execute the CA when the legitimate users have the potential to hop carrier-frequencies with .
Iii Challenges in Mitigating Convolution Attack
Without the attack, i.e., , the complex channel is determined only by the environment. Importantly, the coherence-time of the channel is determined only by the relative velocity of the surrounding objects in the environment. However, with attack, an additional signal component is added to the received signal at Bob as shown in (II-A). A naive way to handle this additional term is by considering it as noise. However, this will naturally lower the signal-to-noise-ratio (SINR), and degrade the error performance when Eve’s power is dominant. Instead, since the additional component contains useful information, it is prudent for Bob to treat it as the signal term in the decoding process. After incorporating Eve’s signals in the decoding process, Bob is forced to view an equivalent channel model, given by
Although Eve is contributing additional signal power into the system, Bob is unsure of how to use this additional power as it may rapidly change every symbol. We now summarize the major changes introduced in the channel model when Eve executes CA with significant power compared to that at Alice: (i) Since can be arbitrarily chosen by Eve, the equivalent channel can be frequency-selective despite using narrowband for communication. (ii) Unlike in traditional channels, the delay-spread of the equivalent frequency-selective channel may change each symbol since could be composed of arbitrary segment of signals every seconds, and finally, (iii) the coherence-time of the equivalent channel can also be controlled by Eve, to the extent that the channel seen across two successive symbols can be uncorrelated. It is worth emphasizing that Eve is able to force abrupt variations in two fundamental characteristics of the channel, namely: frequency-selectivity and Doppler-spread. To bring in these variations, it is necessary for Eve to spend significant power compared to Alice, otherwise the characteristics of the true wireless channel will continue to dominate, and as a result the attack will be ineffective.
From the model in (8), it seems that Alice and Bob can circumvent the CA by employing encoding and decoding mechanisms that do not rely on the knowledge of channel state information (CSI), such as differential-encoding methods and blind detection techniques [29, 30]. However, these methods work under the assumption that some statistics of the channel remain constant for several blocks, and are also known at the receiver. In the case of CA, these techniques are not directly applicable as is completely controlled by Eve. In the case of frequency-selective equivalent channel, the delayed components are contributed only by Eve as the main channel is frequency-flat due to the narrowband assumption. A straightforward way to handle frequency-selectivity is by using OFDM as the modulation scheme. However, the idea of OFDM modulation requires the channel realizations to be fixed for at least one OFDM symbol, and this assumption can also be violated by Eve. Therefore, OFDM is not applicable in this attack scenario.
Iii-a Impact of Convolution Attack
To showcase the impact of CA, we consider a Binary Phase Shift Keying (BPSK) signalling scheme at Alice aided by coherent maximum-likelihood detection at Bob. We present the error-performance of this scheme under the following attacks: (i) Narrowband jamming (NJ): Eve executes narrowband jamming by injecting noise of energy , with , on one of the
bands with uniform distribution, (ii)Wideband Jamming (WJ): Eve executes wideband jamming by uniformly dividing its energy , with , across the narrowbands, and (iii) Convolution attack (CA): Eve executes CA to result in a rapid-fading frequency-flat equivalent channel, with and . The equivalent channel on each carrier, denoted by , changes rapidly to force error-floor behaviour on coherent maximum-likelihood detection. For the experiments, we use and . In Fig. 2, we plot the bit-error-rate (BER) curves of the above schemes against when Bob is equipped with and receive antennas. The plots show that neither narrowband jamming nor wideband jamming is effective in degrading the error-performance at Bob, whereas CA can force severe BER degradation at an attack-ignorant Bob. Thus, even with large values of , it is important for Alice and Bob to identify CA, and then mitigate it by employing an appropriate countermeasure.
Iv Convolution Attack on FH Based On-Off Keying
We study an FH system with non-coherent On-Off Keying (OOK) as the modulation scheme. In this strategy, Alice communicates bit- by transmitting a signal of energy (referred to as ON state), and bit- by switching-off the communication (referred to as OFF state). For exposition, let denote the bit transmitted at the -th time instant. To communicate , Alice encodes it as
before transmitting on the carrier-frequency . From the nature of the attack, it is clear that Eve forwards only the noise component in the active narrowband when Alice switches-off her transmitter. However, when Alice transmits bit-, Eve forwards significant signal power in the active narrowband. With this signal design, Bob can distinguish bit- and bit- by measuring the received energy on each symbol without the knowledge of the channel. In the rest of this paper, we assume that Eve uses which results in a frequency-flat equivalent channel at Bob. For the frequency-selective case, Alice and Bob may handle it by locking onto a carrier-frequency only for one symbol, i.e., , so that Bob may continue to listen to the preceding set of carrier-frequencies for the delayed components. Our inferences on the attack-strategies and countermeasures are only based on the frequency-flat equivalent channel model. Since the users can handle frequency-selectivity by hopping across the carrier-frequencies for one symbol, we do not expect significant deviations in the inferences with the frequency-selective case.
Applying OOK on the frequency-flat model, the received symbol at Bob is given in (9) (top of this page), where is the gain applied by Eve on its received signal. Based on the nature of operations at Eve, we model the complex channel as , where is the channel from Alice to Eve, distributed as , is the channel from Eve to Bob, distributed as , and is a complex random variable of mean zero and unit variance obtained from the waveform . The forwarded additive noise from Eve is , where is distributed as . Henceforth, we denote as , which is the additional signal energy contributed by Eve through CA.
At the receiver side, Bob decodes to an estimate of, denoted by , based on the following rule:
where is an appropriately designed threshold chosen based on the noise component in (9). One of the challenges in designing OOK against the CA is the derivation of the threshold , as fading characteristics of and have to be considered. We address the choice of in Section IV-C.
When decoding OOK, Bob faces two types of error events: (i) when , and (ii) when . While the former event may occur when the threshold is lower than the noise components jointly contributed by Eve and Bob, the latter event captures the case when Eve attempts to force the effective channel to deep fade, i.e., . We represent the associated probability as . In the following section, we propose a mitigation strategy by Bob to reduce .
Iv-a Mitigation Strategy: Large Number of Receive Antennas
In the case of CA, since is completely controlled by Eve, the distribution of the equivalent channel can be changed to affect provided . However, on the defense-side, since the two users hop across a wide range of narrowbands, Eve cannot learn the narrowband, and therefore, she cannot drive the equivalent channel to deep fade with probability one. As a defense mechanism to counter Eve’s strategy, Bob should collect energy from as many independent paths as possible. One such bandwidth-efficient way is to employ multiple receive antennas at Bob. This way, the probability that Eve can drive all the independent channels simultaneously to deep fade can be reduced. If we use to denote the number of receive antennas at Bob, without additive-noise at Eve and Bob, the total signal energy collected across antennas is given by
where and denote the equivalent channels seen by the -th antenna of Bob on the -th symbol. In the event of no attack, we have , and , given by
We start by expanding as in (14), where is the energy accumulated at Bob without the attack. As a result, can be written as in (15). Since is strictly non-negative, the probability in (15) is lower-bounded by (16). This is because we are only considering the events when
). Finally, applying weak law of large numbers[32, Chapter 3] on (17), we get (18) for sufficiently large . This completes the proof.
With massive MIMO in contention for next-generation networks (e.g. 5G), base-stations equipped with hundreds of antennas are likely to be deployed in practice . This implies that Proposition 1 is useful when base-station plays the role of Bob and a UE (user-equipment) plays the role of Alice.
While the above proposition shows the advantage of employing large number of receive antennas to combat the CA, in the rest of this section, we present numerical results to understand the cumulative distribution function (CDF) ofwhen is not large. To generate the numerical results, we assume that the channels and are i.i.d., and are distributed as . We also assume that is distributed as . When the transmitted bit is , let be the average received energy at every antenna of Bob, out of which, be the signal energy contributed by Eve. We define
as the percentage of average energy contributed by Eve when the transmitted bit is . In Fig. 3, we plot the CDFs of the random variable when is Gaussian distributed. For computing the CDFs, we use . The plots in Fig. 3 show that as increases, the CDFs shift towards right, thereby driving the cross-over probability to lower values.
Iv-B Effect of Multiple antennas at Eve
We acknowledge that Bob’s trick to garner energy for detection comes from using multiple antennas. To keep the comparison fair, we study the effect of CA when Eve is also equipped with multiple antennas. Considering , the total energy at Bob without additive noise at Alice and Bob is given by
where denotes the number of antennas at Eve, , which is distributed as , is the scalar used at the -th antenna of Eve, is the channel from the -th antenna at Eve to Bob, and is the channel from Alice to the -th antenna at Eve. In the case of single antenna at Eve, the energy at Bob is
with and with . With , since (26) is the sum of product of three independent Gaussian random variables, we have observed that the CDF of grows much slower than that of , as shown in Fig. 4. As a result, for a given , the probability of decoding bit- as bit- decreases when multiple antennas are used at Eve.
Furthermore, we compute the CDFs of the received energy across the antennas at Bob when and Eve uses the following strategies: (i) single-antenna, (ii) multiple-antenna with spatially randomized waveforms - are independent, and (iii) multiple-antenna with spatially fixed waveforms - . To give advantage to Eve, we have also considered the case when . The CDFs, which are presented in Fig. 5, highlight that employing multiple antennas at Eve does not aggravate the attack effect as multiple antennas assists Bob in receiving more energy than the single-antenna case. Due to lack of closed-form expressions on the CDFs of energy collected at Bob, we do not have concrete theoretical insights on this argument. Nevertheless, based on the simulation results, we advocate the use of single antenna at Eve and multiple antennas at Bob. In Section IV-D, we also present the BER performance of OOK with and without multiple antennas at Eve to reinforce this observation.
Iv-C Design of Threshold
Having studied the energy distributions during the ON state of OOK, we now address the computation of in (11) to optimize the error-performance at Bob. With CA, the signal energy collected across antennas during the ON state is given by (21) (see the top of the next page), where and denote the equivalent channels seen by the -th antenna of Bob on the -th symbol. Similarly, energy collected during the OFF state is
To determine the optimal threshold we need to solve
which in turn requires Bob to measure the Probability Density Functions (PDFs) onand . Towards that direction, we assume that Bob can learn the distributions empirically using pilots, which are periodically transmitted by Alice. Note that the persistent nature of the CRFH attack helps in measuring the energy distributions with attack, otherwise, Bob is forced to employ threshold values based on the energy distribution of , which in turn degrades the error-performance under convolution attack. From and , we observe that are uncorrelated but not necessarily independent. Similarly, the random variables are also uncorrelated but not independent. Due to challenges in obtaining the closed-form expressions on the PDFs of and , we approximate to be statistically independent and Gaussian distributed as , and then arrive at a sub-optimal solution. Similarly, is also assumed i.i.d., where each is distributed as . Using such approximations, the corresponding versions of received energy are given by (22) and (23), where and are Gaussian distributed. We immediately note that can be written as
where is Chi-square distributed with degrees of freedom . Similarly, can be written as
where is also a Chi-square distributed random variable with degrees of freedom . With this, the approximate solution, henceforth denoted as , is computed as in (29),
Iv-D Error-Performance of OOK Against Convolution Attack
In this section, we present simulation results on the error-performance of OOK against the CA. To carry out the experiments, we assume that the channels across the narrowbands are statistically independent and distributed as . Similarly, the sets of channels and are also i.i.d. across the narrowbands, and are distributed as .
To showcase the effect of CA, we present the error-performance of the non-coherent OOK scheme along with the schemes discussed in Section III-A, namely, (i) Narrowband jamming (NJ), and (ii) the CA, on binary phase shift keying (BPSK) with coherent maximum-likelihood detection at Bob. In Fig. 6, we plot the BER curves of the above schemes against for and . For CA on OOK, we use two different threshold values for energy detection, namely: in (28), and in (29), which are computed based on the attack parameters. The plots show that the Gaussian approximation to compute does not result in significant loss in the error-performance. Moreover, the error-performance of OOK is better than that of coherent modulation method under the CA. Similar to the results in Fig. 6, we also present the BER curves of OOK with in Fig. 7. The plots highlight that it is important for Alice and Bob to identify the CA, and then mitigate it by employing OOK based strategy.
Finally, in Fig. 8, we present the error-performance of OOK when Eve is equipped with multiple antennas, and when and . Similar to the observations in Section IV-B, Fig. 8 confirms that multiple antennas at Eve does not aggravate the attack effect. For the simulations, the threshold values for energy detection are computed based on the energy distribution during the ON and the OFF states similar to the one in (28).
The error-performance of OOK, as presented in Fig. 6 and Fig. 7, captures the best-case results from the perspective of Alice and Bob. This is attributed to the assumption that Eve executes the convolution attack persistently on both the pilot symbols and the data symbols with the same parameters and . However, when Eve selectively attacks only the data symbols, then the corresponding estimate of the threshold will be suboptimal, which in turn will result in degraded performance.
Iv-E Limitations of OOK against Wideband Jamming
In this section, we explore the idea of changing Eve’s strategy to wideband jamming (WJ) once Alice and Bob switch to OOK in response to CA. In WJ, Eve uniformly divides her energy across the narrowbands. The rationale behind this switch is to exploit lower threshold values used for energy detection, thereby forcing Bob to decode bit- as bit-. We first capture the consequence of an attack-ignorant detection in Fig. 9 (left-side), which shows the BER performance of OOK when is optimized based on and . Since is chosen based on , and is much lower than , BER increases with large values of jamming energy; this is mainly contributed by the error event of decoding bit- as bit-. In the attack-aware case, Bob measures the jamming energy using the pilots, and then takes it into account when designing (this is possible due to the persistent nature of the attack). The error-performance of such a strategy is also presented in Fig. 9 (right-side), which shows that unlike the case of attack-ignorant detection, the BER experiences error-floor behaviour when is large; this is because the threshold value linearly increases with , thereby saturating the probability of decoding bit- as bit-, and vice versa. In summary, although OOK mitigates CA, a combination of CA followed by WJ can result in degraded error-performance at Bob when is large.
V Convolution Attack on FH based Frequency Shift Keying
In this section, we study the impact of CA on Binary-FSK (BFSK) based FH scheme as an alternate countermeasure. We have chosen BFSK as the modulation scheme as most military and commercial frequency hopping systems use frequency shift keying. Unlike the generic attack in Section II, the objective of CRFH in this case is to create confusion at Bob when decoding the BFSK modulated symbols. In this attack, Eve uses an appropriate baseband signal to forward a frequency-shifted version of the received passband signal so that the tones carrying bit- and bit- have comparable energy levels at Bob.
V-a Signal Model for BFSK without Attack
At Alice, bit- is transmitted by using the carrier-frequency , and bit- is transmitted by using the carrier-frequency , for some (given in Section II). We assume that , where is the spacing between adjacent carrier-frequencies. We use to denote the bit transmitted at the -th symbol-period, and to denote the complement of . To communicate , Alice transmits the tone , given by
where is chosen based on the shared secret-key between Alice and Bob. Overall, the total set of tones used by Alice and Bob is . In this model, we assume and .
Without any attack, the received complex-baseband symbols at Bob are of the form
where and are the symbols received on the tones and depending on . When bit- is transmitted, and correspond to the symbols on the tones and , respectively. Similarly, when bit- is transmitted, and