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An overview of generic tools for informationtheoretic secrecy performance analysis over wiretap fading channels
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 194 (2021)
Abstract
Physical layer security (PLS) has been proposed to afford an extra layer of security on top of the conventional cryptographic techniques. Unlike the conventional complexitybased cryptographic techniques at the upper layers, physical layer security exploits the characteristics of wireless channels, e.g., fading, noise, interference, etc., to enhance wireless security. It is proved that secure transmission can benefit from fading channels. Accordingly, numerous researchers have explored what fading can offer for physical layer security, especially the investigation of physical layer security over wiretap fading channels. Therefore, this paper aims at reviewing the existing and ongoing research works on this topic. More specifically, we present a classification of research works in terms of the four categories of fading models: (i) smallscale, (ii) largescale, (iii) composite, and (iv) cascaded. To elaborate these fading models with a generic and flexible tool, three promising candidates, including the mixture gamma (MG), mixture of Gaussian (MoG), and Fox’s Hfunction distributions, are comprehensively examined and compared. Their advantages and limitations are further demonstrated via security performance metrics, which are designed as vivid indicators to measure how perfect secrecy is ensured. Two clusters of secrecy metrics, namely (i) secrecy outage probability (SOP), and the lower bound of SOP; and (ii) the probability of nonzero secrecy capacity (PNZ), the intercept probability, average secrecy capacity (ASC), and ergodic secrecy capacity, are displayed and, respectively, deployed in passive and active eavesdropping scenarios. Apart from those, revisiting the secrecy enhancement techniques based on Wyner’s wiretap model, the onoff transmission scheme, jamming approach, antenna selection, and security region are discussed.
Introduction
As stated in the latest released statistics by the International Telecommunications Union (ITU) in 2020 [1], COVID19, to some extent, acts as an accelerator that pushes consumers and businesses to largely adopt digital services and technologies, which in return quickens the digital transformation for societies, business, and governments. Examples, including online learning, digital classrooms, contactless payment, zoom meetings, etc., are reshaping everyone’s life pattern. In light of the highly confidential data streams flowing over the wireless transmission medium, the legitimate data transactions enjoy the convenience largely brought by the inherent openness of the wireless transmission medium while facing the vulnerability of being exposed to illegitimate evil parties.
Traditionally, cryptography is an appealing approach to achieve data confidentiality. It is designed to prevent data disclosure to unauthorized devices and malicious users [2]. Although secrecy is guaranteed through the keybased encoding and decoding process and requires additional computing resources, it in fact assumes there exist errorfree links at the physical layer. Such an assumption would be unfeasible for the emerging decentralized networks (e.g., resourcelimited sensors or radiofrequency identification (RFID) networks) due to the high computational complexity and necessary key distribution and management [3]. Besides, the impacts from the impairments of wireless transmission medium on physical layer security, i.e., the randomness of wireless channels, are totally ignorant in cryptography.
Unlike the conventional complexitybased cryptographic techniques at upper layers via encryption, physical layer security (PLS), being a promising technology complementary to cryptography and certainly not as a replacement, takes full advantage of the physical properties of the wireless propagation environment via the combination of signaling and coding mechanism to provide additional secrecy at the bottom layer [4, 5]. It is proved suitable and feasible for achieving informationtheoretic security against eavesdropping attacks. More specifically, under the cover of the randomness of noise, fading, and interference, different users will receive different noisy copies of the private messages. This can enable the confidentiality of legitimate transmissions at the physical layer.
As a promising approach, physical layer security is built on the two pioneering works laid by Shannon [6] and Wyner [7], where the notion of perfect secrecy and the degraded wiretap channel model are introduced, respectively. It is noteworthy to point out that Wyner’s result established the PLS from the system model level, and he considered the threeuser scenario, consisting of a legitimate source (Alice), an intended legitimate user (Bob), and an eavesdropper (Eve) over the discrete memoryless wiretap channel. In [8], Wyner’s wiretap model was extended to the Gaussian wiretap channel by Leung et al., and they found the fundamental basis of secrecy capacity (\(C_s\)), which is defined as the difference between the channel capacity of the main channel (Alice to Bob, i.e., \(C_M\)) and that of the wiretap channel (Alice to Eve, i.e., \(C_W\)), namely, \(C_s = C_M  C_W\). The conceptual implication of secrecy capacity indicates that only when the legitimate link experiences better quality of received signals compared to the wiretap channel, positive secrecy can be surely guaranteed. Inspired by this fundamental work, considerable research efforts have been devoted to investigate the security performance metrics over wiretap fading channels, e.g., [9, 10]. The insights drawn from these works offer mathematical proofs showing that wireless channels’ fading property can be reversely used to enhance secrecy.
Observing the existing books, surveys, and tutorials related to the PLS [2,3,4,5, 11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], numerous researchers from both the wireless communication and signal processing communities summarized the stateoftheart of PLS from the perspective of application scenarios, e.g., 5G wireless networks [25], cooperative networks [26], and ultrareliable and lowlatency communications (URLLC) [27], and secrecy enhancement, including jamming schemes [3, 19, 26], multipleantenna techniques [24], and wiretap coding [14, Chapter 6] [25] (e.g., lowdensity paritycheck (LDPC) codes, polar codes, and lattice codes.) . It is reported in [2] that Zou et al. have classified the PLS technique into four categories: informationtheoretic security, artificialnoise aided security, securityoriented beamforming, security diversity methods, and physical layer secret key generation.
As an indispensable element of PLS techniques, informationtheoretic security has been further classified into three categories according to different wiretap channels: (i) memoryless wiretap channels; (ii) Gaussian wiretap channels; and (iii) fading wiretap channels. However, the majority of informationtheoretic security is centered around the fading wiretap channels, e.g., see references [9, 10, 31]. The pioneering work is laid by Bloch et al. [9], where the authors explored the impacts of fading characteristic of wireless channels on the security issue and proposed two performance metrics, i.e., the average secrecy capacity (ASC) and outage probability of secrecy capacity (equivalently, secrecy outage probability (SOP)), to measure informationtheoretic security. At the same year, Gopala et al. [10] investigated the perfect secrecy capacity over wiretap fading channels for two scenarios: (i) the full channel state information (CSI) is available at the transmitter; and (ii) only the main channel CSI is perfectly known at the transmitter. The former scenario represents the active eavesdropping, to be specific, Eve is a legitimate network participant (e.g., in a timedivision multipleaccess (TDMA) environment). As a result, Alice is capable of accessing Eve’s CSI, as well as Bob’s CSI. Alice can adapt her coding scheme to every channel coefficient realization. Therefore, the ASC is chosen as the security performance metric. In contrast, the latter scenario indicates the presence of a passive eavesdropper. More specifically, Eve is a totally silent network adversary and only capable of wiretaping the AliceBob link. As such, Alice has no CSI knowledge of the wiretap channel, she cannot flexibly adapt her transmission rate to guarantee perfect secrecy. The SOP is correspondingly adopted as the key secrecy metric to evaluate how perfect secrecy is compromised.
Inspired by these fundamental research works, numerous research works focus on analyzing the security performance metrics over a diverse body of fading wiretap channels for the sake of better understanding the impacts of fading characteristic on secure communications, to list some, Rayleigh [9], Nakagamim, Weibull [32], Rician (Nakagamiq) [33, 34], Hoyt (Nakagamin) [35, 36], Lognormal [37], \(\alpha \mu\) (equivalently generalized Gamma or Stacy) [38,39,40,41,42], \(\kappa \mu\) [43,44,45,46], \(\eta \mu\) [47], generalized\({\mathcal {K}}\) (\({\mathcal {K}}_G\)) [48,49,50,51], extend generalized\({\mathcal {K}}\) (EGK) [52], FisherSnedecor \({\mathcal {F}}\) [53, 54], GammaGamma [55], shadowed \(\kappa \mu\) [56], double shadowed Rician [57], Fox’s Hfunction [52], cascaded Rayleigh/Nakagamim/\(\alpha \mu\) [58,59,60], cascaded \(\kappa \mu\) [61], \(\alpha \kappa \mu\)/\(\alpha \eta \mu\) [62], BeaulieuXie [63], \(\alpha \kappa \eta \mu\) [64, 65], twowave with diffuse power (TWDP) [31], Nwave with diffuse power (NWDP) [66], \(\kappa \mu\)/Gamma [67], Fluctuating Beckmann [68], correlated Rayleigh [69], correlated composite Nakagamim/Gamma [70], correlated \(\alpha \mu\) [71], correlated shadowed \(\kappa \mu\) [72], mixed \(\eta \mu\) and Málaga [73], Málaga [74,75,76,77,78], fluctuating tworay (FTR) channels [79, 80]. The usage of these fading channels is examined practical and feasible in various wireless communications, such as, cellular networks [81], cellular devicetodevice (D2D), vehicletovehicle (V2V) communications [44], radio frequencyfree space optical (RFFSO) systems [55], mmWave communications [79], underwater acoustic communications (UAC), frequency diverse array (FDA) communications [82], bodycentric fading channels, unmanned aerial vehicle (UAV) systems, land mobile satellite (LMS) [56, 83], etc.
To the authors’ best knowledge, no survey or tutorial paper has ever focused on analyzing the security performance metrics over wiretap fading channels. To this end, the main contributions of this work are listed as follows:

1.
reviewing the stateoftheart of informationtheoretic security over four kinds of wiretap fading models: (i) smallscale, (ii) largescale, (iii) composite, and (iv) cascaded.

2.
displaying two clusters of security metrics to quantify informationtheoretic security in the presence of active and passive eavesdropping.

3.
summarizing three generic tools, i.e., the mixture Gamma (MG) distribution, the mixture of Gaussian (MoG) distribution, and Fox’s Hfunction distribution, which are used to assist the derivation of security metrics. These three tools are especially advantageous when the main channel and the wiretap channel confront different type of wiretap fading channels, e.g., the mixture of smallscale fading and composite fading models.

4.
presenting the application scenarios, advantages, and limitations of the three aforementioned statistical tools. The insights drawn from the three tools demonstrate their flexibility to largely encompass the existing four kinds of wiretap fading models via adequately configuring fading channel characteristics.

5.
providing four secrecy enhancement techniques, including the onoff transmission scheme, jamming approach (artificial noise (AN) and artificial fast fading (AFF)), antenna selection, and security region for Wyner’s wiretap channel model.
The remainder of this paper is organized as follows: Sect. 2 presents Wyner’s wiretap channel model, followed by Sect. 3, where the security performance metrics are presented. In Sect. 4, we review physical layer security over fading wiretap channels according to the fading channel models and also present three useful and generic tools used to assist the security metrics analysis. In Sect. 5, we introduce the secrecy enhancement schemes based on classic wiretap fading channels. Finally, Sect. 6 concludes this paper.
Wyner’s wiretap channel model
Consider the classic AliceBobEve wiretap channel model, as shown in Fig. 1, where Alice intends to send confidential messages to Bob in the presence of a malicious eavesdropper (Eve). The instantaneous signaltonoise ratio (SNR) at Bob (B) and Eve (E) is expressed as \(\gamma _i = {\bar{\gamma }}_i g_i, i \in \{B,E\}\), where \({\bar{\gamma }}_i\) is the average received SNR, and \(g_i\) is the channel gain, which can be possibly modeled by any fading channel distributions.
Security performance metrics
According to [9], the instantaneous secrecy capacity for one realization of the (\(\gamma _B, \gamma _E\)) pair over quasistatic wiretap fading channels is given by
where \([x]^+ \overset{\triangle }{=} \max (x,0)\).
Based on the definition of the instantaneous secrecy capacity, security performance metrics used to evaluate the PLS over wiretap fading channels are further developed according to the availability of full CSI or partial CSI of Wyner’s wiretap model. In practice, the aforementioned two scenarios correspond to the passive eavesdropping and active eavesdropping, respectively. More specifically, it is highly questionable to have any knowledge of an evil eavesdropper’s CSI. As such, security performance metrics are classified into two categories (i) the SOP and the lower bound of SOP; and (ii) the probability of nonzero secrecy capacity (PNZ) or the intercept probability, ASC, and ergodic secrecy capacity. To this end, the two clusters of security performance metrics are vivid indicators showing whether perfect secrecy can be surely achieved or not, which are shown and compared in Table 1.
Exact security performance metrics
Secrecy outage probability
In the presence of a passive eavesdropper, who only listens to the main channel without sending any probing messages, Alice transmits her private messages at a constant secrecy rate \(R_t\) to Bob. With this in mind, perfect secrecy can be assured only when \(R_t\) falls below the instantaneous secrecy capacity \(C_s\). Strikingly, the SOP is commonly seen as a key secrecy indicator used for passive eavesdropping, it measures the level that how perfect secrecy is compromised. Mathematically speaking, the SOP is the probability that the instantaneous secrecy capacity is lower than a predetermined secrecy rate \(R_t\),
The probability of nonzero secrecy capacity
The PNZ is regarded as another important secrecy metric that measures the existence of positive secrecy capacity with a probability,
where step (a) is subsequently transformed from the SOP metric by setting \(R_t =0\).
Intercept probability
In contrast to the PNZ metric, the intercept probability denotes the probability of the occurrence of an intercept event. In other words, it displays the probability of the occurrence of a negative instantaneous secrecy capacity event, which is mathematically interpreted as
Compared to the PNZ metric, fewer works have investigated the intercept probability [84,85,86,87]. For instance, Zou and Wang in [85] studied the intercept probability of the industrial wireless sensor networks in the presence of an eavesdropping attacker.
Average secrecy capacity
When an active eavesdropper appears, the ASC serves as a critical measurement that guides Alice to adapt her transmission rate based on \(C_M\) and \(C_W\) so as to achieve perfect secrecy. In other words, the ASC is a metric that evaluates how much achievable secrecy rate can be guaranteed. It is mathematically defined as
where \({\mathcal {E}}[\cdot ]\) is the expectation operator.
Security performance bounds
The usage of nonelementary functions is widely used to describe the statistical characteristics of fading models, e.g., the \(\kappa \mu\) distribution with the modified Bessel function of the first kind in its probability density function (PDF) and the generalized Marcum Q function in its cumulative distribution function (CDF), and the EGK distribution with the extended incomplete Gamma function in its PDF. Obviously, the existence of those special functions makes it highly intractable to deduce the security performance metrics embedded with both the PDF and CDF of the instantaneous SNR \(\gamma _i\) simultaneously. As a result, the acquisition of exact security performance metrics with closedform expressions is a challenging issue, security performance bounds, including the lower bound of the SOP and ergodic secrecy capacity, are in turn adopted as effective alternatives in many works.
The lower bound of SOP
The exact SOP can be accurately approximated by its lower bound when (i) the given transmission rate tends to zero, i.e., \(R_t \rightarrow 0\); and (ii) Eve is closely located to Alice, which can be physically interpreted as Eve having an extremely high average received SNR, i.e., \({\bar{\gamma }}_E \rightarrow \infty\). In this context, the lower bound of SOP can be computed as
Such an alternative has been widely investigated (see references [38, 39, 48, 53, 60]), and was shown to provide a fairly tight approximation.
Ergodic secrecy capacity
As an appropriate secrecy measure to characterize the timevarying feature of wireless channels, the ergodic secrecy capacity is consequently utilized to quantify the ergodic features of wireless channels [42, 88,89,90,91]. The ergodic secrecy capacity is mathematically evaluated by averaging the channel capacity over all fading channel realizations, which is mathematically computed as follows,
For instance, the authors in [92] investigated the ergodic secrecy rate of downlink multipleinput multipleoutput (MIMO) systems with limited CSI feedback. Similarly, considering the zeroforcing (ZF) beamforming at Alice and ZF detectors at Bob and Eve, the upper and lower bounds of the ergodic secrecy capacity of MIMO systems were explored in [90].
Secrecy characterization
In wireless communication systems, the transmitted signals are reflected, diffracted, and scattered from objects that are present on their path to the receivers. The received signals experience fading (multipath) and shadowing (signal power attenuation or pathloss) phenomena, which pose destructive and harmful impacts at the receiver sides. The essence of PLS lies in reversely using the impairments of wireless channels as secrecy enhancement means.
Under the assumption that the main and wiretap channels undergo independent fading conditions, this section mainly presents the security performance analysis over wiretap fading channels according to the following four categories.
Exact secrecy analysis
Smallscale fading channels
The random changes in signal amplitude and phase from the spatial positioning between a receiver and a transmitter is referred to smallscale fading. The wellknown smallscale fading models are Rayleigh, Nakagamim, Rician, \(\alpha \mu\), etc. The simple and tractable form of these models makes smallscale fading appealing and popular in the security and reliability performance analysis. Examples can be found in [9, 33, 38,39,40,41], where the SOP, PNZ, and ASC metrics are analyzed with either closedform or highly tight approximated expressions. It is noteworthy of mentioning that the \(\alpha \mu\) distribution can be reduced to Rayleigh (\(\alpha =2,\mu =1\)), Nakagamim (\(\alpha =2,\mu =m\)), Weibull (\(\alpha\) is the fading parameter, \(\mu =1\)), and Gamma (\(\alpha =1\), \(\mu\) is the fading parameter) distributions by properly attributing the values of \(\alpha\) and \(\mu\). To this end, the applicability and flexibility of the \(\alpha \mu\) distribution have been well explored in the literature. Besides, the TWDP fading model is also of high flexibility as it includes Rayleigh, Rician, and hyperRayleigh as special cases. The TWDP model characterizes propagating scenarios where the received signal contains two strong, specular multipath waves, moreover, it can also model a link worse than Rayleigh fading. More importantly, it provides a good fit to the the realworld frequencyselective fading data from wireless sensor networks [93]. The PLS investigation over TWDP wiretap fading channels was studied in [31]. Apart from the aforementioned works, in [94], the authors studied the effect of eavesdroppers’ location uncertainty on the SOP metric, where Eve is located in a ringshaped area around Alice and undergoes Rayleigh fading.
Another interesting direction of PLS over smallscale fading channels lies in the secrecy investigation over correlated fading channels. The correlation is caused due to the distances between Bob and Eve, or the scattering environments. The physical correlation essentially makes the fading statistics, i.e., the mathematical representation of the joint PDF of \(\gamma _B\) and \(\gamma _E\), fairly complex and eventually makes it intractable and highly difficult to obtain exact closedform security performance metrics, instead, secrecy performance bounds are derived (see references [69, 71]).
Largescale fading channels
The socalled largescale fading results from signal attenuation due to signal propagation over large distance and diffraction around large objects, e.g., hills, mountains, forests, billboards, buildings, etc., in the propagation path. One widely studied example of largescale fading channels is the Lognormal distribution. However, its complex mathematical form hinders the derivation of exact reliability and security performance expressions. For instance, Pan et al. [37] investigated the PLS over nonsmall scale fading channels, wherein independent/correlated Lognormal fading channels and composite fading channels were considered and approximated security performance representations were derived.
Composite fading channels
Different from the smallscale (fading) and largescale (shadowing) fading models, composite fading models are proposed to account for the effects of both smallscale and largescale fading simultaneously. For instance, Kumar et al. in [44] presented the SOP, PNZ, and ASC over \(\kappa \mu\) fading channels and explored the obtained results in several wireless communication scenarios, including cellular D2D, body area networks (BAN), and V2V. Moualeu and Hamouda in [46] subsequently extended the results in [44] to the singleinput multipleoutput (SIMO) scenario and derived the ASC and lower bound of SOP. More recently, to elaborate the shadowing effect of wireless channels, the authors in [57, 72] investigated the security performance over the shadowed Rician and \(\kappa \mu\) wiretap fading channels.
Other widely used fading models, e.g., generalized\({\mathcal {K}}\), Rayleigh/Lognormal (RL), Nakagamim/Lognormal (NL), GammaGamma, and FisherSnedecor \({\mathcal {F}}\), are examined in practice to model the channelinduced physical layer dynamics. For example, the FisherSnedecor \({\mathcal {F}}\) fading model was proposed in [95] to characterize D2D communications, where its simplicity and feasibility are compared with the generalized\({\mathcal {K}}\) fading model. Similarly, the GammaGamma, mixed \(\eta \mu\) and Málaga, and Málaga distributions were shown feasible to accurately model the RFFSO links, and the security performance analysis of RFFSO systems over these fading models are explored in [73,74,75,76, 96]. To encompass more special models in one distribution, one can find that [62, 64, 65], respectively, analyzed the security performance metrics over \(\alpha \eta \mu\), \(\alpha \kappa \mu\), and \(\alpha \eta \kappa \mu\) fading models. For instance, the \(\alpha \eta \kappa \mu\) model can be reduced to the Rayleigh, Nakagamim, Rician, \(\kappa \mu\), \(\eta \mu\), \(\alpha \mu\), etc. Those models are highly valuable and flexible. However, its complex mathematical representation of characteristics makes it difficult to derive the exact closedform security metrics.
Cascaded fading channels
Cascaded fading models were found feasible to characterize the multihop nonregenerative amplifyandforward (AF) relaying with fixed gain, the propagation in the presence of keyholes, the keyhole/pinhole phenomena in MIMO systems, and the reconfigurable intelligent surface (RIS)aided wireless systems [97,98,99]. Yang et al. [97] modeled the RISaided main link as a multiplication of two Rayleigh distributed random variables. For vehicular networks, Ai et al. [58] considered the double Rayleigh fading channels and analyzed the ASC metric. Regarding other works over cascaded Nakagamim, cascaded FisherSnedecor \({\mathcal {F}}\), and cascaded \(\alpha \mu\) wiretap fading channels, readers can refer to [58,59,60,61, 87, 100]. As discussed earlier, the cascaded \(\alpha \mu\) fading channel similarly includes the cascaded Rayleigh, cascaded Nakagamim, cascaded Weibull, and cascaded Gamma distributions. The authors of [60] studied the SOP, PNZ, and ASC performances with closedform expressions, which are given in terms of Fox’s Hfunction. The obtained results therein are identical to the exact analytical representations given in [87, 100]. In [61], Tashman et al. considered multiple eavesdroppers and investigated the SOP and PNZ metrics with closedform expressions over cascaded \(\kappa \mu\) wiretap fading channels.
As shown in Table 2, the existing research works focusing on analyzing security performance metrics over wiretap fading channels are summarized and their contributions are highlighted.
Generic secrecy analysis tools
With the above in mind and under the assumption that the main and wiretap channels undergo independent fading conditions, this subsection will present three useful and flexible distributions, which can largely encompass the aforementioned fading channel models by properly attributing their parameters. It is proved in literature that they are general and advantageous to assist the theoretical analysis of security metrics.
Mixture Gamma (MG) distribution
According to [102, 103], the instantaneous received SNR \(\gamma\) over wireless Rayleigh, Nakagamim, NL, \(\kappa \mu\), Hoyt, \(\eta \mu\), Rician, \({\mathcal {K}}\), \({\mathcal {K}}_G\), \(\kappa \mu\)/Gamma, \(\eta \mu\)/Gamma, and \(\alpha \mu\)/Gamma fading channels can be reformulated using the MG distribution, whereas the PDF and CDF of the instantaneous received SNR \(\gamma\) are denoted as \(f(\gamma )\) and \(F(\gamma )\) and given by
where L is the number of terms in the mixture, while \(\alpha _{l}, \beta _{l}\), and \(\zeta _{l}\) are the parameters of the lth Gamma component. \(\Upsilon (\cdot ,\cdot )\) is the lower incomplete Gamma function.
Lei et al. [49] used the MG distribution to assist the informationtheoretic security performance analysis over wiretap generalized\({\mathcal {K}}\) fading channels. Motivated by [36], the security metrics over the FTR and Málaga turbulence fading channels [74, 79] can be similarly derived using the MG distribution.
Mixture of Gaussian (MoG) distribution
Based on the unsupervised expectationmaximization (EM) learning algorithm, the MoG distribution is essentially beneficial when the characteristics of fading channels are unavailable. In [104], the authors modeled the RL, NL, \(\eta \mu\), \(\kappa \mu\), and shadowed \(\kappa \mu\) fading channels using the MoG distribution. The findings of [104] showcase that the MoG distribution is, especially advantageous to approximate any arbitrarily shaped nonGaussian density and can accurately model both composite and noncomposite channels in a simple expression.
Assuming the instantaneous SNR \(\gamma\) follows the MoG distribution, its PDF and CDF are given by
where C represents the number of Gaussian components. \(w_l>0\), \(\mu _l\), and \(\eta _l\) are the lth mixture component’s weight, mean, and variance with \(\sum _l^C w_l = 1\), \(\Phi (x)\) is the CDF of the standard normal distribution.
Fox’s Hfunction distribution
For known fading characteristics, the Fox’s Hfunction distribution is a general and flexible tool. It is reported in [52, 105,106,107] that many wellknown distributions in the literature, e.g., Rayleigh, Exponential, Nakagamim, Weibull, \(\alpha \mu\), Gamma, FisherSnedecor \({\mathcal {F}}\), Chisquare, cascaded Rayleigh/Nakagamim/\(\alpha \mu\), GammaGamma, Málaga, \({\mathcal {K}}_G\), EGK, etc., can be represented using Fox’s Hfunction distribution. Interested readers are suggested to refer to Table 3.
Assuming \(\gamma\) follows Fox’s Hfunction distribution, its PDF and CDF are given by
where \(H_{p,q}^{m,n}[.]\) is the univariate Fox’s Hfunction [108, Eq. (8.4.3.1)], \({\mathcal {K}} > 0\) and \({\mathcal {C}}\) are constants such that \(\int _0^\infty f (\gamma ) d \gamma = 1\). \(A_i > 0\) for \(i=1,\cdots , p\), \(B_l > 0\) for \(l = 1, \cdots , q\), \(0 \le m \le q\), and \(0 \le n \le p\). For notational convenience, let \({\mathfrak {a}} = (a_1,\cdots ,a_p)\), \({\mathscr {A}}= (A_1,\cdots ,A_p)\), \({\mathfrak {b}} = (b_1,\cdots ,b_q)\), and \({\mathscr {B}} = (B_1,\cdots ,B_q)\). Thus, hereafter the Fox’s Hfunction is denoted as \({\mathcal {H}}_{p,q}^{m,n}({\mathcal {K}}, {\mathcal {C}}, {\mathfrak {a}}, {\mathscr {A}}, {\mathfrak {b}}, {\mathscr {B}})\).
To compare the security performance analysis using the three aforementioned approaches, the PNZ metric is taken as an example. Provided that the main and wiretap links undergo the same fading conditions, the PNZ expressions are derived in terms of the Gauss Hypergeometric function [36, Eq. (7)], error function [101, Eq. (9)], and Fox’s Hfunction [52, Eq. (16)]. In Fig. 2, we plotted the PNZ performance versus \({\bar{\gamma }}_B\) for different fading channel models. Their tightness and accuracy have already been individually presented and confirmed in [36, 52, 101].
Remark
Conclusively speaking, the MG, MoG, and Fox’s Hfunction distributions have demonstrated their feasibility and applicability when analyzing security performance metrics. They all are valid when the main channel and wiretap channel are subjected to different wireless fading channels. Their advantages and limitations are listed in Table 4.
Note that the three aforesaid solutions are unfeasible when the main and wiretap channels are correlated.
Outdated and imperfect and correlated CSI
The aforementioned works mainly focus on the scenario that perfect CSI is available at all parties. Such an assumption is unrealistic in practice, since outdated CSI and imperfect CSI are the general cases due to the time varying nature of wireless channels and channel estimation errors.
In [109], the effects of outdated CSI on security performance were investigated over multipleinput singleoutput (MISO) systems when the transmit antenna selection (TAS) scheme is applied at Alice. The obtained analytical results show that the diversity gain of using multiple antenna techniques cannot be achieved when the CSI is outdated during the TAS process. Later on in [110], Hu et al. adopted the onoffbased transmission scheme at Alice to efficiently take advantage of the useful information in the outdated CSI. Alice does transmission only when she has a better link to Bob compared with that to Eve. Perfect knowledge of the main and wiretap channel CSI are always favorable, but the existence of noise in the channel estimation process makes it an unrealistic assumption. The impacts of imperfect CSI have been widely explored in diverse research topics, e.g., imperfect CSI in the ANassisted training and communications [111], imperfect CSI with an active fullduplex eavesdropper [112], imperfect CSI in a mixed RF/FSO system [55], etc.
Apart from the above two scenarios, the correlation between the main channel and wiretap channel also attracts a growing body of research interests. Channel correlation at the physical layer is often observed, which is mainly caused by the antenna deployments (e.g., insufficient antenna spacing in small mobile units equipped with space and polarization antenna diversity), proximity of the legitimate and illegitimate receivers, and random scatters around them [69]. The correlation is mathematically modeled with the correlated wiretap fading channel models. For example, Jeon et al. in [69] used the correlated Rayleigh fading wiretap channel and explored the secrecy capacity bounds. The results quantitatively showcased how much of secrecy capacity is lost due to channel correlation. In continuation of this work, the security performance analysis over correlated Nakagamim, correlated \(\alpha \mu\), correlated shadowed \(\kappa \mu\), and correlated Málaga fading channels are explored in [71, 72, 78, 113].
Secrecy enhancement approaches
The essence of PLS is to utilize the impairments (e.g., fading, noise, interference, and path diversity) of wireless channels to enhance security. In this section, we mainly focus on comparing the existing secrecy enhancement techniques suitable for wiretap channels.
Onoff transmission scheme
Consider the imperfect channel estimation, He and Zhou in [89] first proposed the onoff transmission scheme to improve the reliability and security performance. The principle of onoff transmission lies in the comparison between the estimated instantaneous SNRs at Bob and Eve, i.e., \({\hat{\gamma }}_B\) and \({\hat{\gamma }}_E\), and two given corresponding thresholds i.e., \(\mu _B\) and \(\mu _E\). More specifically, only when the condition \({\hat{\gamma }}_B \ge \mu _B\) and \({\hat{\gamma }}_E \le \mu _E\) meet, the ‘on’ mode at Alice is then activated, otherwise, Alice is in ‘off’ mode. The onoff transmission scheme is an appealing enabler to allow the SOP metric to be arbitrarily small. Building on He’s work, the onoff transmission is thereafter widely investigated in the following works [110, 114,115,116], where the imperfect CSI, outdated CSI, and correlated CSI are considered.
Jamming approach
Assuming the transmitter has more antennas than the eavesdropper, Goel and Negi proposed the concept of artificial noise (AN) [117]. The principle of AN lies in that the transmitter allocates some of its available power to generate AN to confuse passive eavesdroppers. Similarly, Wang et al. in [118] proposed the artificial fast fading (AFF) secrecy enhancement scheme, where the randomized beamforming is employed at the transmitter to ‘upgrade’ the main channel to an AWGN one and degrade the wiretap channel to a fast fading channel.
Unlike the aforesaid transmitting beamformingbased techniques, i.e., AN and AFF, the quality of the wiretap link is further degraded by allocating part of the transmitting resources (i.e., power or antennas) at the transmitter, specifically to Eve. Based on the survey papers [3, 19], one can conclude that jamming is a useful means to enhance the PLS. Considering the threenode wiretap fading channel, jamming can be alternatively realized by a fullduplex Bob, where Bob would receive signals from Alice and send jamming signals (e.g., noise) to Eve in order to reduce Eve’s received SNR’s quality [119]. Bob and Eve usually only act purely as a legitimate receiver or an illegitimate evil eavesdropper. However, in practice, they might behave with multiple roles. For instance, in [112], an active eavesdropper operates in fullduplex mode so that it can send jamming signals to degrade the legitimate receiver’s SNR, while in [120, 121], an untrustworthy relay works as a relay and eavesdropper simultaneously in a bidirectional cooperative network.
Antenna selection technique
In multipleantenna systems, TAS is seen as an effective way for reducing hardware complexity while boosting diversity benefits. In [113, 116, 122,123,124,125,126,127], TAS is deployed as a secrecy enhancement solution in MIMO systems. There exist three kinds of TAS schemes, i.e., (i) the antenna that maximizes the instantaneous output SNR at Bob is selected (see [122, 123]); (ii) more than one single antenna are selected (see [124]); and (iii) a general order of antenna is selected (see [126]).
Unlike the works [122,123,124,125] assuming that the multiantenna channels are independent, quite recently, Si et al. consider antenna correlation in [116], where the exact and asymptotic SOP are derived with consideration of three diversity combining schemes, namely maximal ratio combining (MRC), selection combining (SC), and equal gain combining (EGC) at Bob. This work is extended in [113], where the authors continuously consider the joint antenna and channel correlation, while the relationship between the correlation and the SOP is analytically established.
Protected zones
Protected zones (equivalently, secrecy region) mean a geometrical region (see [91, 128]), defined as the legitimate receiver’s locations having a certain guaranteed level of secrecy, or an area where the set of ordered nodes can safely communicate with typical destination, for a given secrecy outage constraint [42, 129].
Concluding remarks
In this paper, we have comprehensively reviewed the development of PLS over various wiretap fading channels. Based on the characteristics of wireless channels, research works focusing on investigating security performance metrics are thereafter classified into four categories: (i) smallscale fading; (ii) largescale fading; (iii) cascaded fading; and (iv) composite fading models. After comparing some significant existing and ongoing research works, we introduced three valuable and practical approaches, i.e., the MG, MoG, and Fox’s Hfunction distributions, to simplify the analysis of security performance metrics. The three approaches are highly beneficial and advantageous since they can broadly encompass the existing fading models. Besides, we discussed four secrecy enhancement techniques deployed on Wyner’s wiretap channel model, including onoff transmission, jamming approach, TAS technique, and protected zones. Hopefully, this paper can serve as a valuable reference for interested readers on better understanding the physical layer security over wiretap fading channels.
Abbreviations
 AN:

Artificial noise
 AF:

Amplifyandforward
 AFF:

Artificial fast fading
 ASC:

Average secrecy capacity
 D2D:

Devicetodevice
 EGK:

Extended generalized \({\mathcal {K}}\)
 MG:

Mixture of Gamma
 MoG:

Mixture of Gaussian
 SNR:

Signaltonoise ratio
 MIMO:

Multipleinput multipleoutput
 MISO:

Multipleinput singleoutput
 CDF:

Cumulative distribution function
 PDF:

Probability density function
 TAS:

Transmit antenna selection
 SC:

Selection combining
 MRC:

Maximal ratio combining
 EGC:

Equal gain combining
 SOP:

Secrecy outage probability
 PNZ:

Probability of nonzero secrecy capacity
 AWGN:

Additive white Gaussian noise
 ITU:

International Telecommunications Union
 CSI:

Channel state information
 RFFSO:

Radio frequencyfree space optical
 OSI:

Open systems interconnection
 RFID:

Radiofrequency identification
 IRS:

Intelligent reconfigurable surfaces
 TDMA:

Timedivision multipleaccess
 TWDP:

Twowave with diffuse power
 NWDP:

Nwave with diffuse power
 V2V:

Vehicletovehicle
 FTR:

Fluctuating tworay
 UAC:

Underwater acoustic communications
 UAV:

Unmanned aerial vehicle
 LMS:

Land mobile satellite
 PLS:

Physical layer security
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Acknowledgements
The authors appreciate the University of Luxembourg for sponsoring the publication of this article.
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This work was supported in part by the Luxembourg National Research Fund (FNR) projects, titled Exploiting Interference for Physical Layer Security in 5G Networks (CIPHY) and Resource Optimization for integrated Satellite5G nETworks with nonorThogonal multiple Access (ROSETTA).
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Kong, L., Ai, Y., Lei, L. et al. An overview of generic tools for informationtheoretic secrecy performance analysis over wiretap fading channels. J Wireless Com Network 2021, 194 (2021). https://doi.org/10.1186/s13638021020654
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DOI: https://doi.org/10.1186/s13638021020654
Keywords
 Physical layer security (PLS)
 Channel state information (CSI)
 Mixture Gamma (MG)
 Mixture of Gaussian (MoG)
 Fox’s Hfunction
 Artificial noise (AN)
 Artificial fast fading (AFF)
 Wiretap fading model
 Jamming
 Antenna selection