Study of Opportunistic Relaying and Jamming Based on Secrecy-Rate Maximization for Buffer-Aided Relay Systems

Recently, the concept of physical-layer security with multiuser wireless networks has been thoroughly investigated and approaches based on transmit processing and relay techniques have drawn a great deal of attention [9, 10, 11, 12]. Transmit processing relies on intelligent design of precoding and signalling strategies to improve the secrecy rate performance. The use of relays [13] and the exploitation of spatial diversity can also enhance secrecy rates. Moreover, recent advances like buffer-aided relays have gained significant attention [14, 15, 16] as they can provide significant performance advantages over standard relays.

Buffer-aided relay systems with secure constraints have been investigated in half-duplex [14, 15, 17, 16] and full-duplex systems [18]. Opportunistic relay schemes have been examined with buffer-aided systems in [19, 20, 21]. In this context, inter-relay interference cancellation (IC) at relay nodes is a fundamental aspect in opportunistic relay schemes. In [19], IC has been combined with buffer-aided relays and power adjustment to mitigate inter-relay interference (IRI) and minimize the energy expenditure. Furthermore, in [20] a distributed joint relay-pair selection has been proposed with the aim of rate maximization in each time slot using a threshold to avoid increased relay-pair switching and CSI acquisition. In [21] and [22], a jammer selection algorithm and a joint relay and jammer selection technique have been investigated. The studies in [21] and [22] have shown that relaying contributes to a better transmission rate for legitimate users, whereas jamming can deteriorate the transmission to the eavesdropper. Therefore, relaying and jamming lead to an improvement in secrecy rate performance. However, opportunistic buffer-aided relay schemes with jamming techniques for improving physical layer security have not been examined so far.

In this work, we propose an opportunistic relaying and jamming scheme and develop relay selection algorithms for the downlink of multiuser single-antenna and multiple-input multiple-output (MIMO) buffer-aided relay networks that maximize the secrecy rate, which is a challenging task due to the difficulty to obtain CSI of the eavesdroppers. Preliminary results of the proposed techniques have been reported in [23], where relaying and jamming selection have been examined, and in [24], where relay selection based on the secrecy rate has been studied. Here, we devise a relay selection approach for effective secrecy rate (E-SR) maximization that does not require CSI of the eavesdroppers. The proposed relaying and jamming function selection (RJFS) algorithms select multiple relay nodes as well as multiple jamming nodes to help the transmission. We also present an opportunistic relaying and jamming scheme in which relaying or jamming is performed within the same set of relays at different time slots. In the proposed RJFS algorithms, IC is employed to improve the transmission rate to legitimate users and the residual interference is used to amplify the jamming signal to the eavesdroppers. We exploit buffer-aided relays to store the jamming signals at the relay nodes and devise a buffer-aided relaying and jamming function selection (BF-RJFS) algorithm. Greedy RJFS and BF-RJFS algorithms are also developed for relay selection with reduced complexity. Simulations show that the proposed RJFS and BF-RJFS algorithms can outperform previously reported techniques in the absence of CSI of the eavesdroppers. In addition, the greedy RJFS and BF-RJFS algorithms achieve a performance close to that of the exhaustive search-based RJFS and BF-RJFS algorithms, while requiring a much lower computational cost. The main contributions of this work are:

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  The E-SR maximization approach that does not require CSI of the eavesdroppers is proposed.

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  An opportunistic relaying and jamming scheme for single-antenna and MIMO buffer-aided relay systems.

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  Novel RJFS algorithms that maximize the secrecy rate are developed for buffer-aided relay systems.

• •

  Greedy RJFS and BF-RJFS algorithms are developed to reduce the computational complexity of exhaustive search-based RJFS and BF-RJFS algorithms.

• •

  A secrecy rate analysis of the proposed RJFS algorithms.

This paper is organized as follows. In Section II, the system model and problem formulation are introduced. A review of relay selection techniques and a novel relay selection criterion without CSI to the eavesdroppers are included in Section III. The proposed RJFS and BF-RJFS algorithms are introduced in Section IV. In Section V a secrecy analysis is carried out. In Section VI, we present and discuss the simulation results. The conclusions are given in Section VII.

Fig. 1 describes the downlink of an opportunistic multiuser MIMO relay system with $N_{t}$ antennas employed to transmit data streams aided by precoding to $M$ users in the presence of $N$ eavesdroppers. The system is equipped with a total of $S_{\rm total}$ relay nodes and a relay selection scheme that chooses $S$ out of $S_{\rm total}$ relay nodes. Each relay node is equipped with $N_{i}$ antennas and the buffer of each relay can store $L$ data packets. To show the states of the relays, we use the state matrix $\boldsymbol{L}_{\rm state}\in{\mathbb{C}}^{S_{\rm total}N_{i}\times L}$. Each column of the state matrix $\boldsymbol{L}_{\rm state}$ represents the signals stored in the buffers in one time slot. The buffer state matrix is initialized with zeros. Similarly to relay systems with two slots, the transmission can be divided into two parts: Link I and Link II. In Fig. 1, the solid lines represent the transmission of intended signals and the dashed lines denote the transmission of jamming signals. In Link I, the eavesdroppers try to obtain the signals transmitted from the source and the jamming signals will cause interference to the eavesdroppers. Furthermore, in Link II the eavesdroppers attempt to get the information from the selected relays. In this scenario, the jamming signals will be generated and transmitted by the relays. In such opportunistic scheme the relays can be selected to perform different functions in the same time slot. From Fig. 1 in both Link I and Link II, although the eavesdropper can accumulate the information received in both links, the relays selected to perform jamming will always transmit jamming signals to the eavesdroppers. Depending on the buffer size, the opportunistic scheme can be considered in two scenarios:

• •

  Buffer size $L=1$: In the first time slot, only Link I is employed whereas in the second time slot Link II is used. As a result, there is no cooperation between Link I and Link II and the temporal advantage is unavailable in this scenario, which means this scheme is equivalent to a relay scheme without buffers.

• •

  Buffer size $L>1$: The thresholds $\eta_{\rm LinkI}$ and $\eta_{\rm LinkII}$ that indicate the power allocation to the transmitter $\eta_{\rm LinkI}P$ or $\eta_{\rm LinkII}(2-P)$, where $P$ is the power, are calculated separately for Link I and Link II, which determines if the relays perform relaying or jamming.

  • –

    If $\eta_{\rm LinkI}>\eta_{\rm LinkII}$, Link I is active. It indicates that the channels from the source to the relays can provide a better transmission environment. In this scenario, the jamming signals are generated independently at the relays, which are selected to perform the jamming function. The selection of the relays which perform the relaying function can be done according to different relay selection criteria. The jamming signal will also be stored at the buffers.

  • –

    If $\eta_{\rm LinkI}\leq\eta_{\rm LinkII}$, Link II is active. It indicates that the channels from the relays to the users have better links. In this scenario, relays will forward the signals to the destination. The jamming signals in Link II are the stored jamming signals in Link I, which means that the jamming signals in Link II do not need to be generated in Link II.

If CSI remains unchanged or one link is always better than the other than the system will employ a counter, $\eta_{\rm L}$, compare it with a maximum value $\eta_{L_{\rm max}}$ and activate the link that has been inactive for $\eta_{L_{\rm max}}$ transmissions. The value $\eta_{L_{\rm max}}$ is set by the designer. In addition, the change of links makes it more difficult for the eavesdropper to obtain the pattern.

In this system, each relay node is equipped with $N_{i}$ antennas. To indicate when the relays are performing the jamming function, the relay antenna number is represented by $N_{k}$. For one relay we can have $N_{k}=N_{i}$. At the receiver side each user and each eavesdropper is equipped with $N_{r}$ and $N_{e}$ receive antennas. We also assume that the eavesdroppers do not jam the transmission and the data transmitted to each user, relay, jammer and eavesdropper experience a flat-fading MIMO channel. The quantities ${\boldsymbol{H}}_{i}\in{\mathbb{C}}^{N_{i}\times N_{t}}$ and ${\boldsymbol{H}}_{e}\in{\mathbb{C}}^{N_{e}\times N_{t}}$ denote the channel matrices of the ith relay and the eth eavesdropper, respectively. The quantities ${\boldsymbol{H}}_{ke}\in{\mathbb{C}}^{N_{e}\times N_{k}}$ and ${\boldsymbol{H}}_{kr}\in{\mathbb{C}}^{N_{r}\times N_{k}}$ denote the channel matrices of the kth relay to the eth eavesdropper and the kth relay to the rth user, respectively. The channel between the kth relay to the ith relay is represented by ${\boldsymbol{H}}_{ki}\in{\mathbb{C}}^{N_{i}\times N_{k}}$.

To support the transmission of data to $M$ users, the source is equipped with $N_{t}\geqslant N_{r}M$ antennas. The total number of antennas with $S$ relaying function nodes as well as $K$ jamming function nodes should satisfy $N_{i}S\geqslant N_{r}M$ and $N_{k}K\geqslant N_{r}M$, respectively. At the same time we assume that the total number of antennas of the eavesdroppers is $N_{e}N\geqslant N_{r}M$. In order to satisfy the precoding constraints [25], the number of $N_{r}M$ transmit antennas is used to transmit signals to $M$ users. The relays can estimate the channel from the jammers by assuming that there are pilots in the packet structure, that they know the jamming signals and that the eavesdroppers cannot decode the jammers. This is reasonable because the relays also perform jamming and therefore should know the jamming signals. Moreover, we also assume that CSI of the users can be obtained at the transmitter by feedback channels from the relays. Alternatively, advanced parameter estimation and relay techniques can be employed [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. The vector ${\boldsymbol{s}}_{i}^{(t)}\in{\mathbb{C}}^{N_{i}\times 1}$ contains the data symbols of each user to be transmitted in time slot $t$. The transmit signal at the transmitter can be expressed as:

In previous works [39, 40, 41, 42, 43], precoding techniques have been applied to mitigate the interference among users. In this work, we adopt for simplicity linear zero-forcing precoding whose precoding matrix can be described by

with ${\boldsymbol{U}}_{i}\in{\mathbb{C}}^{N_{t}\times N_{i}}$, the total precoding matrix can be expressed as

and the channel matrix to $S$ selected relays is given by

If the number of antennas equipped at each relay and each user are the same, the minimum required number of relays is ${\color[rgb]{0,0,0}S=M}$. The channels of the selected relays forwarding the signals to the $r$th user are described by

and the channels from the relays to the users are described by

The selected relays also perform jamming for Link I’s transmission to the eavesdroppers, whereas the channels of the jammers to the $i$th relay are given by

In each link, if we assume that the total jamming signals are ${\boldsymbol{J}}=[{{\boldsymbol{j}}_{1}}^{T}\quad{{\boldsymbol{j}}_{2}}^{T}\quad\cdots\quad{{\boldsymbol{j}}_{K}}^{T}]^{T}$, the received signal ${\boldsymbol{y}}_{i}^{(t)}\in{\mathbb{C}}^{N_{i}\times 1}$ at each relay node can be expressed by

In (8), $\boldsymbol{n}_{i}\in\mathcal{CN}(0,\sigma^{2}_{n})$ and the superscript $pt$ designates the previous time slot when the signal is stored in the buffer at the relay nodes. The quantity $\sigma^{2}_{n}$ is the noise variance for the channel and ${\boldsymbol{H}}_{Ki}{\boldsymbol{J}}$ is regarded as the IRI among the $i$th relay and the $K$ jammers. The intended relays are selected according to different criteria, which will be explained later on. The received signals are expressed by ${\boldsymbol{y}}^{(pt)}=[{{\boldsymbol{y}}_{1}^{(pt_{1})}}^{T}\quad{{\boldsymbol{y}}_{2}^{(pt_{2})}}^{T}\quad\cdots\quad{{\boldsymbol{y}}_{S}^{(pt_{S})}}^{T}]^{T}$. The superscript $pt$ represents the time slot and due to the characteristics of buffer relay nodes, the values can be different for each relay node. According to the theorem in [19], IRI can be cancelled. The jammers are targeted towards the $e$th eavesdropper channel described by

The received signal at the $e$th eavesdropper is then given by

where $\boldsymbol{n}_{e}\in\mathcal{CN}(0,\sigma^{2}_{n})$ is the noise vector at the eavesdropper. For the eavesdropper, the term ${\boldsymbol{H}}_{Ke}^{(t)}{\boldsymbol{J}}$ acts as the jamming signal, which cannot be removed without CSI knowledge from the kth jammer to the eth eavesdropper.

If we assume that the transmitted signals from the relays to the users are expressed as ${\boldsymbol{r}}^{(t)}$, the received signal at the destination is given by

where $\boldsymbol{n}_{r}\in\mathcal{CN}(0,\sigma^{2}_{n})$ is the noise vector at the users.

In the existing IRI scenario based on (8) when the transmit signals $\boldsymbol{s}$ are statistically independent with unit average energy $\mathbb{E}[\boldsymbol{s}\boldsymbol{s}^{H}]=\boldsymbol{I}$, the SINR at relay node i $\varGamma_{\rm{IRI}-i}^{(t)}$ is expressed by

where $\varphi(K,i)$ is the factor that describes the IC feasibility and $\boldsymbol{\gamma}_{m,n}$ represents the instantaneous received signal power for the links $m\longrightarrow n$ as described by

The SINR at the $e$th eavesdropper node $\varGamma_{e}^{(t)}$ as well as the $r$th legitimate user $\varGamma_{r}^{(t)}$ is described by

and

where the terms in (15) and (16) are given by

and

Depending on the IRI cancelation (IC) at the relay nodes, two type of schemes can be applied. According to [19], if we assume $\mathbb{E}[\boldsymbol{s}\boldsymbol{s}^{H}]=\boldsymbol{I}$ and $\mathbb{E}[{\color[rgb]{0,0,0}{\boldsymbol{y}}^{(pt)}{{\boldsymbol{y}}^{(pt)}}^{H}}]=\boldsymbol{I}$, the feasibility of IC can be described by a factor $\varphi(K,i)$ which is described by

where $\varphi(K,i)=0$ means the interference can be cancelled from the received signal at the relays, whereas $\varphi(K,i)=1$ means IC should not be performed. The quantity $\gamma_{0}$ is the threshold that indicates the feasibility of IC, which is obtained by simulation. We assume that the channels from the relays, which perform jamming, are available at the transmitter.

In the IC scenario, interference mitigation can be performed at the relay nodes by setting $\varphi(K,i)=0$. The SINR expressions at the $i$th relay node, the $e$th eavesdropper and the $r$th receiver are respectively given by

and

Alternative interference mitigation techniques can also be considered [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 54] [32, 55, 56, 57, 58, 59, 60, 50, 61, 62, 63, 64, 65, 66, 34, 67] [68, 69, 70, 71, 31, 33, 46, 48, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 35, 84] [85, 86, 43, 87, 88, 89, 30, 90, 64, 59, 91]

In this subsection, we describe the secrecy rate used in the literature to assess the performance of the proposed algorithms in physical-layer security systems and formulate the problem. The MIMO system secrecy capacity [4] is expressed by

where ${\boldsymbol{Q}}_{s}$ is the covariance matrix associated with the signal and ${\boldsymbol{H}}_{ba}$ and ${\boldsymbol{H}}_{ea}$ represent the links between the source to the users and the eavesdroppers, respectively. For relay systems [21], according to (8) and (11), with equal power $P$ allocated to the transmitter and the relays, the achievable rate of the users is given by

where $\boldsymbol{\Gamma}_{r}^{(t)}$ according to (16) is given by

Similarly, the achievable rate of eavesdroppers is described by

and the $\boldsymbol{\Gamma}_{e}^{(t)}$ according to (15) is described by

where

In (28), $\boldsymbol{\varDelta}$ is the jamming signal to the eavesdropper. Using (25) and (27) the secrecy rate is given by

where $[x]^{+}=\max(0,x)$. In (30), we assume that each eavesdropper will listen to the information transmitted to a particular user. However, the assumption of the availability of global CSI knowledge is impractical, especially for the eavesdroppers. For this reason, we consider partial CSI knowledge to the relays as well as to the users. The problem we are interested in solving is to select the set of relay nodes to perform relaying or jamming based on the maximization of the secrecy rate. Therefore, the proposed optimization problem can be formulated as:

where $\boldsymbol{\Psi}$ represents the collection of relay subsets and $\boldsymbol{\varOmega}^{r}$ and $\boldsymbol{\varOmega}^{m}$ denote the set of selected jamming function nodes and the set of relaying function nodes, respectively.

Conventional relay selection does not take the jamming function of relay nodes into account and the relay nodes are selected with different selection criteria to assist the transmission between the source and the destination with only one eavesdropper [15] or without consideration of eavesdroppers [22, 93].

In [93], a max-min relay selection has been considered as the optimal selection scheme for conventional decode-and-forward (DF) relay setups. In a single-antenna scenario the relay selection is given by

where $h_{S,R_{k}}$ is the channel gain between the source and the relay and $h_{R_{k},D}$ is the channel gain between the relay $k$ and the destination. Similarly, a max-link approach has also been introduced to relax the limitation that the source and the relay transmission must be fixed. The max-link relay selection strategy can be described by

With the consideration of the eavesdropper, a max-ratio selection policy is proposed in [15] and is expressed by

with

The aforementioned relay selection procedure is based on knowledge of CSI.

Since conventional relay selection [22] may not support systems with secrecy constraints, we consider optimal selection (OS) which takes the eavesdropper into consideration. The SINR of OS in the downlink of multiuser MIMO relay systems under consideration can be expressed similarly to (15) and (16), as described by

and

The OS algorithm is given by

In the previously described relay selection algorithms, the availability of CSI to the eavesdroppers is an adopted assumption in the design of relay selection algorithms with secrecy constraints. However, in the optimization problem in (31), the CSI of the eavesdroppers is not available to the transmitter and the users. In order to circumvent this limitation, we propose a novel relay selection criterion that is termed effective secrecy rate (E-SR), which does not require CSI to the eavesdroppers and is incorporated in the multiuser MIMO buffer-aided relay system under study. The proposed E-SR approach is based on the maximization of the secrecy rate and introduces a simplification in the computation of the expression that does not require the knowledge of CSI to the eavesdroppers. The proposed E-SR approach for selecting multiple relays is expressed by

where the covariance matrix of the interference and the signal can be described as $\boldsymbol{R}_{I}=(\boldsymbol{H}_{i})^{-1}(\boldsymbol{H}_{i}^{H})^{-1}+\sum_{j\neq i}\boldsymbol{U}_{j}{\boldsymbol{s}}_{j}^{(t)}{{\boldsymbol{s}}_{j}^{(t)}}^{H}{\boldsymbol{U}_{j}}^{H}$ and $\boldsymbol{R}_{d}=\boldsymbol{U}_{i}{\boldsymbol{s}}_{i}^{(t)}{{\boldsymbol{s}}_{i}^{(t)}}^{H}{\boldsymbol{U}_{i}}^{H}$, respectively. The details of E-SR relay selection criterion are given in the Appendix. In (40), no CSI to the eavesdroppers is required and the E-SR approach only depends on the CSI to the intended receiver and the covariance matrix of the interference and the signal. In the following proposed relaying and jamming schemes, the E-SR technique is applied to circumvent the need for global instantaneous CSI of the eavesdroppers.

We assume that the total number of relay nodes is $S_{\rm total}$ and $\boldsymbol{\Omega}$ is the total relay set. To apply the opportunistic scheme in the system, an initial state is set according to the channel:

where ${\boldsymbol{H}}_{\boldsymbol{\varOmega}^{m}}$ refers to the set of channels examined prior to selection and we assume that in the initial state the relays will not perform the jamming function. $S$ relay nodes are selected according to the criterion as explained in Section II. With the total number of relaying and jamming nodes $S_{\rm total}$ and the number of selected nodes in each group $S$, the selection operation can be expressed as:

where $\boldsymbol{\Psi}$ represents the total number of sets of S combinations and in each set there are $S$ selected relaying or jamming nodes. For a particular set $\boldsymbol{\varOmega}^{m}$, the channel matrix of selected sets can be described by

If the total collection of selected sets is represented by $\boldsymbol{\Psi}_{\rm Relaying}$, then for each set the relay selection is given by

where

and

In (45) and (46), the covariance matrices $\boldsymbol{R}_{I}^{\varOmega^{m}}$ and $\boldsymbol{R}_{d}^{\varOmega^{m}}$ can be obtained in the same way as illustrated in (40). The only difference of the RJFS algorithm resides in the calculation of $\boldsymbol{R}_{I}^{\varOmega^{m}}$, apart from the interference from different users, there is also existing interference from the jamming function relay nodes. With the same distributions of the channels from the jamming function relay nodes to the eavesdropper, $\boldsymbol{R}_{I}^{\varOmega^{m}}$ can be calculated in a similar way to that in (40). In Algorithm 1 the main steps of RJFS are given. Step 1 of Algorithm  1 gives the collection of relay subsets, which contain the combinations of $S$ relay nodes out of $S_{\rm total}$ relay nodes. In our definition, $\boldsymbol{\Psi}_{\rm Relaying}$ is the same as $\boldsymbol{\Psi}$. However, to indicate the differences in the buffer relay system, we use $\boldsymbol{\Psi}_{\rm Relaying}$ instead of $\boldsymbol{\Psi}$. Note that in both RJFS and BF-RJFS algorithms, we use $\boldsymbol{\Psi}_{\rm Relaying}$ as a collection of relay subsets which perform the relaying function. With no buffers implemented at the relay nodes, the RJFS algorithm only selects the relays in Link I. The relays used in Link II are the same as those selected in Link I.

Here we describe the proposed BF-RJFS algorithm, which exploits relays equipped with buffers. Based on the RJFS algorithm, the selection of the $S$ relays used for signal reception is the same as that in the buffer relay scenario. The main difference between the proposed BF-RJFS and RJFS algorithms relies on the selection of the jammer. The selection of the set of jamming and communication relays is performed simultaneously. According to (44), we assume the corresponding threshold is stored in $\boldsymbol{\varGamma}$. Given the total collection of jamming selections $\boldsymbol{\Psi}_{\rm Jamming}$, the remaining relays are selected according to the proposed E-SR criterion as described by

where $\boldsymbol{\Gamma}_{\boldsymbol{\varOmega}^{r,n}}^{(t)}$ is given by

where $\boldsymbol{R}_{d}^{BF}$ is the covariance matrix of the transmit signal from the jamming function relay nodes to the users. The jamming signal is the same as the received signal from the relays in previous time slots. The calculation of $\boldsymbol{R}_{d}^{BF}$ depends on (20) and $\boldsymbol{R}_{I}^{BF}$ relies on (17) and (18). In this procedure, the calculation of $\boldsymbol{\Gamma}_{\boldsymbol{\varOmega}^{r,n}}^{(t)}$ is obtained by

where the relays used for jamming in the next time slot are selected. With the selection of communication relays and jamming relays the system can provide a better secrecy performance as compared to conventional relay systems. In Algorithm 2 the main steps of BF-RJFS are outlined. Steps 1 to 8 of Algorithm 2 eliminate the relay nodes with empty buffers because they cannot perform relaying function. Steps 9 to 20 eliminate relay nodes with a full buffer as the signals from the source cannot be stored in these relay nodes. Steps 21 to 25 return the results of the subset of selected relay nodes.

In both RJFS and BF-RJFS algorithms, exhaustive searches are implemented to select the relaying and jamming nodes. The incorporation of a greedy strategy [30] in both RJFS and BF-RJFS algorithms can significantly reduce the computational cost of the proposed exhaustive search-based RJFS and BF-RJFS algorithms. In an exhaustive search, all possible combinations are investigated to achieve optimal relay selection. Unlike an exhaustive search, a greedy search selects the relay with the best output at every iteration and then repeats the process with the remaining relays. The selection is completed when the desired number of relays are chosen. The total number of relays considered in the search is given by

From (50), we can see that the number of relays considered increases linearly with the total number of relays $S_{\rm total}$ which contributes to the reduction of the computational complexity.

In the following we describe the proposed greedy RJFS algorithm. When the $K$ relays that forward the signals to the users are determined, the relays used for signal reception are chosen based on the E-SR criterion, as given by

where $m$ represents the selected relay and $\boldsymbol{\Gamma}_{m}^{(t)}$ corresponds to the $m$th relay which is calculated based on (12) and given by

whereas $\boldsymbol{\Gamma}_{e}^{(t)}$ is described by

Instead of the exhaustive search of the selected set $\boldsymbol{\varOmega}^{m}$, the $m$th relay is computed with the aim of finding the relay that provides the highest secrecy rate based on $m$. In (52), $\boldsymbol{R}_{d}^{m}$ and $\boldsymbol{R}_{I}^{m}$ are obtained in the same way as in (45) and (46). The main steps are described in Algorithm 3.

Similarly to the BF-RJFS algorithm, the Greedy-BF-RJFS algorithm substitutes the exhaustive search of all combinations with a greedy search of individual relays. The main difference lies in the jamming relay selection. For a particular user $r$, each relay performs a threshold calculation and the relay $k$ with the highest threshold is selected until $S$ relays are selected to forward the signal to all users. The details of the Greedy-BF-RJFS algorithm are given in Algorithm 4.