Joint relay and jammer selection in Internet of Things systems

Abstract

In this article, we consider the physical layer security issue in Internet of Things systems, in which there exist a sensing transceiver pair, a number of candidate nodes, and an eavesdropper. The transceiver pair needs to select a jammer node and a relay node among the candidate nodes so as to preserve the secrecy of the communications. Considering the diversity of candidate channels and the limited available power, it is infeasible to scan all the nodes and find the optimal one. We formulate this jammer and relay selection problem as an optimal stopping problem under a fixed sensing order. Then, through applying dynamic programming solution, we propose a low-complexity approach to obtain the optimal sensing order. The performance of the proposed selection scheme is evaluated through numerical results.

Keywords

Internet of Things systems jammer and relay selection optimal stopping theory physical layer security cooperative jamming

Introduction

The Internet of Things (IoT), an advanced paradigm to support omnipresent connectivity among physical devices (e.g. sensors, actuators, and smart phones), gains its popular since it was first laid by K Ashton.¹ To connect things for exchanging and gathering information, wireless networks (such as wireless sensor networks^2–4) play an integral part in IoT. Yet, the wide adoption and deployment of IoT devices may shadowed by security threat.⁵ Specially, devices in IoT are inherently impressionable to eavesdropping attacks.⁶ Therefore, plenty of work needs to be devoted to improve the secrecy capacity in such an environment. Traditionally, security is considered as an issue in upper layers (e.g. the network layer) using cryptographic methods (e.g. encryption).^7,8 However, these cryptographic methods may be hard to implement in IoT systems since key distribution, encryption, and decryption are costly and complex for low-profile IoT devices.⁶

To solve the security issue in IoT systems, physical layer security has come to our mind. Different from the upper layer security, secret key is not needed in physical layer security, resulting in low complexity and low energy cost, which makes it more suitable for IoT devices. The basic thought of it is to make use of the features of the spectrum to prevent the eavesdropper from correctly decoding the information signals.^9–11 Specially, physical layer security methods attempt to destroy the signal to interference plus noise ratio (SINR) of the eavesdropper to maintain a positive value of secrecy capacity, which is defined as the maximum rate difference between the legitimate link and the transmitter-eavesdropper link.¹² In order to achieve a large secrecy capacity, cooperative jamming has been put forward.^13–15 The main idea of it is to confuse the eavesdropper via adopting artificial noises from a cooperative helper.

However, the existing cooperative jamming schemes may not be suitable for IoT systems since these schemes need the instantaneous channel state information (CSI) of all users (we refer to as GCSI). As we know, the general approach to get GCSI is transmitting training signals for channel estimation between transmitter and receiver. Yet, in IoT systems, due to the restricted energy and the lack of high-rate feedback channels, the channel training opportunities are limited [6]. The acquisition of accurate GCSI is a waste of spectrum access occasions. As a result, it is prohibitively difficult to get the GCSI in IoT systems. Besides, in these schemes, one can notice that energy consumption is not designed as a constraint condition. However, this issue is a matter of great concern in IoT systems [16]. Thus, the problem of how to design a suitable cooperative jamming scheme in such an energy-limited network with only statistical CSI needs to be tackled.

To solve these problems, we first propose a joint jammer and relay selection scheme in an IoT system, in which there exist a sensing transmitter–receiver pair, some other candidate nodes, and an eavesdropper. The source needs to select two candidate nodes which are employed as the jammer and the relay, respectively. In the proposed scheme, the source tests the secrecy capacity of the candidate nodes in a certain sequential order. Due to the time and energy constraint, as we mentioned before, it is scarcely possible to sense all the candidate nodes to select the best relay and jammer. To solve this, we attempt to employ the optimal stopping theory to joint select proper jammer and relay nodes. Specifically, the first two candidate nodes (one acts as the relay and the other acts as the jammer) that satisfy the secrecy capacity thresholds are selected as the relay and the jammer, respectively. The optimal thresholds are calculated according to the probability distribution of candidate nodes’ CSI and available power. Besides, just like Swindlehurst and colleagues,^14,15 we assume the candidate nodes are all equipped with multiple antennas. To cancel out jamming signals at the legitimate receiver, the corresponding jamming vectors are designed. Finally, considering a more general case that each candidate node has its unique probability distribution of CSI and available power, a proper sensing order is needed since it can help the transmitter–receiver pair to find the superior candidate nodes to reduce time and energy cost and meanwhile improve the secrecy capacity. By applying dynamic programming, we propose a low-complexity method to obtain the sensing order.

The rest of the article is organized as follows. Section “Related work” presents the related work. The system model is pointed out in section “System model.” The joint relay and jammer selection scheme is derived in section “Optimal stopping theory–based joint relay and jammer selection scheme.” The performance evaluations are detailed in section “Evaluation.” Section ”Conclusion” concludes the article.

Notations: $(\cdot)^{†}$ and ||·|| denote the Hermitian transpose of a matrix and the Euclidean norm, respectively. $E [\cdot]$ is the statistical expectation, while |·| represents the absolute value. $I$ denotes an identity matrix of corresponding dimension.

Related work

We summarize the related work under the categories of jamming schemes in physical layer security and optimal stopping theory–based schemes in wireless resource allocation.

Existing work on jamming schemes in physical layer security

Dong et al.¹³ considered a three-node topology and used cooperative jamming to confuse the eavesdropper. As for the IoT system, Zhang et al.¹⁷ proposed a jamming strategy among a large number of IoT devices. Huang and Swindlehurst¹⁴ employed cooperative jamming in relay networks. Using convex optimization, the jamming covariance matrices were derived in this work.

Jammer selection plays an important role in the cooperative jamming–based physical layer security methods. To fulfill the security performance requirements, amounts of recent research works have been performed in the respect of jammer selection.^18–22

Chen et al.¹⁸ investigated the joint jammer and relay selection scheme in an amplify-and-forward (AF)-based network. Similar to Chen et al.,¹⁸ Liu et al.¹⁹ proposed a cooperative jamming scheme in a relay network, in which one relay node and one or two jammers are selected. In these works, the number of antennas is assumed to be 1. In the context of multi-antenna networks, Wang et al.²⁰ investigated the jammer selection issue, in which the secrecy capacity is maximized using a null-steering beamforming technique. Different from Wang et al.,²⁰ Hui et al.²¹ introduced another criterion, termed as secrecy outage probability to select jammers. In the work by Hui et al.,²¹ the node that can minimize the secrecy outage probability is selected as the jammer. To choose multiple friendly jammers, Wang et al.²² attempted to select the nodes whose channels are orthogonal to the legitimate channel. In summary, these existing jammer selection schemes mainly pay attention to choose one or more jammers to optimize the security performance, assuming the instantaneous GCSI is known. However, to obtain the instantaneous GCSI, it is necessary to scan all the candidate nodes, resulting in a worse overall throughput since the time utilized for data transmissions is confined by the jammer selection process. And more importantly, as we mentioned before, due to the time and energy constraint in IoT systems, it is scarcely possible to sense all the candidate nodes. With the purpose of saving time and energy, we attempt to use the optimal stopping theory in the selection problem in IoT systems.

Existing work on the optimal stopping theory in wireless resource allocation

As we know, the optimal stopping theory has been well studied in wireless communications such as opportunistic scheduling, relay selection, and spectrum sensing.

For example, the opportunistic scheduling is an important issue in many wireless networks.^23–25 Tan et al.²³ studied the distributed opportunistic scheduling problem through the use of stopping theory. In the scheduling scheme, the authors characterized the optimal scheduling policies under delay constraints. A distributed opportunistic scheduling framework was proposed by Li et al.²⁴

Besides, the optimal stopping theory is employed to investigate the problem of spectrum sensing in cognitive radio networks. Shu and Krunz²⁶ considered the spectrum sensing issue as a stopping theory problem. An optimal decision strategy is suggested to enhance the overall network performance by maximizing the system rewards. Jia et al.²⁷ considered the problem of channel allocation in cognitive radio networks. In the context of relay selection, Jing et al.²⁸ proposed a stopping theory–based selection strategy, in which the node that can maximize the transmission throughput is selected as the relay. As far as we know, using the optimal stopping theory to address the joint jammer and relay selection issue in IoT systems remains a whitespace in existing literature.

System model

We establish a two-hop IoT system (Figure 1), which consists of a sensing transmitter S, a sensing receiver D, an eavesdropper E, and M candidate nodes denoted by $S_{CN} = {C N_{1}, C N_{2}, \dots, C N_{M}}$ . All the candidate nodes are assumed to own multiple antennas, while S, D, and E are equipped with only one antenna. In our model, the transmission process is divided into two phases. Assuming $C N_{i}$ and $C N_{j}$ are selected as the relay and the jammer, respectively, in the first transmission phase, the data signals are sent from S to $C N_{i}$ . To protect the signals from being overlapped by E, $C N_{j}$ performs cooperative jamming to confuse E. In the second phase, $C N_{i}$ transmits data signals to D, while $C N_{j}$ also transmits jamming signals to degrade the SINR at E. Accordingly, the overall system time slot (T) is divided into three phases: relay and jammer selection phase (Phase I), the first transmission phase (Phase II), and the second transmission phase (Phase III), as depicted in Figure 2.

Figure 1.

Network model.

Figure 2.

Time slot structure.

In Phase I, S observes the candidate nodes step by step. In a certain sensing step, S picks and senses two candidate nodes. The time needed for one observation step can be given as t. After sensing these two candidate nodes, S should make a decision regarding whether to select them as the relay and the jammer or to skip to the next sensing step. Suppose in the $k th$ sensing step the corresponding sensed two nodes are selected, the total time cost for Phase I is $T_{1} = kt$ . Then, the selection process is terminated while the data transmission process (i.e. Phase II and Phase III) begins. The duration of Phase II and Phase III can be expressed as $T_{2} = a (T - kt)$ and $T_{3} = (1 - a) (T - kt)$ , respectively, where $0 < a < 1$ .

Data transmission process

Transmission process in phase II

Without loss of generality, we assume $C N_{i}$ and $C N_{j}$ are selected as the relay and the jammer in Phase I. In Phase II, the signals received at $C N_{i}$ and E can be formulated as

y_{C N_{i}} = h_{S, C N_{i}} s_{p} + H_{C N_{j}, C N_{i}} u_{C N_{j}}^{1} s_{j} + n_{C N_{i}}

(1)

y_{E}^{1} = h_{S, E} s_{p} + h_{C N_{j}, E} u_{C N_{j}}^{1} s_{j} + n_{E}

(2)

where $s_{p}$ and $s_{j}$ are the information signals and jamming signals transmitted by S and $C N_{j}$ , respectively. The power constraints are expressed as $E [| s_{p} |^{2}] = P_{S}$ and $E [| s_{j} |^{2}] = P_{C N_{j}}$ , respectively. The beamformer of $C N_{j}$ is given as $u_{C N_{j}}^{1}$ . $n_{C N_{i}}$ and $n_{E}$ denote the noise power at $C N_{i}$ and E, respectively. We also define the following constants, vectors and matrices, and variables: $h_{ij}$ means the channel gain constant of i and j, where $i \in {S, C N_{j}}$ and $j \in {S, C N_{i}}$ , respectively. And so is $H_{ij}$ ( $h_{ij}$ ).

To cancel out the undesired interference, a decoding vector can be designed at $C N_{i}$ by setting $v_{C N_{i}}^{†} H_{C N_{j}, C N_{i}} u_{C N_{j}}^{1} = 0$ . This design is often termed as zero-forcing beamforming (ZFBF).

Then, the received signals at $C N_{i}$ are written as

y_{C N_{i}} = v_{C N_{i}}^{†} h_{S, C N_{i}} s_{p} + v_{C N_{i}}^{†} n_{C N_{i}}

(3)

After using ZFBF, one can see that $C N_{j}$ can inerrably receive and decode the information signals in phase II. As for the jamming beamformer $u_{C N_{j}}^{1}$ , since the undesirable interference caused by $C N_{j}$ can be successfully removed, $u_{C N_{j}}^{1}$ can be simply designed in the same direction of $h_{C N_{j}, E}$ .

Transmission process in phase III

In Phase III, the signals received at the receiver and the eavesdropper can be given as

y_{D} = h_{C N_{i}, D} u_{C N_{i}}^{2} s_{p} + h_{C N_{j}, D} u_{C N_{j}}^{2} s_{j} + n_{D}

(4)

y_{E} = h_{C N_{i}, E} u_{C N_{i}}^{2} s_{p} + h_{C N_{j}, E} u_{C N_{j}}^{2} s_{j} + n_{E}

(5)

Without ambiguity, the information signals and the jamming signals are still denoted by $s_{p}$ and $s_{j}$ sent from $C N_{i}$ and $C N_{j}$ , respectively. The power constraints are denoted as $E [| s_{p} |^{2}] = P_{C N_{i}}$ and $E [| s_{j} |^{2}] = P_{C N_{j}}$ . The beamformers of $C N_{i}$ and $C N_{j}$ are defined as $u_{C N_{i}}^{2}$ and $u_{C N_{j}}^{2}$ . Similar to Phase I, $h_{ij}$ means the channel gain constant of i and j, where $i \in {C N_{i}, C H_{j}}$ and $j \in {D, E}$ , respectively.

Since D only has one single antenna, we deliberately design $u_{C N_{j}}^{2}$ to null out the jamming signals at D, which can be given as

\max | h_{C N_{j}, E} u_{C N_{j}}^{2} |

(6)

s . t . h_{C N_{j}, D} u_{C N_{j}}^{2} = 0

(7)

Using matrix transformation, this problem can be solved and $u_{C N_{j}}^{2}$ can be formulated as

u_{C N_{j}}^{2} = \frac{(I - \frac{h_{C N_{j}, D}^{†} h_{C N_{j}, D}}{∥ h_{C N_{j}, D} ∥^{2}}) h_{C N_{j}, E}^{†}}{‖ (I - \frac{h_{C N_{j}, D}^{†} h_{C N_{j}, D}}{∥ h_{C N_{j}, D} ∥^{2}}) H_{C N_{j}, E}^{†} ‖}

(8)

For more information, one can refer to Gao et al.²⁹ Thus, the signals received at D can be given as

y_{D} = h_{C N_{i}, D} u_{C N_{i}}^{2} s_{p} + n_{D}

(9)

To be concluded, by carefully designing the jamming beamformer and information beamformer, no matter which candidate nodes are selected, the information signals can be perfectly transmitted from S to D since jamming signals are masked.

Optimal stopping theory–based joint relay and jammer selection scheme

In section “System model,” choosing preferable relay and jammer has an important influence on the SINR at E. Intuitively, with the purpose of enhancing the secrecy capacity, the channel gain of the jammer-eavesdropper link needs to be greater, while the channel gain of the jammer-receiver should be smaller. And when it comes to the relay node, the situation is reversed. In this section, we attempt to employ the optimal stopping theory to construct the relay and jammer selection process (i.e. Phase I). With the purpose of maximizing the reward function, the source should decide to stop or to continue according to the comparison of the instantaneous reward and the value of expected reward in the subsequent sensing steps. The concepts of stopping theory are given as follows.

A sequence of random variables (i.e. $X_{1}, X_{2}, \dots$ ) and the joint distribution are known a priori;

A sequence of reward functions, (i.e. $y_{0}, y_{1} (X_{1}), y_{2} (X_{1}, X_{2}), \dots, y_{\infty} (X_{1}, X_{2}, \dots)$ ), which are real-valued functions of the random variables.

To be more specific, given these concepts, the optimal stopping problem can be formulated as follows: for each $n = 1, 2, \dots$ step, after observing $X_{1} = x_{1}, X_{2} = x_{2}, \dots, X_{n} = x_{n}$ , the decision of stopping or continuing should be made by the source in the light of the comparison of the instantaneous reward $y_{0}, y_{1} (x_{1}), y_{2} (x_{1}, x_{2}), \dots, y_{\infty} (x_{1}, x_{2}, \dots)$ and the expected reward. In the next subsection, we attempt to get the expression of the reward function based on the secrecy capacity.

Reward function of secrecy capacity

As mentioned before, we assume $C N_{i}$ and $C N_{j}$ are selected as the relay and the jammer, respectively. In Phase II, the SINR at $C N_{i}$ and E can be calculated by

γ_{R}^{1} (C N_{i}, C N_{j}) = \frac{P_{S} | h_{S, C N_{i}} |^{2}}{σ^{2}}

(10)

γ_{E}^{1} (C N_{i}, C N_{j}) = \frac{P_{S} | h_{S, E} |^{2}}{P_{C N_{j}} | h_{C N_{j}, E} |^{2} + σ^{2}}

(11)

where $γ_{C N_{i}}^{1}$ and $γ_{E}^{1}$ represent the SINR at $C N_{i}$ and E, during Phase II.

According to Huang and Swindlehurst,³⁰ the secrecy capacity in Phase II, denoted by $C_{s}^{1} (C N_{i}, C N_{j})$ , can be calculated as

C_{s}^{1} (C N_{i}, C N_{j}) = \frac{lo g_{2} (1 + γ_{R}^{1} (C N_{i}, C N_{j}))}{lo g_{2} (1 + γ_{E}^{1} (C N_{i}, C N_{j}))}

(12)

Similarly, in Phase III, the SINR at D and E can be calculated by

γ_{D}^{2} (C N_{i}, C N_{j}) = \frac{P_{C N_{i}} | h_{C N_{i}, D} |^{2}}{σ^{2}}

(13)

γ_{E}^{2} (C N_{i}, C N_{j}) = \frac{P_{C N_{i}} | h_{C N_{i}, E} |^{2}}{P_{C N_{j}} | h_{C N_{j}, E} |^{2} + σ^{2}}

(14)

And the secrecy capacity in Phase III, denoted by $C_{s}^{2} (C N_{i}, C N_{j})$ , can be calculated as

C_{s}^{2} (C N_{i}, C N_{j}) = \frac{lo g_{2} (1 + γ_{D}^{2} (C N_{i}, C N_{j}))}{lo g_{2} (1 + γ_{E}^{2} (C N_{i}, C N_{j}))}

(15)

Since the eavesdropping attack can take place in Phase II and Phase III, both of the secrecy capacities $C_{s}^{1} (C N_{i}, C N_{j})$ and $C_{s}^{2} (C N_{i}, C N_{j})$ have to be positive to guarantee the secure transmission. Moreover, since the overall secrecy capacity subjects to the inferior phase, we define the achievable secrecy capacity as the smaller of $C_{s}^{1} (C N_{i}, C N_{j})$ and $C_{s}^{2} (C N_{i}, C N_{j})$ , which can be given as

X (C N_{i}, C N_{j}) = \min {C_{s}^{1} (C N_{i}, C N_{j}), C_{s}^{2} (C N_{i}, C N_{j})}

(16)

Then, we derive the reward function denoted by $Y_{k}$ . We consider a case that the sensing order is fixed. To be more specific, in the $k th$ observation step, the $k th$ candidate node and the $k + 1 th$ candidate node are sensed as the relay and the jammer, respectively. A tuning factor, denoted by $c_{k}$ , represents the stop of the sensing process at the $k th$ observation step. And $c_{k}$ can be expressed as

c_{k} = 1 - \frac{kt}{T}

(17)

According to equation (17), the value of $c_{k}$ decreases with an increase in the k. That is to say, the more CNs that S observed, the less the efficiency of the selection process, resulting in a shorter time for data transmission. The reward function after the $k th$ observation step can be written as

Y_{k} = c_{k} X_{k}

(18)

Optimal selection scheme

In IoT systems, as mentioned before, the instantaneous GCSI is hard to obtain. Therefore, we assume only statistical GCSI is know a priori for S. The channel gains for nodes i and j are assumed to be selected from a set of discrete values $H \overset{Δ}{=} {H_{l}, l = 1, 2, \dots, L_{1}}$ , where the distribution of $| h_{i, j} |^{2}$ is given by $\Pr (| h_{i, j} |^{2} = h_{l}) = p_{i, j, l}, l = 1, 2, \dots, L_{1}$ . However, the instantaneous power of the candidate nodes is also hard to obtain. Similarly, the available power for candidate node i is selected from a finite set $P \overset{Δ}{=} {P_{l}, l = 1, 2, \dots, L_{2}}$ . Let the distribution of $P_{i}$ be $\Pr (P_{i} = P_{l}) = p_{i, l}, l = 1, 2, \dots, L_{2}$

Under these assumptions, we can derive the expected reward for each observation step. Denoted by $V_{k}^{M - 1} (x_{1}, x_{2}, \dots, x_{k})$ , the maximum reward after the $k th$ observation step, which can be expressed as

\begin{matrix} V_{k}^{M - 1} (x_{1}, x_{2}, \dots, x_{k}) = \max {y_{k} (x_{1}, x_{2}, \dots, x_{k}), \\ E [V_{k + 1}^{(M - 1)} (x_{1}, x_{2}, \dots, x_{k}, X_{k + 1}) \\ \times | X_{1} = x_{1}, X_{2} = x_{2}, \dots, X_{k} = x_{k}]} \end{matrix}

(19)

where $y_{k} (x_{1}, x_{2}, \dots, x_{k})$ represents the instantaneous reward if the selection process stops at the $k th$ observation, and $E [V_{k + 1}^{(M - 1)} (x_{1}, x_{2}, \dots, x_{k}, X_{k + 1}) \times | X_{1} = x_{1}, X_{2} = x_{2}, \dots, X_{k} = x_{k}]$ means the expected reward of the $k + 1 th$ observation step. Note that the superscript ${of}_{k}^{M - 1}$ is $M - 1$ , since in this subsection, we assume the $k th$ candidate node and the $k + 1 th$ candidate node are sensed as the relay and the jammer, respectively, in the $k th$ observation step.

One can find that it is suitable to continue the selection process if

\begin{matrix} V_{k}^{M - 1} (x_{1}, x_{2}, \dots, x_{k}) = E [V_{k + 1}^{(M - 1)} (x_{1}, x_{2}, \dots, x_{k}, X_{k + 1}) \\ \times | X_{1} = x_{1}, X_{2} = x_{2}, \dots, X_{k} = x_{k}] \end{matrix}

That is to say, Phase I stops at $k th$ step and these two corresponding candidate nodes are selected if and only if the following condition holds³¹

\begin{matrix} y_{k} (x_{1}, x_{2}, \dots, x_{k}) \geq E [V_{k + 1}^{(M - 1)} (x_{1}, x_{2}, \dots, x_{k}, X_{k + 1}) \\ \times | X_{1} = x_{1}, X_{2} = x_{2}, \dots, X_{k} = x_{k}] \end{matrix}

(20)

In the following, we propose to use the backward induction method to obtain the expected reward for each observation step, since the number of the observation steps is limited to $M - 1$ . $W_{M - 1 - k}$ represents the expected reward $E {V_{k + 1}^{M - 1}}$ if the selection process proceeds to the next observation step, which can be written as

\begin{matrix} W_{M - 1 - k} = & E [V_{k + 1}^{(M - 1)} (x_{1}, x_{2}, \dots, x_{k}, X_{k + 1}) \\ \times | X_{1} = x_{1}, X_{2} = x_{2}, \dots, X_{k} = x_{k}] \end{matrix}

(21)

In this subsection, we simply assume the distribution of the available power and the channel gains of each candidate node are independent and identically distributed (i.i.d.) variables. As a result, $X_{1}, X_{2}, \dots, X_{M - 1}$ can be treated as a series of i.i.d. variables, indicating that $V_{k}^{M - 1}$ is only a function of $X_{k}$ and $W_{M - 1 - k}$ . To get the expression of $W_{M - 1 - k}$ , we begin with the derivation of the final step, that is, the $M - 1 th$ step. In the $M - 1 th$ step, the $M - 1 th$ and the $M th$ candidate nodes have to be selected. Thus, we set $W_{0} = - \infty$ . Then, according to equations (20) and (21), $W_{1}$ can be computed as equation (22). And for $k \geq 1$ , $W_{k + 1}$ can be computed as equation (23).

\begin{matrix} W_{1} & = E [V_{M - 1}^{(M - 1)} (X_{M - 1})] = E [Y_{M - 1}] = E [c_{M - 1} X_{M - 1}] \\ = c_{M - 1} \sum_{l_{1} = 1}^{L_{1}} \sum_{l_{2} = 1}^{L_{1}} \sum_{l_{3} = 1}^{L_{1}} \sum_{l_{4} = 1}^{L_{1}} \sum_{l_{5} = 1}^{L_{2}} \sum_{l_{6} = 1}^{L_{2}} p_{S, C N_{k}, l_{1}} p_{C N_{k}, E, l_{2}} p_{C N_{k + 1}, D, l_{3}} p_{C N_{k + 1}, E, l_{4}} p_{k, l_{5}} p_{k + 1, l_{6}} \\ \min {\log_{2} (1 + \frac{P_{S} h_{l_{1}}}{σ^{2}}) - \log_{2} (1 + \frac{P_{S} | h_{S, E} |^{2}}{P_{l_{6}} h_{l_{2}} + σ^{2}}), \log_{2} (1 + \frac{P_{l_{5}} h_{l_{3}}}{σ^{2}}) - \log_{2} (1 + \frac{P_{l_{5}} h_{l_{4}}}{P_{l_{6}} h_{l_{2}} + σ^{2}})} \end{matrix}

(22)

\begin{matrix} W_{k + 1} = E {Y_{M - 1 - k}, W_{k}} \\ = c_{M - 1 - k} \sum_{l_{1} = 1}^{L_{1}} \sum_{l_{2} = 1}^{L_{1}} \sum_{l_{3} = 1}^{L_{1}} \sum_{l_{4} = 1}^{L_{1}} \sum_{l_{5} = 1}^{L_{2}} \sum_{l_{6} = 1}^{L_{2}} p_{S, C N_{k}, l_{1}} p_{C N_{k}, E, l_{2}} p_{C N_{k + 1}, D, l_{3}} p_{C N_{k + 1}, E, l_{4}} p_{k, l_{5}} p_{k + 1, l_{6}} \\ \max {\min {\log_{2} (1 + \frac{P_{S} h_{l_{1}}}{σ^{2}}) - \log_{2} (1 + \frac{P_{S} | h_{S, E} |^{2}}{P_{l_{6}} h_{l_{2}} + σ^{2}}), \log_{2} (1 + \frac{P_{l_{5}} h_{l_{3}}}{σ^{2}}) - \log_{2} (1 + \frac{P_{l_{5}} h_{l_{4}}}{P_{l_{6}} h_{l_{2}} + σ^{2}})}, W_{k}} \end{matrix}

(23)

\begin{matrix} E [V_{M - 1}^{(M - 1)} (S_{C N_{i, j}}^{M - 1})] = E [c_{M - 1} X_{M - 1}] \\ = c_{M - 1} \sum_{l_{1} = 1}^{L_{2}} \sum_{l_{2} = 1}^{L_{2}} \sum_{l_{3} = 1}^{L_{1}} \sum_{l_{4} = 1}^{L_{1}} \sum_{l_{5} = 1}^{L_{1}} \sum_{l_{6} = 1}^{L_{1}} p_{C N_{i}, l_{1}} p_{C N_{j}, l_{2}} p_{S, C N_{i}, l_{3}} p_{D, C N_{i}, l_{4}} p_{E, C N_{i}, l_{5}} p_{E, C N_{j}, l_{6}} \\ \min {lo g_{2} (1 + \frac{P_{S} h_{S, C N_{i}, l_{3}}}{σ^{2}}) - lo g_{2} (1 + \frac{P_{S} | h_{S, E} |^{2}}{P_{C N_{j}, l_{2}} h_{C N_{i}, E, l_{6}} + σ^{2}}), lo g_{2} (1 + \frac{P_{C N_{i}, l_{1}} h_{C N_{i}, D, l_{4}}}{σ^{2}}) \\ - lo g_{2} (1 + \frac{P_{C N_{i}, l_{1}} h_{C N_{i}, E, l_{5}}}{P_{C N_{j}, l_{2}} h_{C N_{i}, E, l_{6}} + σ^{2}})} \end{matrix}

(24)

\begin{matrix} E [V_{M - 1}^{(M - 1 - k)} (S_{C N_{p, q}}^{k})] \\ = c_{M - 1 - k} \sum_{l_{1} = 1}^{L_{2}} \sum_{l_{2} = 1}^{L_{2}} \sum_{l_{3} = 1}^{L_{1}} \sum_{l_{4} = 1}^{L_{1}} \sum_{l_{5} = 1}^{L_{1}} \sum_{l_{6} = 1}^{L_{1}} p_{C N_{p}, l_{1}} p_{C N_{q}, l_{2}} p_{S, C N_{p}, l_{3}} p_{D, C N_{p}, l_{4}} p_{E, C N_{p}, l_{5}} p_{E, C N_{q}, l_{6}} \\ \max {\min {lo g_{2} (\frac{1 + \frac{P_{S} h_{S, C N_{p}, l_{3}}}{σ^{2}}}{1 + \frac{P_{S} | h_{S, E} |^{2}}{P_{C N_{q}, l_{2}} h_{C N_{p}, E, l_{6}} + σ^{2}}}), lo g_{2} (\frac{1 + \frac{P_{C N_{p}, l_{1}} h_{C N_{p}, D, l_{4}}}{σ^{2}}}{1 + \frac{P_{C N_{p}, l_{1}} h_{C N_{p}, E, l_{5}}}{P_{C N_{q}, l_{2}} h_{C N_{p}, E, l_{6}} + σ^{2}}})}, W_{k}} \end{matrix}

(25)

The process of the proposed selection scheme is shown in Algorithm 1. Similar to Huang and Swindlehurst³⁰ and Ly et al.,³² a common control channel (CCCH) is assumed to be set up for nodes to send the control information. First, S senses the candidate nodes according to a fixed sensing order (Line 1) and obtains the instantaneous reward $y_{k}$ in the $k th$ sensing step (Line 8). Then, S compares the value of $y_{k}$ with the expected reward $W_{M - 1 - k}$ and makes a decision of stopping at $k th$ step if $y_{k} \geq W_{M - 1 - k}$ or continuing the selection process to $k + 1 th$ step (Lines 9–13). Note that if the instantaneous rewards of the first $M - 2$ steps are all smaller than the corresponding expected reward, the selection process has to stop at the $M - 1 th$ step and the $M - 1 th$ and the $M th$ candidate nodes are selected as the relay and the jammer (Line 19).

Algorithm 1: Optimal selection scheme.
1: Construct the observation order, $S_{CN} = {C N_{1}, C N_{2}, \dots, C N_{M}}$ ;
2: Calculate $W_{1}, W_{2}, \dots, W_{M - 1}$ based on (22) and (23);
3: S starts the selection process by sensing $C N_{1}$ and $C N_{2}$ ;
4: for $k = 1$ to $M - 1$ do
5: S sends an eager-to-help (ETH) frame to $C N_{k}$ and $C N_{k + 1}$
6: ifS receives the able-to-help (ATH) frame from $C N_{k}$ and $C N_{k + 1}$ then
7: S gets the instantaneous CSI and the available power for these two nodes;
8: S calculates the instantaneous rewards $y_{k}$ according to (16) - (18);
9: S choose a large value of $y_{k}$ and $W_{M - 1 - k}$ according to (20);
10: if $y_{k} < W_{M - 1 - k}$ then
11: S continues the selection process to $k + 1^{th}$ step;
12: else
13: The selection process stops at the current step and selects $C N_{k}$ and $C N_{k + 1}$ as the relay and the jammer, respectively;
14: end if
15: else
16: Break;
17: end if
18: end for
19: S selects $C N_{M - 1}$ and $C N_{M}$ as the relay and the jammer, respectively.

Optimal sensing order

In the subsection “optimal selection scheme”, we assume the distribution of the channel gains and the available power of each candidate node are i.i.d. variables. However, in a more general and practical case, each candidate node should have its unique probability distributions. Thus, one can find that the sensing sequence can dramatically affect the effectiveness of the selection process. To be more specific, it is easier for S to find superior candidate nodes by constructing an optimal sensing order before the selection process. Inspired by this, in this section, we attempt to find an optimal sensing order to optimize the selection process. Due to the unique characteristic of the candidate node, we redefine the channel gains of i to j exist in a finite set $H_{i} \overset{Δ}{=} {H_{i, j, l}, l = 1, 2, \dots, L_{1}}$ , where the distribution of $| h_{i, j} |^{2}$ is given by $\Pr (| h_{i, j} |^{2} = h_{i, j, l}) = p_{i, j, l}, l = 1, 2, \dots, L_{1}$ . Similarly, the available power for candidate node i is selected from a finite set $P_{i} \overset{Δ}{=} {P_{i, l}, l = 1, 2, \dots, L_{2}}$ . Let the distribution of $P_{i}$ be $\Pr (P_{i} = P_{i, l}) = p_{i, l}, l = 1, 2, \dots, L_{2}$ .

At the $M - 1 th$ step, two candidate nodes are left to be selected. Considering the unique probability distribution of the candidate node, there exist $M^{2} - M$ possible states in this step. A state $S_{C N_{i, j}}^{M - 1}$ is defined as a set of sensing orders, namely

{\underset{(M - 2) CN}{\underset{︸}{CN, CN, \dots, CN, C N_{i}, C N_{j}}}}

Note that the sequence of the first $M - 2$ candidate nodes can be a random sequence. And a state $S_{C N_{i, j}}^{M - 1}$ and another state $S_{C N_{j, i}}^{M - 1}$ are two different states. To be more specific, the former state indicates that in the $M - 1 th$ step, $C N_{i}$ is sensed to be the relay, while $C N_{j}$ is sensed to be the jammer. The latter state means $C N_{i}$ and $C N_{j}$ are sensed as the jammer and the relay, respectively. For state $S_{C N_{i, j}}^{M - 1}$ , the expected reward is given as equation (24).

At the $k th$ observation step, a certain state, say $S_{C N_{p, q}}^{k}$ , indicates S chooses $C N_{p}$ and $C N_{q}$ to sense. The expected reward regarding to $S_{C N_{p, q}}^{k}$ is denoted as equation (25).

At each observation step, S can record which state results in the maximum expected reward. Thus, the optimal sensing order can be obtained based on the optimal state that recorded by S at each observation step. Another approach to derive the optimal sensing order is to use brute force to scan all the possible orders. S can calculate the maximum expected reward of each step with Algorithm 1 in a certain order. One can find that our proposed approach can significantly reduce the computation overhead compared with the brute force search method.

Evaluation

In this section, we evaluate the performance of our proposed scheme through simulation experiments. As described in section “System model,” all the CNs are assumed with four antennas while the system slot is assumed to be 0.2 ms. We assume the distribution of the channel power gain is ${0.5, 1, 1.5, 2, 2.5, 3}$ with probability ${0.43, 0.2, 0.1, 0.11, 0.08, 0.08}$ . And the distribution of the available power is ${1, 2, 4, 6, 8, 10}$ with probability ${0.35, 0.25, 0.15, 0.1, 0.1, 0.05}$ . The transmit power of S is assumed as 5 mW. For the sake of simplicity, the parameter a that has no influence on the selection scheme is set to be $ 0.5$.

Simulation study of fixed sensing order

In this subsection, we pay attention to the case that the sensing order is fixed, namely, the candidate nodes are sensed from 1 to M one by one. The time cost of one observation step is 2 µs with $M = 4, 6, 8, \dots, 20$ .

First, we compare the optimal stopping theory–based scheme with a random selection scheme, in which the jammer and the relay are randomly selected by S. Figure 3 shows the secrecy capacity versus the number of candidate nodes. The proposed selection scheme can remarkably improve the secrecy capacity.

Figure 3.

Secrecy capacity versus the number of candidate nodes.

Moreover, the variation trend of secrecy capacity in the random scheme is observed to change with no rules. However, the proposed scheme performs a continuous growth with an increase in the network size. Besides, one can also notice the secrecy capacity levels off with a large number of the candidate nodes. It can be explained that when the number is small, with the increased number of the candidate nodes, there are more opportunities for S to choose suitable nodes. For a larger number of the candidate nodes, however, a marginal increase in the candidate nodes has little impact on variations of the channel gain and the available power.

In Figure 4, the impact of the sensing time on the secrecy capacity is detailed. With an increase in the sensing time, less time can be used for data transmission, which leads to a poorer performance regardless of the network size. Figure 5 reports the impact of the sensing time on the number of sensing steps, which is observed to be increased with the number of candidate nodes. The reason can be given as with an increase in the candidate nodes, S has more chance to find suitable nodes. According to Figure 4, one can also notice that the secrecy capability is more likely to be affected with a large network size. This can be understood by the fact that with more sensing steps and bigger sensing times, the performance is drastically reduced when the network size increases. In Figure 5, one can also find that in a fixed network size, the sensing time almost has no effect on the number of sensing steps. This demonstrates the convergence property of the proposed selection scheme.

Figure 4.

Secrecy capacity versus sensing times.

Figure 5.

The number of sensing steps versus sensing times.

Simulation study of optimal sensing order

Note that in section “Simulation study of fixed sensing order,” we assume the distributions of the available power and the channel gains are i.i.d. variables for all the candidate nodes. In this subsection, the performance of our scheme is evaluated with the optimal sensing order by changing the distribution of the available power. That is to say, each candidate node has its unique probability distribution of the available power. We compare the proposed optimal sensing order with that of a fixed one, which is defined as the descending order. For example, when the candidate nodes set M as 4, the fixed sensing order can be given as [4,3,2,1].

Figure 6 represents the secrecy capacity achieved by these two sensing orders versus the number of candidate nodes. One can find that the performance of the optimal sensing order outperforms that of the fixed sensing order, which verifies the analysis in section “Optimal sensing order.” The increase in the secrecy capacity is observed to level off with a large number of the candidate nodes. The same conclusion can also be found in Figure 3. In Figure 7, the number of sensing steps versus the number of candidate nodes is plotted. One can see that the optimal sensing order can effectively reduce the sensing time of the proposed selection scheme.

Figure 6.

Secrecy capacity versus the number of candidate nodes with different sensing order.

Figure 7.

The number of sensing steps versus the number of candidate nodes.

Conclusion

In this article, we have investigated the joint jammer and relay selection issue in an IoT system, which consists of a sensing transmitter, a sensing receiver, some candidate nodes, and an eavesdropper. With the purpose of maximizing the secrecy capacity, we have formulated the selection process as an optimal stopping theory by considering the channel gains and the available power. The first two candidate nodes (one acts as the relay and the other acts as the jammer) that satisfy the secrecy capacity thresholds are selected as the relay and the jammer, respectively. The optimal thresholds are calculated according to the probability distribution of candidate nodes’ CSI and available power. Then, considering a more general case that each candidate node has its unique probability distribution of CSI and available power, we have proposed a low-complexity method to obtain the optimal sensing order, by applying dynamic programming.

Footnotes

Academic Editor: Qing Yang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the National Natural Science Foundation of China (61572070, 61272505, 61371069, and 61471028) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20130009110015).

References

Ashton

That “Internet of Things” thing. RFiD J, 1999, http://www.rfidjournal.com/articles/view?4986

Cheng

Cai

. Drawing dominant dataset from big sensory data in wireless sensor networks. In: Proceedings of the 2015 IEEE international conference on computer communications, INFOCOM, Hong Kong, 26 April–1 May 2015, pp.531–539. New York: IEEE.

Cheng

Cai

Curve query processing in wireless sensor networks. IEEE T Veh Technol 2015; 64(11): 5198–5209.

Cai

Cheng

. Approximate aggregation for tracking quantiles and range countings in wireless sensor networks. Theor Comput Sci 2015; 607(3): 381–390.

Kraijak

Tuwanut

. A survey on IoT architectures, protocols, applications, security, privacy, real-world implementation and future trends. In: Proceedings of the WiCOM 2015, Shanghai, China, 21–23 September 2015, pp.1–6. New York: IEEE.

Mukherjee

. Physical-layer security in the internet of things: sensing and communication confidentiality under resource constraints. Proc IEEE 2015; 103(10): 1747–1761.

Diffie

Hellman

New directions in cryptography. IEEE T Inform Theory 1976; 22(6): 644–654.

Schneier

Cryptographic design vulnerabilities. Computer 1998; 31(9): 29–33.

Parada

Blahut

Secrecy capacity of SIMO and slow fading channels. In: Proceedings of IEEE international symposium on information theory, ISIT, Adelaide, SA, Australia, 4–9 September 2005, pp.2152–2155. New York: IEEE.

10.

Liu

Shamai

A note on the secrecy capacity of the multiple-antenna wiretap channel. IEEE T Inform Theory 2009; 55(6): 2547–2553.

11.

Goel

Negi

Guaranteeing secrecy using artificial noise. IEEE T Wirel Commun 2008; 7(6): 2180–2189.

12.

Wyner

The wire-tap channel. Bell Syst Tech J 1975; 54(8): 1355–1387.

13.

Dong

Han

Petropulu

. Improving wireless physical layer security via cooperating relays. IEEE T Signal Proces 2010; 58(3): 1875–1888.

14.

Huang

Swindlehurst

Cooperative jamming for secure communications in MIMO relay networks. IEEE T Signal Proces 2011; 59(10): 4871–4884.

15.

Pei

Swindlehurst

. Adaptive limited feedback for MISO wiretap channels with cooperative jamming. IEEE T Signal Proces 2014; 62(4): 993–1004.

16.

Afzal

Zaidi

SAR

Shakir

. The cognitive internet of things: a unified perspective. Mobile Netw Appl 2015; 20(1): 72–85.

17.

Zhang

. High-rate cooperative beamforming for physical-layer security in wireless cyber-physical systems. In: Proceedings of IEEE international conference on communications (ICC), London, 8–12 June 2015, pp.2622–2626. New York: IEEE.

18.

Chen

Zhang

Song

. Joint relay and jammer selection for secure two-way relay networks. IEEE T Inf Foren Sec 2012; 7(1): 310–320.

19.

Liu

Tan

Low complexity power allocation and joint relay-jammer selection in cooperative jamming DF relay wireless secure networks. In: Proceedings of the international conference on anti-counterfeiting, security and identification, Shanghai, 25–27 October 2013, pp.1–5. New York: IEEE.

20.

Wang

Cho

Liu

Power allocation and jammer selection of a cooperative jamming strategy for physical-layer security. In: Proceedings of the VTC Spring, Seoul, Korea, 18–21 May 2014, pp.1–5. New York: IEEE.

21.

Hui

Swindlehurst

. Secure relay and jammer selection for physical layer security. IEEE Signal Proc Let 2015; 22(8): 1147–1151.

22.

Wang

Xia

. Uncoordinated jammer selection for securing simome wiretap channels: a stochastic geometry approach. IEEE T Wirel Commun 2015; 14(5): 2596–2612.

23.

Tan

Zheng

Zhang

. Distributed opportunistic scheduling for ad-hoc communications under delay constraints. In: Proceedings of IEEE international conference on computer communications, INFOCOM, San Diego, CA, 15–19 March 2010, pp.1–9. New York: IEEE.

24.

Huang

Zhang

. Distributed opportunistic scheduling for energy harvesting based wireless networks: a two-stage probing approach. IEEE ACM T Network 2016; 24(3): 1618–1631.

25.

Han

Yan

Cai

. An exploration of broader influence maximization in timeliness networks with opportunistic selection. J Netw Comput Appl 2016; 63: 39–49.

26.

Shu

Krunz

Throughput-efficient sequential channel sensing and probing in cognitive radio networks under sensing errors. In: Proceedings of ACM MobiCom, Beijing, China, 20–25 September 2009, pp.37–48. New York: ACM.

27.

Jia

Zhang

Shen

XS.

HC-MAC: a hardware-constrained cognitive MAC for efficient spectrum management. IEEE J Sel Area Comm 2008; 26(1): 106–117.

28.

Jing

Zhu

. Cooperative relay selection in cognitive radio networks. In: Proceedings of the IEEE international conference on computer communications, INFOCOM, Turin, 14–19 April 2013, pp.175–179. New York: IEEE.

29.

Gao

Jing

Xing

. Simultaneous energy and information cooperation in MIMO cooperative cognitive radio systems. In: Proceedings of the IEEE WCNC, New Orleans, LA, 9–12 March 2015, pp.351–356. New York: IEEE.

30.

Huang

Swindlehurst

Secure communications via cooperative jamming in two-hop relay systems. In: Proceedings of the IEEE GLOBECOM, Miami, FL, 6–10 December 2010, pp.1–5. New York: IEEE.

31.

Ferguson

. Optimal stopping and applications, 2003, http://www.math.ucla.edu/∼tom/Stopping/

32.

Liu

Liang

Multiple-input multiple-output Gaussian broadcast channels with common and confidential messages. IEEE T Inform Theory 2010; 56(11): 5477–5487.